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1. Conventions

section1.1.1 Frequently used notation1A2835F01724Conventions
section1.1.2 Scope, attribution and conventions61B2D7362CC0Conventions

2. Measures, distributions, and convergence

definitionDefinition 2.2.1901C72102EF9Radon measures and integration
propositionProposition 2.2.2D0882F84DCFCRadon measures and integration
propositionProposition 2.2.39CB694815500Radon measures and integration
lemmaLemma 2.2.4 (Jensen's inequality)9931E90187F6Radon measures and integration
lemmaLemma 2.2.5D0A959AF6D25Radon measures and integration
lemmaLemma 2.2.697A1ECFF2340Radon measures and integration
propositionProposition 2.2.7379F54715B13Radon measures and integration
lemmaLemma 2.2.89F2BA6481FD8Radon measures and integration
equationEquation 2.3.160E30C5BC9BAPositive linear functionals
lemmaLemma 2.3.2BAEFE6D3AE3CPositive linear functionals
theoremTheorem 2.3.3 (Riesz representation)0C614771FB98Positive linear functionals
equationEquation 2.3.4193D84CDC12FPositive linear functionals
propositionProposition 2.3.50BB807E5CBD4Positive linear functionals
definitionDefinition 2.4.155085BC2Test functions and distributions
exampleExample 2.4.2E2DD6B809EB8Test functions and distributions
lemmaLemma 2.5.13C40E6BD1222Convolution and mollifiers
lemmaLemma 2.5.2677586F8712CConvolution and mollifiers
propositionProposition 2.5.30CB24AD25DC3Convolution and mollifiers
equationEquation 2.5.43AA3C6A6F513Convolution and mollifiers
corollaryCorollary 2.5.5DF54B49EC7D4Convolution and mollifiers
theoremTheorem 2.5.66A928E70A605Convolution and mollifiers
propositionProposition 2.6.14DFD880D9E4DWeak convergence of measures
propositionProposition 2.6.2132090FCDF4AWeak convergence of measures
propositionProposition 2.6.354258F39AC23Weak convergence of measures
exampleExample 2.6.481B4A3D092FFWeak convergence of measures
theoremTheorem 2.6.5F9F4241C5BD2Weak convergence of measures
propositionProposition 2.7.14603E36ECC71Nets and compactness
lemmaLemma 2.7.28734A1486521Nets and compactness
theoremTheorem 2.7.3DF30CEF15D17Nets and compactness
propositionProposition 2.7.4 (Arzelà–Ascoli)2963E35FB0D3Nets and compactness
lemmaLemma 2.8.19790D3B21542Densities and differentiation of integrals
corollaryCorollary 2.8.2D30ED2659FC7Densities and differentiation of integrals
theoremTheorem 2.8.3 (Lebesgue differentiation)CF98893DE93EDensities and differentiation of integrals
lemmaLemma 2.8.4CAAD90CD1576Densities and differentiation of integrals
propositionProposition 2.8.5B412D7DC28A3Densities and differentiation of integrals
section2.9.1 Fourier transform and energy4C3D61D53E89Fourier analysis and incidence of complex lines
lemmaLemma 2.9.2A976F203FD66Fourier analysis and incidence of complex lines
equationEquation 2.9.3407772967810Fourier analysis and incidence of complex lines
theoremTheorem 2.9.4 (Plancherel)4C82DCD17551Fourier analysis and incidence of complex lines
propositionProposition 2.9.54B698DACE987Fourier analysis and incidence of complex lines
theoremTheorem 2.9.669D141CB2664Fourier analysis and incidence of complex lines
exampleExample 2.9.7E243D52E8F6AFourier analysis and incidence of complex lines
equationEquation 2.9.8AC1522EBF063Fourier analysis and incidence of complex lines
section2.9.9 Second derivatives and singular integrals4D259A5A576AFourier analysis and incidence of complex lines
lemmaLemma 2.9.100821A7B52A3BFourier analysis and incidence of complex lines
lemmaLemma 2.9.11780F1882B8A5Fourier analysis and incidence of complex lines
theoremTheorem 2.9.12CAB0E7754072Fourier analysis and incidence of complex lines
equationEquation 2.9.1361BC1B5115B3Fourier analysis and incidence of complex lines
lemmaLemma 2.9.14534938A78D42Fourier analysis and incidence of complex lines
lemmaLemma 2.9.1506216156055DFourier analysis and incidence of complex lines
lemmaLemma 2.9.16 (Morrey's inequality)DE7722E667E2Fourier analysis and incidence of complex lines
section2.9.17 Unitarily invariant measures5254DB7AD29FFourier analysis and incidence of complex lines
propositionProposition 2.9.18D01DFDB6AE9AFourier analysis and incidence of complex lines
lemmaLemma 2.9.1980451E036222Fourier analysis and incidence of complex lines
section2.9.20 Circle selection and high-value sets965F35E04B2DFourier analysis and incidence of complex lines
lemmaLemma 2.9.21DC992DBF55C7Fourier analysis and incidence of complex lines
theoremTheorem 2.9.222C0F0197F3BFFourier analysis and incidence of complex lines
section2.9.23 Projection densities and incidence3E5438AF5E6EFourier analysis and incidence of complex lines
lemmaLemma 2.9.247D4E3784406EFourier analysis and incidence of complex lines
equationEquation 2.9.25115E6AD9CC4FFourier analysis and incidence of complex lines
theoremTheorem 2.9.2603F670D5FB5AFourier analysis and incidence of complex lines
theoremTheorem 2.9.27AF10D8C2B9D9Fourier analysis and incidence of complex lines
corollaryCorollary 2.9.28644CAD92A580Fourier analysis and incidence of complex lines

3. Harmonic and subharmonic functions

section3.1.1 The Laplacian and mean values524A2588C8CEHarmonic functions and the Dirichlet problem
equationEquation 3.1.2AAD3B6ABC9B1Harmonic functions and the Dirichlet problem
equationEquation 3.1.34CF38DCFDD7EHarmonic functions and the Dirichlet problem
equationEquation 3.1.4DEF1A85BF9E8Harmonic functions and the Dirichlet problem
propositionProposition 3.1.5620D5E4DD6ECHarmonic functions and the Dirichlet problem
equationEquation 3.1.62B78C72EAD6CHarmonic functions and the Dirichlet problem
theoremTheorem 3.1.7A60BA3B2DBFCHarmonic functions and the Dirichlet problem
equationEquation 3.1.8D63569C18649Harmonic functions and the Dirichlet problem
equationEquation 3.1.9EB7C9EF52470Harmonic functions and the Dirichlet problem
propositionProposition 3.1.10B304785CC98DHarmonic functions and the Dirichlet problem
theoremTheorem 3.1.11 (Weyl's lemma)C9127FBA9AE8Harmonic functions and the Dirichlet problem
section3.1.12 Fundamental solutions981631AD31A1Harmonic functions and the Dirichlet problem
propositionProposition 3.1.13F8A69601BB8CHarmonic functions and the Dirichlet problem
equationEquation 3.1.1449DB76D43578Harmonic functions and the Dirichlet problem
equationEquation 3.1.159448EDDC2E41Harmonic functions and the Dirichlet problem
section3.1.16 Poisson kernels and Harnack inequalities9C3CB4CF29A2Harmonic functions and the Dirichlet problem
equationEquation 3.1.171AF0CC9EF476Harmonic functions and the Dirichlet problem
theoremTheorem 3.1.1808ED9D7CF1CFHarmonic functions and the Dirichlet problem
propositionProposition 3.1.19DAC835A39525Harmonic functions and the Dirichlet problem
equationEquation 3.1.20880B4ABDC24AHarmonic functions and the Dirichlet problem
corollaryCorollary 3.1.21D0BED761B9E7Harmonic functions and the Dirichlet problem
corollaryCorollary 3.1.22F898AC76FE42Harmonic functions and the Dirichlet problem
section3.1.23 Perron envelopes and boundary barriersA3DA602C02CBHarmonic functions and the Dirichlet problem
lemmaLemma 3.1.243F83842AEB11Harmonic functions and the Dirichlet problem
theoremTheorem 3.1.25537ACE84788CHarmonic functions and the Dirichlet problem
lemmaLemma 3.1.2623975478AEBAHarmonic functions and the Dirichlet problem
theoremTheorem 3.1.278FE1B604CDA2Harmonic functions and the Dirichlet problem
definitionDefinition 3.1.2831D179000C4DHarmonic functions and the Dirichlet problem
theoremTheorem 3.1.29C1B72E118338Harmonic functions and the Dirichlet problem
propositionProposition 3.1.30DBDE3CC0301DHarmonic functions and the Dirichlet problem
exampleExample 3.1.318C85C3D68394Harmonic functions and the Dirichlet problem
section3.1.32 Harmonic measure and Green functions56F22305E141Harmonic functions and the Dirichlet problem
propositionProposition 3.1.33C5D6FBB32967Harmonic functions and the Dirichlet problem
theoremTheorem 3.1.34476CFD04AAC6Harmonic functions and the Dirichlet problem
section3.4.1 Mean inequalities and comparisonABE9A23B5282Subharmonic functions
definitionDefinition 3.4.23E4C6A9CCF19Subharmonic functions
equationEquation 3.4.339B65BA580E1Subharmonic functions
lemmaLemma 3.4.436C9571A59E0Subharmonic functions
theoremTheorem 3.4.5384B48086228Subharmonic functions
equationEquation 3.4.67DDC45C194C3Subharmonic functions
propositionProposition 3.4.76D3B30A16AAESubharmonic functions
propositionProposition 3.4.85975EB5B5D7DSubharmonic functions
theoremTheorem 3.4.962E4A7EF2B4FSubharmonic functions
section3.4.10 Riesz measures and pointwise representativesD48BD09AF75ESubharmonic functions
propositionProposition 3.4.111F52C5AF98EFSubharmonic functions
theoremTheorem 3.4.12E76E281548A5Subharmonic functions
equationEquation 3.4.1378BEC0BF5E07Subharmonic functions
theoremTheorem 3.4.14 (Riesz decomposition)F04DBD614238Subharmonic functions
exampleExample 3.4.158DF74A8828CFSubharmonic functions
section3.4.16 Compactness and Hartogs' lemmaAA374DAE79FDSubharmonic functions
lemmaLemma 3.4.1744B1F1B5F95ESubharmonic functions
equationEquation 3.4.188D9820A797E4Subharmonic functions
theoremTheorem 3.4.19F159A5A96A98Subharmonic functions
theoremTheorem 3.4.20 (Hartogs' lemma)5784B80133BFSubharmonic functions
section3.4.21 Upper envelopes and countable selectionF1C0C53624DFSubharmonic functions
lemmaLemma 3.4.22 (Choquet's lemma)B37EED9280E3Subharmonic functions
theoremTheorem 3.4.231C986491E366Subharmonic functions
exampleExample 3.4.244B0199B64A0DSubharmonic functions
section3.4.25 Polar sets and removable singularities7DF84C7DAE9ESubharmonic functions
definitionDefinition 3.4.2603DCBB9E3633Subharmonic functions
theoremTheorem 3.4.278B711E470600Subharmonic functions
section3.3.1 Differentiating compact parameter integrals9767EC7AA70AClassical harmonic analysis
lemmaLemma 3.3.277E9E0297059Classical harmonic analysis
section3.3.3 Regularity of ball averagesAABF9B330CC1Classical harmonic analysis
theoremTheorem 3.3.4 (Ball-average regularity)F549D908E93AClassical harmonic analysis
equationEquation 3.3.56CEBA4319BDEClassical harmonic analysis
section3.3.6 Measures on a compact metric boundaryE0A7F26D42BEClassical harmonic analysis
theoremTheorem 3.3.7 (Compact-metric measure selection)2C91786CB405Classical harmonic analysis
section3.2.1 Inversion and the Kelvin transform9A02E6F8D636Kelvin transforms and half-space potentials
theoremTheorem 3.2.2 (Kelvin transform)97BA27062B9FKelvin transforms and half-space potentials
equationEquation 3.2.30E9BCA99B831Kelvin transforms and half-space potentials
section3.2.4 Nonnegative harmonic functions on a half-space5C12992087C2Kelvin transforms and half-space potentials
equationEquation 3.2.521F1298C637CKelvin transforms and half-space potentials
theoremTheorem 3.2.6 (Half-space representation)6BC86EEF03A9Kelvin transforms and half-space potentials
equationEquation 3.2.7F3ED29BBAD39Kelvin transforms and half-space potentials
equationEquation 3.2.88F7C774A9D89Kelvin transforms and half-space potentials
equationEquation 3.2.92009D781FC4AKelvin transforms and half-space potentials
equationEquation 3.2.10B95428ABB7D8Kelvin transforms and half-space potentials
section3.2.11 Bounded boundary dataFA92650B78EBKelvin transforms and half-space potentials
theoremTheorem 3.2.12 (Half-space Poisson integral)9ADD2DF19642Kelvin transforms and half-space potentials
equationEquation 3.2.13AB2849CD8791Kelvin transforms and half-space potentials
section3.3.8 Radial boundary limits from symmetric densities4C55BC86FBE8Classical harmonic analysis
theoremTheorem 3.3.9CAE5D89698DFClassical harmonic analysis
section3.3.10 Fine coverings and differentiation on a sphere03E23BFDC019Classical harmonic analysis
lemmaLemma 3.3.113FB36EAD186AClassical harmonic analysis
theoremTheorem 3.3.125F363A880A82Classical harmonic analysis
section3.3.13 Nontangential boundary limits of Poisson integrals3412F805AF75Classical harmonic analysis
theoremTheorem 3.3.14EEE7DD56F49DClassical harmonic analysis
section3.3.15 Harmonic reflection across a sphereA9175B0B7BA5Classical harmonic analysis
lemmaLemma 3.3.16B0CFE200676DClassical harmonic analysis
theoremTheorem 3.3.1714F410C2E770Classical harmonic analysis
section3.5.1 Harmonic lowering and greatest harmonic minorantsE0853C6A83C4Green functions and potentials
lemmaLemma 3.5.2AC0C4669DCBFGreen functions and potentials
theoremTheorem 3.5.32791D284428CGreen functions and potentials
section3.5.4 Minimal Green functions on general open sets870251679082Green functions and potentials
theoremTheorem 3.5.5F3C6870EA9A8Green functions and potentials
theoremTheorem 3.5.69E37095B7A8AGreen functions and potentials
section3.5.7 Potentials and global Riesz decomposition47255929FFC0Green functions and potentials
theoremTheorem 3.5.8977ECF0867E0Green functions and potentials
theoremTheorem 3.5.99453A8C660CCGreen functions and potentials
section3.5.10 Continuity of potentials and bounded densities7AECB61AA15FGreen functions and potentials
theoremTheorem 3.5.1164E880580F0EGreen functions and potentials
theoremTheorem 3.5.129CF388990FE4Green functions and potentials
section3.5.13 The generalized Laplacian589DA36A9C17Green functions and potentials
theoremTheorem 3.5.1427ABF8BDD497Green functions and potentials
section3.5.15 A global extension finite outside a prescribed ballB78F0B018AE7Green functions and potentials
lemmaLemma 3.5.167C49056207BEGreen functions and potentials

4. Energy, capacity, and fine topology

section4.1.1 Kernel energies and equilibrium measuresFF9D3CA4D463Potentials, energy and capacity
equationEquation 4.1.2718E110735E0Potentials, energy and capacity
lemmaLemma 4.1.321858171114BPotentials, energy and capacity
equationEquation 4.1.4FD55A94B512FPotentials, energy and capacity
propositionProposition 4.1.509A0D05A868FPotentials, energy and capacity
equationEquation 4.1.6F2F2EC96C26EPotentials, energy and capacity
theoremTheorem 4.1.7AA43DD93EFE4Potentials, energy and capacity
section4.1.8 Continuous potentials and polar sets12EDE2E332B0Potentials, energy and capacity
lemmaLemma 4.1.9468BD94D4D83Potentials, energy and capacity
theoremTheorem 4.1.10872A6887585BPotentials, energy and capacity
propositionProposition 4.1.1164A94A9EE9E3Potentials, energy and capacity
propositionProposition 4.1.12EBEA4A14BCD2Potentials, energy and capacity
section4.1.13 Relative extremal functions and capacity2B6F9623A939Potentials, energy and capacity
lemmaLemma 4.1.14D17D3298467BPotentials, energy and capacity
equationEquation 4.1.15B44D46472BE9Potentials, energy and capacity
propositionProposition 4.1.164A7451727417Potentials, energy and capacity
theoremTheorem 4.1.175B4ACD7E054DPotentials, energy and capacity
propositionProposition 4.1.183CC68CCFAF3APotentials, energy and capacity
theoremTheorem 4.1.19AEE550586C10Potentials, energy and capacity
equationEquation 4.1.200A0CA0153B81Potentials, energy and capacity
exampleExample 4.1.21FC1E9D9B42F0Potentials, energy and capacity
section4.1.22 Capacitability and exceptional setsE98B97E4C96BPotentials, energy and capacity
theoremTheorem 4.1.2362C826EB1B48Potentials, energy and capacity
corollaryCorollary 4.1.24098882E3CBEBPotentials, energy and capacity
lemmaLemma 4.1.2559F83B6EAAB7Potentials, energy and capacity
theoremTheorem 4.1.26E54414B74C32Potentials, energy and capacity
propositionProposition 4.1.27CA3187A3AD4FPotentials, energy and capacity
section4.1.28 Dirichlet energy and quasi-continuityEB60B98F8F53Potentials, energy and capacity
propositionProposition 4.1.295E7FF77DE321Potentials, energy and capacity
theoremTheorem 4.1.306B0925DDDD3APotentials, energy and capacity
section4.2.1 Fine neighborhoods and thinnessE15D6A501F63Fine topology and balayage
definitionDefinition 4.2.2C067468FA3F0Fine topology and balayage
definitionDefinition 4.2.32B94F86FC31DFine topology and balayage
propositionProposition 4.2.4CAAD6BABA60CFine topology and balayage
propositionProposition 4.2.59E787ECDB068Fine topology and balayage
propositionProposition 4.2.6498B2C3E2413Fine topology and balayage
lemmaLemma 4.2.7A1CB0D81584CFine topology and balayage
exampleExample 4.2.84D234E8193ACFine topology and balayage
section4.2.9 Green energy and compact reduction27012C9B879CFine topology and balayage
definitionDefinition 4.2.10CD643C4C40FFFine topology and balayage
propositionProposition 4.2.112BC493B5F349Fine topology and balayage
propositionProposition 4.2.12AF6AC1F168D7Fine topology and balayage
propositionProposition 4.2.13A13D01405A7FFine topology and balayage
theoremTheorem 4.2.1497A5DA659CA1Fine topology and balayage
section4.2.15 Sweeping measures and harmonic measureDB98A10B30F4Fine topology and balayage
theoremTheorem 4.2.16F28E6D45483FFine topology and balayage
equationEquation 4.2.177DCA4DACFD19Fine topology and balayage
theoremTheorem 4.2.180056FED1AF8CFine topology and balayage
equationEquation 4.2.19265580F8DEEAFine topology and balayage
exampleExample 4.2.20372946152BF9Fine topology and balayage
section4.2.21 Countable covers and quasi-open setsA51B9EF16CF8Fine topology and balayage
theoremTheorem 4.2.22730413AAC920Fine topology and balayage
propositionProposition 4.2.23F3EA8F8D8D8FFine topology and balayage
section4.2.24 Complex lines and the choice of topology3AAB9C9B1384Fine topology and balayage
propositionProposition 4.2.252733CF927914Fine topology and balayage
section4.3.1 Global polar functions and exact compact pole sets65321DAEF711Polar sets, capacity and sweeping
theoremTheorem 4.3.2253540032CE5Polar sets, capacity and sweeping
section4.3.3 Boundary minimum principles with polar exceptions5F11D2AF64E9Polar sets, capacity and sweeping
theoremTheorem 4.3.409FD5942CD4FPolar sets, capacity and sweeping
section4.3.5 Reduction and capacity on an admissible open set0416CDC0A00DPolar sets, capacity and sweeping
theoremTheorem 4.3.6C651523E22C3Polar sets, capacity and sweeping
theoremTheorem 4.3.7ABDF7610F7B2Polar sets, capacity and sweeping
section4.4.1 Perron solutions and completed harmonic measureE00A8C634DA6Perron theory and fine boundary limits
theoremTheorem 4.4.209873BDAF3E0Perron theory and fine boundary limits
lemmaLemma 4.4.3621606E7390CPerron theory and fine boundary limits
section4.4.4 Local barriers, cone regularity and regular exhaustions93318C47A3DEPerron theory and fine boundary limits
theoremTheorem 4.4.52F74CBABB2A6Perron theory and fine boundary limits
theoremTheorem 4.4.6123C9AB7C052Perron theory and fine boundary limits
section4.4.7 Green boundary limits, Kellogg's theorem and planar Greenian setsBD2BAD3CFA50Perron theory and fine boundary limits
theoremTheorem 4.4.852535FA416BFPerron theory and fine boundary limits
theoremTheorem 4.4.92C2E1569F3E2Perron theory and fine boundary limits
section4.3.8 Sweeping identities and the domination principle1E913881086DPolar sets, capacity and sweeping
theoremTheorem 4.3.939A7F7684147Polar sets, capacity and sweeping
theoremTheorem 4.3.10C4F5D8B270EBPolar sets, capacity and sweeping
section4.4.10 Exterior Poisson kernels and Helms's point at infinity43AD76172E0CPerron theory and fine boundary limits
theoremTheorem 4.4.11E16C09608A00Perron theory and fine boundary limits
theoremTheorem 4.4.128FA95EB71F47Perron theory and fine boundary limits
lemmaLemma 4.4.1358BE7FB3C948Perron theory and fine boundary limits
section4.4.14 Perron theory on the one-point compactification02718F5DB5E5Perron theory and fine boundary limits
lemmaLemma 4.4.15E0D6C3CD3750Perron theory and fine boundary limits
lemmaLemma 4.4.165D12CB9707B5Perron theory and fine boundary limits
theoremTheorem 4.4.17A7E357C38C8EPerron theory and fine boundary limits
theoremTheorem 4.4.18C07BA2371FA3Perron theory and fine boundary limits
section4.4.19 Thinness and fine limits78ACC7C20E8CPerron theory and fine boundary limits
lemmaLemma 4.4.20201EE1A61AF0Perron theory and fine boundary limits
theoremTheorem 4.4.21D7E4078B4FE9Perron theory and fine boundary limits
theoremTheorem 4.4.222B657EA5CC09Perron theory and fine boundary limits
theoremTheorem 4.4.23DEAF5030EE5EPerron theory and fine boundary limits
section4.4.24 The Wiener criterion and planar continuaED2AAB8DC124Perron theory and fine boundary limits
theoremTheorem 4.4.2504CCB65ED650Perron theory and fine boundary limits
theoremTheorem 4.4.26A856D11F5895Perron theory and fine boundary limits
section4.5.1 Green energy and the complete positive cone57DD989B07A4Green energy and Martin boundaries
theoremTheorem 4.5.2CC3D1AD9179BGreen energy and Martin boundaries
theoremTheorem 4.5.32C4B36B5F3CDGreen energy and Martin boundaries
theoremTheorem 4.5.4CE0D5F4DF7B2Green energy and Martin boundaries
exampleExample 4.5.561A5D2A82793Green energy and Martin boundaries
theoremTheorem 4.5.6B15EFD960FD5Green energy and Martin boundaries
section4.5.7 The Martin boundaryB6BAE13977A4Green energy and Martin boundaries
lemmaLemma 4.5.8D878F27BAB16Green energy and Martin boundaries
theoremTheorem 4.5.99D6B5CF5E0CFGreen energy and Martin boundaries
lemmaLemma 4.5.10B0A113187C30Green energy and Martin boundaries
lemmaLemma 4.5.11FDE7BF6A9927Green energy and Martin boundaries
theoremTheorem 4.5.126736FBE7368CGreen energy and Martin boundaries
theoremTheorem 4.5.130E65F552B1A6Green energy and Martin boundaries

5. Plurisubharmonic functions and positive currents

equationEquation 5.2.1E0E563B56BCEComplex manifolds and differential forms
equationEquation 5.2.2284A415465F7Complex manifolds and differential forms
definitionDefinition 5.2.342A02A1A5EFEComplex manifolds and differential forms
propositionProposition 5.2.4A683A13C69D9Complex manifolds and differential forms
equationEquation 5.2.5B25865120DE7Complex manifolds and differential forms
propositionProposition 5.2.6DAE61F3D77A3Complex manifolds and differential forms
lemmaLemma 5.2.798494B0CCC31Complex manifolds and differential forms
equationEquation 5.2.8E58B935C99BAComplex manifolds and differential forms
equationEquation 5.2.9EFB1A41694DEComplex manifolds and differential forms
equationEquation 5.2.10C44E7362B437Complex manifolds and differential forms
equationEquation 5.3.1976BD2929614Currents and positivity
propositionProposition 5.3.2E58104E974B0Currents and positivity
lemmaLemma 5.3.33D25BAD0C21ECurrents and positivity
lemmaLemma 5.3.48907196C3513Currents and positivity
definitionDefinition 5.3.5A86EC2473CB6Currents and positivity
equationEquation 5.3.63E683990E7C6Currents and positivity
theoremTheorem 5.3.71C212E45A497Currents and positivity
lemmaLemma 5.3.8376F5C171BEDCurrents and positivity
exampleExample 5.3.91176DC9B7704Currents and positivity
equationEquation 5.4.1E64CA5AFE6C4Local potentials for closed (1,1)-currents
lemmaLemma 5.4.27815541A95ADLocal potentials for closed (1,1)-currents
theoremTheorem 5.4.32408CCC1B5A0Local potentials for closed (1,1)-currents
propositionProposition 5.4.4AA6533F0128ALocal potentials for closed (1,1)-currents
lemmaLemma 5.5.1AD4CF8180187Holomorphic zeros and the Poincaré–Lelong formula
theoremTheorem 5.5.2 (Weierstrass preparation)F54A645FF399Holomorphic zeros and the Poincaré–Lelong formula
propositionProposition 5.5.337EEAF275A32Holomorphic zeros and the Poincaré–Lelong formula
equationEquation 5.5.4FF48D92EEC58Holomorphic zeros and the Poincaré–Lelong formula
lemmaLemma 5.5.5116B90FD9752Holomorphic zeros and the Poincaré–Lelong formula
theoremTheorem 5.5.6 (Poincaré–Lelong)8D4BA339507EHolomorphic zeros and the Poincaré–Lelong formula
equationEquation 5.5.79E4726681500Holomorphic zeros and the Poincaré–Lelong formula
exampleExample 5.5.8DB42A2BDBC5BHolomorphic zeros and the Poincaré–Lelong formula
propositionProposition 5.6.1EB9FCD4819C3Maps of currents and local mass
propositionProposition 5.6.236A2806D4E71Maps of currents and local mass
equationEquation 5.6.356D4111AE674Maps of currents and local mass
corollaryCorollary 5.6.4FFF97DDD2CF2Maps of currents and local mass
propositionProposition 5.6.56DAA1675792BMaps of currents and local mass
exampleExample 5.6.65A6B397A8B30Maps of currents and local mass
definitionDefinition 5.8.16070D2D83EC6The line definition and upper semicontinuity
lemmaLemma 5.8.23C70A4ABB14AThe line definition and upper semicontinuity
equationEquation 5.8.3A577EADC63E4The line definition and upper semicontinuity
theoremTheorem 5.8.41A391017E17AThe line definition and upper semicontinuity
theoremTheorem 5.9.12A4F77263F48Positive Hessians and holomorphic maps
propositionProposition 5.9.206C722B7D837Positive Hessians and holomorphic maps
propositionProposition 5.9.3F6423BCE3BB1Positive Hessians and holomorphic maps
lemmaLemma 5.9.4598B886D700EPositive Hessians and holomorphic maps
propositionProposition 5.9.57EBEE004A7FAPositive Hessians and holomorphic maps
theoremTheorem 5.10.1FE144A69076ELimits, envelopes and local integrability
theoremTheorem 5.10.2D32A2FB94F6BLimits, envelopes and local integrability
propositionProposition 5.10.31A6F10A07BA5Limits, envelopes and local integrability
theoremTheorem 5.10.4C2DB8DFE9E2CLimits, envelopes and local integrability
lemmaLemma 5.10.569CE23EA1637Limits, envelopes and local integrability
equationEquation 5.10.66A92D19CE668Limits, envelopes and local integrability
equationEquation 5.10.7C493F766D641Limits, envelopes and local integrability
propositionProposition 5.10.8B979DF329B28Limits, envelopes and local integrability
equationEquation 5.10.98326366018BBLimits, envelopes and local integrability
theoremTheorem 5.10.10F2A3CA649325Limits, envelopes and local integrability
lemmaLemma 5.10.1103676DBFBA61Limits, envelopes and local integrability
propositionProposition 5.10.12219E06AF4230Limits, envelopes and local integrability
equationEquation 5.10.13DE8035B23862Limits, envelopes and local integrability
theoremTheorem 5.11.1C910F1109424Extension across closed exceptional sets
lemmaLemma 5.11.2B88B826B1EDCExtension across closed exceptional sets
theoremTheorem 5.11.376D63A606619Extension across closed exceptional sets
theoremTheorem 5.12.1 (Kiselman's minimum principle)1F87EF4FCF69Kiselman's minimum principle
corollaryCorollary 5.12.22C1A82803889Kiselman's minimum principle
section5.47.1 Pluripolar sets and relative extremal functions27FCEC6F336FPluripolar sets and the plurifine topology
definitionDefinition 5.47.25458CA6608A5Pluripolar sets and the plurifine topology
equationEquation 5.47.3BD894E43C430Pluripolar sets and the plurifine topology
equationEquation 5.47.4CA38D8657CFBPluripolar sets and the plurifine topology
equationEquation 5.47.507012084B5F3Pluripolar sets and the plurifine topology
propositionProposition 5.47.65292FEC65E83Pluripolar sets and the plurifine topology
propositionProposition 5.47.71D1F65765AD9Pluripolar sets and the plurifine topology
theoremTheorem 5.47.89421BA4DC329Pluripolar sets and the plurifine topology
section5.47.9 Global poles on Euclidean spaceAE92E4DE9A4FPluripolar sets and the plurifine topology
theoremTheorem 5.47.10 (Josefson)1822741C4B07Pluripolar sets and the plurifine topology
corollaryCorollary 5.47.118EF0B17C7325Pluripolar sets and the plurifine topology
exampleExample 5.47.12174FD6E0DD10Pluripolar sets and the plurifine topology
section5.47.13 Thinness and plurifine neighborhoods333115F369F9Pluripolar sets and the plurifine topology
definitionDefinition 5.47.1409A00D4B8096Pluripolar sets and the plurifine topology
propositionProposition 5.47.152297E8BC91FFPluripolar sets and the plurifine topology
propositionProposition 5.47.16DD1F66C6842APluripolar sets and the plurifine topology
exampleExample 5.47.177E1189000F74Pluripolar sets and the plurifine topology
exampleExample 5.47.1802396DC16057Pluripolar sets and the plurifine topology
section5.47.19 Countable selections and plurifine locality03B3029EB4B3Pluripolar sets and the plurifine topology
theoremTheorem 5.47.2017D9EEFA42DFPluripolar sets and the plurifine topology
lemmaLemma 5.47.21FA3C3BD45E57Pluripolar sets and the plurifine topology
theoremTheorem 5.47.22C1A6B45ED829Pluripolar sets and the plurifine topology
propositionProposition 5.47.23983478512954Pluripolar sets and the plurifine topology
section5.47.24 Common poles and quasi-extrema6AAE49F7183FPluripolar sets and the plurifine topology
propositionProposition 5.47.256EBEFACB2203Pluripolar sets and the plurifine topology
definitionDefinition 5.47.26FF965EAEB1A3Pluripolar sets and the plurifine topology
corollaryCorollary 5.47.27D6E89A0A4867Pluripolar sets and the plurifine topology
subsection5.13.1 The $1$-dimensional case600FCCB417C9The definition of plurisubharmonic functions
definitionDefinition 5.13.2C3781B231934The definition of plurisubharmonic functions
theoremTheorem 5.13.3FDD1E63B4457The definition of plurisubharmonic functions
subsection5.13.4 The higher dimensional caseEB9F157B42F2The definition of plurisubharmonic functions
captionFigure 5.13.5B2EF8B3CC1DBThe definition of plurisubharmonic functions
exampleExample 5.13.6DD77B8A4B7A7The definition of plurisubharmonic functions
exampleExample 5.13.71C9711BBE591The definition of plurisubharmonic functions
theoremTheorem 5.13.82CB6F65A67EEThe definition of plurisubharmonic functions
propositionProposition 5.13.95DCB508EA9ADThe definition of plurisubharmonic functions
equationEquation 5.13.107D0D660F11F8The definition of plurisubharmonic functions
propositionProposition 5.13.112FF7CC0E8132The definition of plurisubharmonic functions
propositionProposition 5.13.12D8BC10548372The definition of plurisubharmonic functions
subsection5.13.13 The manifold case7BAC2CC7335CThe definition of plurisubharmonic functions
definitionDefinition 5.13.14D072C98719AAThe definition of plurisubharmonic functions
exampleExample 5.13.1567A5152A77BCThe definition of plurisubharmonic functions
exampleExample 5.13.16399FC71A61E3The definition of plurisubharmonic functions
propositionProposition 5.13.17726DDFDDA8F2The definition of plurisubharmonic functions
theoremTheorem 5.13.18FEFFC91695E0The definition of plurisubharmonic functions
theoremTheorem 5.13.19 (Josefson's theorem)DB05CEAF06C1The definition of plurisubharmonic functions
theoremTheorem 5.13.200ED3DB1BD7C2The definition of plurisubharmonic functions
propositionProposition 5.14.10372195E73A0Properties of plurisubharmonic functions
propositionProposition 5.14.201C3654494E2Properties of plurisubharmonic functions
propositionProposition 5.14.3 (Choquet's lemma)1D624ECF05E9Properties of plurisubharmonic functions
propositionProposition 5.14.4E2A24FD63150Properties of plurisubharmonic functions
propositionProposition 5.14.5E10B5F00D3EDProperties of plurisubharmonic functions
propositionProposition 5.14.6FA3AF40B9A0AProperties of plurisubharmonic functions
propositionProposition 5.14.7459E68F98F06Properties of plurisubharmonic functions
corollaryCorollary 5.14.8E23F3D6154ABProperties of plurisubharmonic functions
equationEquation 5.14.902A6B28E8E8FProperties of plurisubharmonic functions
theoremTheorem 5.14.10 (Brelot, Grauert–Remmert)DAE812E1A818Properties of plurisubharmonic functions
equationEquation 5.14.1109298412E7CEProperties of plurisubharmonic functions
captionFigure 5.14.12F6B6F57ED80EProperties of plurisubharmonic functions
equationEquation 5.14.13CD940E826F95Properties of plurisubharmonic functions
lemmaLemma 5.14.14741322DD572FProperties of plurisubharmonic functions
propositionProposition 5.14.15 (Kiselman's principle)BB26476C13FAProperties of plurisubharmonic functions
lemmaLemma 5.14.1649CC56CB484CProperties of plurisubharmonic functions
equationEquation 5.14.17504311AA3518Properties of plurisubharmonic functions
captionFigure 5.14.18660980D172BDProperties of plurisubharmonic functions
equationEquation 5.14.1919FE6DB3F44AProperties of plurisubharmonic functions
corollaryCorollary 5.14.2079C564A098F9Properties of plurisubharmonic functions
propositionProposition 5.14.213EDBDD31B3F1Properties of plurisubharmonic functions
propositionProposition 5.14.22EFFC20773E2EProperties of plurisubharmonic functions
subsection5.15.1 Plurifine topology on domainsC9F4423A543APlurifine topology
propositionProposition 5.15.20E4FD8AF1256Plurifine topology
exampleExample 5.15.328F2383E9245Plurifine topology
definitionDefinition 5.15.4EC0662753F9CPlurifine topology
remarkRemark 5.15.53833CFAB6397Plurifine topology
propositionProposition 5.15.6 (Bedford–Taylor)80A580DBD009Plurifine topology
equationEquation 5.15.7921249630360Plurifine topology
equationEquation 5.15.81BCCC754A81FPlurifine topology
lemmaLemma 5.15.9D3F2BF3D82BDPlurifine topology
theoremTheorem 5.15.10 (H.Cartan)458C93F5F6CDPlurifine topology
theoremTheorem 5.15.114388B85DDD67Plurifine topology
equationEquation 5.15.12E3CEA68870B6Plurifine topology
remarkRemark 5.15.132EDAD1EF1694Plurifine topology
corollaryCorollary 5.15.1449909C9A34E2Plurifine topology
corollaryCorollary 5.15.15C94ADCA6E19DPlurifine topology
corollaryCorollary 5.15.16DF37542BAB03Plurifine topology
subsection5.15.17 Plurifine topology on manifolds2403FAC7083CPlurifine topology
propositionProposition 5.15.18E49CF7960E4CPlurifine topology
corollaryCorollary 5.15.19030D740FC9C6Plurifine topology
lemmaLemma 5.15.204014761D7A8APlurifine topology
corollaryCorollary 5.15.2131032260EF62Plurifine topology
definitionDefinition 5.15.228485EC5E3BC1Plurifine topology
definitionDefinition 5.16.1A8F9CC5C117DLelong numbers and multiplier ideal sheaves
equationEquation 5.16.2D4BCC2E76B62Lelong numbers and multiplier ideal sheaves
remarkRemark 5.16.39ADE309D4F8BLelong numbers and multiplier ideal sheaves
propositionProposition 5.16.461D6716B27F1Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.5F9F65BFBF525Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.6C55AC8943767Lelong numbers and multiplier ideal sheaves
theoremTheorem 5.16.7 (Siu)6BAB6143467ALelong numbers and multiplier ideal sheaves
propositionProposition 5.16.8B9C4FE156D00Lelong numbers and multiplier ideal sheaves
corollaryCorollary 5.16.9F3F797E97E9DLelong numbers and multiplier ideal sheaves
definitionDefinition 5.16.106777DFC31EF1Lelong numbers and multiplier ideal sheaves
definitionDefinition 5.16.11E9EA7799080CLelong numbers and multiplier ideal sheaves
remarkRemark 5.16.124E2D4F8663A3Lelong numbers and multiplier ideal sheaves
propositionProposition 5.16.13 (Nadel)926B8027EE55Lelong numbers and multiplier ideal sheaves
theoremTheorem 5.16.1448A7E9240D09Lelong numbers and multiplier ideal sheaves
propositionProposition 5.16.158EFDC4A2C19ELelong numbers and multiplier ideal sheaves
equationEquation 5.16.16DAEF62372440Lelong numbers and multiplier ideal sheaves
lemmaLemma 5.16.17C85C2748AE0BLelong numbers and multiplier ideal sheaves
theoremTheorem 5.16.18B2A67FDA22D2Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.19C474A65CB982Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.2043490E287BDALelong numbers and multiplier ideal sheaves
theoremTheorem 5.16.21 (Guan–Zhou)917763AE5E35Lelong numbers and multiplier ideal sheaves
propositionProposition 5.16.22DE693424352FLelong numbers and multiplier ideal sheaves
definitionDefinition 5.16.23AA412055CB86Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.24514E110B23C0Lelong numbers and multiplier ideal sheaves
equationEquation 5.16.25229919B31C3ELelong numbers and multiplier ideal sheaves
theoremTheorem 5.16.26 (Ohsawa–Takegoshi)151D57BCA31BLelong numbers and multiplier ideal sheaves
definitionDefinition 5.17.13DC2C56EBB31Quasi-plurisubharmonic functions
remarkRemark 5.17.28B989AE345C3Quasi-plurisubharmonic functions
propositionProposition 5.17.3EC4B3805B84DQuasi-plurisubharmonic functions
remarkRemark 5.17.40CEC6D75820CQuasi-plurisubharmonic functions
corollaryCorollary 5.17.5B5E91AB35965Quasi-plurisubharmonic functions
equationEquation 5.17.688BE4BEDE25DQuasi-plurisubharmonic functions
propositionProposition 5.17.7A1D8E7565DF1Quasi-plurisubharmonic functions
propositionProposition 5.17.8AF2A3F35959CQuasi-plurisubharmonic functions
propositionProposition 5.17.95F9E17A436E9Quasi-plurisubharmonic functions
equationEquation 5.18.190744C0CCE8EAnalytic singularities
propositionProposition 5.18.2AB08C5138082Analytic singularities
definitionDefinition 5.18.3670832106E8EAnalytic singularities
equationEquation 5.18.4C6AC1C84A3B3Analytic singularities
lemmaLemma 5.18.53B54149D4222Analytic singularities
propositionProposition 5.18.64A902295BB8CAnalytic singularities
definitionDefinition 5.18.7239591DA9F90Analytic singularities
theoremTheorem 5.18.8D6BDA051B7C1Analytic singularities
equationEquation 5.18.9EF5655B5F536Analytic singularities
definitionDefinition 5.18.10FFAA7E68EEC1Analytic singularities
theoremTheorem 5.18.119348BF57481BAnalytic singularities
propositionProposition 5.18.1236EC36A38FABAnalytic singularities
definitionDefinition 5.18.1399446B95B55BAnalytic singularities
lemmaLemma 5.18.14D9F25876A735Analytic singularities
lemmaLemma 5.18.152E0432C779A0Analytic singularities
equationEquation 5.18.1635E0DBB664F7Analytic singularities
equationEquation 5.18.17D1A302A4CDD6Analytic singularities
theoremTheorem 5.18.182D610C4DD6ADAnalytic singularities
definitionDefinition 5.19.1DF4FCB370781The space of currents
definitionDefinition 5.19.2AE4F30A1999EThe space of currents
definitionDefinition 5.19.349CF4A9C5660The space of currents
definitionDefinition 5.19.40909012E7F31The space of currents
remarkRemark 5.19.5877B9D0BDD26The space of currents
exampleExample 5.19.6AAD0C8775B34The space of currents
equationEquation 5.19.73E19B5D1CC32The space of currents
definitionDefinition 5.19.84A951ED7EEE8The space of currents
definitionDefinition 5.19.9AEE3AF4068CDThe space of currents
theoremTheorem 5.19.10 (Boucksom)847815EFB4FEThe space of currents
definitionDefinition 5.19.111AA67767FB84The space of currents
lemmaLemma 5.19.12 (Siu's decomposition)F956E74CD8C5The space of currents
equationEquation 5.19.136720366F8D5BThe space of currents
definitionDefinition 5.19.145545837BA9C1The space of currents
definitionDefinition 5.19.154DFBEC43A4B0The space of currents
definitionDefinition 5.19.160EB97717F408The space of currents
propositionProposition 5.19.17191D96316EC3The space of currents
lemmaLemma 5.19.189F062B47A2B4The space of currents
definitionDefinition 5.20.1C5880C6F61FAPlurisubharmonic metrics on line bundles
definitionDefinition 5.20.290891EB78EA0Plurisubharmonic metrics on line bundles
definitionDefinition 5.20.3293669D813B8Plurisubharmonic metrics on line bundles
propositionProposition 5.20.4 (Lelong–Poincaré)0C02CD3B5D59Plurisubharmonic metrics on line bundles
equationEquation 5.20.51322FD7BDF4FPlurisubharmonic metrics on line bundles
exampleExample 5.20.60D9E7B0EE3DCPlurisubharmonic metrics on line bundles
equationEquation 5.20.73C9B623060E7Plurisubharmonic metrics on line bundles
definitionDefinition 5.20.8A43467D63B4APlurisubharmonic metrics on line bundles
equationEquation 5.20.96C8BC3AC251DPlurisubharmonic metrics on line bundles
remarkRemark 5.20.10D2F8C4BB2BD4Plurisubharmonic metrics on line bundles
definitionDefinition 5.20.119834FE8F2B0DPlurisubharmonic metrics on line bundles
exampleExample 5.20.12CC1C409253F3Plurisubharmonic metrics on line bundles
equationEquation 5.20.136B76079BBD9CPlurisubharmonic metrics on line bundles
definitionDefinition 5.20.149FE903B6E5A8Plurisubharmonic metrics on line bundles
definitionDefinition 5.20.158FCD7BEEDD56Plurisubharmonic metrics on line bundles
theoremTheorem 5.20.162985932B2B64Plurisubharmonic metrics on line bundles
propositionProposition 5.20.172DDBBF280789Plurisubharmonic metrics on line bundles
equationEquation 5.20.181EE7A6DB1803Plurisubharmonic metrics on line bundles
definitionDefinition 5.22.19CBA378B705DPlurisubharmonic potentials
propositionProposition 5.22.2887F604607E9Plurisubharmonic potentials
propositionProposition 5.22.3F62C0F30E87CPlurisubharmonic potentials
lemmaLemma 5.22.40DED015E67EFPlurisubharmonic potentials
propositionProposition 5.23.1237930AECE45Changes of background
propositionProposition 5.23.2B16999654254Changes of background
lemmaLemma 5.24.1729BA86BCC7CMaxima and the maximum principle
lemmaLemma 5.24.20CFD6850F9F8Maxima and the maximum principle
lemmaLemma 5.24.3D1DAB1D38637Maxima and the maximum principle
lemmaLemma 5.24.405BCBF7FAC9CMaxima and the maximum principle
lemmaLemma 5.24.5E6AC64216A47Maxima and the maximum principle
lemmaLemma 5.24.616E53426F2DBMaxima and the maximum principle
lemmaLemma 5.24.7F3A890BB2575Maxima and the maximum principle
lemmaLemma 5.24.864CFA1B8522BMaxima and the maximum principle
lemmaLemma 5.24.998E72E23E884Maxima and the maximum principle
lemmaLemma 5.25.145B3FA75B7BDNormalized compactness
theoremTheorem 5.25.2FC59B7D406A3Normalized compactness
corollaryCorollary 5.25.30AD144F69807Normalized compactness
equationEquation 5.26.1D87814C66F39Uniform exponential integrability
lemmaLemma 5.26.288C7A8DCE36EUniform exponential integrability
theoremTheorem 5.27.1 (Bedford–Taylor)8B1218AE4FDCLocal products and capacity
lemmaLemma 5.27.22A60C2B3B4C3Local products and capacity
equationEquation 5.27.3F138661713A9Local products and capacity
definitionDefinition 5.27.49D64513AE343Local products and capacity
theoremTheorem 5.27.5 (Mixed Monge–Ampère inequality)DBC85A3C1608Local products and capacity
equationEquation 5.27.65E980A5776A2Local products and capacity
equationEquation 5.27.7A9D318047BDALocal products and capacity
equationEquation 5.27.8F547D6DF0352Local products and capacity
equationEquation 5.27.9B4EA93FC61EELocal products and capacity
lemmaLemma 5.27.109149838224AELocal products and capacity
lemmaLemma 5.27.11C56C5CBA0E0DLocal products and capacity
equationEquation 5.27.12EBF435BDB6A0Local products and capacity
subsection5.27.13 Polar parts on curves1953F5546AC6Local products and capacity
lemmaLemma 5.27.14B54F79E95767Local products and capacity
corollaryCorollary 5.27.15F9517F96057FLocal products and capacity
equationEquation 5.27.166FC7AD4CB464Local products and capacity
subsection5.27.17 Products at small pole sets486795BA5E5CLocal products and capacity
theoremTheorem 5.27.181BBBF8109419Local products and capacity
lemmaLemma 5.27.19F7D352207205Local products and capacity
subsection5.27.20 Sobolev regularity at divisorial polesC37C319849E5Local products and capacity
lemmaLemma 5.27.21C3F45D193092Local products and capacity
propositionProposition 5.27.224BF5A61AFC51Local products and capacity
propositionProposition 5.27.235B874925BCA1Local products and capacity
exampleExample 5.27.244B607387D3F1Local products and capacity
subsection5.27.25 Spherical Monge–Ampère measuresC95109EA4FF8Local products and capacity
lemmaLemma 5.27.263242D4BA1F13Local products and capacity
subsection5.27.27 Measure concentration under integrable convergence4FE21E50E832Local products and capacity
lemmaLemma 5.27.2855862640C106Local products and capacity
lemmaLemma 5.27.29E2307F7792A2Local products and capacity
equationEquation 5.27.3097724F2A7D19Local products and capacity
propositionProposition 5.27.31026A64B7A5A2Local products and capacity
subsection5.27.32 Adding a prescribed local measure38F5E7BC8433Local products and capacity
lemmaLemma 5.27.330EE345C93AFALocal products and capacity
lemmaLemma 5.27.340950BB9BC118Local products and capacity
propositionProposition 5.27.356C1E4E8060FFLocal products and capacity
theoremTheorem 5.27.36CCB8860FB3B5Local products and capacity
subsection5.27.37 Full cluster sets at the zero potentialC9D7C212A8BFLocal products and capacity
lemmaLemma 5.27.387F112BE621FDLocal products and capacity
theoremTheorem 5.27.39DCCBB28F9C78Local products and capacity
subsection5.27.40 Cluster sets at a bounded backgroundA72166484AB2Local products and capacity
lemmaLemma 5.27.41CD407359DDEALocal products and capacity
theoremTheorem 5.27.42745A94C03209Local products and capacity
equationEquation 5.28.1C87F0ACE9C76Products with smooth backgrounds
propositionProposition 5.28.23EDAB69A7289Products with smooth backgrounds
lemmaLemma 5.28.35087175B2C39Products with smooth backgrounds
lemmaLemma 5.28.4636D22C6199BProducts with smooth backgrounds
propositionProposition 5.28.56CBD4E4A94B6Products with smooth backgrounds
propositionProposition 5.28.62A843EB4B4AEProducts with smooth backgrounds
lemmaLemma 5.28.76C81352D9EE7Products with smooth backgrounds
equationEquation 5.28.8C06268DF902DProducts with smooth backgrounds
equationEquation 5.28.94547B4E0976DProducts with smooth backgrounds
lemmaLemma 5.28.10C5B4B944142CProducts with smooth backgrounds
equationEquation 5.28.11D12EFD5440D2Products with smooth backgrounds
lemmaLemma 5.28.1200AB47766404Products with smooth backgrounds
lemmaLemma 5.28.13D263A93C1D69Products with smooth backgrounds
subsection5.29.1 Capacity estimates1C5C06D2C5D9Hermitian capacity
lemmaLemma 5.29.25CEF121C5391Hermitian capacity
equationEquation 5.29.387E51BAEC1C0Hermitian capacity
equationEquation 5.29.43EF8F920CDD3Hermitian capacity
corollaryCorollary 5.29.5EACB00126847Hermitian capacity
subsection5.29.6 Integration against a Hölder referenceCACF6025C681Hermitian capacity
lemmaLemma 5.29.7D9AF18430F97Hermitian capacity
equationEquation 5.29.816B648CFC04AHermitian capacity
corollaryCorollary 5.29.9F701C1999BDCHermitian capacity
lemmaLemma 5.29.10EF54C7C9A0A8Hermitian capacity
subsection5.29.11 Capacity bounds for Hölder references510FD09C306AHermitian capacity
lemmaLemma 5.29.121017EA20E8D8Hermitian capacity
propositionProposition 5.29.13D00B685B1791Hermitian capacity
subsection5.29.14 Convergence of potentials and measures97C040D849DDHermitian capacity
lemmaLemma 5.29.15DF96BD63B553Hermitian capacity
lemmaLemma 5.29.16F56E1BEB2997Hermitian capacity
lemmaLemma 5.29.177EE2FAA017DEHermitian capacity
lemmaLemma 5.29.18D0C7DB81F772Hermitian capacity
lemmaLemma 5.29.19505602F4CB54Hermitian capacity
lemmaLemma 5.29.2031C5645526CAHermitian capacity
subsection5.29.21 Vanishing area on curvesB092BD05E335Hermitian capacity
lemmaLemma 5.29.22E8D07C38BC7CHermitian capacity
theoremTheorem 5.29.23FD7A4AE0DAC6Hermitian capacity
lemmaLemma 5.30.12319B6057130Comparing local regularizations
equationEquation 5.30.24BCD629A10F1Comparing local regularizations
corollaryCorollary 5.30.3ACA94F989A1EComparing local regularizations
equationEquation 5.31.138E79A7BB294Gluing with a small curvature error
propositionProposition 5.31.2752BA1321BFAGluing with a small curvature error
equationEquation 5.31.3BF9EF9467EBBGluing with a small curvature error
theoremTheorem 5.32.1240BA0465940Regularization in a fixed Hermitian background
propositionProposition 5.33.1EFC5ED339FC3Singularity type and multiplier ideals
propositionProposition 5.33.232843D040902Singularity type and multiplier ideals
exampleExample 5.33.336A8480CE7DFSingularity type and multiplier ideals
propositionProposition 5.34.1AAD8908A0CF9Analytic types and restriction
subsection5.34.2 Lelong level setsE911F3E2C8E9Analytic types and restriction
corollaryCorollary 5.34.3C6F4FBF73B9EAnalytic types and restriction
exampleExample 5.34.4697E97CED780Analytic types and restriction
propositionProposition 5.35.1DBEB8E0E0318The local Bergman estimates
equationEquation 5.35.2BFDE2F85F930The local Bergman estimates
equationEquation 5.35.3F9AE1A865ACDThe local Bergman estimates
lemmaLemma 5.35.4FE03F7C266DDThe local Bergman estimates
theoremTheorem 5.36.1 (Demailly regularization)3CA98F11CC5BThe global curvature bound
corollaryCorollary 5.36.28F2184BBF4C4The global curvature bound
subsection5.36.3 Regularization of closed currentsD4C469F61056The global curvature bound
subsection5.36.4 Smooth remainders after resolution70049B9396E7The global curvature bound
theoremTheorem 5.36.5 (Demailly regularization)88038BD901A0The global curvature bound
lemmaLemma 5.37.1C21A2A6F9E07Multiplier ideals and logarithmic poles
lemmaLemma 5.37.2EBB1890494B1Multiplier ideals and logarithmic poles
theoremTheorem 5.37.33F1FC2F620B4Multiplier ideals and logarithmic poles
lemmaLemma 5.37.4ACC2529A680BMultiplier ideals and logarithmic poles
lemmaLemma 5.37.55929CFA4B87FMultiplier ideals and logarithmic poles
equationEquation 5.37.6582A3343A949Multiplier ideals and logarithmic poles
lemmaLemma 5.37.72FA053FE56D9Multiplier ideals and logarithmic poles
exampleExample 5.37.8E39EAA145A67Multiplier ideals and logarithmic poles
subsection5.37.9 Weighted distance from a sum of multiplier ideals1B01826812A9Multiplier ideals and logarithmic poles
lemmaLemma 5.37.1095217B5924C0Multiplier ideals and logarithmic poles
equationEquation 5.37.11363923246C67Multiplier ideals and logarithmic poles
equationEquation 5.37.1235D5D6699C73Multiplier ideals and logarithmic poles
equationEquation 5.37.1304CF8B360946Multiplier ideals and logarithmic poles
propositionProposition 5.37.14B929E98153F2Multiplier ideals and logarithmic poles
theoremTheorem 5.37.155433D9F2E00CMultiplier ideals and logarithmic poles
equationEquation 5.37.16646E9C7EF985Multiplier ideals and logarithmic poles
subsection5.37.17 The optimal constant in dimension oneA5A50BF6024BMultiplier ideals and logarithmic poles
lemmaLemma 5.37.18750629715507Multiplier ideals and logarithmic poles
equationEquation 5.37.19B6502E786DEEMultiplier ideals and logarithmic poles
lemmaLemma 5.37.209AA5F4F533CDMultiplier ideals and logarithmic poles
equationEquation 5.37.21594319E5AC8FMultiplier ideals and logarithmic poles
theoremTheorem 5.37.22E767909834ADMultiplier ideals and logarithmic poles
equationEquation 5.37.2317133E696650Multiplier ideals and logarithmic poles
subsection5.37.24 Distance for several hyperplane weightsA632F1B7CA50Multiplier ideals and logarithmic poles
exampleExample 5.37.259F227CC9F378Multiplier ideals and logarithmic poles
equationEquation 5.37.264F7B8ACF0243Multiplier ideals and logarithmic poles
lemmaLemma 5.38.13AA98220C897Lelong numbers and divisorial estimates
lemmaLemma 5.38.24296DE167374Lelong numbers and divisorial estimates
equationEquation 5.38.33B6E06712331Lelong numbers and divisorial estimates
lemmaLemma 5.38.47A2D4C65C667Lelong numbers and divisorial estimates
lemmaLemma 5.38.565C35F77CB06Lelong numbers and divisorial estimates
corollaryCorollary 5.38.6F5277854ECB3Lelong numbers and divisorial estimates
lemmaLemma 5.38.7E04E4D6C502ELelong numbers and divisorial estimates
lemmaLemma 5.38.81792691862B8Lelong numbers and divisorial estimates
lemmaLemma 5.38.91525C5219E69Lelong numbers and divisorial estimates
lemmaLemma 5.38.10548A845099FBLelong numbers and divisorial estimates
lemmaLemma 5.38.111D91BF7CA736Lelong numbers and divisorial estimates
lemmaLemma 5.38.123CF548646A75Lelong numbers and divisorial estimates
theoremTheorem 5.39.1230882F1D833Potential regularity and metric distance
equationEquation 5.39.233F399DDA815Potential regularity and metric distance
equationEquation 5.39.3A1062D2A5198Potential regularity and metric distance
subsection5.39.4 Dini moduli and limits of distances4B8CF50FE6DCPotential regularity and metric distance
equationEquation 5.39.506D8010B670CPotential regularity and metric distance
theoremTheorem 5.39.621429B9E1BF3Potential regularity and metric distance
corollaryCorollary 5.39.7380EB17A4350Potential regularity and metric distance
corollaryCorollary 5.39.8D3B517FBCE4FPotential regularity and metric distance
lemmaLemma 5.40.1294BB4EEA84FObstacle envelopes
lemmaLemma 5.40.2AA358481B7F6Obstacle envelopes
propositionProposition 5.40.3630365A8F8B8Obstacle envelopes
definitionDefinition 5.40.465C9C3635D58Obstacle envelopes
lemmaLemma 5.40.544FD5A853A0AObstacle envelopes
propositionProposition 5.40.6F031A16FF76FObstacle envelopes
lemmaLemma 5.40.74798FCFAC293Obstacle envelopes
equationEquation 5.40.828CB2DBBAEF2Obstacle envelopes
subsection5.40.9 A hyperplane Green envelope on the ball765486939AA3Obstacle envelopes
propositionProposition 5.40.10B986B6AE211BObstacle envelopes
equationEquation 5.40.117009D7A8D687Obstacle envelopes
propositionProposition 5.40.121709375DAB71Obstacle envelopes
equationEquation 5.40.13F4D9A718C648Obstacle envelopes
equationEquation 5.40.141BF247F761E6Obstacle envelopes
exampleExample 5.40.15F5EC436555E6Obstacle envelopes
lemmaLemma 5.41.1ADD743D2224CThe minimum principle
corollaryCorollary 5.41.2B98B62E1E7D5The minimum principle
propositionProposition 5.41.3ED1DA0551379The minimum principle
subsection5.41.4 Pointwise Legendre transforms63DEBF0DF568The minimum principle
lemmaLemma 5.41.5742009E66DA3The minimum principle
equationEquation 5.41.60CEA269E6373The minimum principle
equationEquation 5.41.7932E0DC4EFDDThe minimum principle
lemmaLemma 5.41.8FAE98D3A3F69The minimum principle
theoremTheorem 5.41.95EA08D4F62B2The minimum principle
lemmaLemma 5.42.1D38324EEBF88Chern-connection regularization
lemmaLemma 5.42.2490E5EB01AE4Chern-connection regularization
subsection5.42.3 The general curvature modulus45B2E87FDE47Chern-connection regularization
lemmaLemma 5.42.4D0B76A103405Chern-connection regularization
lemmaLemma 5.42.5A340B19CD0CCChern-connection regularization
subsection5.42.6 Hölder reference approximation and weighted smoothing91A88ECF3F18Chern-connection regularization
lemmaLemma 5.42.73AB7626C8759Chern-connection regularization
lemmaLemma 5.42.8AEDE3F4F53ECChern-connection regularization
subsection5.42.9 A linear error for a smooth background23459D626290Chern-connection regularization
lemmaLemma 5.42.1096444124D09CChern-connection regularization
equationEquation 5.42.11E762B356717BChern-connection regularization
subsection5.43.1 Holomorphic disks1F1A3CC0946EPullback and descent
lemmaLemma 5.43.212585F177470Pullback and descent
subsection5.43.3 Compact fibersA5D491DF2CA3Pullback and descent
lemmaLemma 5.43.4074D8F716632Pullback and descent
propositionProposition 5.43.58BFC96F6E5F8Pullback and descent
subsection5.43.6 Maps to a curve3110CB46064EPullback and descent
propositionProposition 5.43.747C6B39BA71APullback and descent
lemmaLemma 5.43.811E6C3E4D36CPullback and descent
corollaryCorollary 5.43.95B3D7BB20499Pullback and descent
lemmaLemma 5.43.10633794C1EFC2Pullback and descent
lemmaLemma 5.43.11CE5106030996Pullback and descent
subsection5.43.12 The curve of fiber components35C176E64D5BPullback and descent
propositionProposition 5.43.13F2BB0D2DE2F4Pullback and descent
subsection5.43.14 Finite maps between curves1C4BCE809D00Pullback and descent
lemmaLemma 5.43.150018DC5A7DD0Pullback and descent
lemmaLemma 5.43.16151A1D744DE7Pullback and descent
subsection5.43.17 ModificationsD643D283EFB3Pullback and descent
propositionProposition 5.43.18A7090F92CA39Pullback and descent
lemmaLemma 5.43.19C07C7BB26EF8Pullback and descent
lemmaLemma 5.43.20D7CDD8CAD2ACPullback and descent
subsection5.43.21 Multiplier ideals under modificationsF75CE9F0BB10Pullback and descent
lemmaLemma 5.43.22599D8A3D42BCPullback and descent
lemmaLemma 5.43.236AA2493173A9Pullback and descent
lemmaLemma 5.43.2483C3A394E5BBPullback and descent
subsection5.43.25 Restriction of analytic approximantsB669E8E3479EPullback and descent
corollaryCorollary 5.43.2650D99B4B0660Pullback and descent
subsection5.44.1 A fixed measure48FF0BFC07F6Integration against non-pluripolar measures
lemmaLemma 5.44.2983B3E6349BBIntegration against non-pluripolar measures
subsection5.44.3 Vanishing exponents7442B45F8567Integration against non-pluripolar measures
propositionProposition 5.44.4A89FF7CB0129Integration against non-pluripolar measures
lemmaLemma 5.44.5D260E972F57CIntegration against non-pluripolar measures
subsection5.44.6 Measures with bounded potentials80117515F812Integration against non-pluripolar measures
lemmaLemma 5.44.727CEBF2AA643Integration against non-pluripolar measures
equationEquation 5.44.8CDF5AC2B6D12Integration against non-pluripolar measures
subsection5.44.9 Smoothing the measure37976A2408E0Integration against non-pluripolar measures
lemmaLemma 5.44.102E102F06172BIntegration against non-pluripolar measures
equationEquation 5.44.11D656A5C5B2DCIntegration against non-pluripolar measures
lemmaLemma 5.44.1288E063207C1BIntegration against non-pluripolar measures
equationEquation 5.44.132D0225A9A28DIntegration against non-pluripolar measures
propositionProposition 5.45.1620B099907ABCurvature of line bundles
lemmaLemma 5.46.18F61C8BDD569Polar curvature
lemmaLemma 5.46.2EB97F93DAD74Polar curvature
propositionProposition 5.46.309F56C9DE42CPolar curvature
lemmaLemma 5.46.43B857D475432Polar curvature
propositionProposition 5.46.55AAB9E80546APolar curvature
equationEquation 5.46.6F51804EDE063Polar curvature
propositionProposition 5.46.74A6AA259F9D8Polar curvature

6. Lelong numbers, multiplier ideals, and regularization

propositionProposition 6.2.1B9A2266E14FEHilbert spaces and closed operator complexes
propositionProposition 6.2.21EF937410C2FHilbert spaces and closed operator complexes
lemmaLemma 6.2.30C82C2C21EE7Hilbert spaces and closed operator complexes
theoremTheorem 6.2.4FDE87BE964C7Hilbert spaces and closed operator complexes
definitionDefinition 6.3.110E4AE8BC531Pseudoconvexity and boundary distance
lemmaLemma 6.3.25898ED42C26FPseudoconvexity and boundary distance
lemmaLemma 6.3.3 (Richberg)52930869627BPseudoconvexity and boundary distance
theoremTheorem 6.3.4EA195CC9592DPseudoconvexity and boundary distance
corollaryCorollary 6.3.5FFB7E3647C0BPseudoconvexity and boundary distance
corollaryCorollary 6.3.6EC970F3402EBPseudoconvexity and boundary distance
equationEquation 6.4.10B44CA843C7BInterior density and the weighted estimate
lemmaLemma 6.4.2F03962A0AC31Interior density and the weighted estimate
equationEquation 6.4.33B7B834E5CD4Interior density and the weighted estimate
propositionProposition 6.4.4866FA0174B79Interior density and the weighted estimate
equationEquation 6.4.5038AC5695B6FInterior density and the weighted estimate
equationEquation 6.4.635C885FDEB1EInterior density and the weighted estimate
theoremTheorem 6.4.7 (Hörmander)F85650C9CE6AInterior density and the weighted estimate
equationEquation 6.4.8D557EE261608Interior density and the weighted estimate
theoremTheorem 6.4.988735CB0C076Interior density and the weighted estimate
lemmaLemma 6.4.10961105705CF9Interior density and the weighted estimate
propositionProposition 6.5.166BCDC65B27DInterpolation and extension across a hyperplane
lemmaLemma 6.5.20822DB927561Interpolation and extension across a hyperplane
lemmaLemma 6.5.3EE99921A3F33Interpolation and extension across a hyperplane
theoremTheorem 6.5.4 (Ohsawa–Takegoshi)D315377B239DInterpolation and extension across a hyperplane
equationEquation 6.5.55C2AD1325732Interpolation and extension across a hyperplane
equationEquation 6.5.6FF641A8A654CInterpolation and extension across a hyperplane
equationEquation 6.5.7F388046B5B2AInterpolation and extension across a hyperplane
corollaryCorollary 6.5.833DEA3F4EDB8Interpolation and extension across a hyperplane
corollaryCorollary 6.5.9BFB355F9DDF9Interpolation and extension across a hyperplane
propositionProposition 6.7.1C79BAEA70FC1Logarithmic growth and the Jensen formula
equationEquation 6.7.2A6BF25586D16Logarithmic growth and the Jensen formula
equationEquation 6.7.3EC4655061F9FLogarithmic growth and the Jensen formula
definitionDefinition 6.7.4D9ACE2E4ECE5Logarithmic growth and the Jensen formula
equationEquation 6.7.5D57591458420Logarithmic growth and the Jensen formula
theoremTheorem 6.7.6 (Jensen–Lelong)BD2098F182CCLogarithmic growth and the Jensen formula
equationEquation 6.7.72D5BD01BDC4DLogarithmic growth and the Jensen formula
equationEquation 6.7.859A9BF2835E9Logarithmic growth and the Jensen formula
propositionProposition 6.7.943A1E607F822Logarithmic growth and the Jensen formula
equationEquation 6.7.10B5D2B6B6970ELogarithmic growth and the Jensen formula
lemmaLemma 6.8.18312218F0503Coherent ideals from weighted integrability
lemmaLemma 6.8.28DDA48CD81CBCoherent ideals from weighted integrability
lemmaLemma 6.8.39B47698DCA71Coherent ideals from weighted integrability
lemmaLemma 6.8.4 (Strong Noetherian property)4C0C502048AACoherent ideals from weighted integrability
definitionDefinition 6.8.5990C506B7D25Coherent ideals from weighted integrability
lemmaLemma 6.8.6E20924A743CACoherent ideals from weighted integrability
equationEquation 6.8.769805453021CCoherent ideals from weighted integrability
equationEquation 6.8.8D76FBA2B036ACoherent ideals from weighted integrability
theoremTheorem 6.8.9FE623FA661C4Coherent ideals from weighted integrability
lemmaLemma 6.8.103B37857D5B64Coherent ideals from weighted integrability
equationEquation 6.8.1111D2B663BB36Coherent ideals from weighted integrability
lemmaLemma 6.9.16947654B2D6FAnalytic Lelong level sets and generic values
equationEquation 6.9.2E47D344FBFC4Analytic Lelong level sets and generic values
corollaryCorollary 6.9.3A3BCC047C1A7Analytic Lelong level sets and generic values
exampleExample 6.9.42AA6A0B94DE7Analytic Lelong level sets and generic values
theoremTheorem 6.9.5 (Siu)946909269108Analytic Lelong level sets and generic values
equationEquation 6.9.675410ECF073EAnalytic Lelong level sets and generic values
propositionProposition 6.9.73E1F53764C07Analytic Lelong level sets and generic values
propositionProposition 6.9.8B720E73C2654Analytic Lelong level sets and generic values
theoremTheorem 6.10.1DFD10C71D607Restriction, products and integral closure
equationEquation 6.10.25BDE1812FA47Restriction, products and integral closure
equationEquation 6.10.301EC6A2D5998Restriction, products and integral closure
equationEquation 6.10.40EA6BB9CDFD8Restriction, products and integral closure
corollaryCorollary 6.10.505791D2DFFAARestriction, products and integral closure
propositionProposition 6.10.6675539E4A76CRestriction, products and integral closure
lemmaLemma 6.11.1642017869F45Strong openness by extension from moving slices
lemmaLemma 6.11.2D7BCCD73BD13Strong openness by extension from moving slices
equationEquation 6.11.37264E59B5291Strong openness by extension from moving slices
theoremTheorem 6.11.4FF9DF4BE3CA8Strong openness by extension from moving slices
equationEquation 6.11.5CF22C9DDEA36Strong openness by extension from moving slices
corollaryCorollary 6.11.6 (Strong openness)EA6067858516Strong openness by extension from moving slices
equationEquation 6.11.7B5400B2F91DAStrong openness by extension from moving slices
exampleExample 6.11.8769FF9D2A9B9Strong openness by extension from moving slices
lemmaLemma 6.12.1BCE8B698A66DBirational change of variables and logarithmic thresholds
lemmaLemma 6.12.223099F8BA856Birational change of variables and logarithmic thresholds
definitionDefinition 6.12.3B5514FFD42D1Birational change of variables and logarithmic thresholds
propositionProposition 6.12.458995E4B098ABirational change of variables and logarithmic thresholds
equationEquation 6.12.5DE847DC56B24Birational change of variables and logarithmic thresholds
equationEquation 6.12.6093593AE066FBirational change of variables and logarithmic thresholds
equationEquation 6.12.785BE6E2D5F01Birational change of variables and logarithmic thresholds
theoremTheorem 6.12.85BA222439952Birational change of variables and logarithmic thresholds
equationEquation 6.12.92874465D0AEFBirational change of variables and logarithmic thresholds
equationEquation 6.12.10784A47900871Birational change of variables and logarithmic thresholds
lemmaLemma 6.12.115E8D84189EA2Birational change of variables and logarithmic thresholds
propositionProposition 6.12.12966A90DCC98EBirational change of variables and logarithmic thresholds
equationEquation 6.12.138D2BB5C6ED3CBirational change of variables and logarithmic thresholds
corollaryCorollary 6.12.14AF618D7A70E2Birational change of variables and logarithmic thresholds
equationEquation 6.12.15D84BB304456DBirational change of variables and logarithmic thresholds
theoremTheorem 6.12.1671AAFAF9C564Birational change of variables and logarithmic thresholds
equationEquation 6.12.17A3A0B2553330Birational change of variables and logarithmic thresholds
equationEquation 6.12.184955871DC2AEBirational change of variables and logarithmic thresholds
exampleExample 6.12.19B10A1F1C864BBirational change of variables and logarithmic thresholds
lemmaLemma 6.14.16B84DAE83456Smooth backgrounds and analytic singularities
lemmaLemma 6.14.287D4C2281D7CSmooth backgrounds and analytic singularities
definitionDefinition 6.14.36A70B026D8EFSmooth backgrounds and analytic singularities
lemmaLemma 6.14.4066780FF9DDDSmooth backgrounds and analytic singularities
definitionDefinition 6.14.56A7E913297D5Smooth backgrounds and analytic singularities
equationEquation 6.14.660D977A9A056Smooth backgrounds and analytic singularities
lemmaLemma 6.14.77110958B611BSmooth backgrounds and analytic singularities
propositionProposition 6.14.8AFD530210C3ASmooth backgrounds and analytic singularities
definitionDefinition 6.14.9D022961325A2Smooth backgrounds and analytic singularities
lemmaLemma 6.14.101088A4C2B5CASmooth backgrounds and analytic singularities
propositionProposition 6.14.111A99E612D41CSmooth backgrounds and analytic singularities
definitionDefinition 6.14.124E814A4EB7D1Smooth backgrounds and analytic singularities
lemmaLemma 6.15.1C2732E72E566Local Bergman kernels and ideal generators
equationEquation 6.15.22D48BBAB29E1Local Bergman kernels and ideal generators
lemmaLemma 6.15.309CBB2B6BA0FLocal Bergman kernels and ideal generators
theoremTheorem 6.15.45AEABFD276A6Local Bergman kernels and ideal generators
equationEquation 6.15.5CDB3DBD2FFAALocal Bergman kernels and ideal generators
equationEquation 6.15.6CD15DA06F0BBLocal Bergman kernels and ideal generators
lemmaLemma 6.15.7365648DE5857Local Bergman kernels and ideal generators
lemmaLemma 6.16.16A23B839899EOverlap interpolation and quantitative patching
lemmaLemma 6.16.282376665E5F7Overlap interpolation and quantitative patching
theoremTheorem 6.16.3E9EBCA5A681DOverlap interpolation and quantitative patching
lemmaLemma 6.17.1EFA6E7FF99E6Smooth cutoff sums and decreasing approximation
equationEquation 6.17.29F558292FE56Smooth cutoff sums and decreasing approximation
lemmaLemma 6.17.3C32B34A0713BSmooth cutoff sums and decreasing approximation
lemmaLemma 6.17.47EFB7FFA3AB8Smooth cutoff sums and decreasing approximation
definitionDefinition 6.17.5165F88FAE668Smooth cutoff sums and decreasing approximation
theoremTheorem 6.17.6080EF8C715CFSmooth cutoff sums and decreasing approximation
lemmaLemma 6.18.1BA49EE8EFFB8Comparison of analytic singularities
propositionProposition 6.18.2649F62FDFBADComparison of analytic singularities
exampleExample 6.18.3E35B96BEFDE2Comparison of analytic singularities
definitionDefinition 6.19.1574EEB86E59FBounded growth ideals
theoremTheorem 6.19.2993CE3FB0997Bounded growth ideals
corollaryCorollary 6.19.3BA0E41A3C821Bounded growth ideals

7. Local Monge–Ampère theory and capacity

definitionDefinition 7.2.1254CDCCCAB5EBounded potentials and recursive products
propositionProposition 7.2.2D9911E035563Bounded potentials and recursive products
equationEquation 7.2.3EDF346A35F76Bounded potentials and recursive products
equationEquation 7.2.4706A589A680CBounded potentials and recursive products
theoremTheorem 7.2.5 (Chern–Levine–Nirenberg)2A84C325AB58Bounded potentials and recursive products
propositionProposition 7.2.62F39A3F97E8EBounded potentials and recursive products
equationEquation 7.3.16A8B7EB02562Decreasing continuity and symmetry
equationEquation 7.3.29789F9473996Decreasing continuity and symmetry
lemmaLemma 7.3.39C6450BF0E32Decreasing continuity and symmetry
lemmaLemma 7.3.4DE1E6EDC0B48Decreasing continuity and symmetry
lemmaLemma 7.3.5C04BF828EE91Decreasing continuity and symmetry
lemmaLemma 7.3.64286535DCE99Decreasing continuity and symmetry
theoremTheorem 7.3.7 (Bedford–Taylor)D87C13B3250ADecreasing continuity and symmetry
corollaryCorollary 7.3.88BD7803A393BDecreasing continuity and symmetry
equationEquation 7.3.98BEFFD328D36Decreasing continuity and symmetry
propositionProposition 7.3.100453AAB39932Decreasing continuity and symmetry
theoremTheorem 7.3.11D0AF90DDC7A5Decreasing continuity and symmetry
corollaryCorollary 7.3.12F2BBEF3BF09BDecreasing continuity and symmetry
theoremTheorem 7.4.17C68264563C0Weighted estimates and Monge–Ampère capacity
definitionDefinition 7.4.2E9E7C75396D2Weighted estimates and Monge–Ampère capacity
propositionProposition 7.4.397EFDB6FDEE3Weighted estimates and Monge–Ampère capacity
propositionProposition 7.4.4815E5BBFC34AWeighted estimates and Monge–Ampère capacity
propositionProposition 7.4.5 (Błocki)EB80820D2146Weighted estimates and Monge–Ampère capacity
equationEquation 7.4.628751508F174Weighted estimates and Monge–Ampère capacity
definitionDefinition 7.5.1A2CACA24130BQuasi-continuity and convergence in capacity
theoremTheorem 7.5.2C39FEC1AB985Quasi-continuity and convergence in capacity
lemmaLemma 7.5.3D127F8356D96Quasi-continuity and convergence in capacity
lemmaLemma 7.5.456B45ADBA326Quasi-continuity and convergence in capacity
definitionDefinition 7.5.5DAD5D0809372Quasi-continuity and convergence in capacity
propositionProposition 7.5.69542ED69E1C1Quasi-continuity and convergence in capacity
lemmaLemma 7.5.7880C30054932Quasi-continuity and convergence in capacity
theoremTheorem 7.5.842828E3BDDFAQuasi-continuity and convergence in capacity
theoremTheorem 7.6.1A4C45B704647Increasing limits and exceptional envelopes
corollaryCorollary 7.6.2D865A8538D7DIncreasing limits and exceptional envelopes
lemmaLemma 7.7.14521AF3E3E5CThe homogeneous Dirichlet problem and balayage
lemmaLemma 7.7.20E5A3E632224The homogeneous Dirichlet problem and balayage
equationEquation 7.7.3332CD5FE501EThe homogeneous Dirichlet problem and balayage
theoremTheorem 7.7.49181532558D7The homogeneous Dirichlet problem and balayage
corollaryCorollary 7.7.52E8B947536C1The homogeneous Dirichlet problem and balayage
definitionDefinition 7.7.6FD8547B922C9The homogeneous Dirichlet problem and balayage
theoremTheorem 7.7.7BCFDDA1FCB7AThe homogeneous Dirichlet problem and balayage
lemmaLemma 7.8.1BB19C89586B1Density lower bounds and mixed products
lemmaLemma 7.8.21BE242B641AADensity lower bounds and mixed products
propositionProposition 7.8.3EB2AFC8FB17CDensity lower bounds and mixed products
corollaryCorollary 7.8.4EADDEAF9FAD9Density lower bounds and mixed products
subsection7.8.5 Several independent factors7E603FDE8D9ADensity lower bounds and mixed products
lemmaLemma 7.8.691F43DBA51F0Density lower bounds and mixed products
equationEquation 7.8.7E27D7A891B69Density lower bounds and mixed products
corollaryCorollary 7.8.8142C8110B540Density lower bounds and mixed products
lemmaLemma 7.9.10FF4AFFE0ED0The ball Dirichlet problem with continuous density
theoremTheorem 7.9.2A092925717E1The ball Dirichlet problem with continuous density
equationEquation 7.9.3B8751A529320The ball Dirichlet problem with continuous density
lemmaLemma 7.10.1C09E42A43E93Bounded measurable densities
theoremTheorem 7.10.2C9116FCEFCD2Bounded measurable densities
definitionDefinition 7.11.1E1129C8F823CLocal energy moments
theoremTheorem 7.11.205F71889C324Local energy moments
equationEquation 7.11.3D1E50BAC729CLocal energy moments
equationEquation 7.11.49778E8C0159ELocal energy moments
propositionProposition 7.11.51151DF7B32E4Local energy moments
equationEquation 7.11.6CC0546ABBCE7Local energy moments
equationEquation 7.11.700E17178E4CBLocal energy moments
subsection7.11.8 Convergence of energy integrals6293A9FE1CA2Local energy moments
lemmaLemma 7.11.9504F4EDE59EFLocal energy moments
equationEquation 7.11.100B2BC6EDF748Local energy moments
theoremTheorem 7.11.1178EB0192D174Local energy moments
equationEquation 7.11.120991A86E50A6Local energy moments
equationEquation 7.11.132AE59414D95CLocal energy moments
equationEquation 7.11.1467A2F5CF1C50Local energy moments
equationEquation 7.11.15166F0CCED71ALocal energy moments
subsection7.11.16 Capacity convergence and truncated measures7449912C4F37Local energy moments
equationEquation 7.11.178748F8035D18Local energy moments
lemmaLemma 7.11.18E1A777307E3DLocal energy moments
lemmaLemma 7.11.19546063653198Local energy moments
equationEquation 7.11.20B1ADA2B95634Local energy moments
equationEquation 7.11.2192D8C50415C8Local energy moments
propositionProposition 7.11.221036F9F7DDDALocal energy moments
equationEquation 7.11.2352F9E68B74CFLocal energy moments
theoremTheorem 7.11.24D25577E2BC26Local energy moments
equationEquation 7.11.258C2FAF9DC64BLocal energy moments
subsection7.11.26 A bounded subsolution and the limiting equation7B2778E2F356Local energy moments
lemmaLemma 7.11.27A2369B2DB317Local energy moments
equationEquation 7.11.285F846CC6D336Local energy moments
corollaryCorollary 7.11.2925E0E9371996Local energy moments
theoremTheorem 7.11.306C02DFB33B7CLocal energy moments
equationEquation 7.11.315038E73ED56BLocal energy moments
subsection7.11.32 Mixed energy and truncationA4D9C4FF9512Local energy moments
theoremTheorem 7.11.33A26D05A19EF5Local energy moments
equationEquation 7.11.34802CC6F04E1FLocal energy moments
equationEquation 7.11.35880A8A0EC0A2Local energy moments
equationEquation 7.11.369F29C0C8E20ELocal energy moments
equationEquation 7.11.3777DC238179EBLocal energy moments
corollaryCorollary 7.11.3811945FC061ECLocal energy moments
theoremTheorem 7.12.1AAD1BC0CC6C4Mixed products for a common measure
equationEquation 7.12.267F72EC709B4Mixed products for a common measure
equationEquation 7.12.385B06455E0FFMixed products for a common measure
equationEquation 7.12.4EF405F42095EMixed products for a common measure
equationEquation 7.12.5E7114DA296F9Mixed products for a common measure
corollaryCorollary 7.12.60B504335901FMixed products for a common measure
equationEquation 7.12.7D13F44EA144AMixed products for a common measure
corollaryCorollary 7.12.87071C55A0E4EMixed products for a common measure
lemmaLemma 7.13.18736729F21D9Bounded complex Hessians and contact sets
lemmaLemma 7.13.2D57C89462C1BBounded complex Hessians and contact sets
propositionProposition 7.13.3C006456D2AFCBounded complex Hessians and contact sets
propositionProposition 7.13.41466731348C1Bounded complex Hessians and contact sets
equationEquation 7.14.19148FE66924CGradients against positive currents
equationEquation 7.14.292EE3E85B474Gradients against positive currents
lemmaLemma 7.14.38B33E7CC2E18Gradients against positive currents
equationEquation 7.14.4F48E1DFF8563Gradients against positive currents
equationEquation 7.14.5E07A2779E1E2Gradients against positive currents
equationEquation 7.14.614D420FC6F06Gradients against positive currents

8. Complex Hessian equations

equationEquation 8.2.1C5A4D9AE83EDThe admissible cone
equationEquation 8.2.2B372916B922FThe admissible cone
lemmaLemma 8.2.3F86594126C69The admissible cone
lemmaLemma 8.2.494C4E018DD25The admissible cone
lemmaLemma 8.2.57710D01DF6F5The admissible cone
lemmaLemma 8.2.6473C52E484B0The admissible cone
lemmaLemma 8.2.7B0643A9A1173The admissible cone
lemmaLemma 8.2.88673FC834527The admissible cone
lemmaLemma 8.2.9365519F85C5AThe admissible cone
lemmaLemma 8.2.10A1FFEE061CF5The admissible cone
lemmaLemma 8.3.189F810E43B3ELiouville and gradient estimates
theoremTheorem 8.3.296D01E7C4F0FLiouville and gradient estimates
propositionProposition 8.3.3B863BB49A505Liouville and gradient estimates
lemmaLemma 8.4.1B276B9DFAC29A trace estimate with a prescribed potential
lemmaLemma 8.4.261C6DAF48B41A trace estimate with a prescribed potential
propositionProposition 8.4.3FDAFFF47B8A3A trace estimate with a prescribed potential
lemmaLemma 8.4.4C357698AC1AFA trace estimate with a prescribed potential
lemmaLemma 8.4.5174BDA603BF2A trace estimate with a prescribed potential
propositionProposition 8.4.623966DC8CDBFA trace estimate with a prescribed potential
corollaryCorollary 8.4.7BEA27E46DAEEA trace estimate with a prescribed potential
lemmaLemma 8.5.189771FD1FD15Bounded complex Hessians and elliptic extensions
lemmaLemma 8.5.21881F6670EF4Bounded complex Hessians and elliptic extensions
lemmaLemma 8.5.3BAC60298D0F2Bounded complex Hessians and elliptic extensions
propositionProposition 8.5.46FBA7BB8AC94Bounded complex Hessians and elliptic extensions
lemmaLemma 8.6.18B310CC2C1A1Proper equations and admissible approximation
lemmaLemma 8.6.2B8367971A40DProper equations and admissible approximation
subsection8.6.3 Decreasing approximation of bounded potentialsABD764A7564FProper equations and admissible approximation
lemmaLemma 8.6.4EBF9BA687F39Proper equations and admissible approximation
lemmaLemma 8.6.5DFE78235C5E9Proper equations and admissible approximation
theoremTheorem 8.6.668BE4F589DCEProper equations and admissible approximation
propositionProposition 8.7.1D2A0F255F625Continuous products, envelopes and domination
lemmaLemma 8.7.221F540C3F7D1Continuous products, envelopes and domination
lemmaLemma 8.7.3C39ADEAC74E3Continuous products, envelopes and domination
propositionProposition 8.7.426DD802A141EContinuous products, envelopes and domination
equationEquation 8.7.55451B220B0B1Continuous products, envelopes and domination
lemmaLemma 8.7.6E7EA08EFFDE7Continuous products, envelopes and domination
propositionProposition 8.7.77E8FF9A60609Continuous products, envelopes and domination
propositionProposition 8.7.8103D0E7E46B0Continuous products, envelopes and domination
lemmaLemma 8.7.9F28CED8C4FAFContinuous products, envelopes and domination
propositionProposition 8.8.15D1E62F09217Continuous equations and regularity
theoremTheorem 8.8.23B8FDE2FF95DContinuous equations and regularity
theoremTheorem 8.8.376376AD88152Continuous equations and regularity
lemmaLemma 8.9.1C35233382EB9Bounded potentials and recursive Hessian currents
equationEquation 8.9.266BDC02D9E25Bounded potentials and recursive Hessian currents
propositionProposition 8.9.3EB2FA13CDDE0Bounded potentials and recursive Hessian currents
equationEquation 8.9.4686912AA777FBounded potentials and recursive Hessian currents
equationEquation 8.9.517FA1BFF2491Bounded potentials and recursive Hessian currents
theoremTheorem 8.9.6846671D5582ABounded potentials and recursive Hessian currents
propositionProposition 8.9.726EFAEC0B76DBounded potentials and recursive Hessian currents
lemmaLemma 8.9.8A1E60FC7AA88Bounded potentials and recursive Hessian currents
exampleExample 8.9.95F56B4D9CC35Bounded potentials and recursive Hessian currents
lemmaLemma 8.10.1EB20E121128ASobolev capacity and two Hessian factors
lemmaLemma 8.10.2FFBF9CD42640Sobolev capacity and two Hessian factors
lemmaLemma 8.10.3A82044827575Sobolev capacity and two Hessian factors
propositionProposition 8.10.4B40742E0DD08Sobolev capacity and two Hessian factors
theoremTheorem 8.10.50241EC758621Sobolev capacity and two Hessian factors
lemmaLemma 8.10.62A6BE9847309Sobolev capacity and two Hessian factors
exampleExample 8.10.7AEB4F2397BBFSobolev capacity and two Hessian factors
lemmaLemma 8.11.1370C4F1C4AF0Local capacities and decreasing products
definitionDefinition 8.11.2FA49D2074193Local capacities and decreasing products
equationEquation 8.11.3A08EAEC63C44Local capacities and decreasing products
lemmaLemma 8.11.49AF7CFAD9A0BLocal capacities and decreasing products
lemmaLemma 8.11.59416072D448FLocal capacities and decreasing products
propositionProposition 8.11.6390B1CE08171Local capacities and decreasing products
equationEquation 8.11.7021C38133493Local capacities and decreasing products
equationEquation 8.11.8CA5C66E53CFALocal capacities and decreasing products
equationEquation 8.11.9A133A0104951Local capacities and decreasing products
lemmaLemma 8.11.10348FF027ECE6Local capacities and decreasing products
lemmaLemma 8.11.11978D54F1054CLocal capacities and decreasing products
theoremTheorem 8.11.12C044CD1C65C4Local capacities and decreasing products
lemmaLemma 8.11.13E64937A67BD2Local capacities and decreasing products
theoremTheorem 8.11.14945FEE7DBDC5Local capacities and decreasing products
propositionProposition 8.12.19989CBEEE646Locality and maxima of bounded potentials
theoremTheorem 8.12.2655B0BBF4F9CLocality and maxima of bounded potentials
lemmaLemma 8.12.3132BB9E1B1F5Locality and maxima of bounded potentials
lemmaLemma 8.12.43A8C156268BFLocality and maxima of bounded potentials
propositionProposition 8.12.5BB9187C844DALocality and maxima of bounded potentials
lemmaLemma 8.13.169692CE9985BHessian mass and global comparison
equationEquation 8.13.2B8626F362B85Hessian mass and global comparison
propositionProposition 8.13.3CCDE5B297235Hessian mass and global comparison
theoremTheorem 8.13.453F870DE5980Hessian mass and global comparison
exampleExample 8.13.50DB9D80DE230Hessian mass and global comparison
subsection8.13.6 Modified comparison10ABC920077EHessian mass and global comparison
lemmaLemma 8.13.70F8C3F6B0122Hessian mass and global comparison
theoremTheorem 8.13.8B83D4B8CE619Hessian mass and global comparison
subsection8.13.9 Domination and proper equationsDBEDA039CF16Hessian mass and global comparison
theoremTheorem 8.13.10A532C77D7D0CHessian mass and global comparison
corollaryCorollary 8.13.116BFB2EB67A74Hessian mass and global comparison
propositionProposition 8.13.12FBE7E9A6BE5EHessian mass and global comparison
corollaryCorollary 8.13.135460C4009B01Hessian mass and global comparison

9. Complex spaces and modifications

equationEquation 9.2.1667706D7A5E8Division and finite analytic modules
propositionProposition 9.2.2 (Weierstrass division)611958EF0AB8Division and finite analytic modules
equationEquation 9.2.3A61080C94D0BDivision and finite analytic modules
propositionProposition 9.2.44B4DFDFC5DBADivision and finite analytic modules
lemmaLemma 9.2.5C47992939550Division and finite analytic modules
lemmaLemma 9.2.64AA3D030C896Division and finite analytic modules
theoremTheorem 9.2.7 (Oka's coherence theorem)F28642698B29Division and finite analytic modules
lemmaLemma 9.2.84D71178D595ADivision and finite analytic modules
lemmaLemma 9.2.9AA4164301998Division and finite analytic modules
propositionProposition 9.3.1477256F93241Analytic sets and their local branches
equationEquation 9.3.28DE25C9B839CAnalytic sets and their local branches
lemmaLemma 9.3.32EC38CE665F6Analytic sets and their local branches
theoremTheorem 9.3.4 (Local parametrization)833B11F874F9Analytic sets and their local branches
equationEquation 9.3.518937662BBAEAnalytic sets and their local branches
corollaryCorollary 9.3.6522466FA1EC1Analytic sets and their local branches
lemmaLemma 9.3.716B19C0E622DAnalytic sets and their local branches
corollaryCorollary 9.3.8 (Analytic Nullstellensatz)BE77FEFA9B1CAnalytic sets and their local branches
propositionProposition 9.3.98D696ED2F2ACAnalytic sets and their local branches
theoremTheorem 9.3.10 (Cartan's coherence theorem)3675C20D7E18Analytic sets and their local branches
equationEquation 9.3.11FC41F95FDD78Analytic sets and their local branches
lemmaLemma 9.3.12D245F9712FB1Analytic sets and their local branches
corollaryCorollary 9.3.13F1291FB3DBB9Analytic sets and their local branches
lemmaLemma 9.3.14 (Constant rank theorem)F802E1301C0FAnalytic sets and their local branches
lemmaLemma 9.3.15A814C831E915Analytic sets and their local branches
corollaryCorollary 9.3.16725857B32C0EAnalytic sets and their local branches
corollaryCorollary 9.3.1727E8771DE67AAnalytic sets and their local branches
corollaryCorollary 9.3.186DDD0384D260Analytic sets and their local branches
lemmaLemma 9.3.19113991711A43Analytic sets and their local branches
theoremTheorem 9.3.20D73EBFFE10D2Analytic sets and their local branches
lemmaLemma 9.4.1F56FCBEF9522Reduced spaces and normalization
lemmaLemma 9.4.226BA2ABFD727Reduced spaces and normalization
propositionProposition 9.4.336422A732754Reduced spaces and normalization
lemmaLemma 9.4.4D73CC9F08C05Reduced spaces and normalization
lemmaLemma 9.4.558E87F89D557Reduced spaces and normalization
theoremTheorem 9.4.6 (Normalization)12A47DB401E2Reduced spaces and normalization
corollaryCorollary 9.4.78D2ABD4CAC6AReduced spaces and normalization
definitionDefinition 9.4.80E1E539882ADReduced spaces and normalization
corollaryCorollary 9.4.9EE1633042D3DReduced spaces and normalization
exampleExample 9.4.1060A14C7A1845Reduced spaces and normalization
propositionProposition 9.5.1A3A9384DC317Analytic curves and finite maps
propositionProposition 9.5.21497A0492453Analytic curves and finite maps
corollaryCorollary 9.5.3F7B8B4ADF874Analytic curves and finite maps
lemmaLemma 9.5.4FDF6E2ABDF34Analytic curves and finite maps
lemmaLemma 9.5.5776A532CD79EAnalytic curves and finite maps
propositionProposition 9.5.60E64BF4DC7C7Analytic curves and finite maps
definitionDefinition 9.6.1757D038DF549Blowups and resolution
propositionProposition 9.6.21CAC94BB1C9DBlowups and resolution
propositionProposition 9.6.3865FE5EC97E6Blowups and resolution
equationEquation 9.6.4E62F169B3F9FBlowups and resolution
equationEquation 9.6.54C48E8A5FA42Blowups and resolution
theoremTheorem 9.6.6 (Resolution of singularities)4CEF0578EFE0Blowups and resolution
theoremTheorem 9.6.7 (Principalization)24330D25741DBlowups and resolution
equationEquation 9.6.8012823948235Blowups and resolution
theoremTheorem 9.7.1 (Remmert–Stein)B1493CF698A6Proper images and relative principalization
theoremTheorem 9.7.2 (Remmert)0FC675348681Proper images and relative principalization
corollaryCorollary 9.7.3390229269CD2Proper images and relative principalization
definitionDefinition 9.7.436CB77777986Proper images and relative principalization
theoremTheorem 9.7.5689628C7AF7FProper images and relative principalization
equationEquation 9.7.6D986C1B2DB45Proper images and relative principalization
equationEquation 9.7.7C8A64E02BF42Proper images and relative principalization
equationEquation 9.7.804648CE42AADProper images and relative principalization
theoremTheorem 9.8.1176C7A4C3D95Connected fibers and resolution models
corollaryCorollary 9.8.26EB72DC4EC68Connected fibers and resolution models
propositionProposition 9.8.32312A839D109Connected fibers and resolution models
definitionDefinition 9.8.40C92F63F8A00Connected fibers and resolution models
lemmaLemma 9.8.55488C1DAFED2Connected fibers and resolution models
theoremTheorem 9.8.6373CA5388E72Connected fibers and resolution models
equationEquation 9.8.7D52CB808B809Connected fibers and resolution models
definitionDefinition 9.9.17A62ADAFC2A7Plurisubharmonic functions on reduced spaces
lemmaLemma 9.9.283D4170FB2B3Plurisubharmonic functions on reduced spaces
propositionProposition 9.9.3D6705783DDADPlurisubharmonic functions on reduced spaces
propositionProposition 9.9.4B43080BD49F3Plurisubharmonic functions on reduced spaces
section9.10.1 Complex spaces089901B2844APluripotential theory on unibranch spaces
definitionDefinition 9.10.2F30317C89D08Pluripotential theory on unibranch spaces
definitionDefinition 9.10.34BA9DE7FEF24Pluripotential theory on unibranch spaces
propositionProposition 9.10.41A7FEC3124D0Pluripotential theory on unibranch spaces
theoremTheorem 9.10.5 (Zariski's main theorem)E707851D5DBAPluripotential theory on unibranch spaces
definitionDefinition 9.10.6C9A9F3CE78E7Pluripotential theory on unibranch spaces
equationEquation 9.10.78D4BB1A6524DPluripotential theory on unibranch spaces
theoremTheorem 9.10.8 (Hironaka's Chow lemma)3CD8BF9158DDPluripotential theory on unibranch spaces
theoremTheorem 9.10.98F66FA207B82Pluripotential theory on unibranch spaces
corollaryCorollary 9.10.10178E0605C99DPluripotential theory on unibranch spaces
section9.10.11 Plurisubharmonic functions04B4AD1F512DPluripotential theory on unibranch spaces
definitionDefinition 9.10.12FCF157D7001APluripotential theory on unibranch spaces
theoremTheorem 9.10.13 (Fornaess–Narasimhan)48473380B913Pluripotential theory on unibranch spaces
theoremTheorem 9.10.14 (Grauert–Remmert)C69D2F0ABA4FPluripotential theory on unibranch spaces
equationEquation 9.10.15B66F47F11DC8Pluripotential theory on unibranch spaces
corollaryCorollary 9.10.162DA8FD7B7A99Pluripotential theory on unibranch spaces
section9.10.17 Extensions of the results in the smooth settingCE1C14C908CBPluripotential theory on unibranch spaces
definitionDefinition 9.10.183BD14AE7D7C8Pluripotential theory on unibranch spaces
section9.11.1 Potentials and disk tests0753019C64C3Potentials and equations on unibranch spaces
definitionDefinition 9.11.276A75EB639CBPotentials and equations on unibranch spaces
lemmaLemma 9.11.3E7163CA56764Potentials and equations on unibranch spaces
section9.11.4 Descent through a resolution65930026730APotentials and equations on unibranch spaces
theoremTheorem 9.11.5F4C06D21AA94Potentials and equations on unibranch spaces
section9.11.6 Envelopes and normalized compactness29C5C6037D7DPotentials and equations on unibranch spaces
propositionProposition 9.11.7622B386B559DPotentials and equations on unibranch spaces
section9.11.8 Ideal envelopes defined on resolutions491CF8180F3FPotentials and equations on unibranch spaces
propositionProposition 9.11.9C43A6C429C4DPotentials and equations on unibranch spaces
subsection9.11.10 Boundary weights on a quotient surface6B078FD09EA1Potentials and equations on unibranch spaces
equationEquation 9.11.110C6FD486D0E4Potentials and equations on unibranch spaces
equationEquation 9.11.12727AAF0F7AFFPotentials and equations on unibranch spaces
lemmaLemma 9.11.13A56815B99E6BPotentials and equations on unibranch spaces
propositionProposition 9.11.14DA5FE7ECD89DPotentials and equations on unibranch spaces
equationEquation 9.11.151099F6DF9D77Potentials and equations on unibranch spaces
subsection9.11.16 Adapted monomial tests on affine toric spaces73BE0E663644Potentials and equations on unibranch spaces
equationEquation 9.11.17A72FB1B5D26FPotentials and equations on unibranch spaces
equationEquation 9.11.184B4F6518339APotentials and equations on unibranch spaces
lemmaLemma 9.11.195D5A847F99ACPotentials and equations on unibranch spaces
theoremTheorem 9.11.20CD69C52538E4Potentials and equations on unibranch spaces
equationEquation 9.11.214CC45280AE88Potentials and equations on unibranch spaces
corollaryCorollary 9.11.22E77092231FA9Potentials and equations on unibranch spaces
propositionProposition 9.11.23511E6797B7CDPotentials and equations on unibranch spaces
section9.11.24 Extended measures and finite damping449B961AB405Potentials and equations on unibranch spaces
equationEquation 9.11.2596D4B9F40B0APotentials and equations on unibranch spaces
theoremTheorem 9.11.26D12D6E199E87Potentials and equations on unibranch spaces
section9.11.27 Equations on projective cones821709E49BC3Potentials and equations on unibranch spaces
equationEquation 9.11.28DF6E62BA10C0Potentials and equations on unibranch spaces
equationEquation 9.11.294720371980C9Potentials and equations on unibranch spaces
subsection9.11.30 Logarithmic poles at the vertexF23187DC5D42Potentials and equations on unibranch spaces
equationEquation 9.11.31593FB9025947Potentials and equations on unibranch spaces
equationEquation 9.11.321F428FF466CAPotentials and equations on unibranch spaces
equationEquation 9.11.3303EEDEF4E91BPotentials and equations on unibranch spaces
theoremTheorem 9.11.343DAEEE3C82F5Potentials and equations on unibranch spaces
equationEquation 9.11.35A8F60A69A51DPotentials and equations on unibranch spaces
equationEquation 9.11.3698BFAEFA11C3Potentials and equations on unibranch spaces
equationEquation 9.11.3748E4877345BDPotentials and equations on unibranch spaces
equationEquation 9.11.38E5F90A48E44APotentials and equations on unibranch spaces
equationEquation 9.11.39FB074E27BCB5Potentials and equations on unibranch spaces
equationEquation 9.11.40DA8E2A2BBA2BPotentials and equations on unibranch spaces
subsection9.11.41 The bounded equation750E47668A22Potentials and equations on unibranch spaces
equationEquation 9.11.42EFA25DCE0A09Potentials and equations on unibranch spaces
theoremTheorem 9.11.434CE6B71B5B25Potentials and equations on unibranch spaces
equationEquation 9.11.44BB19755D6BE9Potentials and equations on unibranch spaces
equationEquation 9.11.459D62C6B08002Potentials and equations on unibranch spaces
exampleExample 9.11.46659B6A5D3749Potentials and equations on unibranch spaces
subsection9.11.47 The quadric cone07FD7FBC796CPotentials and equations on unibranch spaces
equationEquation 9.11.486E82FAF4001BPotentials and equations on unibranch spaces
equationEquation 9.11.49E97B7922C415Potentials and equations on unibranch spaces
section9.11.50 Logarithmic equations on normalized curves7AADE341B54CPotentials and equations on unibranch spaces
lemmaLemma 9.11.5199DF4FD89931Potentials and equations on unibranch spaces
equationEquation 9.11.52E623581B83B2Potentials and equations on unibranch spaces
equationEquation 9.11.530567315FF18BPotentials and equations on unibranch spaces
theoremTheorem 9.11.542489A49CB212Potentials and equations on unibranch spaces
equationEquation 9.11.555C30CCD83A1DPotentials and equations on unibranch spaces
subsection9.11.56 A projective cusp with a degenerate pullback90D7DAC242DEPotentials and equations on unibranch spaces
exampleExample 9.11.5777EE41C1372CPotentials and equations on unibranch spaces
equationEquation 9.11.58E7FF7E63C71EPotentials and equations on unibranch spaces
subsection9.11.59 A logarithmic endpoint for integrable dataE2DBF02EA6E5Potentials and equations on unibranch spaces
theoremTheorem 9.11.60E80960D59FB4Potentials and equations on unibranch spaces
equationEquation 9.11.617B4446B6082EPotentials and equations on unibranch spaces
equationEquation 9.11.62BA74C8891A07Potentials and equations on unibranch spaces
section9.11.63 Nonclosed equations on cuspidal products25867983D80DPotentials and equations on unibranch spaces
equationEquation 9.11.64B5525E6496F5Potentials and equations on unibranch spaces
propositionProposition 9.11.65D0F33FF75830Potentials and equations on unibranch spaces
theoremTheorem 9.11.666EDDAD49D627Potentials and equations on unibranch spaces
equationEquation 9.11.67E79D47E722CFPotentials and equations on unibranch spaces
lemmaLemma 9.11.687426C6119D9FPotentials and equations on unibranch spaces
propositionProposition 9.11.69FEAE8B85485EPotentials and equations on unibranch spaces
equationEquation 9.11.70E6758962B60CPotentials and equations on unibranch spaces
theoremTheorem 9.11.71127BF0018B1APotentials and equations on unibranch spaces
corollaryCorollary 9.11.725F7DB1E779E6Potentials and equations on unibranch spaces

10. Cohomology, positivity, and geometric loci

lemmaLemma 10.2.1D02C1B7CE2FAThe trace equation and global potentials
equationEquation 10.2.2B0418825BEF0The trace equation and global potentials
equationEquation 10.2.30947422F7E16The trace equation and global potentials
lemmaLemma 10.2.4E30254F3E614The trace equation and global potentials
lemmaLemma 10.2.50149A5B39655The trace equation and global potentials
equationEquation 10.2.634CE8ED071CAThe trace equation and global potentials
equationEquation 10.2.7A742A4B9C7C0The trace equation and global potentials
equationEquation 10.2.8F5A36978B25BThe trace equation and global potentials
lemmaLemma 10.2.9410DB42E1141The trace equation and global potentials
equationEquation 10.2.1005E65A6873F9The trace equation and global potentials
lemmaLemma 10.2.11B12EA9BB1DC1The trace equation and global potentials
lemmaLemma 10.2.12C7E6E1462EE1The trace equation and global potentials
equationEquation 10.2.131ED32DA97733The trace equation and global potentials
equationEquation 10.2.14748787149B8AThe trace equation and global potentials
theoremTheorem 10.2.15 (The ddc-lemma)D24B01620C94The trace equation and global potentials
equationEquation 10.2.16A5501BC4794CThe trace equation and global potentials
definitionDefinition 10.2.172EE8F1562C62The trace equation and global potentials
equationEquation 10.2.188E700088C0C4The trace equation and global potentials
corollaryCorollary 10.2.195E8D4AB76812The trace equation and global potentials
propositionProposition 10.3.12984F6438B4BNormalized potentials and minimal singularities
corollaryCorollary 10.3.2384717733620Normalized potentials and minimal singularities
corollaryCorollary 10.3.3C7609FD8FC8CNormalized potentials and minimal singularities
equationEquation 10.3.46B3D5F5D04A5Normalized potentials and minimal singularities
corollaryCorollary 10.3.5B2C28BB645D3Normalized potentials and minimal singularities
corollaryCorollary 10.3.6134C98DC2772Normalized potentials and minimal singularities
definitionDefinition 10.3.79499229CD425Normalized potentials and minimal singularities
lemmaLemma 10.3.85755B2D3C604Normalized potentials and minimal singularities
propositionProposition 10.3.94C3B45D704A9Normalized potentials and minimal singularities
corollaryCorollary 10.3.10E6419D4591F9Normalized potentials and minimal singularities
definitionDefinition 10.4.1CFDE52E42C6BPositivity cones and divisorial decomposition
propositionProposition 10.4.23B6E3B145113Positivity cones and divisorial decomposition
theoremTheorem 10.4.3 (The mixed Hodge index theorem)2AE132953DC7Positivity cones and divisorial decomposition
equationEquation 10.4.4A2FE7F1DA253Positivity cones and divisorial decomposition
corollaryCorollary 10.4.5D52AB2ADBD8CPositivity cones and divisorial decomposition
propositionProposition 10.4.6106B224C1E35Positivity cones and divisorial decomposition
corollaryCorollary 10.4.7856BB9025ACAPositivity cones and divisorial decomposition
theoremTheorem 10.4.8 (Siu's divisorial decomposition)D905BB5BE91DPositivity cones and divisorial decomposition
equationEquation 10.4.9383D7ED25D9EPositivity cones and divisorial decomposition
equationEquation 10.4.10823C22786A3DPositivity cones and divisorial decomposition
definitionDefinition 10.4.11A72955F9A4C1Positivity cones and divisorial decomposition
corollaryCorollary 10.4.12B72607AB8B41Positivity cones and divisorial decomposition
definitionDefinition 10.5.1057A3C697CFDMinimal multiplicities and positivity loci
propositionProposition 10.5.268A6D68513B0Minimal multiplicities and positivity loci
propositionProposition 10.5.32DF07B2D5AA9Minimal multiplicities and positivity loci
theoremTheorem 10.5.4DBDF98C510DCMinimal multiplicities and positivity loci
equationEquation 10.5.5B0C232A59CEFMinimal multiplicities and positivity loci
propositionProposition 10.5.6C811386D768FMinimal multiplicities and positivity loci
lemmaLemma 10.5.7C3C3386FA4FCMinimal multiplicities and positivity loci
propositionProposition 10.5.820DDDD0E7ECBMinimal multiplicities and positivity loci
exampleExample 10.5.9A7A7F69130A0Minimal multiplicities and positivity loci
lemmaLemma 10.6.1DC3C9650BBBECurvature and metrics defined by sections
lemmaLemma 10.6.294EC30DAF1A3Curvature and metrics defined by sections
definitionDefinition 10.6.3A68F1B40919CCurvature and metrics defined by sections
propositionProposition 10.6.495892535D04FCurvature and metrics defined by sections
equationEquation 10.6.5A8F577E012B3Curvature and metrics defined by sections
exampleExample 10.6.601B4AD44BAC3Curvature and metrics defined by sections
exampleExample 10.6.74CE1D9D7D803Curvature and metrics defined by sections
definitionDefinition 10.6.89CC891317C85Curvature and metrics defined by sections
propositionProposition 10.6.9E5F620306821Curvature and metrics defined by sections
lemmaLemma 10.7.1A0C109B78052Compact interpolation and global Bergman kernels
lemmaLemma 10.7.295269D214AF0Compact interpolation and global Bergman kernels
equationEquation 10.7.3EA3E7FCDD007Compact interpolation and global Bergman kernels
lemmaLemma 10.7.46217A93C596FCompact interpolation and global Bergman kernels
lemmaLemma 10.7.5C789A06C06B4Compact interpolation and global Bergman kernels
lemmaLemma 10.7.6D51F6B990A39Compact interpolation and global Bergman kernels
equationEquation 10.7.771ED94BE6393Compact interpolation and global Bergman kernels
lemmaLemma 10.7.8189AC91F2DE3Compact interpolation and global Bergman kernels
theoremTheorem 10.7.9846B917F28CDCompact interpolation and global Bergman kernels
equationEquation 10.7.10F28F18A2C23BCompact interpolation and global Bergman kernels
lemmaLemma 10.7.11F8C4EC9BBA24Compact interpolation and global Bergman kernels
equationEquation 10.7.12E4884F8C8C74Compact interpolation and global Bergman kernels
exampleExample 10.7.13E7DB735CB52FCompact interpolation and global Bergman kernels
lemmaLemma 10.8.1D973E523027EExtension from a compact submanifold
equationEquation 10.8.2C5A4CDD14D8EExtension from a compact submanifold
equationEquation 10.8.382B8AA567C14Extension from a compact submanifold
lemmaLemma 10.8.4ABE17C01E6F9Extension from a compact submanifold
equationEquation 10.8.54762DC697088Extension from a compact submanifold
lemmaLemma 10.8.6833820F8A53DExtension from a compact submanifold
equationEquation 10.8.727D0BD8514EBExtension from a compact submanifold
lemmaLemma 10.8.8B903D314198BExtension from a compact submanifold
equationEquation 10.8.96DDE982B7F19Extension from a compact submanifold
lemmaLemma 10.8.10EB08BA5202E3Extension from a compact submanifold
lemmaLemma 10.8.11F36B5947D011Extension from a compact submanifold
equationEquation 10.8.12A6BC4A01A284Extension from a compact submanifold
equationEquation 10.8.139A43DC5AB656Extension from a compact submanifold
equationEquation 10.8.141B0F446E871AExtension from a compact submanifold
lemmaLemma 10.8.157EB0767C032BExtension from a compact submanifold
equationEquation 10.8.1698030817F4D9Extension from a compact submanifold
equationEquation 10.8.1777D11D5D142DExtension from a compact submanifold
theoremTheorem 10.8.18C34A54E553C3Extension from a compact submanifold
equationEquation 10.8.19D04560F8F2BBExtension from a compact submanifold
equationEquation 10.8.2087C2E41D6EF2Extension from a compact submanifold
equationEquation 10.8.218412620C5C8FExtension from a compact submanifold
corollaryCorollary 10.8.2263196D2455BEExtension from a compact submanifold
subsection10.8.23 Extending strictly positive potentials963CD0DB428CExtension from a compact submanifold
lemmaLemma 10.8.244BEE08C69CE8Extension from a compact submanifold
lemmaLemma 10.8.259C08A2D9AA88Extension from a compact submanifold
theoremTheorem 10.8.26853DA86626EBExtension from a compact submanifold
equationEquation 10.8.27897524437A9BExtension from a compact submanifold
equationEquation 10.8.28007C2EF0BE28Extension from a compact submanifold
equationEquation 10.9.17503A52143DAGlobal poles on compact complex manifolds
equationEquation 10.9.2D00D33B0EA52Global poles on compact complex manifolds
lemmaLemma 10.9.340E5831EBB76Global poles on compact complex manifolds
lemmaLemma 10.9.404C8B0EA1481Global poles on compact complex manifolds
equationEquation 10.9.5F6D79B7B476DGlobal poles on compact complex manifolds
equationEquation 10.9.6FFED91B6BA32Global poles on compact complex manifolds
equationEquation 10.9.70D9ABE7F6E36Global poles on compact complex manifolds
equationEquation 10.9.8E24490F72C36Global poles on compact complex manifolds
equationEquation 10.9.9A8D59EA33A3EGlobal poles on compact complex manifolds
propositionProposition 10.9.10CB97B7394604Global poles on compact complex manifolds
equationEquation 10.9.11232AA283C364Global poles on compact complex manifolds
equationEquation 10.9.121C288D4819F2Global poles on compact complex manifolds
lemmaLemma 10.9.13D94E16CFCA6CGlobal poles on compact complex manifolds
equationEquation 10.9.14E73B66195941Global poles on compact complex manifolds
lemmaLemma 10.9.1529371C8618C2Global poles on compact complex manifolds
propositionProposition 10.9.16A3F29E1887A0Global poles on compact complex manifolds
equationEquation 10.9.17736BC1D7B247Global poles on compact complex manifolds
equationEquation 10.9.18435DB3D5519BGlobal poles on compact complex manifolds
theoremTheorem 10.9.19E2AF1B6D37ECGlobal poles on compact complex manifolds
section10.30.1 Affine classes with prescribed derivative89D86926636AClasses with a nonclosed background
propositionProposition 10.30.25F6B6BF8B562Classes with a nonclosed background
section10.30.3 Residual trace classesC43F5C12FA6FClasses with a nonclosed background
theoremTheorem 10.30.428C8F709D1CCClasses with a nonclosed background
corollaryCorollary 10.30.59157A5A52988Classes with a nonclosed background
section10.30.6 Nef realizations and residual-current limitsAFB10A9AB20BClasses with a nonclosed background
propositionProposition 10.30.7A8BE1F55208FClasses with a nonclosed background
theoremTheorem 10.30.8250F831D6E96Classes with a nonclosed background
section10.30.9 Divisorial classes and infinite mass47AA4DD512F8Classes with a nonclosed background
propositionProposition 10.30.106436A9863EBAClasses with a nonclosed background
propositionProposition 10.11.17F12058085DCClosed classes with Hermitian lower bounds
propositionProposition 10.11.2A8D846A01D78Closed classes with Hermitian lower bounds
definitionDefinition 10.12.172E9AF170CBAMinimal multiplicities and the non-nef locus
propositionProposition 10.12.24075D8C83DE1Minimal multiplicities and the non-nef locus
definitionDefinition 10.13.1A39877CAB682The non-Kähler locus
theoremTheorem 10.13.27B5F62BF9296The non-Kähler locus
propositionProposition 10.13.393492323D1A9The non-Kähler locus
equationEquation 10.14.149BB745D6E17Null loci and numerical restricted volumes
propositionProposition 10.15.1FCA1263B1141Restricted volumes and Hermitian perturbations
equationEquation 10.15.2B03D3D606FEBRestricted volumes and Hermitian perturbations
equationEquation 10.16.163C31A84C03FFinite restricted masses of closed classes
equationEquation 10.16.2E4E79DEE021CFinite restricted masses of closed classes
propositionProposition 10.16.35D137EECD5CCFinite restricted masses of closed classes
equationEquation 10.16.4A4CFA8A325BEFinite restricted masses of closed classes
propositionProposition 10.16.5BA31DCC7CCB7Finite restricted masses of closed classes
equationEquation 10.16.61AB8FDFCA122Finite restricted masses of closed classes
equationEquation 10.16.75827E195F217Finite restricted masses of closed classes
equationEquation 10.16.877857992F798Finite restricted masses of closed classes
propositionProposition 10.16.9FE6CD72EA03AFinite restricted masses of closed classes
corollaryCorollary 10.16.103BC40D26B752Finite restricted masses of closed classes
equationEquation 10.16.1143D7AA1ADEB6Finite restricted masses of closed classes
equationEquation 10.17.1857EF7216413Divisorial traces with bounded remainder
equationEquation 10.17.2522C47F58227Divisorial traces with bounded remainder
lemmaLemma 10.17.321B7909B2B52Divisorial traces with bounded remainder
equationEquation 10.17.4477AB76A3ADCDivisorial traces with bounded remainder
equationEquation 10.17.533AAF0E25AE8Divisorial traces with bounded remainder
theoremTheorem 10.17.62D52D521F2D7Divisorial traces with bounded remainder
propositionProposition 10.17.7719EA7B7039BDivisorial traces with bounded remainder
exampleExample 10.17.8ED565B280C89Divisorial traces with bounded remainder
exampleExample 10.17.9C15ADAE71322Divisorial traces with bounded remainder
exampleExample 10.17.105E3AD3C523ACDivisorial traces with bounded remainder
propositionProposition 10.18.17B0AC7895B1FDivisorial components at the threshold
equationEquation 10.18.2C6FE80302EDEDivisorial components at the threshold
exampleExample 10.18.30C57A6C5BFECDivisorial components at the threshold
corollaryCorollary 10.18.4852EFF5927B9Divisorial components at the threshold
corollaryCorollary 10.18.5A786309EB514Divisorial components at the threshold
lemmaLemma 10.19.1464DA68EB4DACurrents on the elliptic threefold
propositionProposition 10.19.224BCA82D5034Currents on the elliptic threefold
subsection10.19.3 Potentials and envelopes from the base7CFF33409C17Currents on the elliptic threefold
propositionProposition 10.19.42555699F5657Currents on the elliptic threefold
subsection10.19.5 Sections selected by multiplier idealsE99505EBA4B4Currents on the elliptic threefold
propositionProposition 10.19.6CDBA8166D33BCurrents on the elliptic threefold
subsection10.19.7 The boundary volume and section growth714E787B0503Currents on the elliptic threefold
propositionProposition 10.19.8141C0ACF5576Currents on the elliptic threefold
equationEquation 10.19.9E7BC529546A7Currents on the elliptic threefold
subsection10.19.10 Limits of the minimizing Gauduchon forms7128CD1C74CBCurrents on the elliptic threefold
propositionProposition 10.19.11983D2EEDBE68Currents on the elliptic threefold
theoremTheorem 10.20.1C4F23D785797Smooth equations on the ample locus
equationEquation 10.20.26DBC0A4AFE96Smooth equations on the ample locus
equationEquation 10.20.3658C9DBC52EDSmooth equations on the ample locus
propositionProposition 10.21.18D773ECA9ED2Density zeros and metric degeneration
equationEquation 10.21.2EA1615732A9DDensity zeros and metric degeneration
exampleExample 10.21.399FB104FBCD5Density zeros and metric degeneration
equationEquation 10.21.464F6BCAA5AB9Density zeros and metric degeneration
subsection10.21.5 Determinant and normal distanceC03017CE71E3Density zeros and metric degeneration
propositionProposition 10.21.6B9DD61C0FD5EDensity zeros and metric degeneration
equationEquation 10.21.7BC6A5A3DBD7ADensity zeros and metric degeneration
subsection10.21.8 Smoothness of exceptional residuals1B7310DAABFDDensity zeros and metric degeneration
propositionProposition 10.21.986DDC28E0989Density zeros and metric degeneration
exampleExample 10.21.109F3252D2DC77Density zeros and metric degeneration
exampleExample 10.21.11554ACF84A13BDensity zeros and metric degeneration
equationEquation 10.21.12D97EEB42728ADensity zeros and metric degeneration
theoremTheorem 10.22.1A86593744D2DRadial densities and metric completeness
equationEquation 10.22.2E811914EE11DRadial densities and metric completeness
equationEquation 10.22.3F205E7AE4ECCRadial densities and metric completeness
equationEquation 10.22.4B8FF1DABE923Radial densities and metric completeness
equationEquation 10.22.5585DD6478F46Radial densities and metric completeness
equationEquation 10.22.6C19D1AD0E901Radial densities and metric completeness
theoremTheorem 10.22.7FF884313C070Radial densities and metric completeness
equationEquation 10.22.8940248BAAEE2Radial densities and metric completeness
equationEquation 10.22.9CC16B5AD8D47Radial densities and metric completeness
equationEquation 10.22.105C6F9700C172Radial densities and metric completeness
theoremTheorem 10.23.13BB4C60DE3DFEquations on products with a curve
equationEquation 10.23.2464F29927E86Equations on products with a curve
equationEquation 10.23.31DB31DDE4B07Equations on products with a curve
theoremTheorem 10.24.18A2E8673E914An incomplete metric in a big class
equationEquation 10.24.20340B9B3D110An incomplete metric in a big class
equationEquation 10.24.34BD50BB63FAEAn incomplete metric in a big class
equationEquation 10.24.48F4A3AEF872FAn incomplete metric in a big class
equationEquation 10.24.54035514E1789An incomplete metric in a big class
subsection10.24.6 A nef class with an incomplete null end903A353086FCAn incomplete metric in a big class
theoremTheorem 10.24.78D207AFD99F5An incomplete metric in a big class
equationEquation 10.24.8D648DC7F97C0An incomplete metric in a big class
lemmaLemma 10.24.90A057E71255DAn incomplete metric in a big class
equationEquation 10.24.1049DD99303236An incomplete metric in a big class
equationEquation 10.24.11C7CD68519547An incomplete metric in a big class
equationEquation 10.24.1226865AC0337FAn incomplete metric in a big class
subsection10.24.13 An incomplete metric with compatible flow data76505E13E4B8An incomplete metric in a big class
equationEquation 10.24.14349718BCBE33An incomplete metric in a big class
equationEquation 10.24.15FC9FBF94E6A4An incomplete metric in a big class
theoremTheorem 10.24.160AE1F943A249An incomplete metric in a big class
equationEquation 10.24.176ABCE921A6F5An incomplete metric in a big class
lemmaLemma 10.24.18F2C09635072EAn incomplete metric in a big class
equationEquation 10.24.196251B9B58B26An incomplete metric in a big class
equationEquation 10.24.206AD6B6EC392CAn incomplete metric in a big class
equationEquation 10.24.210053482330EEAn incomplete metric in a big class
equationEquation 10.24.2282395B883364An incomplete metric in a big class
equationEquation 10.24.238B89C7F09EDAAn incomplete metric in a big class
propositionProposition 10.25.1D05A6736BCFFNormal length near a divisor
equationEquation 10.25.2C974FBD89E14Normal length near a divisor
equationEquation 10.25.313887090B60FNormal length near a divisor
equationEquation 10.25.4EA79C36F0DB1Normal length near a divisor
corollaryCorollary 10.25.5A12B874B6590Normal length near a divisor
equationEquation 10.25.6086B8F769EBFNormal length near a divisor
subsection10.25.7 The critical exponent on a compact torusD95BCCFE2A18Normal length near a divisor
equationEquation 10.25.8E3D518CD1255Normal length near a divisor
lemmaLemma 10.25.9BCA9F26FE4F4Normal length near a divisor
equationEquation 10.25.10F60812366089Normal length near a divisor
exampleExample 10.25.1136258B2EA152Normal length near a divisor
exampleExample 10.25.122D0F50D560BCNormal length near a divisor
theoremTheorem 10.26.19A07C571BD3FRigid boundary classes on a torus
exampleExample 10.26.25B00F62ACA6BRigid boundary classes on a torus
exampleExample 10.26.387A3AA3B13FDRigid boundary classes on a torus
lemmaLemma 10.27.122EA1C593ACDNonrigid classes on a quotient resolution
propositionProposition 10.27.26055AD1EB4F6Nonrigid classes on a quotient resolution
lemmaLemma 10.28.11959B582751FSemipositive metrics in a four-point blowup family
lemmaLemma 10.28.2510B6B95F8B4Semipositive metrics in a four-point blowup family
lemmaLemma 10.28.3DFE3E0889355Semipositive metrics in a four-point blowup family
theoremTheorem 10.28.4195B9613F5AASemipositive metrics in a four-point blowup family
subsection10.28.5 The image of metric restrictionBBAE9B00E672Semipositive metrics in a four-point blowup family
lemmaLemma 10.28.666D32D3D118DSemipositive metrics in a four-point blowup family
lemmaLemma 10.28.7A52A7B9951F1Semipositive metrics in a four-point blowup family
lemmaLemma 10.28.849F835066112Semipositive metrics in a four-point blowup family
theoremTheorem 10.28.958E2ED8FCEB9Semipositive metrics in a four-point blowup family
corollaryCorollary 10.28.1015454CA8E897Semipositive metrics in a four-point blowup family
subsection10.29.1 The constructionCA83ED725CCAOeljeklaus–Toma manifolds
subsection10.29.2 Potentials of the transverse form992E9A24F805Oeljeklaus–Toma manifolds
lemmaLemma 10.29.372EDF5773FACOeljeklaus–Toma manifolds
theoremTheorem 10.29.4244BCBFB750AOeljeklaus–Toma manifolds
equationEquation 10.29.58DB42C059FB9Oeljeklaus–Toma manifolds
corollaryCorollary 10.29.698BD3A283B8EOeljeklaus–Toma manifolds
corollaryCorollary 10.29.7CBF3E42884D1Oeljeklaus–Toma manifolds
subsection10.29.8 Monge–Ampère masses on threefoldsDF61FAE7D603Oeljeklaus–Toma manifolds
equationEquation 10.29.93C8DE164722COeljeklaus–Toma manifolds
propositionProposition 10.29.10E080ADB5BEF1Oeljeklaus–Toma manifolds
equationEquation 10.29.1162F4D45FFF73Oeljeklaus–Toma manifolds
equationEquation 10.29.1261E93097052DOeljeklaus–Toma manifolds
subsection10.29.13 Hermitian metrics whose $\mathrm{dd}^{\mathrm{c}}$ has a signD5CAAE8D5B60Oeljeklaus–Toma manifolds
propositionProposition 10.29.1448299489DE9DOeljeklaus–Toma manifolds
equationEquation 10.29.1573E1BF5F8F36Oeljeklaus–Toma manifolds
equationEquation 10.31.1A6F2ACEED83FBig backgrounds under bounded mass
section10.31.2 Equations for finite measures71BC8ED98CA7Big backgrounds under bounded mass
theoremTheorem 10.31.32568C6F1B2F7Big backgrounds under bounded mass
section10.31.4 Convergence under a fixed measureD176FBCC507CBig backgrounds under bounded mass
theoremTheorem 10.31.56AC45579AB2EBig backgrounds under bounded mass
equationEquation 10.31.6C0EFA39C2570Big backgrounds under bounded mass
section10.31.7 Relative moments985DB0A9A096Big backgrounds under bounded mass
propositionProposition 10.31.856A4FC216C27Big backgrounds under bounded mass
equationEquation 10.31.9C1417B0EF5E7Big backgrounds under bounded mass
equationEquation 10.31.10525AD3E9DFE4Big backgrounds under bounded mass
theoremTheorem 10.31.114AC0DDA39172Big backgrounds under bounded mass
equationEquation 10.31.12706A43A92BE8Big backgrounds under bounded mass
subsection10.31.13 Higher moments in the minimal singularity type811851E56701Big backgrounds under bounded mass
lemmaLemma 10.31.14E0503BA0806ABig backgrounds under bounded mass
propositionProposition 10.31.153464170481EEBig backgrounds under bounded mass
equationEquation 10.31.169761605EDA18Big backgrounds under bounded mass
equationEquation 10.31.17430EB5E01BC8Big backgrounds under bounded mass
section10.31.18 General measures in the minimal class0AD35E504E60Big backgrounds under bounded mass
propositionProposition 10.31.190BEB0F48780FBig backgrounds under bounded mass
equationEquation 10.31.20CCD1547654BDBig backgrounds under bounded mass
subsection10.31.21 Exponential and normalized equations6720AA8AAF70Big backgrounds under bounded mass
theoremTheorem 10.31.22AC1DBB01A890Big backgrounds under bounded mass
corollaryCorollary 10.31.23C8ACE6D27FABBig backgrounds under bounded mass
section10.31.24 Relative projection and supersolutions4FA2BA45E658Big backgrounds under bounded mass
propositionProposition 10.31.254F5454D09F78Big backgrounds under bounded mass
propositionProposition 10.31.26AABBF080DAD2Big backgrounds under bounded mass
equationEquation 10.31.278D4561FAF76ABig backgrounds under bounded mass
corollaryCorollary 10.31.281B6B55C4EF1EBig backgrounds under bounded mass
subsection10.31.29 Varying model references4A341D55C924Big backgrounds under bounded mass
theoremTheorem 10.31.30558CBE100687Big backgrounds under bounded mass
equationEquation 10.31.312A2692F560B3Big backgrounds under bounded mass
equationEquation 10.31.324D376181A695Big backgrounds under bounded mass
section10.31.33 Weighted measure additionDA040FEBC40CBig backgrounds under bounded mass
lemmaLemma 10.31.3468A4962EA99DBig backgrounds under bounded mass
lemmaLemma 10.31.35F92458E62F62Big backgrounds under bounded mass
equationEquation 10.31.369AD37088F405Big backgrounds under bounded mass
subsection10.31.37 Ordinary types and bounded carriers11CC2B98E362Big backgrounds under bounded mass
theoremTheorem 10.31.389A4A36BF299FBig backgrounds under bounded mass
equationEquation 10.31.398E63C401924FBig backgrounds under bounded mass
equationEquation 10.31.40AFD55911EAC6Big backgrounds under bounded mass
equationEquation 10.31.41146C9C3A2FCABig backgrounds under bounded mass
equationEquation 10.31.42A2C109F52CDCBig backgrounds under bounded mass
section10.31.43 General measures with a model reference10C7F976A14CBig backgrounds under bounded mass
theoremTheorem 10.31.44D5758568A543Big backgrounds under bounded mass
theoremTheorem 10.31.45DDDCE55695E1Big backgrounds under bounded mass
subsection10.31.46 Stability under changes of reference4D69742DF215Big backgrounds under bounded mass
theoremTheorem 10.31.4750B353C1E23BBig backgrounds under bounded mass
subsection10.31.48 Arbitrary weighted sums9E6F276DB6E3Big backgrounds under bounded mass
theoremTheorem 10.31.499A2FDB1A65B5Big backgrounds under bounded mass
equationEquation 10.31.509B190A9E8A47Big backgrounds under bounded mass
corollaryCorollary 10.31.51DE6440832625Big backgrounds under bounded mass
section10.31.52 Relative boundedness for $L^ p$ data2CD3B93A6226Big backgrounds under bounded mass
lemmaLemma 10.31.53F816A9EF0F9DBig backgrounds under bounded mass
propositionProposition 10.31.5436BCEF7C4255Big backgrounds under bounded mass
equationEquation 10.31.559E93E3C5CAEDBig backgrounds under bounded mass
corollaryCorollary 10.31.56B2E2FCD34986Big backgrounds under bounded mass

11. Hermitian geometry, torsion, and Gauduchon duality

section11.1.1 Integration by parts with a nonclosed formFD6E63098594Torsion and the variation of mass
lemmaLemma 11.1.23B0A8DD950ADTorsion and the variation of mass
lemmaLemma 11.1.3BF56C7366FC6Torsion and the variation of mass
section11.1.4 The mass polynomialCA97FF13B1BBTorsion and the variation of mass
theoremTheorem 11.1.52B3316345A4ATorsion and the variation of mass
equationEquation 11.1.6EB2FF7F930E4Torsion and the variation of mass
theoremTheorem 11.1.7 (Chiose)FB90FC741F25Torsion and the variation of mass
propositionProposition 11.1.85C29873687EDTorsion and the variation of mass
equationEquation 11.1.90B024E0A2A47Torsion and the variation of mass
corollaryCorollary 11.1.102C05FFD39C6DTorsion and the variation of mass
section11.1.11 Mass variation on a complex torusD4D9C4F87D5DTorsion and the variation of mass
exampleExample 11.1.12F336FEED9D87Torsion and the variation of mass
equationEquation 11.1.13914CB776FFB8Torsion and the variation of mass
corollaryCorollary 11.1.140C9AAEBBC04CTorsion and the variation of mass
section11.1.15 Comparison and mass invarianceB5345EE28748Torsion and the variation of mass
propositionProposition 11.1.162A8C104899E8Torsion and the variation of mass
equationEquation 11.1.17F5F2F7988675Torsion and the variation of mass
corollaryCorollary 11.1.189578EA44189ATorsion and the variation of mass
section11.1.19 Uniform mass boundsD29B5BFF0A9ATorsion and the variation of mass
propositionProposition 11.1.20682711871907Torsion and the variation of mass
section11.2.1 Primitives of the Monge–Ampère one-form49D0776D0F6BMonge–Ampère energy and torsion
lemmaLemma 11.2.28A9C3139B5B5Monge–Ampère energy and torsion
lemmaLemma 11.2.38535B6F80ED4Monge–Ampère energy and torsion
theoremTheorem 11.2.47D069B347672Monge–Ampère energy and torsion
equationEquation 11.2.5B4B9D0AD937EMonge–Ampère energy and torsion
section11.2.6 Radial energy and closed backgroundsB9357DC44463Monge–Ampère energy and torsion
equationEquation 11.2.71CE478F66A34Monge–Ampère energy and torsion
propositionProposition 11.2.8E345C8B2A48CMonge–Ampère energy and torsion
section11.2.9 First variation in a nonclosed backgroundAEF022344DD6Monge–Ampère energy and torsion
propositionProposition 11.2.104793125897B2Monge–Ampère energy and torsion
equationEquation 11.2.1191AE93E20D38Monge–Ampère energy and torsion
propositionProposition 11.2.120009683E4137Monge–Ampère energy and torsion
equationEquation 11.2.13173E9E78941EMonge–Ampère energy and torsion
section11.2.14 Symplectic completions and potential variationsA10584C44FB2Monge–Ampère energy and torsion
lemmaLemma 11.2.1519D4882CC9D5Monge–Ampère energy and torsion
equationEquation 11.2.167C73F0133C9DMonge–Ampère energy and torsion
propositionProposition 11.2.17AE6CE389C569Monge–Ampère energy and torsion
equationEquation 11.2.18BAFFEA611460Monge–Ampère energy and torsion
theoremTheorem 11.2.1904BD04D6C676Monge–Ampère energy and torsion
equationEquation 11.2.208B6CFC8EA37BMonge–Ampère energy and torsion
corollaryCorollary 11.2.2167DF1C6C0509Monge–Ampère energy and torsion
subsection11.2.22 Completion energy on products of complex toriEF1FC602D899Monge–Ampère energy and torsion
exampleExample 11.2.23ACA94D5D202CMonge–Ampère energy and torsion
exampleExample 11.2.24F9D5DB66D92AMonge–Ampère energy and torsion
definitionDefinition 11.4.170139B6ABDC2Conformal Gauduchon metrics
theoremTheorem 11.4.2 (Gauduchon's theorem)3978AEB15BD7Conformal Gauduchon metrics
propositionProposition 11.5.1828FAD1A35BFConservation of the trace mass
equationEquation 11.5.297DCB5703B01Conservation of the trace mass
corollaryCorollary 11.5.3A9609E0AE7B9Conservation of the trace mass
propositionProposition 11.5.4BFC1330527CAConservation of the trace mass
equationEquation 11.5.51F99A7154650Conservation of the trace mass
equationEquation 11.5.6B78F87EF7411Conservation of the trace mass
theoremTheorem 11.6.1 (The generalized Lamari criterion)88BE3B63D9E5The Gauduchon criterion for positivity
corollaryCorollary 11.6.2F83E15A1F529The Gauduchon criterion for positivity
lemmaLemma 11.6.391B8C8A3C10EThe Gauduchon criterion for positivity
propositionProposition 11.7.1155386134D9ECompactness and duality of Gauduchon currents
equationEquation 11.7.27F6E8313B3F8Compactness and duality of Gauduchon currents
equationEquation 11.7.3EDF0C9D4A983Compactness and duality of Gauduchon currents
corollaryCorollary 11.7.4A97DA8CE2358Compactness and duality of Gauduchon currents
theoremTheorem 11.8.1B0762464DCDCTransverse concentration of Gauduchon tests
propositionProposition 11.8.226C9605156BETransverse concentration of Gauduchon tests
lemmaLemma 11.9.1A592FEE54E20Convex norms and Gauduchon degrees
lemmaLemma 11.9.2FD645EB1208BConvex norms and Gauduchon degrees
lemmaLemma 11.9.3EE774D339C2CConvex norms and Gauduchon degrees
theoremTheorem 11.9.4C222D1BB3C65Convex norms and Gauduchon degrees
exampleExample 11.10.1196112D1B85AHomogeneous convex regularization
lemmaLemma 11.10.2A2189A9C7D5AHomogeneous convex regularization
lemmaLemma 11.10.3047ADFD98AAFHomogeneous convex regularization
theoremTheorem 11.10.43FA7CB52F1E5Homogeneous convex regularization
propositionProposition 11.10.51263BBAE3998Homogeneous convex regularization
corollaryCorollary 11.10.6F8EB48E98C63Homogeneous convex regularization
lemmaLemma 11.11.140D29F4EF95BStrict positivity from the tautological metric
propositionProposition 11.11.268260DAF47D2Strict positivity from the tautological metric
corollaryCorollary 11.11.3EB345AEE34BCStrict positivity from the tautological metric
lemmaLemma 11.12.182E28EBCB904Polar norms of semipositive Finsler metrics
propositionProposition 11.12.2672C05CD55F8Polar norms of semipositive Finsler metrics
lemmaLemma 11.12.3BD7ECB0F942EPolar norms of semipositive Finsler metrics
equationEquation 11.12.47BB708C6E5FDPolar norms of semipositive Finsler metrics
equationEquation 11.12.5E21FC7F3990FPolar norms of semipositive Finsler metrics
corollaryCorollary 11.12.63DE300A9B2ADPolar norms of semipositive Finsler metrics
corollaryCorollary 11.12.72984296C9B5APolar norms of semipositive Finsler metrics
lemmaLemma 11.13.18FE2AABC7334Ample comparison quotients on Hirzebruch surfaces
propositionProposition 11.13.26EA287558571Ample comparison quotients on Hirzebruch surfaces
lemmaLemma 11.13.37055476D483CAmple comparison quotients on Hirzebruch surfaces
propositionProposition 11.13.4B6B9C2094631Ample comparison quotients on Hirzebruch surfaces
propositionProposition 11.14.1A195B6162A49Stable extensions approaching the split comparison bundle
propositionProposition 11.14.2FD54983C4E79Stable extensions approaching the split comparison bundle
propositionProposition 11.14.37215288BDF53Stable extensions approaching the split comparison bundle
lemmaLemma 11.15.10EEC7E1B2FB0Deforming to stable ample bundles
lemmaLemma 11.15.2927EFEB9D586Deforming to stable ample bundles
lemmaLemma 11.15.3EBF7EB73EABEDeforming to stable ample bundles
lemmaLemma 11.15.4319641577F72Deforming to stable ample bundles
theoremTheorem 11.15.5E769519C862EDeforming to stable ample bundles
corollaryCorollary 11.15.6A2CE95B32341Deforming to stable ample bundles
lemmaLemma 11.16.100BCE16BE996A fixed Stein hypersurface for all Finsler metrics
lemmaLemma 11.16.28C1AF91F05D6A fixed Stein hypersurface for all Finsler metrics
theoremTheorem 11.16.30A9CF0366804A fixed Stein hypersurface for all Finsler metrics
corollaryCorollary 11.16.440E5740882E3A fixed Stein hypersurface for all Finsler metrics
lemmaLemma 11.17.17A772B0D8822Concave neighborhoods and dual norms
theoremTheorem 11.17.2854B49CB03B9Concave neighborhoods and dual norms
lemmaLemma 11.17.39020E77B2318Concave neighborhoods and dual norms
corollaryCorollary 11.17.4B53EB4EFC135Concave neighborhoods and dual norms
corollaryCorollary 11.17.5E29138267596Concave neighborhoods and dual norms
lemmaLemma 11.18.15022159B8350Curvature and projective complement exhaustions
theoremTheorem 11.18.2B93CB03C68B7Curvature and projective complement exhaustions
definitionDefinition 11.18.33B58CDB44607Curvature and projective complement exhaustions
corollaryCorollary 11.18.418D33491ABFDCurvature and projective complement exhaustions
corollaryCorollary 11.18.5E2EF3C8673FCCurvature and projective complement exhaustions
propositionProposition 11.19.1FD9DEB3AA8D6Negative bundle metrics and logarithmic weights
corollaryCorollary 11.19.2029DAFB20616Negative bundle metrics and logarithmic weights
corollaryCorollary 11.19.36CD37D30AF59Negative bundle metrics and logarithmic weights
lemmaLemma 11.20.1B46ABA0FEC10External tensor products and complement convexity
theoremTheorem 11.20.20B75DFE662A7External tensor products and complement convexity
corollaryCorollary 11.20.3F8F43BAF4F9AExternal tensor products and complement convexity
propositionProposition 11.21.11204B7FDAE31Homogeneous plurisubharmonic functions and Hopf complements
corollaryCorollary 11.21.239C76A336DDAHomogeneous plurisubharmonic functions and Hopf complements
corollaryCorollary 11.21.3F0FD4C3E5239Homogeneous plurisubharmonic functions and Hopf complements
exampleExample 11.21.48F043288C182Homogeneous plurisubharmonic functions and Hopf complements
propositionProposition 11.21.5F38A90B57ECCHomogeneous plurisubharmonic functions and Hopf complements
lemmaLemma 11.22.1D9A37BB4C36BPluriharmonic currents
lemmaLemma 11.22.225BD2AAB112APluriharmonic currents
lemmaLemma 11.22.340BA141971D9Pluriharmonic currents
lemmaLemma 11.22.43CD7629C6061Pluriharmonic currents
equationEquation 11.23.1BEA90D7B5428The adjoint measure of the linearization
propositionProposition 11.23.20B0003A3A873The adjoint measure of the linearization
equationEquation 11.23.39140CF590948The adjoint measure of the linearization
propositionProposition 11.23.4A914D4D9C6CBThe adjoint measure of the linearization
exampleExample 11.23.528F39F2B7C2CThe adjoint measure of the linearization
subsection11.23.6 Convex entropies along a Monge–Ampère flow5C3C01BC6D13The adjoint measure of the linearization
equationEquation 11.23.7DD9886FF9427The adjoint measure of the linearization
propositionProposition 11.23.8AE4D0608184FThe adjoint measure of the linearization
equationEquation 11.23.9313276789DFDThe adjoint measure of the linearization
equationEquation 11.23.100BC0A4E1D2A4The adjoint measure of the linearization
corollaryCorollary 11.23.111C7A6B39E520The adjoint measure of the linearization
propositionProposition 11.23.12F45FF03B558BThe adjoint measure of the linearization
subsection11.23.13 Total variation between two evolving probabilitiesF9C293B132B8The adjoint measure of the linearization
lemmaLemma 11.23.146C85B622ADCFThe adjoint measure of the linearization
equationEquation 11.23.157732A51ABE6EThe adjoint measure of the linearization
propositionProposition 11.23.16621E75C5A9F4The adjoint measure of the linearization
corollaryCorollary 11.23.1721368BD09EBBThe adjoint measure of the linearization
lemmaLemma 11.24.1DEA6C8EFE702Fredholm theory for a continuous Hermitian form
equationEquation 11.24.2616C6C2B269AFredholm theory for a continuous Hermitian form
equationEquation 11.24.388F9E749216FFredholm theory for a continuous Hermitian form
equationEquation 11.24.40EB6296FE2B1Fredholm theory for a continuous Hermitian form
propositionProposition 11.25.13E5CF25D6E23Limits of adjoint measures
definitionDefinition 11.26.12C6FED9B1909Conformal changes and directional curvature positivity
lemmaLemma 11.26.223EB10AE428CConformal changes and directional curvature positivity
theoremTheorem 11.26.3AFF6DB5A7BE1Conformal changes and directional curvature positivity
equationEquation 11.26.478D702B6F891Conformal changes and directional curvature positivity
equationEquation 11.26.50F0F9E5CC398Conformal changes and directional curvature positivity
exampleExample 11.26.68C86B880454AConformal changes and directional curvature positivity
propositionProposition 11.26.79EFFDFAE03AFConformal changes and directional curvature positivity
lemmaLemma 11.27.100DBBF153DBARicci-flat Kähler surfaces
lemmaLemma 11.27.2594B40F9CD8CRicci-flat Kähler surfaces
equationEquation 11.27.3D65B8F2D9646Ricci-flat Kähler surfaces
theoremTheorem 11.27.48C2D7F0F7808Ricci-flat Kähler surfaces
corollaryCorollary 11.27.59A2FC61929F1Ricci-flat Kähler surfaces
equationEquation 11.28.1BC84EA1F260BHolomorphic sectional curvature
lemmaLemma 11.28.28D640C26806FHolomorphic sectional curvature
propositionProposition 11.28.337B317ABBC08Holomorphic sectional curvature
equationEquation 11.28.47659E6D4E603Holomorphic sectional curvature
corollaryCorollary 11.28.5550E97762605Holomorphic sectional curvature
subsection11.28.6 Rigidity on complex tori61E79198BB98Holomorphic sectional curvature
propositionProposition 11.28.719F2C00A6056Holomorphic sectional curvature
corollaryCorollary 11.28.87AE0B102B55AHolomorphic sectional curvature
lemmaLemma 11.29.1E495F7FE3CE6Projective first jets and positive bundle metrics
subsection11.29.2 The finite-dimensional alternative6243CAB7B202Projective first jets and positive bundle metrics
equationEquation 11.29.3F67D74886176Projective first jets and positive bundle metrics
propositionProposition 11.29.407EC0797039CProjective first jets and positive bundle metrics
subsection11.29.5 Approximation with two derivatives84DD621CC41AProjective first jets and positive bundle metrics
lemmaLemma 11.29.686B1F5EAC034Projective first jets and positive bundle metrics
lemmaLemma 11.29.7944943C60132Projective first jets and positive bundle metrics
subsection11.29.8 The metric and measure alternativesB1D39F6DC8D4Projective first jets and positive bundle metrics
theoremTheorem 11.29.9C334EF3CAF72Projective first jets and positive bundle metrics
corollaryCorollary 11.29.10A86A6F25FF07Projective first jets and positive bundle metrics
lemmaLemma 11.30.18B6FA358A78DRamified section tests and a positive root metric
subsection11.30.2 The weighted family and its first successful level5B78AD45B21ARamified section tests and a positive root metric
propositionProposition 11.30.3A8FAF022D4CARamified section tests and a positive root metric
subsection11.30.4 Relation to the fixed very ample criterion32E68898DD56Ramified section tests and a positive root metric
corollaryCorollary 11.30.59F9E99FDE392Ramified section tests and a positive root metric
section11.31.1 The trace map1699E04AE638Gauduchon traces and the topology of potentials
theoremTheorem 11.31.2160CE026081BGauduchon traces and the topology of potentials
equationEquation 11.31.367B93429179FGauduchon traces and the topology of potentials
propositionProposition 11.31.423E45E6251B0Gauduchon traces and the topology of potentials
section11.31.5 A metric on normalized potentials523982F48F36Gauduchon traces and the topology of potentials
corollaryCorollary 11.31.6AA6333BC6B20Gauduchon traces and the topology of potentials
propositionProposition 11.31.756D7737A1AF0Gauduchon traces and the topology of potentials
exampleExample 11.31.8B1E195FF817DGauduchon traces and the topology of potentials
section11.31.9 Continuous potentials and pluriharmonic traces951E64D10409Gauduchon traces and the topology of potentials
lemmaLemma 11.31.10BD07AFC2CACBGauduchon traces and the topology of potentials
theoremTheorem 11.31.11E0C21CC2C1A9Gauduchon traces and the topology of potentials
equationEquation 11.31.1239820CDDAFC2Gauduchon traces and the topology of potentials
equationEquation 11.33.19D47DB96B6E3The scalar functional
propositionProposition 11.33.22770AB03C0D2The scalar functional
equationEquation 11.33.37BAFE60C3D66The scalar functional
theoremTheorem 11.33.48810E18F852CThe scalar functional
theoremTheorem 11.33.55E3583383688The scalar functional
equationEquation 11.34.14DA0C1CF8861Supporting probability measures
theoremTheorem 11.34.286A7D85266AASupporting probability measures
propositionProposition 11.34.3CB2DC1F2B712Supporting probability measures
subsection11.34.4 Directional derivativesBD1A1A3DE2D3Supporting probability measures
propositionProposition 11.34.5792D6FEF2F3CSupporting probability measures
subsection11.34.6 Generic uniqueness39ED4CCC5AE2Supporting probability measures
theoremTheorem 11.34.7A4850266BE73Supporting probability measures
subsection11.34.8 Gauduchon adjoints9E3723511E88Supporting probability measures
theoremTheorem 11.34.97E60599DA339Supporting probability measures
subsection11.34.10 Continuous positive complex Hessians4C927FD81A61Supporting probability measures
theoremTheorem 11.34.11F4FDB7BC8A03Supporting probability measures
subsection11.34.12 Products with a Hermitian factorC277B7427384Supporting probability measures
propositionProposition 11.34.13D2CCD76890C8Supporting probability measures
exampleExample 11.34.141B8C3827F8D7Supporting probability measures
subsection11.34.15 Divisorial smoothingA54217F2E13DSupporting probability measures
lemmaLemma 11.34.16FADB0C708F54Supporting probability measures
equationEquation 11.34.1769FAA61D9E93Supporting probability measures
propositionProposition 11.34.183A41CC697C2ASupporting probability measures
equationEquation 11.34.19E178383CB355Supporting probability measures
equationEquation 11.34.204D939EBD023FSupporting probability measures
theoremTheorem 11.34.21D68D3638A657Supporting probability measures
equationEquation 11.34.22482173ECFBEFSupporting probability measures
equationEquation 11.34.233B3F6AB16431Supporting probability measures
corollaryCorollary 11.34.241AB249BC820ASupporting probability measures
equationEquation 11.34.255D44ECE3CA31Supporting probability measures
equationEquation 11.35.1FFB47018170BA convex functional on probability measures
propositionProposition 11.35.20C21983AC42DA convex functional on probability measures
equationEquation 11.35.388A1B49C1D77A convex functional on probability measures
theoremTheorem 11.35.46830F8DE9750A convex functional on probability measures
equationEquation 11.35.5FA8EC628AEAEA convex functional on probability measures
theoremTheorem 11.36.199653D7759E2Bounded measurable densities and stability
equationEquation 11.36.2F840087A1B81Bounded measurable densities and stability
equationEquation 11.36.312472E4CCDDCBounded measurable densities and stability
lemmaLemma 11.37.12E4E82F8C3FCEntropy and the closed case
equationEquation 11.37.2803909FECEC0Entropy and the closed case
propositionProposition 11.37.333DF3CE1122AEntropy and the closed case
propositionProposition 11.38.1170451FB65CBEntropy bounds
subsection11.38.2 The mass bounds as entropy differences7BF91F477A54Entropy bounds
propositionProposition 11.38.3A3720120A48CEntropy bounds
subsection11.38.4 Entropy for a prescribed ordinary type562884633BF8Entropy bounds
corollaryCorollary 11.38.5A13D502ACAACEntropy bounds
propositionProposition 11.39.1675DA440E964Changing the prescribed singularities
subsection11.39.2 Uniform comparison of the scalar functionals99DA2DCCD99FChanging the prescribed singularities
equationEquation 11.39.39ED20E9267C7Changing the prescribed singularities
propositionProposition 11.39.47DD0741AE5AAChanging the prescribed singularities
propositionProposition 11.39.580C599ADD5CBChanging the prescribed singularities
corollaryCorollary 11.39.6FCFB63397013Changing the prescribed singularities
theoremTheorem 11.40.1133439CA2DC1Varying positive backgrounds
equationEquation 11.40.22E9A16E3BAFFVarying positive backgrounds
corollaryCorollary 11.40.39B8101BC6CE4Varying positive backgrounds
propositionProposition 11.40.41A47D2E8B6D5Varying positive backgrounds
equationEquation 11.40.5B7921C3AB602Varying positive backgrounds
theoremTheorem 11.41.17281EDB812F1Divisorial types and moving sections
equationEquation 11.41.2345F0935B009Divisorial types and moving sections
theoremTheorem 11.41.3BF6A67E9A40EDivisorial types and moving sections
subsection11.41.4 One-sided divisorial classes and sharp thresholds0B775DCEE678Divisorial types and moving sections
lemmaLemma 11.41.5CC6718D255ACDivisorial types and moving sections
theoremTheorem 11.41.6C87E852ED386Divisorial types and moving sections
corollaryCorollary 11.41.739C3764F36BDDivisorial types and moving sections
exampleExample 11.41.8121EF09A28A5Divisorial types and moving sections
theoremTheorem 11.42.11DCF668EBCCFThe mass determined by the functional
equationEquation 11.42.2B169C3B4AD17The mass determined by the functional
exampleExample 11.42.3C6588E30CFF6The mass determined by the functional
exampleExample 11.42.40B7F3ACBCCC4The mass determined by the functional
subsection11.43.1 Canonical models and their mass6439CBEB8575Convex-body criteria for varying references
lemmaLemma 11.43.2832F0D3BF7D6Convex-body criteria for varying references
lemmaLemma 11.43.359D09F5BD0FBConvex-body criteria for varying references
equationEquation 11.43.4C1218CE2B784Convex-body criteria for varying references
subsection11.43.5 Hausdorff control of the functional3EB28248B606Convex-body criteria for varying references
theoremTheorem 11.43.693F19A4F37D3Convex-body criteria for varying references
equationEquation 11.43.7720678E2362CConvex-body criteria for varying references
equationEquation 11.43.808069BF2F396Convex-body criteria for varying references
subsection11.43.9 A family with nonanalytic isolated poles592E13B7D2EDConvex-body criteria for varying references
propositionProposition 11.43.1070BE47F32080Convex-body criteria for varying references
equationEquation 11.43.1147DDB1E89607Convex-body criteria for varying references
equationEquation 11.43.126654DDDC658FConvex-body criteria for varying references
corollaryCorollary 11.43.13F57433433789Convex-body criteria for varying references
subsection11.43.14 Positive mass with a degenerate smooth residualD4A7E78D4C33Convex-body criteria for varying references
propositionProposition 11.43.15C9CD848ECB1AConvex-body criteria for varying references
equationEquation 11.43.16B2ABC5CB86A3Convex-body criteria for varying references
corollaryCorollary 11.43.17FB3A15C67170Convex-body criteria for varying references
equationEquation 11.44.1CC5ED4480663Selection by a vanishing exponential parameter
theoremTheorem 11.44.23F95A960C6FASelection by a vanishing exponential parameter
equationEquation 11.44.32B9C124418F5Selection by a vanishing exponential parameter
subsection11.44.4 A norm defined by the critical scalarA20C27AA8420Selection by a vanishing exponential parameter
propositionProposition 11.44.591FDE4AB5C22Selection by a vanishing exponential parameter
equationEquation 11.44.68267EBDA38A9Selection by a vanishing exponential parameter
subsection11.44.7 Norming probabilities and integral boundsB51F7B7B7005Selection by a vanishing exponential parameter
equationEquation 11.44.875A33AC0E5F0Selection by a vanishing exponential parameter
propositionProposition 11.44.9AD91D203DE64Selection by a vanishing exponential parameter
equationEquation 11.44.10829BF6353639Selection by a vanishing exponential parameter
equationEquation 11.44.1122DEB3F4D84ASelection by a vanishing exponential parameter
equationEquation 11.44.1269C8D4536D80Selection by a vanishing exponential parameter
equationEquation 11.44.13FEE1099EB406Selection by a vanishing exponential parameter
subsection11.44.14 Bounds for the centered potentialsCE46B4F2D02ASelection by a vanishing exponential parameter
lemmaLemma 11.44.15FA5BBA96CCC3Selection by a vanishing exponential parameter
equationEquation 11.44.16CB3DF4196532Selection by a vanishing exponential parameter
subsection11.44.17 Scalar envelopes of subsolutionsBC341AB6E8A6Selection by a vanishing exponential parameter
equationEquation 11.44.18F0A90E8941FFSelection by a vanishing exponential parameter
lemmaLemma 11.44.190E0F3A0EC129Selection by a vanishing exponential parameter
equationEquation 11.44.20D8630F60DEBDSelection by a vanishing exponential parameter
equationEquation 11.44.21AABB829EFF5FSelection by a vanishing exponential parameter
equationEquation 11.44.22E3635828F0E0Selection by a vanishing exponential parameter
subsection11.45.1 The ordinary model and the greatest solutions27CD354DC77FSelection for logarithmic models on products
equationEquation 11.45.2254FB5175831Selection for logarithmic models on products
propositionProposition 11.45.3DA11AEC1C7C7Selection for logarithmic models on products
lemmaLemma 11.45.4688DA0FB893FSelection for logarithmic models on products
equationEquation 11.45.5D9B5F3712AAESelection for logarithmic models on products
equationEquation 11.45.637E96D4A1D80Selection for logarithmic models on products
subsection11.45.7 The selected limit with varying dataA00D72781B06Selection for logarithmic models on products
equationEquation 11.45.8E96D6BB82F6CSelection for logarithmic models on products
theoremTheorem 11.45.972A7418012F7Selection for logarithmic models on products
equationEquation 11.45.1097B5C8E6D45DSelection for logarithmic models on products
equationEquation 11.45.119E6339686644Selection for logarithmic models on products
subsection11.45.12 The critical coefficient and separated orderCEEDD0460992Selection for logarithmic models on products
corollaryCorollary 11.45.1386F268014BE0Selection for logarithmic models on products
subsection11.45.14 Weak adjoints on a product surface2FFAD548CA54Selection for logarithmic models on products
lemmaLemma 11.45.1585744C3971DCSelection for logarithmic models on products
theoremTheorem 11.45.1623BDF62163E8Selection for logarithmic models on products
equationEquation 11.45.176EEDC24A3093Selection for logarithmic models on products
propositionProposition 11.45.18132C5E1825BFSelection for logarithmic models on products
exampleExample 11.45.193C22B09DCF87Selection for logarithmic models on products
subsection11.46.1 The prescribed ordinary type31F950502C14Residual regularity in radial singularity types
lemmaLemma 11.46.23CE4F16CE935Residual regularity in radial singularity types
equationEquation 11.46.381B05A10FA0AResidual regularity in radial singularity types
equationEquation 11.46.4DE57A4B5C703Residual regularity in radial singularity types
equationEquation 11.46.5AF6A825EA4B9Residual regularity in radial singularity types
equationEquation 11.46.644CC5DC8E895Residual regularity in radial singularity types
subsection11.46.7 Density criterion and exact inverse6ECAAB69A33BResidual regularity in radial singularity types
theoremTheorem 11.46.8F273E69013C1Residual regularity in radial singularity types
equationEquation 11.46.9898B5C1F659DResidual regularity in radial singularity types
equationEquation 11.46.1096AA5EC1C8FCResidual regularity in radial singularity types
equationEquation 11.46.11FEE50B8F4491Residual regularity in radial singularity types
equationEquation 11.46.128992F455B382Residual regularity in radial singularity types
equationEquation 11.46.13E319D6D342BAResidual regularity in radial singularity types
subsection11.46.14 Smooth residuals from flat errors2C18A3499311Residual regularity in radial singularity types
propositionProposition 11.46.1533B8FF26FB4BResidual regularity in radial singularity types
subsection11.46.16 Compact examples and sharp derivative thresholdsFE6A97B82E18Residual regularity in radial singularity types
propositionProposition 11.46.17F4F876167D51Residual regularity in radial singularity types
equationEquation 11.46.18405FC6263C95Residual regularity in radial singularity types
subsection11.46.19 Smooth residuals with an open density-zero annulus309EDE002FAEResidual regularity in radial singularity types
propositionProposition 11.46.207D9147EC36A1Residual regularity in radial singularity types
subsection11.46.21 Products with a zero-lower-mass factor31DB1FA6A441Residual regularity in radial singularity types
corollaryCorollary 11.46.227C248BB2EDD2Residual regularity in radial singularity types
equationEquation 11.47.1AF49CAB59368Envelopes for singular radial obstacles
lemmaLemma 11.47.23CD979065654Envelopes for singular radial obstacles
subsection11.47.3 The convex minorant and its residual2FEAC06EE6DBEnvelopes for singular radial obstacles
equationEquation 11.47.461A86D21F0FEEnvelopes for singular radial obstacles
theoremTheorem 11.47.57444FA94A222Envelopes for singular radial obstacles
equationEquation 11.47.67B822F7684AEEnvelopes for singular radial obstacles
equationEquation 11.47.7E93EE9714FF1Envelopes for singular radial obstacles
corollaryCorollary 11.47.8EA98BEE991F7Envelopes for singular radial obstacles
equationEquation 11.47.92FE2F9FAF60DEnvelopes for singular radial obstacles
subsection11.47.10 Uniform bounds through moving contactCFB34E7820F0Envelopes for singular radial obstacles
theoremTheorem 11.47.11278AD9E56CEAEnvelopes for singular radial obstacles
equationEquation 11.47.12624221938CEBEnvelopes for singular radial obstacles
subsection11.47.13 The full envelope on a Hermitian productC6C2D9DCA88FEnvelopes for singular radial obstacles
propositionProposition 11.47.14E4713F2FEF7DEnvelopes for singular radial obstacles
exampleExample 11.47.151E30DC99E57BEnvelopes for singular radial obstacles

12. Gauduchon variational volume

definitionDefinition 12.2.10C4AF7E2CEBBThe quotient and the smooth equation
propositionProposition 12.2.24AF37F3C8091The quotient and the smooth equation
theoremTheorem 12.2.32DB664A6EFDFThe quotient and the smooth equation
corollaryCorollary 12.2.4A486E36565B6The quotient and the smooth equation
theoremTheorem 12.3.12220E0C47510The equality case of concavity and the first variation
propositionProposition 12.3.294AFD9E43DE0The equality case of concavity and the first variation
propositionProposition 12.4.167FDDBC10DBDSecond variation on background classes
equationEquation 12.4.2E544AEC2FDDBSecond variation on background classes
equationEquation 12.4.388A03DC67E97Second variation on background classes
equationEquation 12.4.46BCEC0329DE5Second variation on background classes
propositionProposition 12.5.14BBB232A044CRecovering a positive class from its Gauduchon derivative
propositionProposition 12.6.188C1708A920CThe Hessian metric and its completion
subsection12.6.2 Product and blowup examples5429610F6E7AThe Hessian metric and its completion
exampleExample 12.6.3316E7687E1CCThe Hessian metric and its completion
subsection12.6.4 Invariant backgrounds on compact Lie-group quotients8D8E5D56DAEFThe Hessian metric and its completion
propositionProposition 12.6.547DDBF953B9CThe Hessian metric and its completion
equationEquation 12.6.6684C5D55AF27The Hessian metric and its completion
equationEquation 12.6.785B17908B21FThe Hessian metric and its completion
equationEquation 12.6.88E5F08796F4EThe Hessian metric and its completion
lemmaLemma 12.6.981EECB4E9700The Hessian metric and its completion
equationEquation 12.6.10C6F155F53713The Hessian metric and its completion
equationEquation 12.6.113C044D127EF7The Hessian metric and its completion
theoremTheorem 12.6.12B01FA2265C81The Hessian metric and its completion
equationEquation 12.6.138B3AE102E667The Hessian metric and its completion
equationEquation 12.6.146D807945F202The Hessian metric and its completion
equationEquation 12.6.15ED6415BD1F26The Hessian metric and its completion
equationEquation 12.6.165EC99A6A3831The Hessian metric and its completion
equationEquation 12.6.17B2704F167832The Hessian metric and its completion
exampleExample 12.6.18704C3B45EDAFThe Hessian metric and its completion
equationEquation 12.6.198B7DB948FBBBThe Hessian metric and its completion
equationEquation 12.6.20269EE4BAAD07The Hessian metric and its completion
subsubsection12.6.21 Masses and reference densities in dimension threeD70657252686The Hessian metric and its completion
propositionProposition 12.6.227294934ED49CThe Hessian metric and its completion
equationEquation 12.6.2318897CCC762FThe Hessian metric and its completion
equationEquation 12.6.242DDABE7168CAThe Hessian metric and its completion
corollaryCorollary 12.6.259BB432CC1FE6The Hessian metric and its completion
theoremTheorem 12.6.267AA419EEDABEThe Hessian metric and its completion
equationEquation 12.6.273E1F104EFFA9The Hessian metric and its completion
equationEquation 12.6.2898DEB3AF443EThe Hessian metric and its completion
equationEquation 12.6.298859AB23548EThe Hessian metric and its completion
equationEquation 12.6.303861E23FD396The Hessian metric and its completion
subsection12.6.31 Warped product cones and flat completionC0584AC307C7The Hessian metric and its completion
theoremTheorem 12.6.3240DF235D735CThe Hessian metric and its completion
equationEquation 12.6.3349FA03FEEBDFThe Hessian metric and its completion
equationEquation 12.6.340989D826EF96The Hessian metric and its completion
equationEquation 12.6.359C1715A1F2EDThe Hessian metric and its completion
propositionProposition 12.6.3602AE1B3B4027The Hessian metric and its completion
equationEquation 12.6.3773FE713DE9E4The Hessian metric and its completion
subsubsection12.6.38 Lengths of the two ends96431078A25BThe Hessian metric and its completion
lemmaLemma 12.6.391EF963CF5EA2The Hessian metric and its completion
equationEquation 12.6.40DB02B7FF941AThe Hessian metric and its completion
theoremTheorem 12.6.4172BDB634BD0EThe Hessian metric and its completion
subsubsection12.6.42 Current cones and endpoint equations9FC10688E696The Hessian metric and its completion
propositionProposition 12.6.4367B12ACF7773The Hessian metric and its completion
propositionProposition 12.6.4410BD5355ECF4The Hessian metric and its completion
equationEquation 12.6.45ED7129134319The Hessian metric and its completion
subsubsection12.6.46 Limits of the base potentialsFDA6B324FD1BThe Hessian metric and its completion
propositionProposition 12.6.47C7D8502937F2The Hessian metric and its completion
subsubsection12.6.48 Traces of minimizing metrics9C8B56E18A60The Hessian metric and its completion
propositionProposition 12.6.4994A159D3DBDBThe Hessian metric and its completion
equationEquation 12.6.50F61793B0A157The Hessian metric and its completion
propositionProposition 12.6.512AC49E96484CThe Hessian metric and its completion
subsubsection12.6.52 Projective and torus examplesDDBDE1BD78C7The Hessian metric and its completion
exampleExample 12.6.5327A22321E298The Hessian metric and its completion
exampleExample 12.6.54CD9181602A11The Hessian metric and its completion
equationEquation 12.7.1A99038CD10B4A compact dual problem over Gauduchon currents
lemmaLemma 12.7.2FB6C2382E066A compact dual problem over Gauduchon currents
theoremTheorem 12.7.30537569F9F52A compact dual problem over Gauduchon currents
equationEquation 12.7.4053F006B114FA compact dual problem over Gauduchon currents
lemmaLemma 12.8.147843166E777Supporting probabilities represented by maximizing currents
theoremTheorem 12.8.2E9E119AAA269Supporting probabilities represented by maximizing currents
theoremTheorem 12.9.1BF52D98B114FIntegrability of Gauduchon determinant measures
theoremTheorem 12.9.2E7F890105D42Integrability of Gauduchon determinant measures
corollaryCorollary 12.9.3893A41D78AF5Integrability of Gauduchon determinant measures
subsection12.9.4 Uniform bounds near a smooth densityA0340E674DC4Integrability of Gauduchon determinant measures
equationEquation 12.9.5A7F3B3DF78D5Integrability of Gauduchon determinant measures
lemmaLemma 12.9.69F57BA26805AIntegrability of Gauduchon determinant measures
theoremTheorem 12.9.76A30AAE398CDIntegrability of Gauduchon determinant measures
equationEquation 12.9.88926A66CEBEFIntegrability of Gauduchon determinant measures
equationEquation 12.9.9CED2026424BDIntegrability of Gauduchon determinant measures
corollaryCorollary 12.9.107877D1AB98D3Integrability of Gauduchon determinant measures
subsection12.9.11 Uniformly controlled complex Hessians5603574CC3A6Integrability of Gauduchon determinant measures
lemmaLemma 12.9.12F7130F1BDC30Integrability of Gauduchon determinant measures
equationEquation 12.9.13011F0AF31DBFIntegrability of Gauduchon determinant measures
equationEquation 12.9.14DA143DCFAFABIntegrability of Gauduchon determinant measures
propositionProposition 12.9.159D81484FC945Integrability of Gauduchon determinant measures
equationEquation 12.9.1682DFE47055F1Integrability of Gauduchon determinant measures
equationEquation 12.9.17BB82503DF712Integrability of Gauduchon determinant measures
theoremTheorem 12.9.18AD62F7DA0E68Integrability of Gauduchon determinant measures
equationEquation 12.9.19FEAB45A97679Integrability of Gauduchon determinant measures
equationEquation 12.9.202B0B40A18948Integrability of Gauduchon determinant measures
exampleExample 12.9.21CAEF64468276Integrability of Gauduchon determinant measures
theoremTheorem 12.10.141D17749328BIntegrable densities in the compact Gauduchon problem
equationEquation 12.10.2025E3287544BIntegrable densities in the compact Gauduchon problem
equationEquation 12.10.3B5DC9A4ABEECIntegrable densities in the compact Gauduchon problem
subsection12.10.4 The tail condition and perturbations of the density5210ECD5227FIntegrable densities in the compact Gauduchon problem
propositionProposition 12.10.5249445C21482Integrable densities in the compact Gauduchon problem
propositionProposition 12.10.6B62E802ED631Integrable densities in the compact Gauduchon problem
subsection12.10.7 Supporting probabilities for an integrable density516A524E7A2AIntegrable densities in the compact Gauduchon problem
equationEquation 12.10.8C174F8B6FDCFIntegrable densities in the compact Gauduchon problem
propositionProposition 12.10.9CCE6CFE41CBFIntegrable densities in the compact Gauduchon problem
subsection12.10.10 Trace inequalities at the subsolution threshold07A010666AF5Integrable densities in the compact Gauduchon problem
propositionProposition 12.10.1108C2A578ACE2Integrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.12F5E70269636CIntegrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.132C73D81273C0Integrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.146362B224AE69Integrable densities in the compact Gauduchon problem
exampleExample 12.10.1536B0532E29A1Integrable densities in the compact Gauduchon problem
subsection12.10.16 A smooth maximizing form with an analytic degeneracy set8D604EFF48DFIntegrable densities in the compact Gauduchon problem
theoremTheorem 12.10.1751F452E9F8D1Integrable densities in the compact Gauduchon problem
equationEquation 12.10.18E32DBBA007AFIntegrable densities in the compact Gauduchon problem
subsection12.10.19 The limit of the exponential equations604052B5061FIntegrable densities in the compact Gauduchon problem
propositionProposition 12.10.20D804C5144A2DIntegrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.21D179EF5E5E25Integrable densities in the compact Gauduchon problem
subsection12.10.22 Envelope criteria and extension across the degeneracy setDC394A61A477Integrable densities in the compact Gauduchon problem
theoremTheorem 12.10.2330380C32B7A7Integrable densities in the compact Gauduchon problem
propositionProposition 12.10.2462BDDC322780Integrable densities in the compact Gauduchon problem
subsubsection12.10.25 A globally smooth root of rank at least two8EACEED9426FIntegrable densities in the compact Gauduchon problem
theoremTheorem 12.10.267449627F15ECIntegrable densities in the compact Gauduchon problem
equationEquation 12.10.27BD4046661AE4Integrable densities in the compact Gauduchon problem
subsubsection12.10.28 Two attained instances244D24A4396BIntegrable densities in the compact Gauduchon problem
exampleExample 12.10.29E97964925666Integrable densities in the compact Gauduchon problem
exampleExample 12.10.304C03E42B184DIntegrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.312A3A58A80CB4Integrable densities in the compact Gauduchon problem
subsection12.10.32 A full-mass point pole without a smooth maximizing testB4100C32D89AIntegrable densities in the compact Gauduchon problem
propositionProposition 12.10.33DB722F1D363AIntegrable densities in the compact Gauduchon problem
equationEquation 12.10.3433BC8E188D93Integrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.35D0692BB5EDD8Integrable densities in the compact Gauduchon problem
subsection12.10.36 Quantitative convergence of the currents5B8A428F9007Integrable densities in the compact Gauduchon problem
propositionProposition 12.10.37A98696DE3546Integrable densities in the compact Gauduchon problem
subsection12.10.38 The classes of the divisorial components622292D98711Integrable densities in the compact Gauduchon problem
corollaryCorollary 12.10.391E2D5DAE5C6CIntegrable densities in the compact Gauduchon problem
subsection12.10.40 A branched projective exampleBE9EC228849DIntegrable densities in the compact Gauduchon problem
exampleExample 12.10.412A68BBAE5086Integrable densities in the compact Gauduchon problem
subsection12.10.42 A moment condition on maximizing tests47D5B2CA6B21Integrable densities in the compact Gauduchon problem
theoremTheorem 12.10.43A528C756F776Integrable densities in the compact Gauduchon problem
equationEquation 12.10.44797F5974140EIntegrable densities in the compact Gauduchon problem
theoremTheorem 12.11.19ECB2B21B601Nef limits and nonattainment
propositionProposition 12.11.2854AAD5CF1CANef limits and nonattainment
equationEquation 12.12.13B788E18394ESubtraction with a closed nef core
theoremTheorem 12.12.2026E84610646Subtraction with a closed nef core
equationEquation 12.12.32FB723BB4645Subtraction with a closed nef core
propositionProposition 12.12.4FBFFC53EC40FSubtraction with a closed nef core
equationEquation 12.12.591D85F52923DSubtraction with a closed nef core
equationEquation 12.12.6AB7D89842D2ASubtraction with a closed nef core
exampleExample 12.12.7D37A528BDFB8Subtraction with a closed nef core
theoremTheorem 12.13.1B5C9469E9BB3Subtraction on warped products
equationEquation 12.13.2503F3EDAE05ASubtraction on warped products
equationEquation 12.13.3C0A4C9A1103CSubtraction on warped products
propositionProposition 12.13.46E8682D7FD89Subtraction on warped products
equationEquation 12.13.5F4721C1DF1D8Subtraction on warped products
propositionProposition 12.13.6172CF285DEA0Subtraction on warped products
equationEquation 12.13.71B46207D92EBSubtraction on warped products
equationEquation 12.14.171973446DB97Positive volume without bigness
lemmaLemma 12.14.2C0A6E438A11DPositive volume without bigness
equationEquation 12.14.3471F6FE31EB0Positive volume without bigness
equationEquation 12.14.4C34D8558189EPositive volume without bigness
equationEquation 12.14.58523C0528005Positive volume without bigness
theoremTheorem 12.14.6ECAB6BFB33FDPositive volume without bigness
equationEquation 12.14.73D231D52B18APositive volume without bigness
equationEquation 12.14.8F9FEC7FAC55EPositive volume without bigness
subsection12.14.9 Volume tests and threshold testsB51E860FD3F7Positive volume without bigness
propositionProposition 12.14.100E9506DB4E70Positive volume without bigness
equationEquation 12.14.11662BE6BF9C61Positive volume without bigness
equationEquation 12.14.12F4D7155E150DPositive volume without bigness
propositionProposition 12.14.13F8EA88FD522EPositive volume without bigness
equationEquation 12.14.14F8E2395BD557Positive volume without bigness
propositionProposition 12.15.137E6FFC01EDBVarying the probability measure
propositionProposition 12.15.2D26C76DEC922Varying the probability measure
subsection12.15.3 Upper and lower masses53B6AFE7C911Varying the probability measure
theoremTheorem 12.15.4B6CA5669E402Varying the probability measure
equationEquation 12.15.5AE720210AADDVarying the probability measure
propositionProposition 12.15.687CB9377C0BBVarying the probability measure
lemmaLemma 12.16.1744D35F9BE16Curvature conditions for mass extrema
propositionProposition 12.16.2CE7BE3992C5FCurvature conditions for mass extrema
theoremTheorem 12.16.326629526444FCurvature conditions for mass extrema
equationEquation 12.16.4D5CA82478984Curvature conditions for mass extrema
subsection12.17.1 The bounded cubic identityEDD3B0732B25Curvature pairs and positive lower mass
lemmaLemma 12.17.23BBBC535BFA2Curvature pairs and positive lower mass
equationEquation 12.17.3649C88380858Curvature pairs and positive lower mass
lemmaLemma 12.17.473E13B1F9B7ACurvature pairs and positive lower mass
equationEquation 12.17.56344C84663CFCurvature pairs and positive lower mass
subsection12.17.6 The curvature criterionC72702069454Curvature pairs and positive lower mass
theoremTheorem 12.17.77EC7E0C62DF9Curvature pairs and positive lower mass
equationEquation 12.17.8D97FFD43BE5DCurvature pairs and positive lower mass
propositionProposition 12.17.9A60EEF389BCACurvature pairs and positive lower mass
subsection12.17.10 Negative-line-bundle quotients49FAED65F993Curvature pairs and positive lower mass
propositionProposition 12.17.113E19D4A0EA58Curvature pairs and positive lower mass
equationEquation 12.17.1286B14E8734DCCurvature pairs and positive lower mass
subsection12.17.13 A Calabi–Eckmann family without LCK metrics51CD5F251745Curvature pairs and positive lower mass
propositionProposition 12.17.14119B81D2AF8FCurvature pairs and positive lower mass
equationEquation 12.17.15D04D662546BACurvature pairs and positive lower mass
corollaryCorollary 12.18.140D45A12B192Geometric bounds and determinant concentration
propositionProposition 12.18.289AE8B534538Geometric bounds and determinant concentration
equationEquation 12.18.3827DBF9602F3Geometric bounds and determinant concentration
equationEquation 12.18.4A89800F03DB4Geometric bounds and determinant concentration
subsection12.18.5 Intrinsic determinant measures and the complete profile5613B7E2253BGeometric bounds and determinant concentration
lemmaLemma 12.18.65771D42397D7Geometric bounds and determinant concentration
equationEquation 12.18.7DCAEAF642D39Geometric bounds and determinant concentration
theoremTheorem 12.18.83EA495FF9D1AGeometric bounds and determinant concentration
equationEquation 12.18.9290803738499Geometric bounds and determinant concentration
equationEquation 12.18.10AF0F557495C3Geometric bounds and determinant concentration
equationEquation 12.18.11B462DE2DBA4CGeometric bounds and determinant concentration
subsection12.18.12 Fixed determinant data with degenerating ellipticity95D23527B325Geometric bounds and determinant concentration
theoremTheorem 12.18.13BDB7ADF7845BGeometric bounds and determinant concentration
subsection12.18.14 The same obstruction for nonclosed pluriclosed metrics10AA5741C871Geometric bounds and determinant concentration
theoremTheorem 12.18.1517E4315391CCGeometric bounds and determinant concentration
subsection12.18.16 Concentration at exact maximizers for varying rootsA4F04193B5B9Geometric bounds and determinant concentration
theoremTheorem 12.18.176072DDDAB57FGeometric bounds and determinant concentration
theoremTheorem 12.19.138D9206B1A2FVanishing lower mass
subsection12.19.2 The elliptic quotient1DD06E8B2862Vanishing lower mass
equationEquation 12.19.3444A78EADE47Vanishing lower mass
lemmaLemma 12.19.4669B72559DF9Vanishing lower mass
equationEquation 12.19.5388DD0213040Vanishing lower mass
subsection12.19.6 Averaging the trace and compact curves57556869C08AVanishing lower mass
equationEquation 12.19.7E994975CCA57Vanishing lower mass
equationEquation 12.19.879DFF8AF8DCAVanishing lower mass
equationEquation 12.19.9BE92C16DAFFDVanishing lower mass
lemmaLemma 12.19.1040DB573E7C1EVanishing lower mass
equationEquation 12.19.113DC335D33654Vanishing lower mass
subsection12.19.12 The volume of the positive root92006A543CD2Vanishing lower mass
lemmaLemma 12.19.13BD9C00A32F59Vanishing lower mass
subsection12.19.14 Consequences for dimension and equations8E3D46779F5FVanishing lower mass
corollaryCorollary 12.19.15435BD74C27A9Vanishing lower mass
corollaryCorollary 12.19.16940F7BB9270CVanishing lower mass
subsection12.19.17 A balanced metric and a finite upper massDEA97C6D08EBVanishing lower mass
propositionProposition 12.19.1894417AD14985Vanishing lower mass
equationEquation 12.19.199E38044EBFA0Vanishing lower mass
propositionProposition 12.19.203290D6E85075Vanishing lower mass
equationEquation 12.19.2187EE58E6037CVanishing lower mass
propositionProposition 12.19.2269BB70EA26D9Vanishing lower mass
propositionProposition 12.20.1EBFCF76A5A51Orders of degeneration at the nef boundary
propositionProposition 12.20.2883742E77C19Orders of degeneration at the nef boundary
propositionProposition 12.21.1DE2BA13B1A92Elliptic surface volumes and order spectra
equationEquation 12.21.25C3988BE0A59Elliptic surface volumes and order spectra
equationEquation 12.21.3A72D1E62A49BElliptic surface volumes and order spectra
propositionProposition 12.21.458574FAEC5B7Elliptic surface volumes and order spectra
propositionProposition 12.21.541097C9AFBD2Elliptic surface volumes and order spectra
equationEquation 12.21.612515D2FB078Elliptic surface volumes and order spectra
equationEquation 12.21.7CFDC623C2AF7Elliptic surface volumes and order spectra
lemmaLemma 12.22.10A017BFF9468Comparison under an arbitrary modification
equationEquation 12.22.21427E28847CBComparison under an arbitrary modification
lemmaLemma 12.22.351797AE0DE1FComparison under an arbitrary modification
equationEquation 12.22.43145A1C49A8AComparison under an arbitrary modification
propositionProposition 12.22.58AB4DFD2719FComparison under an arbitrary modification
equationEquation 12.22.62D24229F8C67Comparison under an arbitrary modification
equationEquation 12.22.7C7A393B9C4B8Comparison under an arbitrary modification
equationEquation 12.22.81419E4EEEE39Comparison under an arbitrary modification
propositionProposition 12.23.11E30136A67F6Strict changes under one blowup
equationEquation 12.23.248EE58E16325Strict changes under one blowup
equationEquation 12.23.361B56EB11C84Strict changes under one blowup
equationEquation 12.23.4F24A79D3D80EStrict changes under one blowup
equationEquation 12.23.5FE92C82454E6Strict changes under one blowup
equationEquation 12.23.685080CF9B131Strict changes under one blowup
propositionProposition 12.24.17CE9EDC18591Positive-current subtraction and divisorial residuals
equationEquation 12.24.22818CCA10129Positive-current subtraction and divisorial residuals
propositionProposition 12.24.3D15FEB2692E7Positive-current subtraction and divisorial residuals
equationEquation 12.24.4D74CAC12DD59Positive-current subtraction and divisorial residuals
corollaryCorollary 12.24.59860E4D04E6CPositive-current subtraction and divisorial residuals
lemmaLemma 12.25.1DAA29135206DInteger orders in fixed projective cones
equationEquation 12.25.2ED15CA2DA3FCInteger orders in fixed projective cones
equationEquation 12.25.3022F404D6957Integer orders in fixed projective cones
subsubsection12.25.4 Linear blowups and the fixed generators6EBDFF04B7CAInteger orders in fixed projective cones
equationEquation 12.25.500C606303647Integer orders in fixed projective cones
equationEquation 12.25.6DE6D982F2E21Integer orders in fixed projective cones
equationEquation 12.25.7EFE1F66045F4Integer orders in fixed projective cones
theoremTheorem 12.25.83F58257FA940Integer orders in fixed projective cones
equationEquation 12.25.979FA42084398Integer orders in fixed projective cones
subsubsection12.25.10 Equality of orders under modificationsA34399DFCD46Integer orders in fixed projective cones
propositionProposition 12.25.1157BC641DA62EInteger orders in fixed projective cones
subsubsection12.25.12 Subtracting effective divisorsA528EDFC5F96Integer orders in fixed projective cones
lemmaLemma 12.25.135DAA9B186CEEInteger orders in fixed projective cones
theoremTheorem 12.25.14F79036F002E1Integer orders in fixed projective cones
exampleExample 12.25.15EB3698E8A068Integer orders in fixed projective cones
propositionProposition 12.26.194DF4FEA70D3A maximum-rank comparison criterion
corollaryCorollary 12.26.287EBA7661A8AA maximum-rank comparison criterion
exampleExample 12.26.3698A2684C823A maximum-rank comparison criterion
lemmaLemma 12.27.1030B497D5253Flat anticanonical curves
lemmaLemma 12.27.21C23EA256441Flat anticanonical curves
theoremTheorem 12.27.358A77103A732Flat anticanonical curves
equationEquation 12.27.4C58D63577C3FFlat anticanonical curves
propositionProposition 12.27.59877ACAAFD41Flat anticanonical curves
corollaryCorollary 12.27.63CC0EFB19764Flat anticanonical curves
propositionProposition 12.27.79124282E25D3Flat anticanonical curves
lemmaLemma 12.28.1A3034D6EAAEFFlat divisors and Hartogs extension
propositionProposition 12.28.2A37F03BEA004Flat divisors and Hartogs extension
theoremTheorem 12.28.3784DD48DA619Flat divisors and Hartogs extension
corollaryCorollary 12.28.49FC7A89CD619Flat divisors and Hartogs extension
lemmaLemma 12.29.12BAAA2E519DEPunctured line bundles and elliptic quotients
exampleExample 12.29.2423CBD89023EPunctured line bundles and elliptic quotients
propositionProposition 12.29.301B10D72D8CFPunctured line bundles and elliptic quotients
lemmaLemma 12.30.1D4DAF3EEB0E9Cotangent torsors on split elliptic ruled surfaces
lemmaLemma 12.30.268FEC9C7A0BCCotangent torsors on split elliptic ruled surfaces
lemmaLemma 12.30.31F38B2F1E2A7Cotangent torsors on split elliptic ruled surfaces
theoremTheorem 12.30.4E1118C06F481Cotangent torsors on split elliptic ruled surfaces
lemmaLemma 12.31.12604E6F7AC76Symmetric tensors on split elliptic ruled surfaces
lemmaLemma 12.31.2711AF6E8C0D8Symmetric tensors on split elliptic ruled surfaces
theoremTheorem 12.31.3654B70C2AFD7Symmetric tensors on split elliptic ruled surfaces
theoremTheorem 12.31.43B7C9C492833Symmetric tensors on split elliptic ruled surfaces
lemmaLemma 12.32.146169CD2ACA2Section algebras of cotangent extensions
theoremTheorem 12.32.277C35E15A510Section algebras of cotangent extensions
corollaryCorollary 12.32.337D7893F0652Section algebras of cotangent extensions
lemmaLemma 12.33.1C3F367EB396ECotangent extensions with zero ruling component
propositionProposition 12.33.2A78CB65BF75ECotangent extensions with zero ruling component
propositionProposition 12.33.30414E2D53266Cotangent extensions with zero ruling component
problemProblem 12.33.4AD465D07A4C1Cotangent extensions with zero ruling component
theoremTheorem 12.34.1FC99385C18B8Current tests for smooth positivity
propositionProposition 12.34.2635FE1D11798Current tests for smooth positivity
subsection12.34.3 Maximal time of the Chern–Ricci flow355CACEF1F3CCurrent tests for smooth positivity
theoremTheorem 12.34.4B6FA0B70406ACurrent tests for smooth positivity
equationEquation 12.34.500A6644760C0Current tests for smooth positivity
equationEquation 12.34.6A5486C5A27CFCurrent tests for smooth positivity
corollaryCorollary 12.34.790A4213A9E87Current tests for smooth positivity
subsection12.34.8 A positive-volume endpoint that is not big382AAED0F384Current tests for smooth positivity
equationEquation 12.34.9C648779A3DE9Current tests for smooth positivity
theoremTheorem 12.34.10AFAF4566FE5DCurrent tests for smooth positivity
equationEquation 12.34.11348CDC5BE9A1Current tests for smooth positivity
propositionProposition 12.34.1283512AFDC859Current tests for smooth positivity
subsection12.34.13 Twisting and distance boundsFFD5AB006AEFCurrent tests for smooth positivity
exampleExample 12.34.14706929790BE5Current tests for smooth positivity
propositionProposition 12.35.1C43B33AA4A18Divisorial constraints and residual volume
equationEquation 12.35.2D8ACD0FADFBDDivisorial constraints and residual volume
equationEquation 12.35.344DAB02F7C7CDivisorial constraints and residual volume
theoremTheorem 12.35.47B32818E6010Divisorial constraints and residual volume
equationEquation 12.35.53EE856593807Divisorial constraints and residual volume
subsection12.35.6 Varying divisorial coefficients2AD000453E7CDivisorial constraints and residual volume
propositionProposition 12.35.7280181A9DE22Divisorial constraints and residual volume
subsection12.35.8 A ruled residual and its subtraction costE45BF47B8568Divisorial constraints and residual volume
propositionProposition 12.35.9E45E9799C813Divisorial constraints and residual volume
equationEquation 12.35.10DA1A40EA1D3ADivisorial constraints and residual volume
corollaryCorollary 12.35.11F402CC8C0A7EDivisorial constraints and residual volume
equationEquation 12.35.1210918993D19DDivisorial constraints and residual volume
exampleExample 12.35.1376B0694EBA2FDivisorial constraints and residual volume
equationEquation 12.35.14AFBB384B6340Divisorial constraints and residual volume
propositionProposition 12.36.13A78333ED147A divisorial threshold on the Hopf threefold

13. Non-pluripolar measures and mass

lemmaLemma 13.2.16632853A40C9Truncation and plurifine locality
lemmaLemma 13.2.2D59675D9F6D8Truncation and plurifine locality
equationEquation 13.2.3E88E33ACAD5BTruncation and plurifine locality
definitionDefinition 13.2.4D79952D7E272Truncation and plurifine locality
propositionProposition 13.2.51FAE60CC76F9Truncation and plurifine locality
propositionProposition 13.2.67F7123F92478Truncation and plurifine locality
equationEquation 13.2.7491D0E94E07ATruncation and plurifine locality
equationEquation 13.2.8591B39D2BEA0Truncation and plurifine locality
exampleExample 13.2.96F4C920FEDCATruncation and plurifine locality
theoremTheorem 13.3.18A125B338F90Closedness and cutoff gradients
lemmaLemma 13.4.1263D0F5AEE0ABounded factors and the first-degree product
lemmaLemma 13.4.25D93FCFBA57FBounded factors and the first-degree product
propositionProposition 13.4.390E0E28D9A95Bounded factors and the first-degree product
lemmaLemma 13.4.4CA3A887474EEBounded factors and the first-degree product
lemmaLemma 13.4.54AD0BFC1BFCDBounded factors and the first-degree product
equationEquation 13.4.67F078C7DCBDDBounded factors and the first-degree product
propositionProposition 13.4.73553558C2E27Bounded factors and the first-degree product
equationEquation 13.4.86DEB3D875595Bounded factors and the first-degree product
equationEquation 13.4.971D090723F15Bounded factors and the first-degree product
propositionProposition 13.4.10FB5ACFB46DB8Bounded factors and the first-degree product
exampleExample 13.4.11E9D7DF5798B0Bounded factors and the first-degree product
propositionProposition 13.5.106A36B9354A5Global finiteness and bounded cohomological mass
lemmaLemma 13.5.205FB10FAF0D8Global finiteness and bounded cohomological mass
equationEquation 13.5.381F1166426A9Global finiteness and bounded cohomological mass
theoremTheorem 13.5.43783DE0B4D00Global finiteness and bounded cohomological mass
equationEquation 13.5.5896BDD434F7FGlobal finiteness and bounded cohomological mass
exampleExample 13.5.63D09B5B530AAGlobal finiteness and bounded cohomological mass
subsection13.5.7 Mixed lower bounds1BA47696B5FBGlobal finiteness and bounded cohomological mass
propositionProposition 13.5.8A93A2F4C95C2Global finiteness and bounded cohomological mass
equationEquation 13.5.96B41F6267BFAGlobal finiteness and bounded cohomological mass
corollaryCorollary 13.5.107FDDF6925B80Global finiteness and bounded cohomological mass
corollaryCorollary 13.5.1143C93EF28279Global finiteness and bounded cohomological mass
corollaryCorollary 13.6.1BEC54934FC2CFiber integration and modifications
equationEquation 13.6.2F6B4982710D8Fiber integration and modifications
propositionProposition 13.6.30811FFDF15B7Fiber integration and modifications
lemmaLemma 13.7.1AB4F8DB9BF3ECapacity convergence and loss of mass
definitionDefinition 13.7.2019A8D37A03FCapacity convergence and loss of mass
lemmaLemma 13.7.3274B700496A8Capacity convergence and loss of mass
lemmaLemma 13.7.488D1663D1111Capacity convergence and loss of mass
theoremTheorem 13.7.567C4F7EC316ECapacity convergence and loss of mass
exampleExample 13.7.6ACAEA8D8BE87Capacity convergence and loss of mass
subsection13.7.7 Ordered error sets, monotone nets and weights64A0C33C8690Capacity convergence and loss of mass
propositionProposition 13.7.84B60F7F33590Capacity convergence and loss of mass
corollaryCorollary 13.7.9D73416504271Capacity convergence and loss of mass
lemmaLemma 13.8.1663159428707Mass comparison and projective fibers
lemmaLemma 13.8.2984FFB015C24Mass comparison and projective fibers
equationEquation 13.8.3FE05CF9EAD76Mass comparison and projective fibers
equationEquation 13.8.44EF00D8CA3C9Mass comparison and projective fibers
propositionProposition 13.8.5A00B4940B406Mass comparison and projective fibers
equationEquation 13.8.656FDDE3E50E0Mass comparison and projective fibers
theoremTheorem 13.8.760DE4F681F75Mass comparison and projective fibers
corollaryCorollary 13.8.8A8BA63307F6CMass comparison and projective fibers
corollaryCorollary 13.8.956037F3B58D7Mass comparison and projective fibers
definitionDefinition 13.9.14F64B29C315EMaximal mass and comparison
propositionProposition 13.9.244F242B6965FMaximal mass and comparison
definitionDefinition 13.9.32AF01ACCF985Maximal mass and comparison
corollaryCorollary 13.9.445601E7EC1EEMaximal mass and comparison
propositionProposition 13.9.5FEA17FAE6BAAMaximal mass and comparison
corollaryCorollary 13.9.6D91589CC5A89Maximal mass and comparison
equationEquation 13.1.182D27D8CAD36Non-pluripolar products
definitionDefinition 13.10.1F6CEAE1224B7Bedford–Taylor theory
remarkRemark 13.10.2918938F52375Bedford–Taylor theory
propositionProposition 13.10.3E1938BA48A2FBedford–Taylor theory
theoremTheorem 13.10.4032A3F745585Bedford–Taylor theory
lemmaLemma 13.10.510BB181E99E0Bedford–Taylor theory
equationEquation 13.10.6E7F106829576Bedford–Taylor theory
definitionDefinition 13.11.11A52175C3D64The non-pluripolar products
equationEquation 13.11.2826BE13A141DThe non-pluripolar products
equationEquation 13.11.363B22DDAC51CThe non-pluripolar products
propositionProposition 13.11.49E81D36B1290The non-pluripolar products
equationEquation 13.11.5C3F3FF3F504BThe non-pluripolar products
equationEquation 13.11.67E91D53462ACThe non-pluripolar products
equationEquation 13.11.7A1F77BA03A26The non-pluripolar products
exampleExample 13.11.800950EDCBA2CThe non-pluripolar products
equationEquation 13.11.99F35F663186DThe non-pluripolar products
definitionDefinition 13.11.1005C17B17762FThe non-pluripolar products
equationEquation 13.11.113FE189D717D7The non-pluripolar products
remarkRemark 13.11.1276807540FC41The non-pluripolar products
remarkRemark 13.11.13873B08B81AC9The non-pluripolar products
propositionProposition 13.11.14FB457F67F566The non-pluripolar products
corollaryCorollary 13.11.15AE7DB6D35BE9The non-pluripolar products
equationEquation 13.11.16091EC34E5A75The non-pluripolar products
equationEquation 13.11.178D9958903619The non-pluripolar products
equationEquation 13.11.1832D177AC1071The non-pluripolar products
definitionDefinition 13.12.1C96BE5099B6AQuasi-continuous functions
theoremTheorem 13.12.2C5C10F2AB708Quasi-continuous functions
definitionDefinition 13.12.3BA13EE505D9BQuasi-continuous functions
theoremTheorem 13.12.49F9E57CC1422Quasi-continuous functions
exampleExample 13.12.5472C675FCEA8Quasi-continuous functions
definitionDefinition 13.12.6F7490AE48EACQuasi-continuous functions
propositionProposition 13.12.7D56D6D4A5E94Quasi-continuous functions
theoremTheorem 13.12.8272876B00528Quasi-continuous functions
equationEquation 13.13.144C4E200865BProperties of non-pluripolar products
remarkRemark 13.13.2218A02F6B9E0Properties of non-pluripolar products
propositionProposition 13.13.3C0B519026258Properties of non-pluripolar products
propositionProposition 13.13.4365687A6A087Properties of non-pluripolar products
theoremTheorem 13.13.57F7BE8292E76Properties of non-pluripolar products
remarkRemark 13.13.622841C35D44BProperties of non-pluripolar products
equationEquation 13.13.76F52E700CD98Properties of non-pluripolar products
lemmaLemma 13.13.8F77E450F00FDProperties of non-pluripolar products
theoremTheorem 13.13.9 (Semicontinuity theorem)B3DC3180FF35Properties of non-pluripolar products
equationEquation 13.13.108EB996A1BAE1Properties of non-pluripolar products
theoremTheorem 13.13.11 (Monotonicity theorem)0E5F6133B048Properties of non-pluripolar products
corollaryCorollary 13.13.1237623491DEDCProperties of non-pluripolar products
equationEquation 13.13.1318A95DBDA47AProperties of non-pluripolar products
lemmaLemma 13.13.14F56651914DE1Properties of non-pluripolar products
equationEquation 13.13.1537A13FA36078Properties of non-pluripolar products
corollaryCorollary 13.13.167CB7A3C51640Properties of non-pluripolar products
equationEquation 13.13.17D30B25375476Properties of non-pluripolar products
definitionDefinition 13.13.18434FC63292E2Properties of non-pluripolar products
equationEquation 13.13.19B0CB5D62EE36Properties of non-pluripolar products
equationEquation 13.13.205F5458A60B41Properties of non-pluripolar products
lemmaLemma 13.13.212904B26253E2Properties of non-pluripolar products
equationEquation 13.13.220D2A1EE39CF8Properties of non-pluripolar products
equationEquation 13.13.23BB745DB26624Properties of non-pluripolar products
theoremTheorem 13.13.24A60C61FA5696Properties of non-pluripolar products
equationEquation 13.13.25426FF7E9832CProperties of non-pluripolar products
equationEquation 13.13.261F8BEEBD94D3Properties of non-pluripolar products
equationEquation 13.13.272745CA7D012BProperties of non-pluripolar products
equationEquation 13.13.286BC72E6A227CProperties of non-pluripolar products
equationEquation 13.13.29B774EF560F8BProperties of non-pluripolar products
lemmaLemma 13.13.301CC7CDBE65A9Properties of non-pluripolar products
equationEquation 13.13.31D1F48E296303Properties of non-pluripolar products
equationEquation 13.13.32CD0083D5C7BCProperties of non-pluripolar products
equationEquation 13.13.3323948CFE8AEEProperties of non-pluripolar products
equationEquation 13.13.348921383DCB12Properties of non-pluripolar products
equationEquation 13.13.3500E5A7157825Properties of non-pluripolar products
equationEquation 13.13.368E31B7D70F87Properties of non-pluripolar products
equationEquation 13.13.37CEDD9EE9573EProperties of non-pluripolar products
equationEquation 13.13.3887D0AE27AE70Properties of non-pluripolar products
equationEquation 13.13.392263D1F9CA2FProperties of non-pluripolar products
equationEquation 13.13.401C8D5126D276Properties of non-pluripolar products
equationEquation 13.13.41675086676ABEProperties of non-pluripolar products
corollaryCorollary 13.13.42992A84CBAC5DProperties of non-pluripolar products
theoremTheorem 13.13.43 (Domination principle)B798B54E6F2EProperties of non-pluripolar products
equationEquation 13.13.4492DC05470A2EProperties of non-pluripolar products
equationEquation 13.13.4599773FAEB14EProperties of non-pluripolar products
remarkRemark 13.13.46B80857C56A45Properties of non-pluripolar products
lemmaLemma 13.13.473AA4A74713F0Properties of non-pluripolar products
lemmaLemma 13.13.4872A62384B9ECProperties of non-pluripolar products
theoremTheorem 13.13.49 (Tosatti, Chu–Zhou)FE0721C3DE01Properties of non-pluripolar products
propositionProposition 13.13.50 (Darvas–Rubinstein)4BEA034735FDProperties of non-pluripolar products
lemmaLemma 13.13.519453F71CEE8FProperties of non-pluripolar products
equationEquation 13.13.5293FB8527BF9BProperties of non-pluripolar products
equationEquation 13.13.5312862DAB63ABProperties of non-pluripolar products
equationEquation 13.13.54A9866D01984DProperties of non-pluripolar products
equationEquation 13.13.556FB38BF4B16CProperties of non-pluripolar products
equationEquation 13.13.5677FFDFF752EBProperties of non-pluripolar products
theoremTheorem 13.13.57 (Di Nezza–Trapani)6245F87DD88EProperties of non-pluripolar products
equationEquation 13.13.58F5A7BEAB8003Properties of non-pluripolar products
equationEquation 13.13.591FE4032CE3EEProperties of non-pluripolar products
equationEquation 13.13.6088AC8C43CD82Properties of non-pluripolar products
corollaryCorollary 13.13.6177CAD5891555Properties of non-pluripolar products
theoremTheorem 13.13.62 (Di Nezza–Trapani)700E44C7083BProperties of non-pluripolar products
section13.14.1 Truncation and non-pluripolar measures600C2F850586Non-pluripolar Monge–Ampère measures
lemmaLemma 13.14.2D36A46B4DD86Non-pluripolar Monge–Ampère measures
theoremTheorem 13.14.3 (Extended non-pluripolar measure)99C7ACBB2589Non-pluripolar Monge–Ampère measures
equationEquation 13.14.4E6AA45ACEA33Non-pluripolar Monge–Ampère measures
section13.14.5 Contact sets900BB92BD0E3Non-pluripolar Monge–Ampère measures
propositionProposition 13.14.617AF37734177Non-pluripolar Monge–Ampère measures
section13.14.7 Weighted lower semicontinuity74BA8C79EB05Non-pluripolar Monge–Ampère measures
lemmaLemma 13.14.864174941A7D1Non-pluripolar Monge–Ampère measures
section13.14.9 Density and trace inequalities7A696CA7D25BNon-pluripolar Monge–Ampère measures
lemmaLemma 13.14.107051A18CAB21Non-pluripolar Monge–Ampère measures
equationEquation 13.14.11DA998466C80CNon-pluripolar Monge–Ampère measures
lemmaLemma 13.14.122CA81432B327Non-pluripolar Monge–Ampère measures
section13.14.13 The bounded mass propertyD919086FF578Non-pluripolar Monge–Ampère measures
definitionDefinition 13.14.14EB45638CBB2CNon-pluripolar Monge–Ampère measures
propositionProposition 13.14.1545F622BE862ANon-pluripolar Monge–Ampère measures
propositionProposition 13.14.161342466C71A9Non-pluripolar Monge–Ampère measures
section13.14.17 Infinite mass with trivial multiplier ideals54BB8D4D734DNon-pluripolar Monge–Ampère measures
theoremTheorem 13.14.18AEE72111FEDANon-pluripolar Monge–Ampère measures
corollaryCorollary 13.14.1960F41A586D0FNon-pluripolar Monge–Ampère measures
section13.14.20 Integrable Hessians and infinite mass25C7A8032994Non-pluripolar Monge–Ampère measures
propositionProposition 13.14.21D889AEAF2866Non-pluripolar Monge–Ampère measures
corollaryCorollary 13.14.221D7A78F6BD02Non-pluripolar Monge–Ampère measures
corollaryCorollary 13.14.23BED3510B9193Non-pluripolar Monge–Ampère measures
section13.15.1 Range and restricted massBD96324E19C7Weighted Monge–Ampère measures
propositionProposition 13.15.257A94BD59310Weighted Monge–Ampère measures
lemmaLemma 13.15.3357B2DE7779EWeighted Monge–Ampère measures
equationEquation 13.15.4F3E7262939D2Weighted Monge–Ampère measures
section13.15.5 Exponential damping569E9BAF1CA2Weighted Monge–Ampère measures
propositionProposition 13.15.68B51BF8706B7Weighted Monge–Ampère measures
theoremTheorem 13.15.74394319FDE01Weighted Monge–Ampère measures
section13.15.8 Decreasing weights and capacity continuity1FE9FE4F07DEWeighted Monge–Ampère measures
theoremTheorem 13.15.9EE93EE2035F6Weighted Monge–Ampère measures
equationEquation 13.15.10D2588BCCC6F7Weighted Monge–Ampère measures
theoremTheorem 13.15.11B72AF5B37CB7Weighted Monge–Ampère measures
section13.15.12 Uniform tails and undamped convergence0D2E3E7B4DFDWeighted Monge–Ampère measures
theoremTheorem 13.15.138B2A9680933CWeighted Monge–Ampère measures
corollaryCorollary 13.15.147312F05A4396Weighted Monge–Ampère measures
propositionProposition 13.15.15ACA3FA133A3BWeighted Monge–Ampère measures
section13.15.16 Mass concentrationA75B7FC1B6D9Weighted Monge–Ampère measures
exampleExample 13.15.17446B53338035Weighted Monge–Ampère measures
corollaryCorollary 13.15.185804300415EAWeighted Monge–Ampère measures
subsection13.15.19 Intermediate currents of a shrinking sequence0117A750C1BEWeighted Monge–Ampère measures
propositionProposition 13.15.20708664A6CF93Weighted Monge–Ampère measures
equationEquation 13.15.2113C111AE3D65Weighted Monge–Ampère measures
section13.15.22 Damping for a smooth backgroundED067EDD9B9FWeighted Monge–Ampère measures
theoremTheorem 13.15.2318CDC482A9DDWeighted Monge–Ampère measures
theoremTheorem 13.17.1D7019DA65027Comparison of singularity volumes
propositionProposition 13.17.21C00C4C5A656Comparison of singularity volumes
lemmaLemma 13.18.192ABF533A916Volume deficits and relative full mass
equationEquation 13.18.28F79AE2F6948Volume deficits and relative full mass
propositionProposition 13.18.3169DBA6BAF3BVolume deficits and relative full mass
definitionDefinition 13.18.4FC98C471363BVolume deficits and relative full mass
theoremTheorem 13.18.5D73DFBAE1EC1Volume deficits and relative full mass
theoremTheorem 13.19.1B390F0847B3ADomination and ordinary envelopes
equationEquation 13.19.250B3C0245F90Domination and ordinary envelopes
corollaryCorollary 13.19.385CE662E8010Domination and ordinary envelopes
lemmaLemma 13.19.450A87754E163Domination and ordinary envelopes
theoremTheorem 13.19.5CB29AE1DFDA3Domination and ordinary envelopes
theoremTheorem 13.19.6C0F29EF85157Domination and ordinary envelopes
equationEquation 13.19.7F93B2C2EE801Domination and ordinary envelopes
corollaryCorollary 13.19.8B0CCD40EE4B7Domination and ordinary envelopes
lemmaLemma 13.20.1EDD01F5B65E5Strict barriers and background variation
equationEquation 13.20.2080E895C193DStrict barriers and background variation
corollaryCorollary 13.20.3D51B82B18D30Strict barriers and background variation
propositionProposition 13.20.40729BE10D92EStrict barriers and background variation
equationEquation 13.20.5F65E6B4224EAStrict barriers and background variation
propositionProposition 13.20.6F4C607C7C503Strict barriers and background variation
theoremTheorem 13.21.1453D4DA50352Monotone limits
theoremTheorem 13.21.2BE27C58EE55BMonotone limits
propositionProposition 13.21.3EE58E71478F6Monotone limits
corollaryCorollary 13.21.4D1480FEAFF87Monotone limits
corollaryCorollary 13.21.5293D12D027CCMonotone limits
subsection13.21.6 Recovery of envelope systems237A3786C187Monotone limits
theoremTheorem 13.21.79EA752566EF2Monotone limits
definitionDefinition 13.22.1A220D62C8412Cauchy convergence of model potentials
equationEquation 13.22.2451874D2170DCauchy convergence of model potentials
exampleExample 13.22.34B14FA731306Cauchy convergence of model potentials
propositionProposition 13.22.438359A611D4CCauchy convergence of model potentials
propositionProposition 13.22.5412DBAD823B9Cauchy convergence of model potentials
equationEquation 13.22.6DBE56A98BF3ECauchy convergence of model potentials
theoremTheorem 13.22.762ECCE0A41E6Cauchy convergence of model potentials
exampleExample 13.22.8AB2E8992474CCauchy convergence of model potentials
lemmaLemma 13.23.12B568CD72F9DRooftops and capacity convergence
propositionProposition 13.23.21542F41C8A51Rooftops and capacity convergence
theoremTheorem 13.23.37D5F8FB5CFABRooftops and capacity convergence
equationEquation 13.23.400C1CF12A199Rooftops and capacity convergence
corollaryCorollary 13.24.1C72B68A062F4Volumes of multiplier-ideal envelopes
propositionProposition 13.24.271B65CF5AD10Volumes of multiplier-ideal envelopes
theoremTheorem 13.24.3E9FC137BBE4EVolumes of multiplier-ideal envelopes
equationEquation 13.24.434D208A53BBDVolumes of multiplier-ideal envelopes
theoremTheorem 13.24.58BB867A0263FVolumes of multiplier-ideal envelopes
subsection13.24.6 A countable divisorial positivity criterion1D7E7321D20CVolumes of multiplier-ideal envelopes
lemmaLemma 13.24.7487BF774470FVolumes of multiplier-ideal envelopes
propositionProposition 13.24.820027E32C14EVolumes of multiplier-ideal envelopes
equationEquation 13.24.9027648EDCB33Volumes of multiplier-ideal envelopes
exampleExample 13.24.10E76D9C1B44B6Volumes of multiplier-ideal envelopes
problemProblem 13.24.11D0837305A3CFVolumes of multiplier-ideal envelopes
propositionProposition 13.25.161209D2B61BDModel comparison under background changes
propositionProposition 13.25.2F274299A5A71Model comparison under background changes
corollaryCorollary 13.25.31ED63BC3072BModel comparison under background changes
equationEquation 13.25.49EE86F88EB92Model comparison under background changes
theoremTheorem 13.25.5CBA354870D02Model comparison under background changes
lemmaLemma 13.26.1D5C96B9222A4Two-sided background perturbations
equationEquation 13.26.22826105ED7B3Two-sided background perturbations
theoremTheorem 13.26.3218C2CE831D1Two-sided background perturbations
equationEquation 13.26.4D190CC4FAB40Two-sided background perturbations
equationEquation 13.26.55E725A41F096Two-sided background perturbations
equationEquation 13.26.614679059DAF2Two-sided background perturbations
theoremTheorem 13.26.7279D427FE3F2Two-sided background perturbations
exampleExample 13.27.1A1CA1768E392A complete metric on positive-volume models
propositionProposition 13.27.244E1935166ECA complete metric on positive-volume models
equationEquation 13.27.3160186BB8FD9A complete metric on positive-volume models
theoremTheorem 13.27.4126E2D85CA62A complete metric on positive-volume models
equationEquation 13.27.5C3F22D1C1644A complete metric on positive-volume models
equationEquation 13.27.6207990AB4803A complete metric on positive-volume models
equationEquation 13.27.73DFF4A358B9CA complete metric on positive-volume models
equationEquation 13.27.8D335BA8A7131A complete metric on positive-volume models
corollaryCorollary 13.27.95AF48DFCEE5AA complete metric on positive-volume models
lemmaLemma 13.28.12422DD1AB3ABModel potentials on curves
equationEquation 13.28.2B922F9AF307BModel potentials on curves
equationEquation 13.28.3C63C9A28E474Model potentials on curves
theoremTheorem 13.28.474646DD62F50Model potentials on curves
equationEquation 13.28.5FB9F740F1579Model potentials on curves
equationEquation 13.28.6E72F3C33D0D0Model potentials on curves
corollaryCorollary 13.28.7FB89848D2500Model potentials on curves
theoremTheorem 13.28.83F90EA1D636AModel potentials on curves
equationEquation 13.28.9E5C7D16CF970Model potentials on curves
exampleExample 13.28.1098D940133E3BModel potentials on curves
propositionProposition 13.28.1103E01F83CA7AModel potentials on curves
equationEquation 13.28.122C63A2647FACModel potentials on curves
equationEquation 13.28.1357E1FB41A4EBModel potentials on curves
exampleExample 13.28.14730582377A86Model potentials on curves
subsection13.28.15 The ordinary-type topology on a curve7E4C501C1CA6Model potentials on curves
equationEquation 13.28.16D1D81F67E5DBModel potentials on curves
theoremTheorem 13.28.171BAAC809D02CModel potentials on curves
subsection13.28.18 Concentration of diffuse polar curvature0F287C888CF7Model potentials on curves
propositionProposition 13.28.194651CC70BDA5Model potentials on curves
subsection13.28.20 Convergence for all obstacles70267E3AB91DModel potentials on curves
equationEquation 13.28.210B5C0EA6F405Model potentials on curves
propositionProposition 13.28.22E7DAF71126F6Model potentials on curves
corollaryCorollary 13.28.23017BC9CAC896Model potentials on curves
propositionProposition 13.28.244805035AEC54Model potentials on curves
propositionProposition 13.29.1C9C98321BEFCBackground changes on curves
corollaryCorollary 13.29.202F73F083FCEBackground changes on curves
theoremTheorem 13.29.3151BAA2FA1ABBackground changes on curves
equationEquation 13.29.47ADEC26E190ABackground changes on curves
propositionProposition 13.29.5D21EE85B9F1FBackground changes on curves
equationEquation 13.29.6C555907478C9Background changes on curves
corollaryCorollary 13.29.721F64887A53EBackground changes on curves
equationEquation 13.29.80B648D2C5A55Background changes on curves
equationEquation 13.30.18E9F949AEF0DA complete space over varying curve backgrounds
equationEquation 13.30.2685F813E23E8A complete space over varying curve backgrounds
theoremTheorem 13.30.3C5551396F93BA complete space over varying curve backgrounds
equationEquation 13.30.434DDD9664A6DA complete space over varying curve backgrounds
corollaryCorollary 13.30.58918C681057EA complete space over varying curve backgrounds

14. Monge–Ampère equations and stability

equationEquation 14.1.1C5FA7F078BB6Smooth Monge–Ampère equations
theoremTheorem 14.2.1 (Tosatti–Weinkove)F33DA8C9DF9FScalar bounds and uniqueness
propositionProposition 14.2.2853DC10A2ADAScalar bounds and uniqueness
equationEquation 14.2.3EF5B417A3208Scalar bounds and uniqueness
propositionProposition 14.2.4366AA2E61158Scalar bounds and uniqueness
lemmaLemma 14.3.1E2F31793530DThe potential bound from weighted integration
lemmaLemma 14.3.2D570CDA4FA55The potential bound from weighted integration
equationEquation 14.3.3B38AB5EB6BF0The potential bound from weighted integration
equationEquation 14.3.42D25F9B7C178The potential bound from weighted integration
theoremTheorem 14.3.5EEF74F1DEBB8The potential bound from weighted integration
equationEquation 14.3.60F99936A9B52The potential bound from weighted integration
propositionProposition 14.4.12AD3CD874EA4The linearized equation and smooth dependence
lemmaLemma 14.5.1BC4847C0CD7DSecond derivatives and ellipticity
equationEquation 14.5.2CAD281B47B5ASecond derivatives and ellipticity
equationEquation 14.5.396DF02AD3F7CSecond derivatives and ellipticity
equationEquation 14.5.4827F0F57BBACSecond derivatives and ellipticity
theoremTheorem 14.5.5A61645907A4DSecond derivatives and ellipticity
equationEquation 14.5.68A4325B57E04Second derivatives and ellipticity
lemmaLemma 14.5.7E5E1B9237922Second derivatives and ellipticity
corollaryCorollary 14.5.836B42C74EE0BSecond derivatives and ellipticity
propositionProposition 14.7.1C52E44D9C563The smooth exponential equation
theoremTheorem 14.7.297F5C96646CCThe smooth exponential equation
theoremTheorem 14.8.1C611B2CA264AVariation of the normalization scalar
equationEquation 14.8.211F81D837A44Variation of the normalization scalar
equationEquation 14.8.3578022E3660AVariation of the normalization scalar
corollaryCorollary 14.8.4B9C312CB7069Variation of the normalization scalar
propositionProposition 14.9.1CA9F2EF1D149Normalization on the Hopf threefold
equationEquation 14.9.2028038652BCCNormalization on the Hopf threefold
equationEquation 14.9.39CEA8A3A68F0Normalization on the Hopf threefold
corollaryCorollary 14.9.4BFCF1480CC8BNormalization on the Hopf threefold
theoremTheorem 14.10.1 (Székelyhidi–Tosatti–Weinkove)76AF759A4024Prescribing a Gauduchon volume form
propositionProposition 14.11.160FD5A56EE10Bounds for the Gauduchon potential
equationEquation 14.11.28F909795AE16Bounds for the Gauduchon potential
propositionProposition 14.12.1706EB82AB6A2The adjoint equation for Gauduchon volume
equationEquation 14.12.262F6382D2CACThe adjoint equation for Gauduchon volume
propositionProposition 14.12.3029A1D30A55EThe adjoint equation for Gauduchon volume
equationEquation 14.12.4BD3712C889A6The adjoint equation for Gauduchon volume
equationEquation 14.12.5195CC5F57ABEThe adjoint equation for Gauduchon volume
equationEquation 14.13.1D281466AD7B7The matrix structure of the Gauduchon equation
lemmaLemma 14.13.26168D5D4EF96The matrix structure of the Gauduchon equation
equationEquation 14.13.3A6EC7DDEACFBThe matrix structure of the Gauduchon equation
equationEquation 14.13.41525F23C956CThe matrix structure of the Gauduchon equation
lemmaLemma 14.13.5DBC2334A4FC0The matrix structure of the Gauduchon equation
equationEquation 14.13.6051408386360The matrix structure of the Gauduchon equation
equationEquation 14.13.72B456F0E5D2BThe matrix structure of the Gauduchon equation
equationEquation 14.13.8677E6AB0568AThe matrix structure of the Gauduchon equation
theoremTheorem 14.14.12763FB514B34The largest-eigenvalue estimate
paragraph14.14.2 Differentiating the largest eigenvalue.9994CF39B2B0The largest-eigenvalue estimate
equationEquation 14.14.3E4B294710238The largest-eigenvalue estimate
equationEquation 14.14.41E20E0CF9A10The largest-eigenvalue estimate
equationEquation 14.14.5143977813E90The largest-eigenvalue estimate
paragraph14.14.6 The third derivatives and the torsion term.D5E11D27FAAAThe largest-eigenvalue estimate
equationEquation 14.14.7BE0A6E37B652The largest-eigenvalue estimate
equationEquation 14.14.85D951DE4F0FDThe largest-eigenvalue estimate
equationEquation 14.14.9EACA66A12E82The largest-eigenvalue estimate
equationEquation 14.14.10D807F632DAD7The largest-eigenvalue estimate
equationEquation 14.14.1135CF4803162EThe largest-eigenvalue estimate
paragraph14.14.12 Cancellation when the smallest eigenvalue is controlled.DA6F05F8F203The largest-eigenvalue estimate
equationEquation 14.14.137B27AFD05F7EThe largest-eigenvalue estimate
equationEquation 14.14.149CFA88376131The largest-eigenvalue estimate
paragraph14.14.15 The remaining eigenvalue configuration.57C7A82680F3The largest-eigenvalue estimate
lemmaLemma 14.15.16E3495B1A225A Liouville theorem for the limiting cone
theoremTheorem 14.15.2AC61FE141521A Liouville theorem for the limiting cone
equationEquation 14.15.3B2C22D2540A1A Liouville theorem for the limiting cone
equationEquation 14.15.4C26DE70CAF39A Liouville theorem for the limiting cone
equationEquation 14.15.589711B0B590AA Liouville theorem for the limiting cone
propositionProposition 14.16.1EFE7268223F4The gradient bound by rescaling
equationEquation 14.16.2977E6E7B264DThe gradient bound by rescaling
equationEquation 14.16.33DCC81BE4D1EThe gradient bound by rescaling
lemmaLemma 14.17.119E4D601D907Existence for the Gauduchon ansatz
lemmaLemma 14.18.11ECE72025D8ENormalization for the Gauduchon equation
theoremTheorem 14.18.2F90961841CA0Normalization for the Gauduchon equation
exampleExample 14.18.38AC1A3B0F2AFNormalization for the Gauduchon equation
section14.19.1 Continuous solvability49F0FF1AED46Continuous solutions and normalization
theoremTheorem 14.19.2 (Kołodziej–Nguyen)C8D798C2D2F0Continuous solutions and normalization
theoremTheorem 14.19.3 (Nguyen)CFA579B45DF9Continuous solutions and normalization
equationEquation 14.19.4C01B1AA0FBCAContinuous solutions and normalization
section14.19.5 The modified comparison principle88B9A2247D18Continuous solutions and normalization
theoremTheorem 14.19.6492E047184B3Continuous solutions and normalization
equationEquation 14.19.763BE08CDB80CContinuous solutions and normalization
equationEquation 14.19.83A0FB1FFA628Continuous solutions and normalization
equationEquation 14.19.9C197626AD934Continuous solutions and normalization
theoremTheorem 14.19.10C0DEE578BC5DContinuous solutions and normalization
corollaryCorollary 14.19.117C5FAB3BA5DAContinuous solutions and normalization
corollaryCorollary 14.19.12AB62A0258D5DContinuous solutions and normalization
section14.19.13 Capacity estimates and uniform compactnessEF206AD1915FContinuous solutions and normalization
lemmaLemma 14.19.14370AD0860DADContinuous solutions and normalization
equationEquation 14.19.15F937FDFBF165Continuous solutions and normalization
lemmaLemma 14.19.169C1A40570E78Continuous solutions and normalization
equationEquation 14.19.173CFE977F80E0Continuous solutions and normalization
equationEquation 14.19.18623E09972ED8Continuous solutions and normalization
theoremTheorem 14.19.19EA32B8891E2BContinuous solutions and normalization
equationEquation 14.19.20F7083E3F1991Continuous solutions and normalization
theoremTheorem 14.19.21BF849D15DF41Continuous solutions and normalization
subsection14.19.22 Stability under a power capacity bound042B61EBF79DContinuous solutions and normalization
propositionProposition 14.19.233B70BD39881DContinuous solutions and normalization
equationEquation 14.19.24B0DB641874CEContinuous solutions and normalization
section14.19.25 Construction of continuous solutionsFEE6FC2B120FContinuous solutions and normalization
lemmaLemma 14.19.26B2B0EF9B5807Continuous solutions and normalization
equationEquation 14.19.2774778A2A41BFContinuous solutions and normalization
lemmaLemma 14.19.288068BC80F22DContinuous solutions and normalization
corollaryCorollary 14.19.29AEA7DCEDBA8BContinuous solutions and normalization
section14.19.30 Smoothness of bounded solutions6CE6BC329E93Continuous solutions and normalization
lemmaLemma 14.19.3155D3763AB355Continuous solutions and normalization
theoremTheorem 14.19.32B9E6E7C4706BContinuous solutions and normalization
section14.19.33 The vanishing exponent and scalar uniqueness27FAAF143258Continuous solutions and normalization
propositionProposition 14.19.34880520C4A2B1Continuous solutions and normalization
equationEquation 14.19.353B8781BD8BF5Continuous solutions and normalization
propositionProposition 14.19.3692CEFF8A020BContinuous solutions and normalization
section14.19.37 Normalization on the Hopf threefold4D6684B2CEDBContinuous solutions and normalization
propositionProposition 14.19.3856B19584AE9EContinuous solutions and normalization
equationEquation 14.19.395789C77F73D2Continuous solutions and normalization
propositionProposition 14.19.4021F8F4F18D0AContinuous solutions and normalization
exampleExample 14.19.41F8EB82A5AD2BContinuous solutions and normalization
section14.19.42 Stability for concave equationsA73917A5BC62Continuous solutions and normalization
lemmaLemma 14.19.43C3049F567279Continuous solutions and normalization
propositionProposition 14.19.44877A402699F6Continuous solutions and normalization
theoremTheorem 14.19.450F1A655FC651Continuous solutions and normalization
corollaryCorollary 14.19.46E9194EDCB1FFContinuous solutions and normalization
section14.19.47 Gradient bounds for concave equationsF77131C8889AContinuous solutions and normalization
lemmaLemma 14.19.48D049243EB2FAContinuous solutions and normalization
lemmaLemma 14.19.49BD78CDB91D24Continuous solutions and normalization
propositionProposition 14.19.50F13386AF12E0Continuous solutions and normalization
theoremTheorem 14.19.51979B56319AA1Continuous solutions and normalization
equationEquation 14.20.12EA7CA32AE91Uniqueness for degenerate equations
equationEquation 14.21.193A49A38E101Extrema and density sublevels
equationEquation 14.21.2E391B921798AExtrema and density sublevels
theoremTheorem 14.21.36EB399397505Extrema and density sublevels
equationEquation 14.21.47732F2B90E2AExtrema and density sublevels
lemmaLemma 14.21.559B8A4C8DA05Extrema and density sublevels
lemmaLemma 14.21.696D40C4808C2Extrema and density sublevels
equationEquation 14.21.757588F1C3107Extrema and density sublevels
equationEquation 14.21.8144125DEDD1CExtrema and density sublevels
equationEquation 14.21.9CEEE110C45FEExtrema and density sublevels
equationEquation 14.21.10D7E23A564A00Extrema and density sublevels
equationEquation 14.21.11C5CDE7D4BA13Extrema and density sublevels
subsection14.21.12 A square-root threshold95F071B4092FExtrema and density sublevels
theoremTheorem 14.21.1361851D05B072Extrema and density sublevels
equationEquation 14.21.14B72083672909Extrema and density sublevels
equationEquation 14.21.15BB22D6F8FAF9Extrema and density sublevels
theoremTheorem 14.22.1919310C73DBEPolarized cofactors and propagation
equationEquation 14.22.224A83D889133Polarized cofactors and propagation
subsection14.22.3 Weighted energy1961884C0812Polarized cofactors and propagation
equationEquation 14.22.4271FE8625215Polarized cofactors and propagation
equationEquation 14.22.5A2BD70AF14ECPolarized cofactors and propagation
equationEquation 14.22.6A832621F5643Polarized cofactors and propagation
equationEquation 14.22.7A81FFABB4E8DPolarized cofactors and propagation
subsection14.22.8 The real symbol and exterior-ball propagation1FF2DD5C252CPolarized cofactors and propagation
equationEquation 14.22.904DC9E828E52Polarized cofactors and propagation
equationEquation 14.22.107A7369E544A8Polarized cofactors and propagation
equationEquation 14.22.112692BD3333CAPolarized cofactors and propagation
subsection14.22.12 The unweighted energy defectB7138A0B3C69Polarized cofactors and propagation
propositionProposition 14.22.13F563CACF5D7DPolarized cofactors and propagation
equationEquation 14.22.14475D4376A313Polarized cofactors and propagation
corollaryCorollary 14.22.15EAC3CF41BD99Polarized cofactors and propagation
theoremTheorem 14.23.1AF9D036E600FIntegral rigidity in dimensions two and three
equationEquation 14.23.26127DB916FC9Integral rigidity in dimensions two and three
equationEquation 14.23.3CB4E143A2628Integral rigidity in dimensions two and three
equationEquation 14.23.4899D523AFD3CIntegral rigidity in dimensions two and three
equationEquation 14.23.5CFA2A8BCECDAIntegral rigidity in dimensions two and three
equationEquation 14.23.681191FBC0D41Integral rigidity in dimensions two and three
corollaryCorollary 14.23.73DF5F7F34405Integral rigidity in dimensions two and three
theoremTheorem 14.24.198EC31947E4CBounded uniqueness and connectedness
propositionProposition 14.24.22C3AA01F259BBounded uniqueness and connectedness
equationEquation 14.24.3C7BC3EE4D543Bounded uniqueness and connectedness
subsection14.24.4 The energy of the difference42826D458068Bounded uniqueness and connectedness
propositionProposition 14.24.525FABE38E60BBounded uniqueness and connectedness
equationEquation 14.24.65EA1FD990FD5Bounded uniqueness and connectedness
equationEquation 14.24.7B3915B7974CABounded uniqueness and connectedness
equationEquation 14.24.8C45BFBEEAF7FBounded uniqueness and connectedness
equationEquation 14.24.9D3E407319B3EBounded uniqueness and connectedness
equationEquation 14.24.107EF7549DDD02Bounded uniqueness and connectedness
lemmaLemma 14.24.114FB9051BD00EBounded uniqueness and connectedness
subsection14.24.12 Concentrating smooth test potentialsE597C8D6FBF4Bounded uniqueness and connectedness
propositionProposition 14.24.137F590FD94846Bounded uniqueness and connectedness
equationEquation 14.24.146928CCD52E25Bounded uniqueness and connectedness
equationEquation 14.24.15768EAB7723D9Bounded uniqueness and connectedness
equationEquation 14.24.167044361ECDC0Bounded uniqueness and connectedness
corollaryCorollary 14.24.17527FB17149BBBounded uniqueness and connectedness
equationEquation 14.25.10B3AF72C81CBComparison and invariant mass
theoremTheorem 14.25.227DED19DBC11Comparison and invariant mass
theoremTheorem 14.25.3FE27B72E2DB5Comparison and invariant mass
equationEquation 14.25.4A25B3D75AF64Comparison and invariant mass
equationEquation 14.25.5A548048D0134Comparison and invariant mass
equationEquation 14.25.60024EEF19B40Comparison and invariant mass
remarkRemark 14.25.7EFB13673C92AComparison and invariant mass
theoremTheorem 14.26.1050594389138Smooth nonuniqueness for a degenerate density
subsection14.26.2 Three rank-one directions0CFD28034339Smooth nonuniqueness for a degenerate density
equationEquation 14.26.36970E9123EB5Smooth nonuniqueness for a degenerate density
equationEquation 14.26.471DFC7D0861CSmooth nonuniqueness for a degenerate density
equationEquation 14.26.51C52D8C53893Smooth nonuniqueness for a degenerate density
equationEquation 14.26.6DE71ED12948BSmooth nonuniqueness for a degenerate density
subsection14.26.7 The scalar coefficients68DF63E92045Smooth nonuniqueness for a degenerate density
equationEquation 14.26.86C8664C40E89Smooth nonuniqueness for a degenerate density
subsection14.26.9 Two forms with the same cubeDBEDC79E97CDSmooth nonuniqueness for a degenerate density
subsection14.26.10 Restoring strict positivity of the background2D54B6D6C7ADSmooth nonuniqueness for a degenerate density
equationEquation 14.26.110711E4CCA0FESmooth nonuniqueness for a degenerate density
subsection14.26.12 A continuum of radial solutions6FDC265145E3Smooth nonuniqueness for a degenerate density
equationEquation 14.26.13381112F03D1ESmooth nonuniqueness for a degenerate density
propositionProposition 14.26.143A73F825D91BSmooth nonuniqueness for a degenerate density
equationEquation 14.26.15D2FC90D7554BSmooth nonuniqueness for a degenerate density
equationEquation 14.26.162EC8417EFF1ESmooth nonuniqueness for a degenerate density
theoremTheorem 14.27.1E16A6F31A061Nonuniqueness with a zero set of empty interior
equationEquation 14.27.242198F5D282ANonuniqueness with a zero set of empty interior
lemmaLemma 14.27.3C96C68E6BB33Nonuniqueness with a zero set of empty interior
equationEquation 14.27.4FE5D8F02666FNonuniqueness with a zero set of empty interior
equationEquation 14.27.5FCE58E25CF39Nonuniqueness with a zero set of empty interior
equationEquation 14.27.6DDB2BCDB3552Nonuniqueness with a zero set of empty interior
lemmaLemma 14.27.731CA87ECD1DFNonuniqueness with a zero set of empty interior
equationEquation 14.27.85C8FF3274F30Nonuniqueness with a zero set of empty interior
theoremTheorem 14.28.19C11A9E88B60Quadratic density zeros
equationEquation 14.28.2ACD0920D4D78Quadratic density zeros
lemmaLemma 14.28.3624B2CF7FBA2Quadratic density zeros
equationEquation 14.28.43D1CCE4952BAQuadratic density zeros
equationEquation 14.28.586148D5BD1E9Quadratic density zeros
equationEquation 14.28.694F5CC68CD22Quadratic density zeros
theoremTheorem 14.29.1DD001DBBE659A real analytic density
equationEquation 14.29.2B358F4101DEDA real analytic density
equationEquation 14.29.307B68FE9B8BCA real analytic density
subsection14.29.4 Prescribing the determinantCB8BE6BADB1AA real analytic density
equationEquation 14.29.53FD3DD269BA8A real analytic density
equationEquation 14.29.614762A8A84D0A real analytic density
equationEquation 14.29.77FEE08922F33A real analytic density
equationEquation 14.29.8E916650EE31AA real analytic density
equationEquation 14.29.9025CE4264442A real analytic density
equationEquation 14.29.1017DDFD1D68B2A real analytic density
subsection14.29.11 Extension over the projective endsEA62A44D2D9FA real analytic density
subsection14.29.12 A positive equation background38D1973448C7A real analytic density
equationEquation 14.29.131FC5DA40EAF0A real analytic density
subsection14.29.14 Zeros and higher dimensionsE82948C4DB2AA real analytic density
equationEquation 14.29.15E3CA723897C6A real analytic density
equationEquation 14.29.16C72BCF7334A2A real analytic density
remarkRemark 14.29.17DF533F76B4CEA real analytic density

15. Monge–Ampère equations with prescribed singularities

lemmaLemma 15.2.1E9045CC3F2A3Domination with a smaller background
theoremTheorem 15.2.2D70D1E469F9DDomination with a smaller background
corollaryCorollary 15.2.3AF890AD49AC5Domination with a smaller background
lemmaLemma 15.3.1D716F01D638BBounded equations in a semipositive big background
subsection15.3.2 The exponential equation0853ABB454FCBounded equations in a semipositive big background
theoremTheorem 15.3.396A341EAF1D4Bounded equations in a semipositive big background
subsection15.3.4 The normalized equation727A21D220CEBounded equations in a semipositive big background
theoremTheorem 15.3.5A972C11565EDBounded equations in a semipositive big background
equationEquation 15.3.69D2B0195702FBounded equations in a semipositive big background
theoremTheorem 15.4.172397EB33655A quantitative stability estimate
equationEquation 15.4.2E617EAC91766A quantitative stability estimate
corollaryCorollary 15.4.3E429FB866D08A quantitative stability estimate
theoremTheorem 15.5.1CA29182325B0Hölder continuity from bounded stability
equationEquation 15.5.22D7C0ED421F7Hölder continuity from bounded stability
subsection15.5.3 Regularity for a measure with a Hölder reference9A0897CE06B0Hölder continuity from bounded stability
theoremTheorem 15.5.456C5E320D330Hölder continuity from bounded stability
theoremTheorem 15.6.102C5F4AFC7FFLocal continuity in a semipositive big background
lemmaLemma 15.7.15C54B7FA3A8BHomogeneous replacement on a coordinate ball
theoremTheorem 15.7.2B0FDBEE3B071Homogeneous replacement on a coordinate ball
equationEquation 15.7.36815D885EDC4Homogeneous replacement on a coordinate ball
theoremTheorem 15.8.1F6FD255985FFContact support in a big background
equationEquation 15.8.2EE69BB255211Contact support in a big background
subsection15.8.3 Quasi-continuous obstacles5E57580F8E39Contact support in a big background
propositionProposition 15.8.429A74E6E8A18Contact support in a big background
propositionProposition 15.8.5BDE06B833BC8Contact support in a big background
theoremTheorem 15.8.66F087EC35E2EContact support in a big background
equationEquation 15.8.75ACCF67CCE26Contact support in a big background
theoremTheorem 15.9.1A15C1A7CB4F8Compactness with a common lower barrier
equationEquation 15.9.2B172EB5C34CBCompactness with a common lower barrier
propositionProposition 15.9.3EFAF50966FDFCompactness with a common lower barrier
corollaryCorollary 15.9.4F856D22ADF4CCompactness with a common lower barrier
theoremTheorem 15.10.1BA5CB37AAAC1A bounded subsolution theorem
equationEquation 15.10.225E63CEC282CA bounded subsolution theorem
theoremTheorem 15.11.17A54D08FEB18Adding weighted Monge–Ampère measures
theoremTheorem 15.12.17AFA75E73767Equations in a prescribed ordinary type
lemmaLemma 15.12.2D0B5A6CE28E9Equations in a prescribed ordinary type
theoremTheorem 15.12.3413B92CD2B72Equations in a prescribed ordinary type
lemmaLemma 15.12.45A797D7301BBEquations in a prescribed ordinary type
subsection15.12.5 Existence for a positive exponent3EFAC1693CBDEquations in a prescribed ordinary type
subsection15.12.6 The normalized equation21A9095B1925Equations in a prescribed ordinary type
lemmaLemma 15.13.1E6E5D71344DDComparison above a strict singular barrier
theoremTheorem 15.13.2F71978C42841Comparison above a strict singular barrier
corollaryCorollary 15.13.324CD508AEB8DComparison above a strict singular barrier
lemmaLemma 15.14.1869D3138741DSubsolutions and the Gauduchon trace
propositionProposition 15.14.20A5580561320Subsolutions and the Gauduchon trace
equationEquation 15.14.3B100E71E56DESubsolutions and the Gauduchon trace
theoremTheorem 15.15.1668D07ADFF44Stability above a strict barrier
equationEquation 15.15.26B6C204B9D9EStability above a strict barrier
corollaryCorollary 15.15.3131F17206F6AStability above a strict barrier
lemmaLemma 15.16.1154DE11154EEMinimal singularities in an arbitrary big background
theoremTheorem 15.16.2352C3567BAADMinimal singularities in an arbitrary big background
equationEquation 15.16.3A8E86D90F114Minimal singularities in an arbitrary big background
equationEquation 15.16.4B66C3AA2C051Minimal singularities in an arbitrary big background
lemmaLemma 15.16.5DA20F7BB0188Minimal singularities in an arbitrary big background
equationEquation 15.16.627B45CD3E8B0Minimal singularities in an arbitrary big background
theoremTheorem 15.16.7B7C31259154BMinimal singularities in an arbitrary big background
theoremTheorem 15.17.1D662421F3436Local Hölder continuity in a big background
subsection15.17.2 Limits of normalized exponential solutionsB0C9769ACDCDLocal Hölder continuity in a big background
propositionProposition 15.17.3D11439C1393FLocal Hölder continuity in a big background
corollaryCorollary 15.17.464893D9DABFELocal Hölder continuity in a big background
theoremTheorem 15.18.143AC522E5ED8Interior smoothness on the regular locus
subsection15.18.2 Local comparison7A246CDE0880Interior smoothness on the regular locus
propositionProposition 15.18.3D2F8F8FD5A17Interior smoothness on the regular locus
subsection15.18.4 Bounded traces and smoothness5FD96F70C06EInterior smoothness on the regular locus
propositionProposition 15.18.57DF8FE8A18D7Interior smoothness on the regular locus
equationEquation 15.18.67A41D9785A7CInterior smoothness on the regular locus
subsection15.18.7 Smooth exact approximantsE9933083FDD7Interior smoothness on the regular locus
lemmaLemma 15.18.83EBB652C42D7Interior smoothness on the regular locus
equationEquation 15.18.959253C0B8B5AInterior smoothness on the regular locus
equationEquation 15.18.108A24C1B85BE1Interior smoothness on the regular locus
equationEquation 15.18.111894AFF4178EInterior smoothness on the regular locus
equationEquation 15.18.129C906B419E9FInterior smoothness on the regular locus
paragraph15.18.13 A polynomial boundary trace estimate94BAFACA36D1Interior smoothness on the regular locus
propositionProposition 15.18.146586F9B3B128Interior smoothness on the regular locus
equationEquation 15.18.15F89A683ED708Interior smoothness on the regular locus
equationEquation 15.18.16A09942AD8934Interior smoothness on the regular locus
equationEquation 15.18.17F834489897B3Interior smoothness on the regular locus
equationEquation 15.18.1813E61CA163BFInterior smoothness on the regular locus
equationEquation 15.18.194327E0F53426Interior smoothness on the regular locus
paragraph15.18.20 Exact approximationDF626CDAD5D9Interior smoothness on the regular locus
propositionProposition 15.18.2159A5B60FAB74Interior smoothness on the regular locus
equationEquation 15.18.229F54A2578D16Interior smoothness on the regular locus
equationEquation 15.18.23082B0A6218F2Interior smoothness on the regular locus
subsection15.18.24 Localization near the nonregular setB0A92E91F8D0Interior smoothness on the regular locus
propositionProposition 15.18.25F9CDEC120FA0Interior smoothness on the regular locus
equationEquation 15.18.2687542664E1D7Interior smoothness on the regular locus
equationEquation 15.18.27E603AE0642F8Interior smoothness on the regular locus
equationEquation 15.18.28D7F225F56CDCInterior smoothness on the regular locus
equationEquation 15.18.29BDAA917CBC24Interior smoothness on the regular locus
equationEquation 15.18.30EBBDF08FE74BInterior smoothness on the regular locus
propositionProposition 15.18.3145A010B2A5AFInterior smoothness on the regular locus
equationEquation 15.18.32D534513E3EFAInterior smoothness on the regular locus
equationEquation 15.18.334E46369984C7Interior smoothness on the regular locus
subsection15.18.34 Derivative estimates and bounded traces163F19CEE4A0Interior smoothness on the regular locus
propositionProposition 15.18.35ADEA37AAF3BBInterior smoothness on the regular locus
equationEquation 15.18.36E4388ADCCF56Interior smoothness on the regular locus
equationEquation 15.18.373DC50D5C4903Interior smoothness on the regular locus
equationEquation 15.18.382F9977F68923Interior smoothness on the regular locus
equationEquation 15.18.398CAF7DF1496DInterior smoothness on the regular locus
equationEquation 15.18.4040CD1B311623Interior smoothness on the regular locus
equationEquation 15.18.41FA7DC597E9CBInterior smoothness on the regular locus
equationEquation 15.18.429EDD6E774993Interior smoothness on the regular locus
equationEquation 15.18.434261D49DE46DInterior smoothness on the regular locus
equationEquation 15.18.444DCF25F09DA4Interior smoothness on the regular locus
equationEquation 15.18.4597A8DB2D4465Interior smoothness on the regular locus
propositionProposition 15.18.4629094CECD90FInterior smoothness on the regular locus
equationEquation 15.18.477B1658CB459DInterior smoothness on the regular locus
equationEquation 15.18.4866EE1806AFB7Interior smoothness on the regular locus
subsection15.18.49 Proof of the main theorem6E838824045AInterior smoothness on the regular locus
corollaryCorollary 15.18.50AD8A02A6088DInterior smoothness on the regular locus
equationEquation 15.19.1C397B905A723Degenerating semipositive backgrounds
equationEquation 15.19.28C6C5AF468B2Degenerating semipositive backgrounds
lemmaLemma 15.19.34A0FCD6D9FCDDegenerating semipositive backgrounds
lemmaLemma 15.19.4C390030D09AEDegenerating semipositive backgrounds
equationEquation 15.19.5279BD7D5BB42Degenerating semipositive backgrounds
equationEquation 15.19.62D587C24F80BDegenerating semipositive backgrounds
theoremTheorem 15.19.7E00C93820852Degenerating semipositive backgrounds
subsection15.19.8 Small resolutions of nodes8C7192C73C68Degenerating semipositive backgrounds
equationEquation 15.19.95C6AB7F124F9Degenerating semipositive backgrounds
lemmaLemma 15.19.10C6428C798D5FDegenerating semipositive backgrounds
equationEquation 15.19.11AC4599B20814Degenerating semipositive backgrounds
corollaryCorollary 15.19.12A1536B3C6F39Degenerating semipositive backgrounds
subsection15.19.13 Measures and bounded potentials at a nodeDA6098A35E46Degenerating semipositive backgrounds
lemmaLemma 15.19.149225EE111CC9Degenerating semipositive backgrounds
equationEquation 15.19.15DE65B074D310Degenerating semipositive backgrounds
equationEquation 15.19.169FFE9C060E33Degenerating semipositive backgrounds
theoremTheorem 15.19.17E621E67CCA9ADegenerating semipositive backgrounds
subsection15.19.18 Distances across finitely many contracting regionsF6016CBBF3C5Degenerating semipositive backgrounds
definitionDefinition 15.19.190EB9DF723176Degenerating semipositive backgrounds
lemmaLemma 15.19.20022849340F28Degenerating semipositive backgrounds
lemmaLemma 15.19.21E3C9C96E314EDegenerating semipositive backgrounds
propositionProposition 15.19.2265049DB7855EDegenerating semipositive backgrounds
subsubsection15.19.23 Curve areas and radial tube lengths83F8EFD496FFDegenerating semipositive backgrounds
lemmaLemma 15.19.24240B8EBF50C3Degenerating semipositive backgrounds
exampleExample 15.19.25F7F939140E90Degenerating semipositive backgrounds
propositionProposition 15.19.264A271B6E1DDBDegenerating semipositive backgrounds
exampleExample 15.19.2711425F03EC01Degenerating semipositive backgrounds
propositionProposition 15.20.11750C7E75315Symmetry and the selected projective limit
corollaryCorollary 15.20.2D0124640DBA7Symmetry and the selected projective limit
subsection15.20.3 The invariant projective solutionD8114BC528BBSymmetry and the selected projective limit
theoremTheorem 15.20.472B88B8B1639Symmetry and the selected projective limit
lemmaLemma 15.20.55D3F7173CA27Symmetry and the selected projective limit
equationEquation 15.20.6B34A7F7705B9Symmetry and the selected projective limit
subsection15.20.7 Second derivatives at positive exponentBDDD67E926DCSymmetry and the selected projective limit
propositionProposition 15.20.8517F0BB04B80Symmetry and the selected projective limit
subsection15.21.1 Smooth boundary traces93E0127DCAFFLocal density and boundary control
theoremTheorem 15.21.29826C9FA5510Local density and boundary control
equationEquation 15.21.380E9773F7495Local density and boundary control
exampleExample 15.21.4DF2D43B90C47Local density and boundary control
equationEquation 15.21.5334EFF96259FLocal density and boundary control
subsection15.21.6 A bounded discontinuous potentialAD0BD024C9A0Local density and boundary control
equationEquation 15.21.76321AD313CEDLocal density and boundary control
lemmaLemma 15.21.85E22269BEE55Local density and boundary control
equationEquation 15.21.969692418B3EELocal density and boundary control
equationEquation 15.21.10331160A2E4B5Local density and boundary control
subsection15.21.11 Compact gluingAABABE424D5CLocal density and boundary control
theoremTheorem 15.21.125FC2412C22E9Local density and boundary control
equationEquation 15.21.13477A59B42B24Local density and boundary control
equationEquation 15.21.14BAE98470F15DLocal density and boundary control
exampleExample 15.21.1541D1398D77FCLocal density and boundary control
equationEquation 15.22.11B942D170060Equations with resolved logarithmic poles
theoremTheorem 15.22.23B5397268E44Equations with resolved logarithmic poles
equationEquation 15.22.34669708E978CEquations with resolved logarithmic poles
equationEquation 15.22.491C08F2AC112Equations with resolved logarithmic poles
equationEquation 15.22.5BBA63F88F7F7Equations with resolved logarithmic poles
equationEquation 15.22.67DF899E20ABDEquations with resolved logarithmic poles
exampleExample 15.22.70362BBF11A0BEquations with resolved logarithmic poles
equationEquation 15.22.8853BE83A0787Equations with resolved logarithmic poles
subsection15.22.9 Residual mass12F28DB77548Equations with resolved logarithmic poles
propositionProposition 15.22.10D1ADF0EFDD30Equations with resolved logarithmic poles
equationEquation 15.22.1100D02AC76853Equations with resolved logarithmic poles
equationEquation 15.23.104B0CB069F1CMinimal singularities on blowups
propositionProposition 15.23.2E42F32DA29D8Minimal singularities on blowups
equationEquation 15.23.3C7554C964203Minimal singularities on blowups
equationEquation 15.23.4282A5F737F88Minimal singularities on blowups
theoremTheorem 15.23.566FFA0C36433Minimal singularities on blowups
equationEquation 15.23.670493856D26CMinimal singularities on blowups
equationEquation 15.23.77E7FF8A6A54EMinimal singularities on blowups
equationEquation 15.23.82BE1216FD9F5Minimal singularities on blowups
exampleExample 15.23.96C568E91A192Minimal singularities on blowups
theoremTheorem 15.23.1037D35D78F0C6Minimal singularities on blowups
equationEquation 15.23.11749BE96956FEMinimal singularities on blowups
propositionProposition 15.24.1A46357C92C12Representing measures in an ordinary type
equationEquation 15.24.2036B1477ABB4Representing measures in an ordinary type
equationEquation 15.24.33F73A2DB6267Representing measures in an ordinary type
lemmaLemma 15.24.4F9088D3DDEBBRepresenting measures in an ordinary type
theoremTheorem 15.25.18CE230CD94D4The homogeneous equation with a Hölder reference
equationEquation 15.26.1938F2C76C18ASingular measures on shrinking spheres
propositionProposition 15.26.28F3C95237950Singular measures on shrinking spheres
corollaryCorollary 15.26.3AB5B44EC4BDDSingular measures on shrinking spheres
theoremTheorem 15.26.46076D6F3256CSingular measures on shrinking spheres
lemmaLemma 15.27.1CE5CA1071A3CLower mass and solvability for all measures
lemmaLemma 15.27.2366F6832AB46Lower mass and solvability for all measures
theoremTheorem 15.27.3C7AA7377C144Lower mass and solvability for all measures
equationEquation 15.27.48D5EB0B8CCD7Lower mass and solvability for all measures
corollaryCorollary 15.27.5810CC86832BFLower mass and solvability for all measures
corollaryCorollary 15.27.605B92E5A4684Lower mass and solvability for all measures
equationEquation 15.28.1E3AA9E3C0D2DSubsolutions for an individual density
equationEquation 15.28.24A9175A29709Subsolutions for an individual density
lemmaLemma 15.28.38298FD96E154Subsolutions for an individual density
theoremTheorem 15.28.4CC13A9B1D927Subsolutions for an individual density
subsection15.28.5 The greatest normalized threshold subsolutionE09A140F700DSubsolutions for an individual density
propositionProposition 15.28.6D42D2088B421Subsolutions for an individual density
propositionProposition 15.28.769BB1D568698Subsolutions for an individual density
subsection15.28.8 Smooth comparison testsA89D44842BA5Subsolutions for an individual density
theoremTheorem 15.28.94AC87EC7D266Subsolutions for an individual density
theoremTheorem 15.29.1346B39C3F0B6Gauduchon tests for integrable densities
equationEquation 15.29.277553DB34B46Gauduchon tests for integrable densities
equationEquation 15.29.33FECA4E31049Gauduchon tests for integrable densities
equationEquation 15.29.4C6012A389791Gauduchon tests for integrable densities
propositionProposition 15.29.5429BB0FF98A9Gauduchon tests for integrable densities
lemmaLemma 15.30.1B30B4EF45772Exact equations on products with zero lower mass
lemmaLemma 15.30.240B884870886Exact equations on products with zero lower mass
theoremTheorem 15.30.3B36BAA6B4AA0Exact equations on products with zero lower mass
corollaryCorollary 15.30.48C4A9CEF04C8Exact equations on products with zero lower mass
subsection15.30.5 Closed factors and arbitrary integrable densitiesEB3E71C5CDCBExact equations on products with zero lower mass
theoremTheorem 15.30.6AB3D52EC73EFExact equations on products with zero lower mass
equationEquation 15.31.155B17A6945D4Singular measures on products with a curve
subsection15.31.2 The logarithmic potential and its measure trace06937C5027C2Singular measures on products with a curve
lemmaLemma 15.31.32CA6CA22B712Singular measures on products with a curve
lemmaLemma 15.31.4F5E3175FA178Singular measures on products with a curve
equationEquation 15.31.5DB0D4C618F78Singular measures on products with a curve
equationEquation 15.31.679703FA8B5E7Singular measures on products with a curve
subsection15.31.7 Exponential equations and uniform selection on the curve9002CCE9B34FSingular measures on products with a curve
theoremTheorem 15.31.8346FA2C8C90ASingular measures on products with a curve
equationEquation 15.31.9C82032A970D1Singular measures on products with a curve
equationEquation 15.31.10C8BB18387ED9Singular measures on products with a curve
equationEquation 15.31.1191D68636194FSingular measures on products with a curve
subsection15.31.12 The product equation and its maximal coefficientC004782057EESingular measures on products with a curve
propositionProposition 15.31.13BE532EB6E98CSingular measures on products with a curve
equationEquation 15.31.140CA3FF665601Singular measures on products with a curve
theoremTheorem 15.31.155F0343E9A338Singular measures on products with a curve
propositionProposition 15.31.16DD2C5D7B64ECSingular measures on products with a curve
theoremTheorem 15.32.1D988D47D5E0EStability of the subsolution threshold
equationEquation 15.32.2ACC15559958BStability of the subsolution threshold
subsection15.32.3 Finite sums of densities3DD22F1D6AC3Stability of the subsolution threshold
propositionProposition 15.32.4D697A55DE07CStability of the subsolution threshold
propositionProposition 15.32.54C06D1B94BD1Stability of the subsolution threshold
propositionProposition 15.32.6C926F63790E3Stability of the subsolution threshold
subsection15.32.7 Approximation by bounded density roots1BD556522380Stability of the subsolution threshold
equationEquation 15.32.8CA92BD48A42DStability of the subsolution threshold
propositionProposition 15.32.92210D4123A64Stability of the subsolution threshold
subsection15.32.10 Density roots from a closed factor6436CD7911BDStability of the subsolution threshold
propositionProposition 15.32.117A2CC1A3258EStability of the subsolution threshold
theoremTheorem 15.33.135816840F731Power capacity bounds and threshold equations
equationEquation 15.33.27DAF690D3482Power capacity bounds and threshold equations
equationEquation 15.33.3E5CF15D0251APower capacity bounds and threshold equations
equationEquation 15.33.41A3F6C7B811BPower capacity bounds and threshold equations
subsection15.33.5 A logarithmic density with critical coefficient oneB8C5C5469618Power capacity bounds and threshold equations
theoremTheorem 15.33.6B37DD22EFF75Power capacity bounds and threshold equations
equationEquation 15.33.7E4AFA559F42CPower capacity bounds and threshold equations
equationEquation 15.33.8F1C776CF9796Power capacity bounds and threshold equations
equationEquation 15.33.9F69DF10A862EPower capacity bounds and threshold equations
subsection15.34.1 A positive limiting exponent00CDC2A31380Limits of exponential equations
theoremTheorem 15.34.29F0AB366D34BLimits of exponential equations
equationEquation 15.35.1A708922257FBExponential equations for an individual density
theoremTheorem 15.35.29016C2906071Exponential equations for an individual density
corollaryCorollary 15.35.3B2672237F244Exponential equations for an individual density
subsection15.35.4 An error bound for the scalar limit36B99A272B8FExponential equations for an individual density
propositionProposition 15.35.5AC5E6031A12CExponential equations for an individual density
equationEquation 15.35.6747F5A0D1221Exponential equations for an individual density
subsection15.35.7 Exponential equations at zero threshold49C07144356FExponential equations for an individual density
lemmaLemma 15.35.8C9E4D8D9901AExponential equations for an individual density
theoremTheorem 15.35.98FF7FBD67552Exponential equations for an individual density
theoremTheorem 15.35.10092A98EE953DExponential equations for an individual density
subsection15.35.11 Solvable exponents and collapse at zero threshold4B9655E85450Exponential equations for an individual density
propositionProposition 15.35.12B2DB6BD3AC4DExponential equations for an individual density
propositionProposition 15.35.13C791D4F48DECExponential equations for an individual density
subsection15.35.14 Positive endpoints and one-sided limits44A2CB44D7CBExponential equations for an individual density
lemmaLemma 15.35.15A9F34BD594E1Exponential equations for an individual density
propositionProposition 15.35.169E30269224CDExponential equations for an individual density
propositionProposition 15.35.17B1974D837A1AExponential equations for an individual density
propositionProposition 15.35.180535F55900CBExponential equations for an individual density
propositionProposition 15.35.195E68200DBADDExponential equations for an individual density
subsection15.36.1 Separation and the radial flux870F2FA3F30ARadial selection for integrable densities
lemmaLemma 15.36.291B8E02B5759Radial selection for integrable densities
equationEquation 15.36.3C1A1056DFED6Radial selection for integrable densities
subsection15.36.4 The normalized limitCE0514438603Radial selection for integrable densities
theoremTheorem 15.36.5AACA90B5B52BRadial selection for integrable densities
equationEquation 15.36.6FF104C305C42Radial selection for integrable densities
subsection15.36.7 Scalar centering and its moment boundaryC00DFB603383Radial selection for integrable densities
propositionProposition 15.36.8704167C1A830Radial selection for integrable densities
equationEquation 15.36.924634670B4B5Radial selection for integrable densities
subsection15.36.10 Order in the separated classA874D9C61D74Radial selection for integrable densities
corollaryCorollary 15.36.11E75339160354Radial selection for integrable densities
equationEquation 15.37.1E4532A30340FRelative equations under positive lower mass
equationEquation 15.37.2C37817AD17BARelative equations under positive lower mass
section15.37.3 Equations for finite measures89AC64C97CA7Relative equations under positive lower mass
theoremTheorem 15.37.4C568B2958DB9Relative equations under positive lower mass
section15.37.5 DominationF894544905D6Relative equations under positive lower mass
theoremTheorem 15.37.65023B8CBC940Relative equations under positive lower mass
equationEquation 15.37.7F1593AD00D2BRelative equations under positive lower mass
subsection15.37.8 Domination for extended measures6EC4D5B55F22Relative equations under positive lower mass
lemmaLemma 15.37.9FBB6DCAAD3E7Relative equations under positive lower mass
theoremTheorem 15.37.10FB5C83859579Relative equations under positive lower mass
corollaryCorollary 15.37.11267BAD315E3CRelative equations under positive lower mass
section15.37.12 Common envelopes and dominated continuityBD25F1249EB6Relative equations under positive lower mass
lemmaLemma 15.37.13E5ED88B4042ARelative equations under positive lower mass
equationEquation 15.37.14EEB0F315C2A3Relative equations under positive lower mass
equationEquation 15.37.159D97DEE5D482Relative equations under positive lower mass
equationEquation 15.37.165FB1B43EDC20Relative equations under positive lower mass
propositionProposition 15.37.17D220D741C684Relative equations under positive lower mass
subsection15.37.18 Convergence under a fixed measure905590071E2FRelative equations under positive lower mass
theoremTheorem 15.37.196FBF04A6D4DERelative equations under positive lower mass
section15.37.20 Moving bounded carrier measures0BEAB476F799Relative equations under positive lower mass
propositionProposition 15.37.2169ED083168C9Relative equations under positive lower mass
equationEquation 15.37.22D519538AEA33Relative equations under positive lower mass
equationEquation 15.37.2353CEBC73788BRelative equations under positive lower mass
corollaryCorollary 15.37.24D3170E7413ECRelative equations under positive lower mass
subsection15.37.25 Convergence with a common envelope997CC67968A6Relative equations under positive lower mass
theoremTheorem 15.37.26C73D2354DDB2Relative equations under positive lower mass
section15.37.27 General measures in the minimal envelope class0E0ED0471018Relative equations under positive lower mass
theoremTheorem 15.37.28D23EB7EFA5E3Relative equations under positive lower mass
equationEquation 15.37.29AFE1DD54089DRelative equations under positive lower mass
section15.37.30 Equations with a model reference1E3C2CEF22BFRelative equations under positive lower mass
section15.37.31 Normalization at exponent zero643A9037F89CRelative equations under positive lower mass
corollaryCorollary 15.37.3274BE83BA167DRelative equations under positive lower mass
subsection15.37.33 Nonuniqueness in the minimal envelope classE6EC3A0A407ARelative equations under positive lower mass
corollaryCorollary 15.37.3413248E1C4143Relative equations under positive lower mass

16. Singularity types, envelopes, and models

lemmaLemma 16.11.133A40EB62C98Obstacles and rooftops
lemmaLemma 16.11.21BA71B27F698Obstacles and rooftops
definitionDefinition 16.11.3A4F165851008Obstacles and rooftops
propositionProposition 16.11.4641F3FEC623DObstacles and rooftops
definitionDefinition 16.11.5F80B4C1B3032Obstacles and rooftops
propositionProposition 16.11.6046401771DABObstacles and rooftops
exampleExample 16.11.7908479CB91B3Obstacles and rooftops
lemmaLemma 16.12.1EC77D6F30D3DMeasures on contact sets
lemmaLemma 16.13.115A418789279Concentration on the contact set
lemmaLemma 16.13.2691B1A212F1AConcentration on the contact set
lemmaLemma 16.13.3175BA4BB6A7CConcentration on the contact set
theoremTheorem 16.13.438E18A798D6BConcentration on the contact set
corollaryCorollary 16.13.57FE9391D7A51Concentration on the contact set
equationEquation 16.14.1B47D0EFD195AObstacles with bounded metric Laplacian
exampleExample 16.14.274EA4AAF7D98Obstacles with bounded metric Laplacian
equationEquation 16.14.30304C463D49FObstacles with bounded metric Laplacian
lemmaLemma 16.14.475E3F7AC3FA0Obstacles with bounded metric Laplacian
equationEquation 16.14.5D02E2F9A7872Obstacles with bounded metric Laplacian
propositionProposition 16.14.6CB0B256B92D7Obstacles with bounded metric Laplacian
lemmaLemma 16.14.7A64885BC3406Obstacles with bounded metric Laplacian
equationEquation 16.14.8487F3CA23BA4Obstacles with bounded metric Laplacian
propositionProposition 16.14.9F9586D89B24FObstacles with bounded metric Laplacian
equationEquation 16.14.10F8F2AE7A2F29Obstacles with bounded metric Laplacian
corollaryCorollary 16.14.11E3B684F226D1Obstacles with bounded metric Laplacian
equationEquation 16.15.110B72FBD0FD9Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.21559308396CCPenalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.323CF8D2A3054Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.4F24675566E64Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.50B31488A28BBPenalized equations and metric-Laplacian envelopes
equationEquation 16.15.60805E02FBF79Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.7EA811ACFD26EPenalized equations and metric-Laplacian envelopes
equationEquation 16.15.845972FF63040Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.9F97763A1D077Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.10905710F73D03Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.11EFC27557FD59Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.125041A51B4C58Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.13A7AB2F207A57Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.14295683EFEB30Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.1514AD13D091C3Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.16BB3FD42D191CPenalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.17E43893840E09Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.187C2F75E247E5Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.197E8949E72BB2Penalized equations and metric-Laplacian envelopes
theoremTheorem 16.15.20AA7242F2CF85Penalized equations and metric-Laplacian envelopes
propositionProposition 16.15.21E43C0935B545Penalized equations and metric-Laplacian envelopes
equationEquation 16.15.2241CA49F209DAPenalized equations and metric-Laplacian envelopes
lemmaLemma 16.15.237FBC12CCC033Penalized equations and metric-Laplacian envelopes
theoremTheorem 16.15.24A9F89B404AD6Penalized equations and metric-Laplacian envelopes
lemmaLemma 16.16.1C8D21E3D949AReal Hessian bounds for Kähler envelopes
equationEquation 16.16.2FA70E3BD3873Real Hessian bounds for Kähler envelopes
equationEquation 16.16.3D5B65BED5E46Real Hessian bounds for Kähler envelopes
equationEquation 16.16.4E052CD3153AFReal Hessian bounds for Kähler envelopes
equationEquation 16.16.5AADD4CDBAE91Real Hessian bounds for Kähler envelopes
equationEquation 16.16.625C8519515ADReal Hessian bounds for Kähler envelopes
theoremTheorem 16.16.74B6432E0991BReal Hessian bounds for Kähler envelopes
lemmaLemma 16.17.1B48E71C5ACA1Smoothing and its joint complex Hessian
equationEquation 16.17.2B9894E658BB3Smoothing and its joint complex Hessian
equationEquation 16.17.340EB67D1192CSmoothing and its joint complex Hessian
equationEquation 16.17.47573AD3BEC72Smoothing and its joint complex Hessian
equationEquation 16.17.5632F7FE2389BSmoothing and its joint complex Hessian
equationEquation 16.17.6F605A97417F4Smoothing and its joint complex Hessian
equationEquation 16.17.7478D40DEB163Smoothing and its joint complex Hessian
equationEquation 16.17.89997CDF02FD6Smoothing and its joint complex Hessian
equationEquation 16.17.9692788279CD1Smoothing and its joint complex Hessian
equationEquation 16.17.104F98A99005B0Smoothing and its joint complex Hessian
equationEquation 16.17.1124D0E54E3561Smoothing and its joint complex Hessian
equationEquation 16.17.12910B6450F3B7Smoothing and its joint complex Hessian
equationEquation 16.17.13841C6F5C788ASmoothing and its joint complex Hessian
equationEquation 16.17.14E593DD5093A9Smoothing and its joint complex Hessian
equationEquation 16.17.15DEF49E73E180Smoothing and its joint complex Hessian
equationEquation 16.17.16BA95304D82DBSmoothing and its joint complex Hessian
equationEquation 16.17.17050736A1D09BSmoothing and its joint complex Hessian
equationEquation 16.17.1803D658719ADASmoothing and its joint complex Hessian
lemmaLemma 16.18.193E8FFBE7FEBA minimum with a varying upper radius
theoremTheorem 16.19.12F8D75599A90Big-class envelopes on the ample locus
equationEquation 16.19.27291A43E569DBig-class envelopes on the ample locus
equationEquation 16.19.3CAB66005ED26Big-class envelopes on the ample locus
equationEquation 16.19.4BC60EB07467BBig-class envelopes on the ample locus
equationEquation 16.19.53A2D9D67C59BBig-class envelopes on the ample locus
equationEquation 16.19.6CE7BA779A40FBig-class envelopes on the ample locus
equationEquation 16.19.7A6E0B033C065Big-class envelopes on the ample locus
equationEquation 16.19.808F075EA4AE4Big-class envelopes on the ample locus
equationEquation 16.19.948A59B42EEB3Big-class envelopes on the ample locus
equationEquation 16.19.1081A46C775A33Big-class envelopes on the ample locus
equationEquation 16.19.1139D742E3F114Big-class envelopes on the ample locus
equationEquation 16.19.126B8D59FE8970Big-class envelopes on the ample locus
lemmaLemma 16.20.1BB171C00B874Contact with an arbitrary singular potential
theoremTheorem 16.20.2F6E7514B9542Contact with an arbitrary singular potential
equationEquation 16.20.3F55FDC7BB706Contact with an arbitrary singular potential
corollaryCorollary 16.20.46EFE0C8DBCE3Contact with an arbitrary singular potential
definitionDefinition 16.21.199168E437529Singularity envelopes
propositionProposition 16.21.2037CA226D141Singularity envelopes
propositionProposition 16.21.35ADA09DD223ESingularity envelopes
lemmaLemma 16.22.17420706BAE2FExtrapolation and model singularities
lemmaLemma 16.22.24D5FAF80D021Extrapolation and model singularities
lemmaLemma 16.22.3B36FB6000F3BExtrapolation and model singularities
theoremTheorem 16.22.4FCFA7EB59B34Extrapolation and model singularities
equationEquation 16.22.54801296ED5A2Extrapolation and model singularities
equationEquation 16.22.6C646467A8FDDExtrapolation and model singularities
equationEquation 16.22.7B2A56AF9922EExtrapolation and model singularities
equationEquation 16.22.8246B560461D1Extrapolation and model singularities
lemmaLemma 16.22.952ED2F2B0A63Extrapolation and model singularities
theoremTheorem 16.22.10 (Domination principle)0F4D01059518Extrapolation and model singularities
theoremTheorem 16.22.114A1DC8C4E020Extrapolation and model singularities
exampleExample 16.22.12246496245694Extrapolation and model singularities
definitionDefinition 16.22.13D3EBF9A66D06Extrapolation and model singularities
propositionProposition 16.22.147E199392108CExtrapolation and model singularities
propositionProposition 16.23.13499B9FD713CRooftops and mass inequalities
propositionProposition 16.23.2E33C504A684FRooftops and mass inequalities
propositionProposition 16.23.337241FAAC9EFRooftops and mass inequalities
lemmaLemma 16.23.4F4294ED919CCRooftops and mass inequalities
theoremTheorem 16.23.5641CB44C78E9Rooftops and mass inequalities
propositionProposition 16.24.1585A4246348BDivisorial constraints under envelopes
lemmaLemma 16.24.2FFB5A7083041Divisorial constraints under envelopes
definitionDefinition 16.25.13558A6E66BA8Singularity constraints and relative capacity
propositionProposition 16.25.2467197AADC50Singularity constraints and relative capacity
propositionProposition 16.25.34338492503DCSingularity constraints and relative capacity
propositionProposition 16.25.4C32431185924Singularity constraints and relative capacity
definitionDefinition 16.25.5E9BD67F94210Singularity constraints and relative capacity
propositionProposition 16.25.61587DBB4654FSingularity constraints and relative capacity
propositionProposition 16.25.7CDA70099F425Singularity constraints and relative capacity
theoremTheorem 16.26.164513E83F854Quantitative rooftop extrapolation
lemmaLemma 16.27.149691FF58A24Comparison of relative capacities
lemmaLemma 16.27.2A73E0CBD9FABComparison of relative capacities
equationEquation 16.27.3E0D5BEEFADA4Comparison of relative capacities
equationEquation 16.27.4E22DEE0D88EEComparison of relative capacities
lemmaLemma 16.27.548CC217E0795Comparison of relative capacities
equationEquation 16.27.66B1343C4F9EDComparison of relative capacities
equationEquation 16.27.77D6FB72AF287Comparison of relative capacities
propositionProposition 16.27.8C745DD03819CComparison of relative capacities
equationEquation 16.27.97D19131D8004Comparison of relative capacities
corollaryCorollary 16.27.109C70131AD005Comparison of relative capacities
equationEquation 16.27.118CF36AE13615Comparison of relative capacities
propositionProposition 16.27.1204DE884FD6DBComparison of relative capacities
equationEquation 16.27.133F70CF1B1F96Comparison of relative capacities
equationEquation 16.27.14391CDC0D1779Comparison of relative capacities
lemmaLemma 16.27.15A285A0B9F4B7Comparison of relative capacities
equationEquation 16.27.16106A989F297DComparison of relative capacities
theoremTheorem 16.27.17337360EF765BComparison of relative capacities
propositionProposition 16.27.187EC4B0FF8858Comparison of relative capacities
definitionDefinition 16.7.13E4D57445698Comparison of singularities
equationEquation 16.3.16AE4825E4C6FRooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.2C1971A5F3DFDRooftop operator and the definition of the $P$-envelope
lemmaLemma 16.3.34CF7246E6E5ARooftop operator and the definition of the $P$-envelope
equationEquation 16.3.4E08D87E98114Rooftop operator and the definition of the $P$-envelope
definitionDefinition 16.3.5395DB7860949Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.67E001B03F24ARooftop operator and the definition of the $P$-envelope
equationEquation 16.3.7FA68BB354359Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.8268F8CAEF7E2Rooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.9EF5AC4B19C4DRooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.10E58386609911Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.110F7C5A9FEE2DRooftop operator and the definition of the $P$-envelope
theoremTheorem 16.3.12553335F185CERooftop operator and the definition of the $P$-envelope
equationEquation 16.3.135CEAE61A4277Rooftop operator and the definition of the $P$-envelope
corollaryCorollary 16.3.14B6EF8456606FRooftop operator and the definition of the $P$-envelope
theoremTheorem 16.3.157ADF30509E4ERooftop operator and the definition of the $P$-envelope
equationEquation 16.3.162B585EFD6781Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.17A43EDA3894B0Rooftop operator and the definition of the $P$-envelope
definitionDefinition 16.3.186623E38E5FC2Rooftop operator and the definition of the $P$-envelope
remarkRemark 16.3.19ED7781211B7BRooftop operator and the definition of the $P$-envelope
corollaryCorollary 16.3.20E10CA97A11EFRooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.214C5C62135747Rooftop operator and the definition of the $P$-envelope
exampleExample 16.3.22E548BFB3BF04Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.23B2DFFDD6A0F3Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.24C825DA5CEC8CRooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.25F7C0BACE770DRooftop operator and the definition of the $P$-envelope
equationEquation 16.3.26E6B68D406648Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.27E47535691487Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.2874C044AB05CBRooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.29B9461C380E5CRooftop operator and the definition of the $P$-envelope
equationEquation 16.3.308997CB842099Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.317E981960ACBCRooftop operator and the definition of the $P$-envelope
lemmaLemma 16.3.32BCC1B8FD1BEDRooftop operator and the definition of the $P$-envelope
equationEquation 16.3.33725175FB4F3CRooftop operator and the definition of the $P$-envelope
equationEquation 16.3.34500FC88C1939Rooftop operator and the definition of the $P$-envelope
theoremTheorem 16.3.355B751E9DFBC1Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.36020126C445C3Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.3735F5A4FBAC68Rooftop operator and the definition of the $P$-envelope
equationEquation 16.3.38BCB4335E3D6ERooftop operator and the definition of the $P$-envelope
definitionDefinition 16.3.39B791A3D8205DRooftop operator and the definition of the $P$-envelope
exampleExample 16.3.4017FC69E7E25ERooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.41BE692352C556Rooftop operator and the definition of the $P$-envelope
propositionProposition 16.3.421F5A8A58FB4DRooftop operator and the definition of the $P$-envelope
propositionProposition 16.4.1055CAE14F80FProperties of the $P$-envelope
propositionProposition 16.4.2CF5FECD1B547Properties of the $P$-envelope
propositionProposition 16.4.38EB68D9BFB64Properties of the $P$-envelope
propositionProposition 16.4.4048490BEAC8CProperties of the $P$-envelope
equationEquation 16.4.5B1E48C713966Properties of the $P$-envelope
equationEquation 16.4.6243BB43027AEProperties of the $P$-envelope
equationEquation 16.4.7C4257CD1FB0BProperties of the $P$-envelope
corollaryCorollary 16.4.8953C40C3B9B8Properties of the $P$-envelope
propositionProposition 16.4.93654A10A0CCCProperties of the $P$-envelope
equationEquation 16.4.10DC0519B3B917Properties of the $P$-envelope
definitionDefinition 16.5.16444F0A28464Relative full mass classes
equationEquation 16.5.2A8BA75F03A09Relative full mass classes
remarkRemark 16.5.3226D72040A60Relative full mass classes
propositionProposition 16.5.4557595391303Relative full mass classes
propositionProposition 16.5.5 (Comparison principle)FDFD0A6F6287Relative full mass classes
equationEquation 16.5.6EAAF267DE59ERelative full mass classes
equationEquation 16.5.7D3168D446564Relative full mass classes
theoremTheorem 16.5.86F1A42885C28Relative full mass classes
theoremTheorem 16.5.90E437FB98B97Relative full mass classes
definitionDefinition 16.5.108A07570966F8Relative full mass classes
equationEquation 16.5.11AB28CEAE8B4ARelative full mass classes
equationEquation 16.5.12899E1F34D705Relative full mass classes
equationEquation 16.5.13BC7744A40FECRelative full mass classes
lemmaLemma 16.5.1493210731F7FDRelative full mass classes
equationEquation 16.5.15F7FEEE32D362Relative full mass classes
lemmaLemma 16.5.168D9EA7C5CC7CRelative full mass classes
equationEquation 16.5.176E4685F8B030Relative full mass classes
lemmaLemma 16.5.18B42B448077CFRelative full mass classes
equationEquation 16.5.19806CE06D337BRelative full mass classes
propositionProposition 16.5.2037860FDCEAB5Relative full mass classes
equationEquation 16.5.21BBEAC6CFC4B5Relative full mass classes
propositionProposition 16.5.22E2221B6939BERelative full mass classes
corollaryCorollary 16.5.23555B1CEACB0FRelative full mass classes
equationEquation 16.5.243BD672AA942BRelative full mass classes
corollaryCorollary 16.5.25B5D23DBE0536Relative full mass classes
lemmaLemma 16.5.268C85369857E4Relative full mass classes
equationEquation 16.5.276786C4A2C71BRelative full mass classes
corollaryCorollary 16.5.2861DB6FF8D92FRelative full mass classes
equationEquation 16.5.29987A6BCB813ERelative full mass classes
propositionProposition 16.5.304280CB108A7ERelative full mass classes
equationEquation 16.5.31F8225060CBACRelative full mass classes
equationEquation 16.5.329D55DDF004D1Relative full mass classes
equationEquation 16.5.33989753C108DDRelative full mass classes
equationEquation 16.5.340BA9923E2163Relative full mass classes
equationEquation 16.5.35F913F2DBE96CRelative full mass classes
propositionProposition 16.5.3651CFFEE97C34Relative full mass classes
propositionProposition 16.5.373DA562555236Relative full mass classes
equationEquation 16.5.3860942A58F028Relative full mass classes
equationEquation 16.5.397AAAE969D14BRelative full mass classes
lemmaLemma 16.5.404C2A27A8F97ARelative full mass classes
equationEquation 16.5.414E22C768C562Relative full mass classes
equationEquation 16.5.425B35DF529607Relative full mass classes
lemmaLemma 16.5.43BAAC43C43BD5Relative full mass classes
equationEquation 16.5.447058C51CA443Relative full mass classes
lemmaLemma 16.5.457BA1823201EARelative full mass classes
equationEquation 16.5.462E33CEABAB85Relative full mass classes
equationEquation 16.5.47B09C34234351Relative full mass classes
equationEquation 16.5.48053AD38249AFRelative full mass classes
propositionProposition 16.5.49AAD590A053BBRelative full mass classes
equationEquation 16.5.50081622986279Relative full mass classes
equationEquation 16.5.51B0A9B146545ERelative full mass classes
propositionProposition 16.5.52420BBF2E1362Relative full mass classes
propositionProposition 16.5.53322796EE46DERelative full mass classes
propositionProposition 16.5.5475FA2659491DRelative full mass classes
lemmaLemma 16.5.5588E2DBCC3C35Relative full mass classes
equationEquation 16.5.56F8B858957CF5Relative full mass classes
lemmaLemma 16.5.578B9E394D5AE5Relative full mass classes
equationEquation 16.5.58B09DAFFD8EA2Relative full mass classes
corollaryCorollary 16.5.594A849C02F386Relative full mass classes
equationEquation 16.5.60B424028202D9Relative full mass classes
subsection16.6.1 $\mathcal{I}$-equivalence6C0625A38157Multiplier-ideal envelopes
propositionProposition 16.6.28769D844C274Multiplier-ideal envelopes
definitionDefinition 16.6.33B668848773EMultiplier-ideal envelopes
propositionProposition 16.6.4BBF7977859F1Multiplier-ideal envelopes
propositionProposition 16.6.526D170DCA6A1Multiplier-ideal envelopes
subsection16.6.6 The definition of the $\mathcal{I}$-envelope39E4D656BD09Multiplier-ideal envelopes
definitionDefinition 16.6.7E94E1B7F9193Multiplier-ideal envelopes
equationEquation 16.6.8FA4D3373FBAEMultiplier-ideal envelopes
propositionProposition 16.6.9708B2E9B2EF0Multiplier-ideal envelopes
propositionProposition 16.6.10E535E4DC5384Multiplier-ideal envelopes
propositionProposition 16.6.11363E28FF4A61Multiplier-ideal envelopes
equationEquation 16.6.125888DEC3C930Multiplier-ideal envelopes
definitionDefinition 16.6.131D9577F4ECC2Multiplier-ideal envelopes
propositionProposition 16.6.148DF15E9E1928Multiplier-ideal envelopes
equationEquation 16.6.153535B0853CBCMultiplier-ideal envelopes
definitionDefinition 16.6.16202451F22A2AMultiplier-ideal envelopes
propositionProposition 16.6.1712DD9B82F232Multiplier-ideal envelopes
propositionProposition 16.6.188AE79B76D6D2Multiplier-ideal envelopes
equationEquation 16.6.190BC2A1135095Multiplier-ideal envelopes
equationEquation 16.6.20EA49AC910A56Multiplier-ideal envelopes
corollaryCorollary 16.6.21FB21F0385323Multiplier-ideal envelopes
propositionProposition 16.6.2210F9DDE25703Multiplier-ideal envelopes
equationEquation 16.6.23B621808C9E65Multiplier-ideal envelopes
equationEquation 16.6.24D91D171A0296Multiplier-ideal envelopes
equationEquation 16.6.259169FBAAFD41Multiplier-ideal envelopes
subsection16.6.26 Properties of the $\mathcal{I}$-envelope4FA5F6B29982Multiplier-ideal envelopes
propositionProposition 16.6.27F7C47263CB3CMultiplier-ideal envelopes
lemmaLemma 16.6.282A3BCF353C03Multiplier-ideal envelopes
propositionProposition 16.6.292D7AF550FFF7Multiplier-ideal envelopes
propositionProposition 16.6.300C703BE43035Multiplier-ideal envelopes
remarkRemark 16.6.310F2E59A9BE30Multiplier-ideal envelopes
subsection16.8.1 The definitions of the preordersF7493F03C4C2The $P$ and $\mathcal{I}$-preorders
definitionDefinition 16.8.24E959BD934C4The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.3D3383E5F872DThe $P$ and $\mathcal{I}$-preorders
lemmaLemma 16.8.4A3C93A5CDBF4The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.57E95C2DD4B7DThe $P$ and $\mathcal{I}$-preorders
exampleExample 16.8.614285897F9E7The $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.7C7C64ABB4647The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.8DF3EF73A9B89The $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.9EE590FE41862The $P$ and $\mathcal{I}$-preorders
definitionDefinition 16.8.105C5512F45CC2The $P$ and $\mathcal{I}$-preorders
lemmaLemma 16.8.114A5CF38738F2The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.12295DE82294BAThe $P$ and $\mathcal{I}$-preorders
lemmaLemma 16.8.13005A077A0167The $P$ and $\mathcal{I}$-preorders
corollaryCorollary 16.8.14FE600747C81EThe $P$ and $\mathcal{I}$-preorders
corollaryCorollary 16.8.15564E8A0E68F9The $P$ and $\mathcal{I}$-preorders
corollaryCorollary 16.8.1669E7C4B20FAEThe $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.172E8F613E7CBEThe $P$ and $\mathcal{I}$-preorders
exampleExample 16.8.18C7A2AD00BFF3The $P$ and $\mathcal{I}$-preorders
exampleExample 16.8.19325A49CBCF74The $P$ and $\mathcal{I}$-preorders
definitionDefinition 16.8.2033F50367BB68The $P$ and $\mathcal{I}$-preorders
subsection16.8.21 Properties of the preordersF4304BE7EF91The $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.22 (Monotonicity theorem)82DCD8E3B1BAThe $P$ and $\mathcal{I}$-preorders
theoremTheorem 16.8.2368E00C8C52A1The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.243BF94E656512The $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.259690BB77C920The $P$ and $\mathcal{I}$-preorders
lemmaLemma 16.8.26FB1663F2B6E1The $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.27C42706880DEDThe $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.281C6DED516857The $P$ and $\mathcal{I}$-preorders
propositionProposition 16.8.298C181599D520The $P$ and $\mathcal{I}$-preorders
corollaryCorollary 16.8.3021CC5153F56FThe $P$ and $\mathcal{I}$-preorders
equationEquation 16.8.3142D232E5CD85The $P$ and $\mathcal{I}$-preorders
theoremTheorem 16.8.321CA975F04423The $P$ and $\mathcal{I}$-preorders
corollaryCorollary 16.8.334EE47D80CC25The $P$ and $\mathcal{I}$-preorders
subsection16.9.1 The definition of the $d_ S$-pseudometricBB740CB4BE99The $d_ S$-pseudometric
definitionDefinition 16.9.2240B1899916CThe $d_ S$-pseudometric
propositionProposition 16.9.31ABAE06071CDThe $d_ S$-pseudometric
lemmaLemma 16.9.42490CFA511ECThe $d_ S$-pseudometric
equationEquation 16.9.5FE54EE184B7AThe $d_ S$-pseudometric
propositionProposition 16.9.6C1D8A9BD3B3BThe $d_ S$-pseudometric
lemmaLemma 16.9.703FCA1AB2751The $d_ S$-pseudometric
corollaryCorollary 16.9.8255A623EA393The $d_ S$-pseudometric
corollaryCorollary 16.9.9B29DCA70A9E6The $d_ S$-pseudometric
equationEquation 16.9.100AC35440957FThe $d_ S$-pseudometric
corollaryCorollary 16.9.1194B2CC4C30BBThe $d_ S$-pseudometric
corollaryCorollary 16.9.1277738CEC8071The $d_ S$-pseudometric
corollaryCorollary 16.9.13D47345F0C3C8The $d_ S$-pseudometric
corollaryCorollary 16.9.14F84D53FDA599The $d_ S$-pseudometric
equationEquation 16.9.15E95AEF6203E2The $d_ S$-pseudometric
equationEquation 16.9.16FC1B825A9516The $d_ S$-pseudometric
propositionProposition 16.9.171DEED63B61B0The $d_ S$-pseudometric
equationEquation 16.9.182902F1B6C4A6The $d_ S$-pseudometric
equationEquation 16.9.1967C255B50420The $d_ S$-pseudometric
equationEquation 16.9.203B96A5372D55The $d_ S$-pseudometric
equationEquation 16.9.2179C50CD3AC3EThe $d_ S$-pseudometric
equationEquation 16.9.229F803C97D29BThe $d_ S$-pseudometric
equationEquation 16.9.23A82EF3572727The $d_ S$-pseudometric
corollaryCorollary 16.9.2456D1BC5A2B85The $d_ S$-pseudometric
lemmaLemma 16.9.257F9EDAFF39F9The $d_ S$-pseudometric
subsection16.9.26 Convergence theorems0BF3C2E4260BThe $d_ S$-pseudometric
lemmaLemma 16.9.2751EA136E15FAThe $d_ S$-pseudometric
equationEquation 16.9.28D81A5AAFDA65The $d_ S$-pseudometric
lemmaLemma 16.9.292E78B4FEFFF9The $d_ S$-pseudometric
theoremTheorem 16.9.30713B9E26113BThe $d_ S$-pseudometric
equationEquation 16.9.313F8CB8C5B9A5The $d_ S$-pseudometric
equationEquation 16.9.32DDD39F87B553The $d_ S$-pseudometric
corollaryCorollary 16.9.336C532715D3CDThe $d_ S$-pseudometric
equationEquation 16.9.34F98DF8D2E20DThe $d_ S$-pseudometric
equationEquation 16.9.35217CFC062068The $d_ S$-pseudometric
corollaryCorollary 16.9.363EB846B21817The $d_ S$-pseudometric
equationEquation 16.9.37B509AA3C2576The $d_ S$-pseudometric
corollaryCorollary 16.9.38B1559025EC82The $d_ S$-pseudometric
corollaryCorollary 16.9.39300ED38851D0The $d_ S$-pseudometric
equationEquation 16.9.40EDB32DFBA6A5The $d_ S$-pseudometric
lemmaLemma 16.9.41F0C01F4F75EBThe $d_ S$-pseudometric
propositionProposition 16.9.42FDEBDB1E8A22The $d_ S$-pseudometric
equationEquation 16.9.4376B405D47467The $d_ S$-pseudometric
propositionProposition 16.9.44362688C3DDC9The $d_ S$-pseudometric
theoremTheorem 16.9.45666CDCEAA1BEThe $d_ S$-pseudometric
equationEquation 16.9.46FF05887CF686The $d_ S$-pseudometric
propositionProposition 16.9.47A2F662346EB5The $d_ S$-pseudometric
equationEquation 16.9.4817876C1905F0The $d_ S$-pseudometric
theoremTheorem 16.9.4988D575A3C46AThe $d_ S$-pseudometric
lemmaLemma 16.9.50CC25BA40D9FEThe $d_ S$-pseudometric
equationEquation 16.9.5111D8415D35F8The $d_ S$-pseudometric
equationEquation 16.9.52E3D4ADBA54B9The $d_ S$-pseudometric
equationEquation 16.9.534D05EB795FC2The $d_ S$-pseudometric
subsection16.9.54 Continuity of invariantsF6973807BD19The $d_ S$-pseudometric
theoremTheorem 16.9.5589B76134A3A3The $d_ S$-pseudometric
equationEquation 16.9.56D68651624DC9The $d_ S$-pseudometric
theoremTheorem 16.9.57CAB6F4E23080The $d_ S$-pseudometric
equationEquation 16.9.588D9C25DA2712The $d_ S$-pseudometric
theoremTheorem 16.9.59E1576893E02BThe $d_ S$-pseudometric
equationEquation 16.9.60ACB2722EA044The $d_ S$-pseudometric
equationEquation 16.9.61337542A87DFBThe $d_ S$-pseudometric
section16.28.1 Contact sets840B0858DE95Envelopes and volumes of singularities
propositionProposition 16.28.2295F6C861B41Envelopes and volumes of singularities
propositionProposition 16.28.3181D96E0C805Envelopes and volumes of singularities
theoremTheorem 16.28.445D3A8D823A8Envelopes and volumes of singularities
theoremTheorem 16.28.569A5902CF8C5Envelopes and volumes of singularities
propositionProposition 16.28.6107D3E29D21BEnvelopes and volumes of singularities
propositionProposition 16.28.7F0B070E89AE8Envelopes and volumes of singularities
subsection16.28.8 Affine growth and projective envelopes6EE0CD52DB46Envelopes and volumes of singularities
lemmaLemma 16.28.95C02C050C6B5Envelopes and volumes of singularities
propositionProposition 16.28.10B1F342A8772AEnvelopes and volumes of singularities
equationEquation 16.28.11E8DF064EB371Envelopes and volumes of singularities
lemmaLemma 16.28.127A591F1F87FAEnvelopes and volumes of singularities
corollaryCorollary 16.28.130C3E3295B18FEnvelopes and volumes of singularities
theoremTheorem 16.28.146D209F2CFC94Envelopes and volumes of singularities
equationEquation 16.28.155251FBE769B2Envelopes and volumes of singularities
subsubsection16.28.16 Growth slopes and supporting probabilitiesFBAA435459A9Envelopes and volumes of singularities
equationEquation 16.28.17C5CFC7F7F4AEEnvelopes and volumes of singularities
lemmaLemma 16.28.1850B3044AA630Envelopes and volumes of singularities
equationEquation 16.28.1911BD756B4F41Envelopes and volumes of singularities
propositionProposition 16.28.2022685BEFF668Envelopes and volumes of singularities
equationEquation 16.28.2189B57A4461F9Envelopes and volumes of singularities
equationEquation 16.28.22A8634DE3239CEnvelopes and volumes of singularities
equationEquation 16.28.2329BB5B35B07BEnvelopes and volumes of singularities
corollaryCorollary 16.28.240D01C895AF55Envelopes and volumes of singularities
equationEquation 16.28.25F611672B3A6DEnvelopes and volumes of singularities
equationEquation 16.28.260E7EEE36B8D9Envelopes and volumes of singularities
theoremTheorem 16.28.27865AACD2F96AEnvelopes and volumes of singularities
equationEquation 16.28.282AF7902848CFEnvelopes and volumes of singularities
theoremTheorem 16.28.294044625470CCEnvelopes and volumes of singularities
equationEquation 16.28.30926674359DBAEnvelopes and volumes of singularities
subsubsection16.28.31 Variation of continuous weights485B15BBE554Envelopes and volumes of singularities
equationEquation 16.28.32F40A32D819F9Envelopes and volumes of singularities
lemmaLemma 16.28.33A143B4417F78Envelopes and volumes of singularities
equationEquation 16.28.34FB1602967EA5Envelopes and volumes of singularities
propositionProposition 16.28.35FDDE0F20FBB2Envelopes and volumes of singularities
equationEquation 16.28.36ED5EC6D9783AEnvelopes and volumes of singularities
corollaryCorollary 16.28.377D611867F702Envelopes and volumes of singularities
equationEquation 16.28.3863336B576658Envelopes and volumes of singularities
corollaryCorollary 16.28.39DF0FD081C3EFEnvelopes and volumes of singularities
exampleExample 16.28.409167A00E8F32Envelopes and volumes of singularities
equationEquation 16.28.41B2CEEE021C0EEnvelopes and volumes of singularities
subsubsection16.28.42 Local regularity and strict weights5323E094C77DEnvelopes and volumes of singularities
lemmaLemma 16.28.432A500AE8E8B3Envelopes and volumes of singularities
definitionDefinition 16.28.440288BA4D7A38Envelopes and volumes of singularities
lemmaLemma 16.28.45F64C3174FCDBEnvelopes and volumes of singularities
theoremTheorem 16.28.4609FE648CAD9AEnvelopes and volumes of singularities
equationEquation 16.28.478D81382EAE2CEnvelopes and volumes of singularities
corollaryCorollary 16.28.48DECDD98C3557Envelopes and volumes of singularities
subsubsection16.28.49 A regularization defect along a complex disk924715C5EA1BEnvelopes and volumes of singularities
lemmaLemma 16.28.50C78FAC45EABBEnvelopes and volumes of singularities
propositionProposition 16.28.51DDC2E6D4F75FEnvelopes and volumes of singularities
equationEquation 16.28.5209A6A171E444Envelopes and volumes of singularities
equationEquation 16.28.53617AF346ABCFEnvelopes and volumes of singularities
section16.28.54 Bounded domination in a semipositive background7A794726C8E7Envelopes and volumes of singularities
propositionProposition 16.28.559BBE1DE257CBEnvelopes and volumes of singularities
theoremTheorem 16.28.56 (Guedj–Lu)C67CEBABC968Envelopes and volumes of singularities
section16.28.57 Upper and lower volumes879D5A786C1CEnvelopes and volumes of singularities
definitionDefinition 16.28.5852560A3D59BDEnvelopes and volumes of singularities
propositionProposition 16.28.59A8ED96D18B4AEnvelopes and volumes of singularities
section16.28.60 Relative envelope classesE04112C46C03Envelopes and volumes of singularities
definitionDefinition 16.28.617EF949B941B4Envelopes and volumes of singularities
propositionProposition 16.28.6249C69232C604Envelopes and volumes of singularities
lemmaLemma 16.28.63C172793BD93FEnvelopes and volumes of singularities
propositionProposition 16.28.6446B23EB58AABEnvelopes and volumes of singularities
propositionProposition 16.28.6547066B2F8269Envelopes and volumes of singularities
section16.28.66 Model references and positive mass97D4C0CD199EEnvelopes and volumes of singularities
equationEquation 16.28.67237D346B8CBAEnvelopes and volumes of singularities
lemmaLemma 16.28.68F644817639FDEnvelopes and volumes of singularities
lemmaLemma 16.28.6918D2E34A9D42Envelopes and volumes of singularities
theoremTheorem 16.28.700EA7F76611B5Envelopes and volumes of singularities
section16.28.71 Infinite mass inside the envelope classA211FB2FDF28Envelopes and volumes of singularities
propositionProposition 16.28.72AA248AFFA815Envelopes and volumes of singularities
corollaryCorollary 16.28.73944620474A50Envelopes and volumes of singularities
section16.29.1 Curves constant on a left half-line14D1184651D4Envelope systems over Hermitian perturbations
definitionDefinition 16.29.2F0A985779829Envelope systems over Hermitian perturbations
propositionProposition 16.29.3160B15BF08FEEnvelope systems over Hermitian perturbations
section16.29.4 Transition maps and envelope systemsFDD92852ACF2Envelope systems over Hermitian perturbations
propositionProposition 16.29.57DD8E9433762Envelope systems over Hermitian perturbations
definitionDefinition 16.29.6949E1D79FF19Envelope systems over Hermitian perturbations
equationEquation 16.29.7101FB6DC67C6Envelope systems over Hermitian perturbations
section16.29.8 The limiting curve and recovery955DE01C09E9Envelope systems over Hermitian perturbations
theoremTheorem 16.29.9E94DE568F212Envelope systems over Hermitian perturbations
equationEquation 16.29.1044684C69B3B6Envelope systems over Hermitian perturbations
propositionProposition 16.29.1193A1136C4F72Envelope systems over Hermitian perturbations
section16.29.12 Changes of representative and limiting massEE5B374433B6Envelope systems over Hermitian perturbations
propositionProposition 16.29.135B2E6F75BA98Envelope systems over Hermitian perturbations
propositionProposition 16.29.143B3109886F9FEnvelope systems over Hermitian perturbations
section16.29.15 Failure of recovery in a positive background6596E22FF71BEnvelope systems over Hermitian perturbations
lemmaLemma 16.29.16F6C7CDB99C6EEnvelope systems over Hermitian perturbations
theoremTheorem 16.29.17761E4875D2B4Envelope systems over Hermitian perturbations
propositionProposition 16.29.187F133849FE58Envelope systems over Hermitian perturbations
section16.29.19 Recovery through the Hopf fibration6DDF231D4303Envelope systems over Hermitian perturbations
propositionProposition 16.29.20D9BF9081FC38Envelope systems over Hermitian perturbations
lemmaLemma 16.29.212478F9CE0209Envelope systems over Hermitian perturbations
theoremTheorem 16.29.222DB4FF5FB5C1Envelope systems over Hermitian perturbations
subsection16.29.23 A nonanalytic family4FE41BF0F18EEnvelope systems over Hermitian perturbations
exampleExample 16.29.240631D73C4C88Envelope systems over Hermitian perturbations
subsection16.29.25 Exact pullback and smooth changes of backgroundE5CE4F64BF58Envelope systems over Hermitian perturbations

17. Energy, geodesics, rays, and test curves

equationEquation 17.2.16B17AADE0338Integration by parts for bounded differences
lemmaLemma 17.2.25F05F95AAF1FIntegration by parts for bounded differences
lemmaLemma 17.2.3ECFE5ECCB550Integration by parts for bounded differences
equationEquation 17.2.4D023DB1B1039Integration by parts for bounded differences
theoremTheorem 17.2.53E698B84D63DIntegration by parts for bounded differences
equationEquation 17.2.693ACF4FD4967Integration by parts for bounded differences
lemmaLemma 17.2.752C5A516B8C8Integration by parts for bounded differences
definitionDefinition 17.3.18F92EE4FC37CRelative Monge–Ampère energy
definitionDefinition 17.3.2240EF1DE5889Relative Monge–Ampère energy
equationEquation 17.3.3F19AAFA93177Relative Monge–Ampère energy
propositionProposition 17.3.49086ABF36774Relative Monge–Ampère energy
equationEquation 17.3.546190A7C1B2FRelative Monge–Ampère energy
equationEquation 17.3.6BC6BC1B194C2Relative Monge–Ampère energy
equationEquation 17.3.773519BE96F0ERelative Monge–Ampère energy
equationEquation 17.3.825D7DD3C6946Relative Monge–Ampère energy
propositionProposition 17.3.9356D15D0C316Relative Monge–Ampère energy
equationEquation 17.3.1026A7B21B6397Relative Monge–Ampère energy
definitionDefinition 17.3.11FC02DD85845CRelative Monge–Ampère energy
propositionProposition 17.3.1292E951B4916ARelative Monge–Ampère energy
propositionProposition 17.3.1369D86F9D2F90Relative Monge–Ampère energy
theoremTheorem 17.3.1469E7D1FEF207Relative Monge–Ampère energy
equationEquation 17.3.15AE35DCCFE783Relative Monge–Ampère energy
exampleExample 17.3.166EAA0093766CRelative Monge–Ampère energy
corollaryCorollary 17.3.174DCEFD8869DDRelative Monge–Ampère energy
lemmaLemma 17.3.18620127F44043Relative Monge–Ampère energy
propositionProposition 17.3.1998BD01BE1363Relative Monge–Ampère energy
equationEquation 17.3.2003F8F41E7068Relative Monge–Ampère energy
equationEquation 17.3.21474BE0C5FED2Relative Monge–Ampère energy
propositionProposition 17.4.1049F75F5547DContinuity and compactness
equationEquation 17.4.2D1604CAB7F82Continuity and compactness
lemmaLemma 17.4.3369EDE8DEEC5Continuity and compactness
propositionProposition 17.4.42FF0E77E9188Continuity and compactness
corollaryCorollary 17.4.585CFF04E2CECContinuity and compactness
propositionProposition 17.4.62FA1EC20E3A2Continuity and compactness
propositionProposition 17.4.71A67C7F5BE56Continuity and compactness
corollaryCorollary 17.4.8838D06267D86Continuity and compactness
propositionProposition 17.4.969A8D9FFB919Continuity and compactness
lemmaLemma 17.4.10F62E3A64C1EEContinuity and compactness
exampleExample 17.4.1127DF9596E862Continuity and compactness
subsection17.5.1 Existence by maximization06665CC7749CVariational Monge–Ampère equations
definitionDefinition 17.5.237BEFE25B673Variational Monge–Ampère equations
lemmaLemma 17.5.3B408410BD203Variational Monge–Ampère equations
propositionProposition 17.5.42099099BAE21Variational Monge–Ampère equations
theoremTheorem 17.5.5011162B56B40Variational Monge–Ampère equations
theoremTheorem 17.5.68972585EBF32Variational Monge–Ampère equations
propositionProposition 17.5.75BDA01022188Variational Monge–Ampère equations
equationEquation 17.5.820471D0DB1A7Variational Monge–Ampère equations
lemmaLemma 17.5.9B4AA6C4C1C94Variational Monge–Ampère equations
lemmaLemma 17.5.10DD59ECAA0656Variational Monge–Ampère equations
lemmaLemma 17.5.11B5DC04E7729FVariational Monge–Ampère equations
corollaryCorollary 17.5.12F89C0E4EBAC2Variational Monge–Ampère equations
theoremTheorem 17.5.13109ED281DA22Variational Monge–Ampère equations
lemmaLemma 17.5.144D025B7103EAVariational Monge–Ampère equations
theoremTheorem 17.5.15373141BD8F91Variational Monge–Ampère equations
subsection17.5.16 Uniqueness by concentration of mass096A13B0C62EVariational Monge–Ampère equations
theoremTheorem 17.5.17777B716B7208Variational Monge–Ampère equations
equationEquation 17.5.18EAFAB682B4E1Variational Monge–Ampère equations
subsection17.5.19 Bounded differences from the referenceE5281BDCDF1FVariational Monge–Ampère equations
theoremTheorem 17.5.20F21AE677873AVariational Monge–Ampère equations
equationEquation 17.5.212646B3091195Variational Monge–Ampère equations
equationEquation 17.5.22856E2E7BF89AVariational Monge–Ampère equations
equationEquation 17.5.23E2AC3FD3AA08Variational Monge–Ampère equations
equationEquation 17.5.24A41B6BAF1A25Variational Monge–Ampère equations
corollaryCorollary 17.5.256DAA97EFFAAFVariational Monge–Ampère equations
corollaryCorollary 17.5.26DAF36530C8C1Variational Monge–Ampère equations
subsection17.5.27 Positive mass and bigness1080E9C054BAVariational Monge–Ampère equations
lemmaLemma 17.5.28 (Lamari's criterion)D70C0784CA6EVariational Monge–Ampère equations
equationEquation 17.5.2999EE27FCDCB1Variational Monge–Ampère equations
equationEquation 17.5.30C39A9A97E83CVariational Monge–Ampère equations
equationEquation 17.5.31316C32F396FAVariational Monge–Ampère equations
equationEquation 17.5.321FC0F0E3114EVariational Monge–Ampère equations
lemmaLemma 17.5.33EB970BD6E881Variational Monge–Ampère equations
equationEquation 17.5.340511A794D8D0Variational Monge–Ampère equations
equationEquation 17.5.35B73555CD9C1DVariational Monge–Ampère equations
equationEquation 17.5.3693DD9AEE16DCVariational Monge–Ampère equations
theoremTheorem 17.5.37F3340D970A43Variational Monge–Ampère equations
corollaryCorollary 17.5.38C78DFCF82004Variational Monge–Ampère equations
corollaryCorollary 17.5.3992C6B04981BAVariational Monge–Ampère equations
subsection17.5.40 Mixed masses and concavity69E4E21B68CAVariational Monge–Ampère equations
theoremTheorem 17.5.410A392CD9486EVariational Monge–Ampère equations
equationEquation 17.5.42AC065DF59BDDVariational Monge–Ampère equations
propositionProposition 17.6.1DC1056C1A4A1Singularity envelopes at arbitrary mass
equationEquation 17.6.2C6464A625C78Singularity envelopes at arbitrary mass
equationEquation 17.6.3A5C05836DEF9Singularity envelopes at arbitrary mass
definitionDefinition 17.6.40C3B4338E463Singularity envelopes at arbitrary mass
theoremTheorem 17.6.5C6C3A19AAB0DSingularity envelopes at arbitrary mass
definitionDefinition 17.7.1B0604A42ABDFSubgeodesics and Perron segments
lemmaLemma 17.7.20702EBBE9C4CSubgeodesics and Perron segments
lemmaLemma 17.7.3ADE91FED84D6Subgeodesics and Perron segments
propositionProposition 17.7.461A0C5DD7757Subgeodesics and Perron segments
lemmaLemma 17.7.5F4AAE999A6C4Subgeodesics and Perron segments
lemmaLemma 17.7.67B309BFF1A4DSubgeodesics and Perron segments
definitionDefinition 17.7.74A0AFDAD6156Subgeodesics and Perron segments
theoremTheorem 17.7.82FECBE87C487Subgeodesics and Perron segments
propositionProposition 17.7.9DD04D189A4D3Subgeodesics and Perron segments
propositionProposition 17.7.106DDFC01E2315Subgeodesics and Perron segments
lemmaLemma 17.8.1950F051CF55AEnergy along weak geodesics
lemmaLemma 17.8.23D0F1309E327Energy along weak geodesics
theoremTheorem 17.8.3678A18D6D406Energy along weak geodesics
equationEquation 17.8.42B3BEF6C25C1Energy along weak geodesics
propositionProposition 17.9.10583958ACA6DExtremal slopes and long Perron segments
propositionProposition 17.9.2A6533ED6CC68Extremal slopes and long Perron segments
section17.10.1 SubgeodesicsF069E255307EGeodesic rays in the space of potentials
definitionDefinition 17.10.28699FCF63094Geodesic rays in the space of potentials
remarkRemark 17.10.38F5460F49D02Geodesic rays in the space of potentials
remarkRemark 17.10.48353EDEFC366Geodesic rays in the space of potentials
propositionProposition 17.10.52042C299A40EGeodesic rays in the space of potentials
exampleExample 17.10.62F494BEED8F5Geodesic rays in the space of potentials
propositionProposition 17.10.71880C1FBD88FGeodesic rays in the space of potentials
equationEquation 17.10.8214F303DA2CCGeodesic rays in the space of potentials
corollaryCorollary 17.10.9C645EE3517E1Geodesic rays in the space of potentials
equationEquation 17.10.103DF2DEE2C546Geodesic rays in the space of potentials
corollaryCorollary 17.10.11E31D1E7DDF87Geodesic rays in the space of potentials
equationEquation 17.10.12F238807D334FGeodesic rays in the space of potentials
propositionProposition 17.10.135ECD747A7B0BGeodesic rays in the space of potentials
equationEquation 17.10.14BA313A9B1A1FGeodesic rays in the space of potentials
propositionProposition 17.10.15E5064A497AB8Geodesic rays in the space of potentials
lemmaLemma 17.10.163BEBB86C7F79Geodesic rays in the space of potentials
definitionDefinition 17.10.17872270015CE7Geodesic rays in the space of potentials
remarkRemark 17.10.18E63B316F0AB8Geodesic rays in the space of potentials
section17.10.19 Geodesics in the space of potentialsBE861874FCE4Geodesic rays in the space of potentials
definitionDefinition 17.10.20774DAAE2CF9EGeodesic rays in the space of potentials
equationEquation 17.10.21D0FE4F9BEB1BGeodesic rays in the space of potentials
lemmaLemma 17.10.22BDEA4F06B7F1Geodesic rays in the space of potentials
exampleExample 17.10.23C16140E8B047Geodesic rays in the space of potentials
propositionProposition 17.10.248AB36536E36DGeodesic rays in the space of potentials
equationEquation 17.10.25991B9E91FD84Geodesic rays in the space of potentials
equationEquation 17.10.26C45355A6B32EGeodesic rays in the space of potentials
equationEquation 17.10.27431F43B008A4Geodesic rays in the space of potentials
lemmaLemma 17.10.288A0F90757653Geodesic rays in the space of potentials
propositionProposition 17.10.29DE479E2BD255Geodesic rays in the space of potentials
equationEquation 17.10.308AE427F2EF68Geodesic rays in the space of potentials
captionFigure 17.10.31E48DC51BE08DGeodesic rays in the space of potentials
equationEquation 17.10.32FEE6016E13FFGeodesic rays in the space of potentials
equationEquation 17.10.3318547F185BC3Geodesic rays in the space of potentials
equationEquation 17.10.3416DA31D36BD2Geodesic rays in the space of potentials
equationEquation 17.10.351B22CF257482Geodesic rays in the space of potentials
equationEquation 17.10.365825ABE1D62BGeodesic rays in the space of potentials
propositionProposition 17.10.379335B48D7458Geodesic rays in the space of potentials
definitionDefinition 17.10.382E335019C3A7Geodesic rays in the space of potentials
propositionProposition 17.10.39FD1B5378AA06Geodesic rays in the space of potentials
section17.10.40 The metrics on the spaces of potentials and rays770FD61FCF7BGeodesic rays in the space of potentials
definitionDefinition 17.10.4108EFAEA550B1Geodesic rays in the space of potentials
equationEquation 17.10.42D8C8C0BB6585Geodesic rays in the space of potentials
propositionProposition 17.10.4361DD221912DEGeodesic rays in the space of potentials
equationEquation 17.10.44DCF5BF05B577Geodesic rays in the space of potentials
lemmaLemma 17.10.45488B7E9EA42CGeodesic rays in the space of potentials
definitionDefinition 17.10.46D926E3A0E83AGeodesic rays in the space of potentials
lemmaLemma 17.10.47CB87AA08E67CGeodesic rays in the space of potentials
theoremTheorem 17.10.481178F95447B1Geodesic rays in the space of potentials
equationEquation 17.10.4959E3B0356FFDGeodesic rays in the space of potentials
equationEquation 17.10.50C3DBA22EBCCDGeodesic rays in the space of potentials
corollaryCorollary 17.10.5193BC4D45D9EFGeodesic rays in the space of potentials
equationEquation 17.10.529D4858FD76E4Geodesic rays in the space of potentials
lemmaLemma 17.10.53B3BB9434F2D9Geodesic rays in the space of potentials
equationEquation 17.10.54FC79763AA6F5Geodesic rays in the space of potentials
definitionDefinition 17.10.553FB52FA474A3Geodesic rays in the space of potentials
definitionDefinition 17.10.56602E107F6064Geodesic rays in the space of potentials
equationEquation 17.10.578F3EC66A0258Geodesic rays in the space of potentials
lemmaLemma 17.10.58DB740351ED65Geodesic rays in the space of potentials
propositionProposition 17.10.598F175566EB00Geodesic rays in the space of potentials
theoremTheorem 17.10.60C89C58354E49Geodesic rays in the space of potentials
equationEquation 17.10.6125452B178A81Geodesic rays in the space of potentials
lemmaLemma 17.10.62ACEE2B318CA0Geodesic rays in the space of potentials
theoremTheorem 17.10.63D84A6072F1A6Geodesic rays in the space of potentials
equationEquation 17.10.641C0180D1496EGeodesic rays in the space of potentials
propositionProposition 17.10.650C8741A27906Geodesic rays in the space of potentials
equationEquation 17.10.66ECB123C0AED3Geodesic rays in the space of potentials
equationEquation 17.10.674470A045802CGeodesic rays in the space of potentials
equationEquation 17.10.6809167137EDA8Geodesic rays in the space of potentials
equationEquation 17.10.69EC42C56857FFGeodesic rays in the space of potentials
equationEquation 17.10.7073AB644A26B4Geodesic rays in the space of potentials
equationEquation 17.10.71223A0DF1B850Geodesic rays in the space of potentials
equationEquation 17.10.72ABBD36F9E5C6Geodesic rays in the space of potentials
equationEquation 17.10.732DD90D6C83CEGeodesic rays in the space of potentials
equationEquation 17.10.740F481A9AC521Geodesic rays in the space of potentials
equationEquation 17.10.75563CCD9BFEFFGeodesic rays in the space of potentials
equationEquation 17.10.7664A3E8566CC1Geodesic rays in the space of potentials
propositionProposition 17.10.777ADCF3A953E5Geodesic rays in the space of potentials
equationEquation 17.10.78D9CCCF0D8552Geodesic rays in the space of potentials
propositionProposition 17.10.795408F4A1486EGeodesic rays in the space of potentials
definitionDefinition 17.10.806945CDF5DE8AGeodesic rays in the space of potentials
theoremTheorem 17.10.81B30037E401C6Geodesic rays in the space of potentials
definitionDefinition 17.10.8289AFFBB3978EGeodesic rays in the space of potentials
propositionProposition 17.10.833A33DCAAE061Geodesic rays in the space of potentials
equationEquation 17.10.8467CD28C301F2Geodesic rays in the space of potentials
definitionDefinition 17.10.85F06EA26AA808Geodesic rays in the space of potentials
propositionProposition 17.10.86B40B93A4363DGeodesic rays in the space of potentials
equationEquation 17.10.87CA6EEFF029A5Geodesic rays in the space of potentials
equationEquation 17.10.88947F65850DBCGeodesic rays in the space of potentials
equationEquation 17.10.89EF4F3DA38F04Geodesic rays in the space of potentials
lemmaLemma 17.10.906C6200F67BD7Geodesic rays in the space of potentials
equationEquation 17.10.91FE9D22CC2BA6Geodesic rays in the space of potentials
exampleExample 17.10.9202BD88048323Geodesic rays in the space of potentials
equationEquation 17.10.93BF8F51ADD4CBGeodesic rays in the space of potentials
equationEquation 17.10.94795CA33E2D29Geodesic rays in the space of potentials
equationEquation 17.10.95277396283395Geodesic rays in the space of potentials
equationEquation 17.10.96AAACF2ED8350Geodesic rays in the space of potentials
equationEquation 17.10.97287169AF175CGeodesic rays in the space of potentials
definitionDefinition 17.12.1C8399F09D86DThe notion of test curves
equationEquation 17.12.2E3D67CAF2F52The notion of test curves
remarkRemark 17.12.33D73AB2E32CBThe notion of test curves
captionFigure 17.12.49BE4C00F2E15The notion of test curves
equationEquation 17.12.55EFADBE6069EThe notion of test curves
lemmaLemma 17.12.6348ACDBCB24EThe notion of test curves
equationEquation 17.12.7F4B7B1DBBE89The notion of test curves
captionFigure 17.12.820FE43CB91F9The notion of test curves
propositionProposition 17.12.9B7CD1B90CFDAThe notion of test curves
definitionDefinition 17.12.1069529F41AA18The notion of test curves
equationEquation 17.12.113578A2265BDCThe notion of test curves
equationEquation 17.12.12942AC971E484The notion of test curves
exampleExample 17.12.131A17E2EECB2DThe notion of test curves
propositionProposition 17.12.14B015D2EA4B4FThe notion of test curves
propositionProposition 17.12.152059587BD91BThe notion of test curves
propositionProposition 17.12.167AF1F81E928CThe notion of test curves
equationEquation 17.12.17E703D3F2D79DThe notion of test curves
definitionDefinition 17.12.1837432F25600EThe notion of test curves
propositionProposition 17.12.19724BF0CAE002The notion of test curves
definitionDefinition 17.13.12105331E66C7Ross–Witt Nyström correspondence
equationEquation 17.13.222C9794122C8Ross–Witt Nyström correspondence
equationEquation 17.13.31824FEE5F9FFRoss–Witt Nyström correspondence
remarkRemark 17.13.4ABB931E35703Ross–Witt Nyström correspondence
propositionProposition 17.13.51F48BF7602CARoss–Witt Nyström correspondence
equationEquation 17.13.624A2034A1287Ross–Witt Nyström correspondence
lemmaLemma 17.13.7BC74B8EA334FRoss–Witt Nyström correspondence
equationEquation 17.13.8DBF41B839C63Ross–Witt Nyström correspondence
corollaryCorollary 17.13.94B87EF03E3EARoss–Witt Nyström correspondence
lemmaLemma 17.13.10C6092031D5B3Ross–Witt Nyström correspondence
lemmaLemma 17.13.119C6538C4298ERoss–Witt Nyström correspondence
equationEquation 17.13.12E5C60BE9B124Ross–Witt Nyström correspondence
equationEquation 17.13.13BD13D6DEBB93Ross–Witt Nyström correspondence
equationEquation 17.13.1410B32891AA98Ross–Witt Nyström correspondence
lemmaLemma 17.13.157B83BD1D5FB9Ross–Witt Nyström correspondence
equationEquation 17.13.165BF1777D23C2Ross–Witt Nyström correspondence
equationEquation 17.13.176F01D0840C12Ross–Witt Nyström correspondence
equationEquation 17.13.1859C9610616F1Ross–Witt Nyström correspondence
equationEquation 17.13.1925F16D030FA9Ross–Witt Nyström correspondence
equationEquation 17.13.20CBA3E07AB082Ross–Witt Nyström correspondence
theoremTheorem 17.13.21B28F0A28CB3ERoss–Witt Nyström correspondence
equationEquation 17.13.226D5BA4827FF5Ross–Witt Nyström correspondence
equationEquation 17.13.23AE510D6D0C2FRoss–Witt Nyström correspondence
equationEquation 17.13.244884EC5C8FA9Ross–Witt Nyström correspondence
equationEquation 17.13.251C55A624E61ERoss–Witt Nyström correspondence
equationEquation 17.13.2685E0D04C585DRoss–Witt Nyström correspondence
equationEquation 17.13.27877135CB19C2Ross–Witt Nyström correspondence
equationEquation 17.13.2823E6778F003ARoss–Witt Nyström correspondence
equationEquation 17.13.29DBCF50A40C12Ross–Witt Nyström correspondence
remarkRemark 17.13.30E10A1DEDC7AFRoss–Witt Nyström correspondence
corollaryCorollary 17.13.3182AB68443E01Ross–Witt Nyström correspondence
exampleExample 17.13.32E2332A16AF00Ross–Witt Nyström correspondence
equationEquation 17.13.33D50F33814868Ross–Witt Nyström correspondence
equationEquation 17.13.34F5A9BEF8F7BARoss–Witt Nyström correspondence
propositionProposition 17.13.35 (He–Testorf–Wang)E6915C7339CDRoss–Witt Nyström correspondence
equationEquation 17.13.36DA238F3ECC6FRoss–Witt Nyström correspondence
equationEquation 17.13.3777E18B7AC795Ross–Witt Nyström correspondence
propositionProposition 17.13.38 (Hisamoto)3746B0074A79Ross–Witt Nyström correspondence
equationEquation 17.13.3916A5FF346A3BRoss–Witt Nyström correspondence
equationEquation 17.13.400893138962D7Ross–Witt Nyström correspondence
equationEquation 17.13.417F6505CF081CRoss–Witt Nyström correspondence
corollaryCorollary 17.13.42F147312B98B3Ross–Witt Nyström correspondence
equationEquation 17.13.43DA97CC2A78DCRoss–Witt Nyström correspondence
definitionDefinition 17.14.16CC72AA0EA90-model test curves
propositionProposition 17.14.2683BF9BB4D63-model test curves
propositionProposition 17.14.3A9AB5CE3BCAB-model test curves
propositionProposition 17.14.492E487FB0DA8-model test curves
definitionDefinition 17.14.503BED679B3AD-model test curves
propositionProposition 17.14.6BF40FB71C62C-model test curves
definitionDefinition 17.14.76C763D5B798A-model test curves
definitionDefinition 17.14.8D6CC0BBA6D35-model test curves
equationEquation 17.14.9F76F341D25A8-model test curves
exampleExample 17.14.1020A8C7BFE0BB-model test curves
equationEquation 17.14.1179C9CAB675E4-model test curves
equationEquation 17.14.129BF5C28825D8-model test curves
equationEquation 17.14.13EA5B88AFDFF1-model test curves
remarkRemark 17.14.14D6FC5A3046FA-model test curves
remarkRemark 17.14.15497927DD4340-model test curves
definitionDefinition 17.15.1DF47E4B2060AOperations on test curves
equationEquation 17.15.2B3E2E25A3B00Operations on test curves
lemmaLemma 17.15.3CA55D62C74E0Operations on test curves
definitionDefinition 17.15.4DAA4083947C1Operations on test curves
equationEquation 17.15.56C3AF1760F9FOperations on test curves
lemmaLemma 17.15.65BE2CC04AAD0Operations on test curves
propositionProposition 17.15.7FBC25DBE6122Operations on test curves
equationEquation 17.15.822BECA7DAF92Operations on test curves
lemmaLemma 17.15.98571F1340B16Operations on test curves
definitionDefinition 17.15.107610A81B88AAOperations on test curves
propositionProposition 17.15.116DDF81D7BEFFOperations on test curves
definitionDefinition 17.15.12D347D46F3537Operations on test curves
equationEquation 17.15.13CFA9EE6DC1EDOperations on test curves
lemmaLemma 17.15.1477520F48ECEBOperations on test curves
equationEquation 17.15.15B75DA94B0995Operations on test curves
equationEquation 17.15.161A2FC8833C52Operations on test curves
corollaryCorollary 17.15.17C8D1BA787705Operations on test curves
equationEquation 17.15.189AB9F5A23D95Operations on test curves
lemmaLemma 17.15.195DB0F974F922Operations on test curves
definitionDefinition 17.15.204CC845CC4171Operations on test curves
equationEquation 17.15.21A6BDA825C9B7Operations on test curves
propositionProposition 17.15.2288665EBBD399Operations on test curves
equationEquation 17.15.2375480BC1C778Operations on test curves
lemmaLemma 17.15.240B89BAEA6D39Operations on test curves
definitionDefinition 17.15.25FA768FFB9E17Operations on test curves
equationEquation 17.15.2680730711D47COperations on test curves
equationEquation 17.15.278A27FFEA5E96Operations on test curves
propositionProposition 17.15.28C4A77DA880D9Operations on test curves
equationEquation 17.15.293758587BAFAAOperations on test curves
lemmaLemma 17.15.30B6D60981A394Operations on test curves
propositionProposition 17.15.314B67ABC0083DOperations on test curves
propositionProposition 17.15.324D160D08BDC8Operations on test curves
definitionDefinition 17.15.33F6690BE17BAFOperations on test curves
propositionProposition 17.15.34148305D820BCOperations on test curves
propositionProposition 17.15.35A44C0F3CE359Operations on test curves
lemmaLemma 17.15.369469C1C052AAOperations on test curves
definitionDefinition 17.15.37BDCF69FED30FOperations on test curves
equationEquation 17.15.3848E9DB47C37COperations on test curves
equationEquation 17.15.39C2C47DDCADA4Operations on test curves
propositionProposition 17.15.4082DA4D86CC78Operations on test curves
lemmaLemma 17.15.41E141137C118EOperations on test curves
equationEquation 17.15.421D25957452B7Operations on test curves
section17.16.1 Subgeodesics and endpoint bounds8B8AAB4B5DD3Perron interpolation and the geodesic equation
propositionProposition 17.16.21A5A4BBF82E3Perron interpolation and the geodesic equation
propositionProposition 17.16.315E98759958FPerron interpolation and the geodesic equation
equationEquation 17.16.43E2013A813CDPerron interpolation and the geodesic equation
section17.16.5 Perron segments872589A6FBF3Perron interpolation and the geodesic equation
propositionProposition 17.16.65D2A9C4885B8Perron interpolation and the geodesic equation
theoremTheorem 17.16.766E38862017FPerron interpolation and the geodesic equation
section17.16.8 The smooth product equationA928C3A148CFPerron interpolation and the geodesic equation
propositionProposition 17.16.95885FD8F9550Perron interpolation and the geodesic equation
equationEquation 17.16.10F12E6A9111B5Perron interpolation and the geodesic equation
section17.16.11 The $L^2$ action and torsionD4C2C057EC75Perron interpolation and the geodesic equation
theoremTheorem 17.16.127A7AE56CA513Perron interpolation and the geodesic equation
equationEquation 17.16.13DCC8DA6D179EPerron interpolation and the geodesic equation
corollaryCorollary 17.16.14BE24BBA15991Perron interpolation and the geodesic equation
section17.16.15 Basic finite energy potentials on a Hopf manifold5C818268C38EPerron interpolation and the geodesic equation
propositionProposition 17.16.160290ABD0E348Perron interpolation and the geodesic equation
equationEquation 17.16.174A3521DF9F72Perron interpolation and the geodesic equation
equationEquation 17.16.186D09A14331E6Perron interpolation and the geodesic equation
theoremTheorem 17.16.19B0CA576B1836Perron interpolation and the geodesic equation
equationEquation 17.16.20D40B24211305Perron interpolation and the geodesic equation
equationEquation 17.16.2158F89D956FBFPerron interpolation and the geodesic equation
propositionProposition 17.16.222B828BD01DBBPerron interpolation and the geodesic equation
section17.16.23 Negative directions of the raw action124D4C83F5E6Perron interpolation and the geodesic equation
equationEquation 17.16.24905CCA72845BPerron interpolation and the geodesic equation
theoremTheorem 17.16.25944944D6FF68Perron interpolation and the geodesic equation
equationEquation 17.16.26F20F3B72C244Perron interpolation and the geodesic equation
equationEquation 17.16.2700E6402ACAB5Perron interpolation and the geodesic equation
section17.16.28 Explicit basic geodesics and singular potentials1DFB5A5992ADPerron interpolation and the geodesic equation
propositionProposition 17.16.29A3E893F6A601Perron interpolation and the geodesic equation
propositionProposition 17.16.309355B610E2C3Perron interpolation and the geodesic equation
section17.16.31 Model projection and test curves67EBAA104FDDPerron interpolation and the geodesic equation
propositionProposition 17.16.32DE987FD1A354Perron interpolation and the geodesic equation
section17.17.1 Envelope test curves and rays22C33F2E9103Test curves and Legendre duality
definitionDefinition 17.17.27D4FF1072CEFTest curves and Legendre duality
definitionDefinition 17.17.3ECDB430D0021Test curves and Legendre duality
section17.17.4 Perron maximalityFA178FC8EDEDTest curves and Legendre duality
theoremTheorem 17.17.5548B64628B04Test curves and Legendre duality
equationEquation 17.17.6DD25DD31A00BTest curves and Legendre duality
section17.17.7 Recovering the singularity envelopesDB66318C651ATest curves and Legendre duality
theoremTheorem 17.17.8062DEC0E6527Test curves and Legendre duality
section17.17.9 A logarithmic pole on the projective line58086AECC383Test curves and Legendre duality
equationEquation 17.17.103B7363400CA2Test curves and Legendre duality
propositionProposition 17.17.11377F0AD20B28Test curves and Legendre duality
equationEquation 17.17.129BA75154C113Test curves and Legendre duality
captionFigure 17.17.13FFC4E521CBE1Test curves and Legendre duality

18. Multiplier-ideal models and partial equilibrium

section18.1.1 The notion of $\mathcal{I}$-good singularities466586A864A3I-good singularities
theoremTheorem 18.1.2F9085BB700B0I-good singularities
equationEquation 18.1.36F6AB284E081I-good singularities
equationEquation 18.1.44730AA1B3287I-good singularities
definitionDefinition 18.1.5AE67290569F6I-good singularities
equationEquation 18.1.6A9EABC721827I-good singularities
remarkRemark 18.1.711FBD440944DI-good singularities
lemmaLemma 18.1.822BB587E13F8I-good singularities
corollaryCorollary 18.1.94662D2032EC0I-good singularities
exampleExample 18.1.1012795171A873I-good singularities
exampleExample 18.1.11E3DE8BCA0BA1I-good singularities
exampleExample 18.1.124C522F2A65F0I-good singularities
exampleExample 18.1.136212EEC14B63I-good singularities
corollaryCorollary 18.1.14F1CD22EF1353I-good singularities
corollaryCorollary 18.1.1536E768E33AFBI-good singularities
section18.1.16 Properties of $\mathcal{I}$-good singularitiesCE8C594C13B9I-good singularities
propositionProposition 18.1.17AC437613D2C1I-good singularities
exampleExample 18.1.183689F6A0A9ECI-good singularities
propositionProposition 18.1.19FD679E0ECD3DI-good singularities
section18.1.20 Mixed volumes80A2DBEB2B97I-good singularities
definitionDefinition 18.1.21EDCDF35668C4I-good singularities
equationEquation 18.1.2265AD0F03CD20I-good singularities
lemmaLemma 18.1.2320316D4886CFI-good singularities
equationEquation 18.1.247F052390F186I-good singularities
definitionDefinition 18.1.257731FA7D8BFDI-good singularities
equationEquation 18.1.260141EC0ADB1EI-good singularities
propositionProposition 18.1.27DF8A923E8F3EI-good singularities
equationEquation 18.1.281BAF68507D97I-good singularities
equationEquation 18.1.295EAAF326E66AI-good singularities
equationEquation 18.1.30A35A4555844EI-good singularities
lemmaLemma 18.1.311AEDF3A60767I-good singularities
theoremTheorem 18.1.32A320A575BBEBI-good singularities
equationEquation 18.1.33BCAEFDCBCBE3I-good singularities
corollaryCorollary 18.1.343305393E4400I-good singularities
equationEquation 18.1.3555BF10D53948I-good singularities
propositionProposition 18.1.36BB7441317AAFI-good singularities
propositionProposition 18.1.3724049C5AD956I-good singularities
lemmaLemma 18.1.3830F0E1037DAEI-good singularities
corollaryCorollary 18.1.39F7A7811A159EI-good singularities
equationEquation 18.1.404A8B3C0B6CE6I-good singularities
corollaryCorollary 18.1.411F0FA3B61EBCI-good singularities
theoremTheorem 18.1.42763BE634A054I-good singularities
equationEquation 18.1.43055370851A43I-good singularities
section18.1.44 The volumes of Hermitian pseudo-effective line bundles91B566E4374EI-good singularities
definitionDefinition 18.1.454B65B5AA614DI-good singularities
theoremTheorem 18.1.46 (Asymptotic Riemann–Roch formula)FEA63C5AE11FI-good singularities
equationEquation 18.1.47A09BDD8C8D10I-good singularities
propositionProposition 18.1.48A51A6DE92117I-good singularities
equationEquation 18.1.4917BF54CBA55BI-good singularities
equationEquation 18.1.501375552B142AI-good singularities
equationEquation 18.1.5156B91126C3D7I-good singularities
corollaryCorollary 18.1.52DA9EAE05F341I-good singularities
equationEquation 18.1.538448EC5F2FB3I-good singularities
equationEquation 18.1.549E9E2CC55B5AI-good singularities
equationEquation 18.1.5518274B986369I-good singularities
exampleExample 18.1.5635A211AC3973I-good singularities
section18.28.1 Partial envelopes9E595DC6361BPartial Bergman kernels
definitionDefinition 18.28.228F090ED5EFAPartial Bergman kernels
equationEquation 18.28.319BBFEFEB969Partial Bergman kernels
equationEquation 18.28.41F7034C27B5APartial Bergman kernels
exampleExample 18.28.5D0D0EA7C293DPartial Bergman kernels
propositionProposition 18.28.6C9561913A3E8Partial Bergman kernels
lemmaLemma 18.28.70F9B18404274Partial Bergman kernels
equationEquation 18.28.896B9EFFFBDA7Partial Bergman kernels
lemmaLemma 18.28.9A86A369D4CBFPartial Bergman kernels
equationEquation 18.28.10226F49B219B2Partial Bergman kernels
corollaryCorollary 18.28.11712499DFC37FPartial Bergman kernels
propositionProposition 18.28.12A1C36A6249C7Partial Bergman kernels
corollaryCorollary 18.28.1372097BEFEC59Partial Bergman kernels
corollaryCorollary 18.28.1496B842B9BF11Partial Bergman kernels
lemmaLemma 18.28.1518E5130ACB80Partial Bergman kernels
equationEquation 18.28.16EC4A564BADABPartial Bergman kernels
corollaryCorollary 18.28.17CE1417F951E3Partial Bergman kernels
equationEquation 18.28.18A99869D43853Partial Bergman kernels
section18.28.19 Partial equilibrium measuresB13AC61C497DPartial Bergman kernels
definitionDefinition 18.28.20E4ED4F8B05A8Partial Bergman kernels
lemmaLemma 18.28.21D7B3984EBF7APartial Bergman kernels
equationEquation 18.28.222F195F41BD70Partial Bergman kernels
equationEquation 18.28.23D22CB4C7F0CCPartial Bergman kernels
corollaryCorollary 18.28.249790F808BF37Partial Bergman kernels
equationEquation 18.28.25FFCF7AC9E703Partial Bergman kernels
definitionDefinition 18.28.2612F5599D8900Partial Bergman kernels
propositionProposition 18.28.27765C3766A729Partial Bergman kernels
equationEquation 18.28.280B2DE1BB9170Partial Bergman kernels
equationEquation 18.28.2994F4F91642C8Partial Bergman kernels
equationEquation 18.28.308353FF09478EPartial Bergman kernels
lemmaLemma 18.28.31D21332713DB8Partial Bergman kernels
equationEquation 18.28.3232CDA8E3EB57Partial Bergman kernels
equationEquation 18.28.33A58ACF533BAEPartial Bergman kernels
equationEquation 18.28.3411BA40C76F9BPartial Bergman kernels
propositionProposition 18.28.359724C0B32032Partial Bergman kernels
equationEquation 18.28.361BDE5E6D4627Partial Bergman kernels
section18.28.37 Quantization of partial equilibrium measures0211344889F8Partial Bergman kernels
subsection18.28.38 Bernstein–Markov measuresB2D64A3DB08EPartial Bergman kernels
definitionDefinition 18.28.39E365D575C322Partial Bergman kernels
equationEquation 18.28.4046ED22785DA5Partial Bergman kernels
equationEquation 18.28.41F1111E54880EPartial Bergman kernels
lemmaLemma 18.28.42E97173FEFAFDPartial Bergman kernels
definitionDefinition 18.28.4348A0DBADBA86Partial Bergman kernels
equationEquation 18.28.44FE92486453EBPartial Bergman kernels
propositionProposition 18.28.45A86573651304Partial Bergman kernels
definitionDefinition 18.28.46C3C647253911Partial Bergman kernels
remarkRemark 18.28.47129D3B333F18Partial Bergman kernels
equationEquation 18.28.48889A68CCEC83Partial Bergman kernels
theoremTheorem 18.28.49EA3E101DCA5EPartial Bergman kernels
equationEquation 18.28.50A6F9F2DF4A0DPartial Bergman kernels
propositionProposition 18.28.514C368CFA09EBPartial Bergman kernels
equationEquation 18.28.524DF720497AC6Partial Bergman kernels
equationEquation 18.28.5314347C41585EPartial Bergman kernels
equationEquation 18.28.541B7624ED3973Partial Bergman kernels
equationEquation 18.28.55C700200CB6BEPartial Bergman kernels
equationEquation 18.28.56CE5263973CB7Partial Bergman kernels
equationEquation 18.28.57725395815923Partial Bergman kernels
definitionDefinition 18.28.58A0F71B40AAE1Partial Bergman kernels
propositionProposition 18.28.59F547344ACE10Partial Bergman kernels
equationEquation 18.28.60D29819040D75Partial Bergman kernels
equationEquation 18.28.6104790CD84F49Partial Bergman kernels
lemmaLemma 18.28.62922E6C36BCC8Partial Bergman kernels
equationEquation 18.28.63C3210B1F6DE6Partial Bergman kernels
corollaryCorollary 18.28.644CE59F3C4805Partial Bergman kernels
lemmaLemma 18.28.655C6401E18EB3Partial Bergman kernels
equationEquation 18.28.662DBAE7A20C99Partial Bergman kernels
equationEquation 18.28.673504DA07F50FPartial Bergman kernels
propositionProposition 18.28.68F431300C59C4Partial Bergman kernels
equationEquation 18.28.69FFF9581DFDD2Partial Bergman kernels
equationEquation 18.28.705AAAC502B9E5Partial Bergman kernels
equationEquation 18.28.71AC1B69834497Partial Bergman kernels
corollaryCorollary 18.28.72B1F17F151050Partial Bergman kernels
propositionProposition 18.28.73AC691D964EDBPartial Bergman kernels
propositionProposition 18.28.740DFF36A22512Partial Bergman kernels
equationEquation 18.28.757EA15066EE63Partial Bergman kernels
equationEquation 18.28.76CBCF54FCD801Partial Bergman kernels
equationEquation 18.28.77FD0D14628CD8Partial Bergman kernels
equationEquation 18.28.788FD19D2B07C7Partial Bergman kernels
remarkRemark 18.28.79E9D132FDBF95Partial Bergman kernels
definitionDefinition 18.4.1F1F202AEE8FDIdeal projection
theoremTheorem 18.4.27F20FD198864Ideal projection
propositionProposition 18.4.39AE7825754C3Ideal projection
propositionProposition 18.4.42506B7AD6F19Ideal projection
corollaryCorollary 18.4.5B853F2E6C70BIdeal projection
propositionProposition 18.4.6EAFF378DAF48Ideal projection
subsection18.4.7 Analytic and divisorial singularities89E4AF40A761Ideal projection
propositionProposition 18.4.8EB95710FF4D7Ideal projection
corollaryCorollary 18.4.9B5BE9775FBF5Ideal projection
propositionProposition 18.5.107B0D783C3F0Changes of background and envelope mass
propositionProposition 18.5.26E8985733CBCChanges of background and envelope mass
subsection18.5.3 Envelope equality and infinite mass1A21A12C70DEChanges of background and envelope mass
equationEquation 18.5.4410A03897E45Changes of background and envelope mass
propositionProposition 18.5.5A2602BC06784Changes of background and envelope mass
propositionProposition 18.6.147A414940A74geometry of ideal classes
equationEquation 18.6.2CE9BB18A9158geometry of ideal classes
subsection18.6.3 Completion on a curveEFFAC36297D5geometry of ideal classes
theoremTheorem 18.6.4A55ADED04CFDgeometry of ideal classes
subsection18.6.5 Mass and capacity convergenceD9DB58D85DFBgeometry of ideal classes
propositionProposition 18.6.68B21A468DD06geometry of ideal classes
subsection18.6.7 Comparison with scalar normalization0F1DD25AE72Egeometry of ideal classes
exampleExample 18.6.80F685C3DE85Fgeometry of ideal classes
propositionProposition 18.7.1B727D2CB45A6Limits of analytic envelopes
corollaryCorollary 18.7.25AE1D9158EE3Limits of analytic envelopes
exampleExample 18.7.3504A5958272FLimits of analytic envelopes
subsection18.7.4 Dyadic regularization3A07CE3E627ALimits of analytic envelopes
equationEquation 18.7.5E93CF05C13A8Limits of analytic envelopes
theoremTheorem 18.7.6EA60F3EAF248Limits of analytic envelopes
corollaryCorollary 18.7.7E941DFE18F1DLimits of analytic envelopes
subsection18.7.8 Absorbing the curvature error13DB6841F63CLimits of analytic envelopes
corollaryCorollary 18.7.9A08D826CFAD4Limits of analytic envelopes
lemmaLemma 18.8.18FA80F8B6F5AA topology on ordinary singularity types
equationEquation 18.8.2D93497D2DEF1A topology on ordinary singularity types
equationEquation 18.8.391C58D64DE57A topology on ordinary singularity types
propositionProposition 18.8.4B919F0D14EE2A topology on ordinary singularity types
theoremTheorem 18.9.1867EFC7DCD8DAnalytic closure
equationEquation 18.9.227F9EF4DC38BAnalytic closure
propositionProposition 18.10.13B87EFC632CFSeparation and mass
propositionProposition 18.11.1760C8726BAE2Transport with an obstacle
propositionProposition 18.11.28B44DB3F02D3Transport with an obstacle
propositionProposition 18.12.1DAA4A0418197All smooth obstacles
propositionProposition 18.12.284E9AB1A05B4All smooth obstacles
theoremTheorem 18.12.326068B87D99AAll smooth obstacles
propositionProposition 18.12.4625C5E3FF1B3All smooth obstacles
theoremTheorem 18.13.16529F0A74D79Countable poles and all obstacles
equationEquation 18.13.2F52CC5DA3364Countable poles and all obstacles
equationEquation 18.13.31696AB66FF1DCountable poles and all obstacles
propositionProposition 18.13.4AB19DE43A7F6Countable poles and all obstacles
propositionProposition 18.14.11195E1B9B18DRelative divisors and fiber obstacles
equationEquation 18.14.2EDA13205415CRelative divisors and fiber obstacles
equationEquation 18.14.3336B6FB1D279Relative divisors and fiber obstacles
propositionProposition 18.14.4A399EFC10B9ERelative divisors and fiber obstacles
corollaryCorollary 18.14.50B566FB6A5E6Relative divisors and fiber obstacles
exampleExample 18.14.68119F7E76588Relative divisors and fiber obstacles
exampleExample 18.14.7A467F501617FRelative divisors and fiber obstacles
exampleExample 18.14.868E397A3AF5CRelative divisors and fiber obstacles
exampleExample 18.14.975FBCB1F7EC3Relative divisors and fiber obstacles
corollaryCorollary 18.14.1020390B807B79Relative divisors and fiber obstacles
propositionProposition 18.15.1C083614DAC5BDivisorial products and zero volume
equationEquation 18.15.295879292B693Divisorial products and zero volume
exampleExample 18.15.387F63F4EB505Divisorial products and zero volume
exampleExample 18.15.46D425566CDE7Divisorial products and zero volume
exampleExample 18.15.5CB61780C4135Divisorial products and zero volume
definitionDefinition 18.15.6F217B40D9074Divisorial products and zero volume
equationEquation 18.15.77DFF38FB6056Divisorial products and zero volume
propositionProposition 18.15.86DEB26AFE67EDivisorial products and zero volume
corollaryCorollary 18.15.9DFCE5415EAB5Divisorial products and zero volume
corollaryCorollary 18.15.1013B4DE048053Divisorial products and zero volume
lemmaLemma 18.16.1D9950FF0BE1CSeveral curve factors
propositionProposition 18.16.29E5A9E7C5770Several curve factors
equationEquation 18.16.3F57341993FE4Several curve factors
equationEquation 18.16.48749420FBA66Several curve factors
propositionProposition 18.16.5B61109A06541Several curve factors
corollaryCorollary 18.16.6A5CCF14B62D6Several curve factors
lemmaLemma 18.17.19040CE491D09Varying the curve coefficients
equationEquation 18.17.2AB4DC460D46EVarying the curve coefficients
propositionProposition 18.17.34A625E059F4CVarying the curve coefficients
subsubsection18.17.4 Comparing bounded model spaces723A61B34308Varying the curve coefficients
propositionProposition 18.17.525F2A9E20F21Varying the curve coefficients
equationEquation 18.17.6E8AFCE9DFA82Varying the curve coefficients
lemmaLemma 18.18.17EF34A8A42E5Arbitrary curve backgrounds
equationEquation 18.18.26A9618543A2FArbitrary curve backgrounds
propositionProposition 18.18.3347997EB7F21Arbitrary curve backgrounds
equationEquation 18.18.4657599C0AEFDArbitrary curve backgrounds
corollaryCorollary 18.18.515AC4263456BArbitrary curve backgrounds
lemmaLemma 18.19.1AFE454C19DE7Equality below all obstacles on separated products
propositionProposition 18.19.236E07E6B83DAEquality below all obstacles on separated products
corollaryCorollary 18.19.39E4426687FC0Equality below all obstacles on separated products
lemmaLemma 18.20.1F61F12847624Auxiliary compact complex factors
propositionProposition 18.20.285AC8B179E4BAuxiliary compact complex factors
corollaryCorollary 18.20.31B6E50A3B70BAuxiliary compact complex factors
corollaryCorollary 18.20.447BA03C63283Auxiliary compact complex factors
exampleExample 18.21.100A4B8521863A polar metric in a background with unbounded mass
lemmaLemma 18.22.1717F85615B2AModels over a curve
lemmaLemma 18.22.24A14B90776C1Models over a curve
propositionProposition 18.22.3AD848996B76CModels over a curve
corollaryCorollary 18.22.4DFA6018C6C86Models over a curve
propositionProposition 18.23.16235BDB38282Finite curve factors
exampleExample 18.23.247BCB7DF51C8Finite curve factors
propositionProposition 18.24.1424A6AE6F169Models pulled back by a finite curve map
propositionProposition 18.24.2C8EE003DDDC9Models pulled back by a finite curve map
corollaryCorollary 18.24.3F204E0019263Models pulled back by a finite curve map
exampleExample 18.24.42C40EC6336ADModels pulled back by a finite curve map
exampleExample 18.25.16CA9B76E77A0Hopf surface fibers and unbounded ambient mass
subsubsection18.26.1 Polar curvature in higher dimension83EE9021272BComplete model topologies
propositionProposition 18.26.2088A7D40B1D1Complete model topologies
equationEquation 18.26.355D42653A018Complete model topologies
corollaryCorollary 18.26.491E1AFBF43BEComplete model topologies
corollaryCorollary 18.26.5D9BE589BBC6EComplete model topologies
subsubsection18.26.6 Gauduchon polar trace comparisons6C8B6C6B86C3Complete model topologies
equationEquation 18.26.7B3860123F073Complete model topologies
propositionProposition 18.26.8CB31D6224569Complete model topologies
propositionProposition 18.26.9B48EB7517450Complete model topologies
corollaryCorollary 18.26.10EA11DD744637Complete model topologies
corollaryCorollary 18.26.11179E3F0D00D8Complete model topologies
propositionProposition 18.26.127C9BBE3DA7BEComplete model topologies
propositionProposition 18.26.136457B1027E36Complete model topologies
subsubsection18.26.14 Diffuse product poles2F85E85CE543Complete model topologies
lemmaLemma 18.26.15D3CEEE21DA5CComplete model topologies
lemmaLemma 18.26.164C3ACB006A19Complete model topologies
propositionProposition 18.26.175F93C15013C0Complete model topologies
corollaryCorollary 18.26.182D7E515584AAComplete model topologies
propositionProposition 18.26.196396018F7822Complete model topologies
corollaryCorollary 18.26.20C441ECFEBCE3Complete model topologies
propositionProposition 18.26.215988333AF401Complete model topologies
corollaryCorollary 18.26.22065C30217B81Complete model topologies
propositionProposition 18.26.236136A0B71CE1Complete model topologies
corollaryCorollary 18.26.249829FC8151FBComplete model topologies
propositionProposition 18.26.25085AE6785656Complete model topologies
propositionProposition 18.26.26FC2CCCE62C86Complete model topologies
problemProblem 18.26.2707524033446CComplete model topologies
equationEquation 18.27.14A552AB10BA9Small curvature and strict ordinary types
lemmaLemma 18.27.296F4F5465740Small curvature and strict ordinary types
lemmaLemma 18.27.3E5C4A193363DSmall curvature and strict ordinary types
theoremTheorem 18.27.40E6CABEB995ASmall curvature and strict ordinary types
propositionProposition 18.31.15E124FAEA311A local Bernstein–Markov estimate
equationEquation 18.32.1B5A5A1951FC8Partial envelopes on a weighted compact set
theoremTheorem 18.32.204BABC8924B6Partial envelopes on a weighted compact set
propositionProposition 18.32.3ABD0D0DB843CPartial envelopes on a weighted compact set
theoremTheorem 18.33.1336ACC796329An upper bound for partial Fubini–Study limits
propositionProposition 18.34.1A1D3D26087E7The determinant functional
corollaryCorollary 18.34.2E8DF257514D4The determinant functional
subsection18.34.3 Dependence on the continuous weight6D1351EF2D7CThe determinant functional
equationEquation 18.34.4389325A12314The determinant functional
propositionProposition 18.34.581FC6A666EC9The determinant functional
theoremTheorem 18.34.6F82E8E685FEAThe determinant functional
equationEquation 18.34.759C2D7B608B5The determinant functional
equationEquation 18.34.81BF37D25A2C8The determinant functional
corollaryCorollary 18.34.97733E1B20E7BThe determinant functional
equationEquation 18.34.10F73FC2DA9CA6The determinant functional
equationEquation 18.34.11CDF008B0B2C7The determinant functional
exampleExample 18.34.125DA140DF64C6The determinant functional
equationEquation 18.34.138F78E08AD393The determinant functional
subsection18.34.14 Stability of normalized kernels2BFA87C25210The determinant functional
equationEquation 18.34.158B5886BCBD7BThe determinant functional
lemmaLemma 18.34.1666CB6C16A6B7The determinant functional
equationEquation 18.34.17A9FF4DD46DA7The determinant functional
propositionProposition 18.34.184920A08E02F1The determinant functional
equationEquation 18.34.195CBC01E88DA5The determinant functional
equationEquation 18.34.20E8E6AD8629D0The determinant functional
exampleExample 18.34.21A118D36549FEThe determinant functional
propositionProposition 18.34.224F99D774A693The determinant functional
equationEquation 18.34.23961FF4CC03F6The determinant functional
exampleExample 18.34.2400C65C3DCEEBThe determinant functional
equationEquation 18.34.2553B83B2F665DThe determinant functional
propositionProposition 18.35.1609A9325CEC9Partial kernels under a modification
equationEquation 18.35.27BB9F79BDB65Partial kernels under a modification
propositionProposition 18.35.3FDE01F2B4B01Partial kernels under a modification
subsection18.35.4 Convergence with a Kähler modification1B62021184B4Partial kernels under a modification
theoremTheorem 18.35.5B187DFAF6D97Partial kernels under a modification
corollaryCorollary 18.35.69C43863CE5F1Partial kernels under a modification
propositionProposition 18.36.17E88736EF73BKernels on a fibration
subsection18.36.2 The quotient equilibrium measure29138044FA48Kernels on a fibration
theoremTheorem 18.36.387C196C13DDEKernels on a fibration
subsection18.36.4 The Hopf equilibrium measure16C99F55F2A7Kernels on a fibration
equationEquation 18.36.5C843534C9B58Kernels on a fibration
lemmaLemma 18.37.122D696DB398EIsolated logarithmic poles on a fibration
equationEquation 18.37.2DA25AD427AC7Isolated logarithmic poles on a fibration
propositionProposition 18.37.3BEF6F7AD320CIsolated logarithmic poles on a fibration
theoremTheorem 18.37.4D9B0D46D42B2Isolated logarithmic poles on a fibration
equationEquation 18.38.151F07D2D0831A point pole on the Hopf manifold
propositionProposition 18.38.22FF3CE9DCBFEA point pole on the Hopf manifold
equationEquation 18.38.3B9A41484DEE3A point pole on the Hopf manifold
theoremTheorem 18.38.4C55B3CFB4B5CA point pole on the Hopf manifold
captionFigure 18.38.5C3609B8E3DF2A point pole on the Hopf manifold
propositionProposition 18.38.6B73567245C4FA point pole on the Hopf manifold
lemmaLemma 18.39.1EF14C0C90C6CSingularities along a section over a divisor
equationEquation 18.39.2145C74FFDD88Singularities along a section over a divisor
theoremTheorem 18.39.3AE701AEF7FDESingularities along a section over a divisor
exampleExample 18.39.4FE69C8A3CBBFSingularities along a section over a divisor
corollaryCorollary 18.39.5115614FB1F9DSingularities along a section over a divisor
subsection18.39.6 A bounded divisorial band from fiber integrability14F47AA5A61BSingularities along a section over a divisor
lemmaLemma 18.39.7BBA6C67F74CFSingularities along a section over a divisor
theoremTheorem 18.39.8A79139E2A3C6Singularities along a section over a divisor
corollaryCorollary 18.39.9094794C14195Singularities along a section over a divisor
exampleExample 18.39.10EB221EFD409ASingularities along a section over a divisor
subsection18.39.11 Several crossing base divisors9C902F002FA9Singularities along a section over a divisor
lemmaLemma 18.39.1246F892FF98FESingularities along a section over a divisor
theoremTheorem 18.39.13A7CF6A5A6A5CSingularities along a section over a divisor
exampleExample 18.39.14411D2D5FFC5DSingularities along a section over a divisor
subsection18.39.15 Fiber integrability on a modified baseA6E0B69D347FSingularities along a section over a divisor
theoremTheorem 18.39.161FC6712A00F0Singularities along a section over a divisor
exampleExample 18.39.1717FFBB442DEBSingularities along a section over a divisor
subsection18.39.18 Semipositive residuals with positive volume27FE241587D2Singularities along a section over a divisor
corollaryCorollary 18.39.19AF408F696ADFSingularities along a section over a divisor
exampleExample 18.39.20C695A6659495Singularities along a section over a divisor
subsection18.39.21 Fixed exceptional orders and coefficient thresholds64330C0C7BF6Singularities along a section over a divisor
propositionProposition 18.39.22F3005ECDDA69Singularities along a section over a divisor
subsection18.39.23 A semipositive residual with nonzero defect5AF8FED16874Singularities along a section over a divisor
lemmaLemma 18.39.24822490631827Singularities along a section over a divisor
exampleExample 18.39.2547ABA9EAB173Singularities along a section over a divisor
equationEquation 18.40.1EB8A6E301949Horizontal poles over projective space
theoremTheorem 18.40.2ED8E35B86C82Horizontal poles over projective space
equationEquation 18.40.30AED85F8B0A4Horizontal poles over projective space
equationEquation 18.40.42E8D8F74367AHorizontal poles over projective space
subsection18.40.5 The domination envelope6419DC6AC605Horizontal poles over projective space
subsection18.40.6 Binomial kernels3220889B229AHorizontal poles over projective space
equationEquation 18.40.71480BE1C8578Horizontal poles over projective space
subsection18.40.8 Changing the integration volumeDF2D0299BE89Horizontal poles over projective space
propositionProposition 18.41.185EDCA2D872CDegenerate curvature and growing vanishing orders
equationEquation 18.41.233E9A1B502F8Degenerate curvature and growing vanishing orders
equationEquation 18.41.3EF4D2AABC4A8Degenerate curvature and growing vanishing orders
subsection18.41.4 A uniform tail estimateEB75AE06118DDegenerate curvature and growing vanishing orders
lemmaLemma 18.41.544E50E6CAEB1Degenerate curvature and growing vanishing orders
propositionProposition 18.41.679BA8CB51C0EDegenerate curvature and growing vanishing orders
equationEquation 18.41.769B0FF116418Degenerate curvature and growing vanishing orders
subsection18.41.8 The limiting envelope and the transition scale3F7E1C843ABBDegenerate curvature and growing vanishing orders
theoremTheorem 18.41.9313505FAE88EDegenerate curvature and growing vanishing orders
equationEquation 18.41.10F26D2556A21EDegenerate curvature and growing vanishing orders
equationEquation 18.41.1180FF36412E91Degenerate curvature and growing vanishing orders
equationEquation 18.41.122C986E64C32EDegenerate curvature and growing vanishing orders
propositionProposition 18.41.132E997A26D38ADegenerate curvature and growing vanishing orders
theoremTheorem 18.42.1F3F808FEE505Localization on curves
equationEquation 18.42.2C94B49B08D68Localization on curves
equationEquation 18.42.3FAC71C0D28AALocalization on curves
equationEquation 18.42.47549BC9C0F5CLocalization on curves
equationEquation 18.42.5E98FEA6D6D19Localization on curves
equationEquation 18.42.61C00E2972164Localization on curves
equationEquation 18.42.769F27719E20ALocalization on curves
equationEquation 18.42.8D74927004F98Localization on curves
subsection18.42.9 Curvature quadrature domains3CBD30511EFALocalization on curves
lemmaLemma 18.42.1058D1933573E2Localization on curves
equationEquation 18.42.118F00DDA42676Localization on curves
subsection18.42.12 Gaussian transitions at curvature boundaries77CA2588A10ELocalization on curves
lemmaLemma 18.42.13FBB9C424D045Localization on curves
equationEquation 18.42.148673DB454C7FLocalization on curves
theoremTheorem 18.42.15B396160A7145Localization on curves
equationEquation 18.42.16CACEDF5E5E52Localization on curves
equationEquation 18.42.17989892B2688BLocalization on curves
equationEquation 18.42.180C736EBAAED9Localization on curves
equationEquation 18.42.19719C724FC716Localization on curves
equationEquation 18.42.209C31FC6AD112Localization on curves
equationEquation 18.42.21891748D548F0Localization on curves
equationEquation 18.42.2248CB44544A07Localization on curves
equationEquation 18.42.235A8714984B04Localization on curves
equationEquation 18.42.242801C6E5C2B2Localization on curves
equationEquation 18.42.255803A22CF101Localization on curves
subsection18.42.26 Growth ranges and compact examplesE7C597B62D6DLocalization on curves
theoremTheorem 18.42.275C5A28D66191Localization on curves
equationEquation 18.42.284AA63CACA524Localization on curves
equationEquation 18.42.29168E8A4B3B56Localization on curves
subsection18.42.30 Corrections from the smooth weightBDE9EA5F5F58Localization on curves
equationEquation 18.42.3149C316E4C1B2Localization on curves
equationEquation 18.42.32FBDC714894A5Localization on curves
equationEquation 18.42.33065262EB036BLocalization on curves
lemmaLemma 18.43.199A4F9F834BFA logarithmic kernel scale in two dimensions
equationEquation 18.43.2AAC0258C6A86A logarithmic kernel scale in two dimensions
propositionProposition 18.43.3BDAF8AE299D2A logarithmic kernel scale in two dimensions
equationEquation 18.44.17FB2622EF514Anisotropic point jets and isolated poles on fibrations
theoremTheorem 18.44.27840E247F9D5Anisotropic point jets and isolated poles on fibrations
equationEquation 18.44.31BC32B08A00FAnisotropic point jets and isolated poles on fibrations
equationEquation 18.44.44219416A5981Anisotropic point jets and isolated poles on fibrations
equationEquation 18.44.5238CE4ED10CDAnisotropic point jets and isolated poles on fibrations
subsection18.44.6 Corrected and homogeneous centers39F4BF549DB7Anisotropic point jets and isolated poles on fibrations
equationEquation 18.44.7452067A980A7Anisotropic point jets and isolated poles on fibrations
propositionProposition 18.44.8443408C4E29AAnisotropic point jets and isolated poles on fibrations
subsection18.44.9 An isolated selecting potential on a submersion68C553A45799Anisotropic point jets and isolated poles on fibrations
equationEquation 18.44.10A73831DBE999Anisotropic point jets and isolated poles on fibrations
theoremTheorem 18.44.11154EDD25787AAnisotropic point jets and isolated poles on fibrations
equationEquation 18.44.121D6E0EF28851Anisotropic point jets and isolated poles on fibrations
equationEquation 18.45.14515165DA915Newton selection and moment-shell limits
subsection18.45.2 The growth constraint and its envelope5E7D89CCF8C5Newton selection and moment-shell limits
lemmaLemma 18.45.3F0A638931204Newton selection and moment-shell limits
equationEquation 18.45.41887637715F3Newton selection and moment-shell limits
equationEquation 18.45.525029B67571ANewton selection and moment-shell limits
subsection18.45.6 Norms, lattice growth and the lower bound53C9DB71324ENewton selection and moment-shell limits
theoremTheorem 18.45.72EC775894117Newton selection and moment-shell limits
equationEquation 18.45.82BD2A5FCA69DNewton selection and moment-shell limits
equationEquation 18.45.92A24E3900F00Newton selection and moment-shell limits
subsection18.45.10 The limit measure and the ambient metric894EB16B432ENewton selection and moment-shell limits
theoremTheorem 18.45.11C6669A63F760Newton selection and moment-shell limits
equationEquation 18.45.12BAC3E7E057E7Newton selection and moment-shell limits
subsection18.45.13 A rank-one selection with nonclosed sampling8461E7D0D299Newton selection and moment-shell limits
exampleExample 18.45.149600E351F18BNewton selection and moment-shell limits
equationEquation 18.46.1C673111EE95FRounding in multiplier-ideal kernel families
theoremTheorem 18.46.2318BC0622217Rounding in multiplier-ideal kernel families
equationEquation 18.46.3E123E52E2308Rounding in multiplier-ideal kernel families
equationEquation 18.46.47E8D4542A29BRounding in multiplier-ideal kernel families
equationEquation 18.46.5F401F8EC9D7ARounding in multiplier-ideal kernel families
lemmaLemma 18.47.1511A1D146037Three monomial bands from a multiplier-ideal selection
lemmaLemma 18.47.22BCFCC9C8808Three monomial bands from a multiplier-ideal selection
theoremTheorem 18.47.31D58A228EF5AThree monomial bands from a multiplier-ideal selection
propositionProposition 18.47.4AD8335312F45Three monomial bands from a multiplier-ideal selection
equationEquation 18.47.5CA62D9FD1FF2Three monomial bands from a multiplier-ideal selection
equationEquation 18.47.68DAD90BE35BEThree monomial bands from a multiplier-ideal selection
theoremTheorem 18.47.7A0E408FAE124Three monomial bands from a multiplier-ideal selection
equationEquation 18.47.8FCE7FB3D2455Three monomial bands from a multiplier-ideal selection
equationEquation 18.47.939BC26BE8655Three monomial bands from a multiplier-ideal selection

19. Sections, graded series, and Okounkov bodies

equationEquation 19.2.1A3A5C96E566BAlmost semigroups
equationEquation 19.2.241423E2E726FAlmost semigroups
definitionDefinition 19.2.3923345ACA44AAlmost semigroups
definitionDefinition 19.2.4B62535F66376Almost semigroups
theoremTheorem 19.2.537D86D761E81Almost semigroups
equationEquation 19.2.64E701FB8EA13Almost semigroups
equationEquation 19.2.7A57FF2EE0305Almost semigroups
corollaryCorollary 19.2.89FB3E8CE18F7Almost semigroups
definitionDefinition 19.2.978B565EC56CFAlmost semigroups
theoremTheorem 19.2.102534E787147EAlmost semigroups
equationEquation 19.2.1164FC27991307Almost semigroups
equationEquation 19.2.12882FE1F63B29Almost semigroups
corollaryCorollary 19.2.13511230BE9AE5Almost semigroups
equationEquation 19.2.144552E8E96799Almost semigroups
subsection19.3.1 The algebraic setting07F4BAAD0C8EFlags and valuations
definitionDefinition 19.3.2F1312787FCC5Flags and valuations
remarkRemark 19.3.35AA6A085DBB0Flags and valuations
subsection19.3.4 The transcendental settingCA8B57CE6237Flags and valuations
definitionDefinition 19.3.546E1CFFE3502Flags and valuations
definitionDefinition 19.3.678704BB19535Flags and valuations
propositionProposition 19.3.7FE522E15C1BAFlags and valuations
exampleExample 19.3.8534C64D15BE3Flags and valuations
propositionProposition 19.3.9EF8E476D98F4Flags and valuations
equationEquation 19.3.1098CF17AA36ADFlags and valuations
equationEquation 19.3.1161539109DEC3Flags and valuations
definitionDefinition 19.3.12E494C67EB4B8Flags and valuations
equationEquation 19.3.1363511F34173EFlags and valuations
propositionProposition 19.3.14D094DD00A592Flags and valuations
equationEquation 19.3.15579570889FCEFlags and valuations
equationEquation 19.3.1692459CED0BC1Flags and valuations
propositionProposition 19.3.178D2910CD1CECFlags and valuations
equationEquation 19.3.181C158885A628Flags and valuations
theoremTheorem 19.3.19B00AB5301F7AFlags and valuations
definitionDefinition 19.3.200887D96E8350Flags and valuations
equationEquation 19.3.219DE3455479DEFlags and valuations
lemmaLemma 19.3.2224E66EB949E4Flags and valuations
remarkRemark 19.3.2353FF6E808B22Flags and valuations
subsection19.4.1 The spaces of sections41E099D72C80Algebraic partial Okounkov bodies
definitionDefinition 19.4.20F5DB984097BAlgebraic partial Okounkov bodies
definitionDefinition 19.4.3DCCC835656D0Algebraic partial Okounkov bodies
equationEquation 19.4.43A54C829B8CCAlgebraic partial Okounkov bodies
definitionDefinition 19.4.551F245B40203Algebraic partial Okounkov bodies
subsection19.4.6 Algebraic Okounkov bodiesA81BAAF5B45FAlgebraic partial Okounkov bodies
propositionProposition 19.4.7632B1323B741Algebraic partial Okounkov bodies
definitionDefinition 19.4.80114DCC2E021Algebraic partial Okounkov bodies
propositionProposition 19.4.9572B003B2680Algebraic partial Okounkov bodies
corollaryCorollary 19.4.108494DFD8DF4DAlgebraic partial Okounkov bodies
propositionProposition 19.4.11AE88967EEE84Algebraic partial Okounkov bodies
propositionProposition 19.4.121ED4AF0D9A23Algebraic partial Okounkov bodies
propositionProposition 19.4.13AE9EBC9C5252Algebraic partial Okounkov bodies
propositionProposition 19.4.142B94C07754DDAlgebraic partial Okounkov bodies
subsection19.4.15 Construction of partial Okounkov bodies2A53BABD6994Algebraic partial Okounkov bodies
theoremTheorem 19.4.16A653D079A694Algebraic partial Okounkov bodies
definitionDefinition 19.4.17FAA070EFD0CCAlgebraic partial Okounkov bodies
equationEquation 19.4.18076205529197Algebraic partial Okounkov bodies
corollaryCorollary 19.4.1956E59F7A3150Algebraic partial Okounkov bodies
equationEquation 19.4.203DAADFC49A31Algebraic partial Okounkov bodies
equationEquation 19.4.212AE36B08DED5Algebraic partial Okounkov bodies
equationEquation 19.4.22653C3FE433F4Algebraic partial Okounkov bodies
remarkRemark 19.4.23555C64B2D134Algebraic partial Okounkov bodies
equationEquation 19.4.24A33E386F12D6Algebraic partial Okounkov bodies
subsection19.4.25 Basic properties of partial Okounkov bodies1FAA4BD1CBCBAlgebraic partial Okounkov bodies
propositionProposition 19.4.2697A03B9F1F87Algebraic partial Okounkov bodies
equationEquation 19.4.27C90692A83F74Algebraic partial Okounkov bodies
propositionProposition 19.4.281934C9D19460Algebraic partial Okounkov bodies
equationEquation 19.4.2944ACC02FAF04Algebraic partial Okounkov bodies
theoremTheorem 19.4.3030EEF3A985B7Algebraic partial Okounkov bodies
equationEquation 19.4.316AAC6A3D90A2Algebraic partial Okounkov bodies
propositionProposition 19.4.32CFE94018A80AAlgebraic partial Okounkov bodies
propositionProposition 19.4.3386E922DD5140Algebraic partial Okounkov bodies
equationEquation 19.4.3470AAFA1F54ABAlgebraic partial Okounkov bodies
theoremTheorem 19.4.3510DA5B8BC320Algebraic partial Okounkov bodies
equationEquation 19.4.36E375090F5B43Algebraic partial Okounkov bodies
propositionProposition 19.4.37300A5A5F84BBAlgebraic partial Okounkov bodies
equationEquation 19.4.38FD8FCE0E872CAlgebraic partial Okounkov bodies
propositionProposition 19.4.3963CE446DCB89Algebraic partial Okounkov bodies
subsection19.4.40 The Hausdorff convergence propertyB6A67594E2A0Algebraic partial Okounkov bodies
theoremTheorem 19.4.4176132CC6940CAlgebraic partial Okounkov bodies
equationEquation 19.4.42958BCE448264Algebraic partial Okounkov bodies
lemmaLemma 19.4.43FD338F03BE48Algebraic partial Okounkov bodies
lemmaLemma 19.4.44F3D0645BDF93Algebraic partial Okounkov bodies
equationEquation 19.4.456A18B89932A8Algebraic partial Okounkov bodies
equationEquation 19.4.468D5878FDF4FDAlgebraic partial Okounkov bodies
equationEquation 19.4.47DA123199D17CAlgebraic partial Okounkov bodies
equationEquation 19.4.48CFA7D4F4C2EBAlgebraic partial Okounkov bodies
equationEquation 19.4.4905403B615855Algebraic partial Okounkov bodies
subsection19.4.50 Recover Lelong numbers from partial Okounkov bodies6EC7E7A8EF44Algebraic partial Okounkov bodies
theoremTheorem 19.4.517CC4ADAC32DCAlgebraic partial Okounkov bodies
equationEquation 19.4.52CA1290F03811Algebraic partial Okounkov bodies
captionFigure 19.4.535C7E1472EE82Algebraic partial Okounkov bodies
equationEquation 19.4.540450C5C72C83Algebraic partial Okounkov bodies
corollaryCorollary 19.4.55C3CA9BD5D216Algebraic partial Okounkov bodies
corollaryCorollary 19.4.56C08CC234CAB9Algebraic partial Okounkov bodies
equationEquation 19.4.573C67C9063373Algebraic partial Okounkov bodies
subsection19.5.1 The traditional approach to the Okounkov body problemAB52CD73DBE6Transcendental partial Okounkov bodies
definitionDefinition 19.5.2AD7EC900552CTranscendental partial Okounkov bodies
equationEquation 19.5.35F949352BAF6Transcendental partial Okounkov bodies
theoremTheorem 19.5.4E839B4A35A2ATranscendental partial Okounkov bodies
subsection19.5.5 Definitions of partial Okounkov bodies232CC36315A0Transcendental partial Okounkov bodies
subsubsection19.5.6 The case of analytic singularities99ECDE21F9AATranscendental partial Okounkov bodies
definitionDefinition 19.5.731DAEE985011Transcendental partial Okounkov bodies
equationEquation 19.5.89AA6F51E3D18Transcendental partial Okounkov bodies
equationEquation 19.5.9326D63AB3B8CTranscendental partial Okounkov bodies
lemmaLemma 19.5.1032DCABF4D374Transcendental partial Okounkov bodies
lemmaLemma 19.5.11F1FC05EF0A7CTranscendental partial Okounkov bodies
lemmaLemma 19.5.125F65C388181ATranscendental partial Okounkov bodies
equationEquation 19.5.13B544FB9D12E6Transcendental partial Okounkov bodies
subsubsection19.5.14 The case of Kähler currents8B697E317627Transcendental partial Okounkov bodies
definitionDefinition 19.5.1524F2B706F011Transcendental partial Okounkov bodies
lemmaLemma 19.5.1608C830D7FB38Transcendental partial Okounkov bodies
corollaryCorollary 19.5.17A07A6093F1F3Transcendental partial Okounkov bodies
propositionProposition 19.5.18F190AF72550BTranscendental partial Okounkov bodies
equationEquation 19.5.194DF6CA86E7CDTranscendental partial Okounkov bodies
lemmaLemma 19.5.201493F0A1407ETranscendental partial Okounkov bodies
equationEquation 19.5.217D755FB49272Transcendental partial Okounkov bodies
equationEquation 19.5.222F017272546DTranscendental partial Okounkov bodies
subsubsection19.5.23 The general case101012300A5FTranscendental partial Okounkov bodies
definitionDefinition 19.5.2475695A1B0FB3Transcendental partial Okounkov bodies
equationEquation 19.5.25C022629C737BTranscendental partial Okounkov bodies
theoremTheorem 19.5.26944BC14A8246Transcendental partial Okounkov bodies
equationEquation 19.5.27E0BB6985CB53Transcendental partial Okounkov bodies
equationEquation 19.5.280696155A37C7Transcendental partial Okounkov bodies
corollaryCorollary 19.5.2968273CCE4042Transcendental partial Okounkov bodies
equationEquation 19.5.3046A0E609C71ETranscendental partial Okounkov bodies
subsection19.5.31 The valuative characterizationBB37B398D505Transcendental partial Okounkov bodies
lemmaLemma 19.5.32082A647C2B55Transcendental partial Okounkov bodies
equationEquation 19.5.3366140D1891ECTranscendental partial Okounkov bodies
lemmaLemma 19.5.34627CFC8D8850Transcendental partial Okounkov bodies
equationEquation 19.5.3577CE629D93F6Transcendental partial Okounkov bodies
remarkRemark 19.5.3649B696578E15Transcendental partial Okounkov bodies
theoremTheorem 19.5.37EF3A66F3D2C0Transcendental partial Okounkov bodies
equationEquation 19.5.386C6AAB28398FTranscendental partial Okounkov bodies
corollaryCorollary 19.5.390EE135D07034Transcendental partial Okounkov bodies
theoremTheorem 19.5.4014121F7F4FC5Transcendental partial Okounkov bodies
equationEquation 19.5.41FA3799E6A644Transcendental partial Okounkov bodies
equationEquation 19.5.42E7704026C99ATranscendental partial Okounkov bodies
remarkRemark 19.5.43A8051310CBABTranscendental partial Okounkov bodies
corollaryCorollary 19.5.44E7520E0649C0Transcendental partial Okounkov bodies
equationEquation 19.5.454DF707F26C95Transcendental partial Okounkov bodies
definitionDefinition 19.6.1ACD722F45231Okounkov test curves
equationEquation 19.6.21DD8B9478031Okounkov test curves
equationEquation 19.6.3DE0B0A1BC57COkounkov test curves
propositionProposition 19.6.429A7DDA38CA2Okounkov test curves
definitionDefinition 19.6.5F0D4BAA63D8AOkounkov test curves
equationEquation 19.6.66B100556700AOkounkov test curves
definitionDefinition 19.6.77B20D9977149Okounkov test curves
equationEquation 19.6.8CEDFEF97CB74Okounkov test curves
lemmaLemma 19.6.921BC38898B4AOkounkov test curves
equationEquation 19.6.10B1CF3E81D382Okounkov test curves
equationEquation 19.6.11910A774B0CAEOkounkov test curves
equationEquation 19.6.128B16FED9C73COkounkov test curves
theoremTheorem 19.6.132584C29ED70FOkounkov test curves
propositionProposition 19.6.147B4552909292Okounkov test curves
equationEquation 19.6.15E0766167D753Okounkov test curves
propositionProposition 19.6.16D70EFA3100C9Okounkov test curves
equationEquation 19.6.17FDE24E7623A9Okounkov test curves
definitionDefinition 19.6.1851940EF88F4FOkounkov test curves
propositionProposition 19.6.19F397AFA38927Okounkov test curves
equationEquation 19.6.20FCEE1C9E921FOkounkov test curves
equationEquation 19.6.21842F6248A5C4Okounkov test curves
lemmaLemma 19.6.229C05354365F4Okounkov test curves
lemmaLemma 19.6.2322B2EF8281A7Okounkov test curves
theoremTheorem 19.6.2474133E3920A2Okounkov test curves
remarkRemark 19.6.25731294EE36CAOkounkov test curves
section19.7.1 Convex bodiesA0D1AD337F41Almost semigroups
definitionDefinition 19.7.29A491602434DAlmost semigroups
definitionDefinition 19.7.35FD89093BE0BAlmost semigroups
theoremTheorem 19.7.4 (Blaschke selection theorem)BA1F1C1CF20FAlmost semigroups
theoremTheorem 19.7.566339C0E3C21Almost semigroups
theoremTheorem 19.7.6AFBD46E7C0A4Almost semigroups
lemmaLemma 19.7.75C4A19C5ED0BAlmost semigroups
equationEquation 19.7.892B22000DFB2Almost semigroups
lemmaLemma 19.7.9806DFD35CCE5Almost semigroups
lemmaLemma 19.7.1009E448752D43Almost semigroups
lemmaLemma 19.7.11127837115DD3Almost semigroups
definitionDefinition 19.7.12FBDC5E2EBC8AAlmost semigroups
propositionProposition 19.7.13F9F053A774FEAlmost semigroups
theoremTheorem 19.7.14 (Brunn–Minkowski)72547EEEBFE4Almost semigroups
section19.7.15 The Okounkov bodies of almost semigroupsA31E0D5C79D3Almost semigroups
subsection19.7.16 Generalities on semigroups3A76517E84BFAlmost semigroups
equationEquation 19.7.17642E2D6850DFAlmost semigroups
lemmaLemma 19.7.187887065F3C7CAlmost semigroups
lemmaLemma 19.7.19680A0890C1EEAlmost semigroups
theoremTheorem 19.7.20 (Kaveh–Khovanskii)85641516C8CAAlmost semigroups
lemmaLemma 19.7.21DCB8E8464FB6Almost semigroups
equationEquation 19.7.221009A092673CAlmost semigroups
lemmaLemma 19.7.23EC9CD63CAB3AAlmost semigroups
subsection19.7.24 Okounkov bodies of semigroupsA6ED691E2495Almost semigroups
lemmaLemma 19.7.25E41D68AC6F7CAlmost semigroups
definitionDefinition 19.7.260270FA2C03BDAlmost semigroups
lemmaLemma 19.7.27B1ED8431B1B9Almost semigroups
lemmaLemma 19.7.28323C6B6B61D0Almost semigroups
lemmaLemma 19.7.29B63FC87B8DDFAlmost semigroups
subsection19.7.30 Okounkov bodies of almost semigroups10CE70BA2854Almost semigroups
definitionDefinition 19.7.31B0C65454CC85Almost semigroups
theoremTheorem 19.7.321551B679B8B7Almost semigroups
equationEquation 19.7.331B239C5F9599Almost semigroups
equationEquation 19.7.34558EBEE0E3F0Almost semigroups
equationEquation 19.7.35F23C8DBEAF80Almost semigroups
equationEquation 19.7.3616236975E9F1Almost semigroups
remarkRemark 19.7.37219FB2B3C530Almost semigroups
section19.8.1 Weighted sections and Bergman currents5F8226DC1A97Holomorphic sections and Hermitian mass
propositionProposition 19.8.2F2503E6C53F2Holomorphic sections and Hermitian mass
propositionProposition 19.8.39E7A6E9D52A6Holomorphic sections and Hermitian mass
section19.8.4 Gauduchon degree and vanishingE5AAFD4B6553Holomorphic sections and Hermitian mass
theoremTheorem 19.8.51DFDCCEF325DHolomorphic sections and Hermitian mass
section19.8.6 Restricted sections and trace dataB7AD9A465882Holomorphic sections and Hermitian mass
theoremTheorem 19.8.7165328A043C0Holomorphic sections and Hermitian mass
section19.8.8 Bounded weighted sectionsC9CFDB0867F5Holomorphic sections and Hermitian mass
exampleExample 19.8.9D1A2270A67A3Holomorphic sections and Hermitian mass
section19.8.10 Section growth on a curve25780F9C24E0Holomorphic sections and Hermitian mass
theoremTheorem 19.8.111E6D5AA611CCHolomorphic sections and Hermitian mass
section19.8.12 Homogeneous sections on a Hopf manifold82379C382500Holomorphic sections and Hermitian mass
propositionProposition 19.8.134ABE18AC08D2Holomorphic sections and Hermitian mass
section19.8.14 Section counts and infinite Hermitian massB32BAEE663D4Holomorphic sections and Hermitian mass
propositionProposition 19.8.156400AAE4BA1EHolomorphic sections and Hermitian mass
section19.9.1 Flag valuations and degree bounds954DDD998705Flags, convex bodies and section growth
lemmaLemma 19.9.22B103C3B777AFlags, convex bodies and section growth
theoremTheorem 19.9.3EA8340BC8D54Flags, convex bodies and section growth
equationEquation 19.9.4E150336DE571Flags, convex bodies and section growth
section19.9.5 The body of a graded section algebraE9A3FA804369Flags, convex bodies and section growth
propositionProposition 19.9.6E6B4502ECB3BFlags, convex bodies and section growth
corollaryCorollary 19.9.7071721B08FC2Flags, convex bodies and section growth
section19.9.8 Intrinsic dimension, lattice and growth72085A67E2A6Flags, convex bodies and section growth
theoremTheorem 19.9.9CC2BF9294FA3Flags, convex bodies and section growth
propositionProposition 19.9.10E32D710594E0Flags, convex bodies and section growth
section19.9.11 Moishezon manifolds and bounded massF8F1527B70F2Flags, convex bodies and section growth
lemmaLemma 19.9.12976C06708A77Flags, convex bodies and section growth
theoremTheorem 19.9.130F8B7237A1BAFlags, convex bodies and section growth
section19.9.14 Flags on a point blowupD45322F36C36Flags, convex bodies and section growth
lemmaLemma 19.9.15744C8C462232Flags, convex bodies and section growth
theoremTheorem 19.9.161EBEF54DB76DFlags, convex bodies and section growth
corollaryCorollary 19.9.17EECB2B1DF1A4Flags, convex bodies and section growth
section19.9.18 Curvature rank and section growth1936F96341B0Flags, convex bodies and section growth
propositionProposition 19.9.19C8B2CB1B6819Flags, convex bodies and section growth
corollaryCorollary 19.9.2014480FFA5B57Flags, convex bodies and section growth
lemmaLemma 19.9.21FCAD080F93F6Flags, convex bodies and section growth
corollaryCorollary 19.9.22CA93F87E4E8DFlags, convex bodies and section growth
propositionProposition 19.9.235C0C7B20BA7AFlags, convex bodies and section growth
corollaryCorollary 19.9.24D22F138F2F08Flags, convex bodies and section growth
exampleExample 19.9.25DE28BE657AA4Flags, convex bodies and section growth
section19.9.26 The Hopf flag and its lower-dimensional body9B3BD06F2362Flags, convex bodies and section growth
propositionProposition 19.9.27987D754E15F6Flags, convex bodies and section growth
captionFigure 19.9.28555D3252DD76Flags, convex bodies and section growth

20. B-divisors and restriction

section20.2.1 The definition of the trace operatorE3F1F75C84E9The trace operator
exampleExample 20.2.2482E170F0825The trace operator
exampleExample 20.2.3DC6BE2600DA0The trace operator
propositionProposition 20.2.482453E358B30The trace operator
equationEquation 20.2.52B91479A9BE3The trace operator
definitionDefinition 20.2.6ADF6E265888FThe trace operator
equationEquation 20.2.77AF21EBA8F4EThe trace operator
propositionProposition 20.2.845EA06425921The trace operator
exampleExample 20.2.975F5F5ACD2FAThe trace operator
definitionDefinition 20.2.105239E9ADD0FFThe trace operator
remarkRemark 20.2.11F5CDFD8547BCThe trace operator
propositionProposition 20.2.12B293F74D2022The trace operator
equationEquation 20.2.13D2EC0F645F1CThe trace operator
equationEquation 20.2.149C0808B7B91FThe trace operator
exampleExample 20.2.15917FA516701FThe trace operator
exampleExample 20.2.1650B7DFD48C9DThe trace operator
exampleExample 20.2.172272C87A04CEThe trace operator
equationEquation 20.2.184411B718BF44The trace operator
remarkRemark 20.2.19ECCF7648C028The trace operator
section20.2.20 Properties of the trace operator4A91513F4829The trace operator
propositionProposition 20.2.21F2C3DD7D449EThe trace operator
propositionProposition 20.2.2258627BFD2457The trace operator
exampleExample 20.2.23531FFD6D4B64The trace operator
lemmaLemma 20.2.24DDEF49346283The trace operator
equationEquation 20.2.253B7501D970A6The trace operator
propositionProposition 20.2.26DB90B2D3ADEBThe trace operator
equationEquation 20.2.27E9AC29FB30B6The trace operator
theoremTheorem 20.2.2871AECE2E70EDThe trace operator
section20.2.29 Relation to the classical restricted volumesA437169E35B2The trace operator
equationEquation 20.2.30D4594BE77AFCThe trace operator
equationEquation 20.2.3103AF8A775E35The trace operator
propositionProposition 20.2.32AFC4F39F65CFThe trace operator
equationEquation 20.2.33DA6F88B54E6FThe trace operator
theoremTheorem 20.2.347EFDFA26519BThe trace operator
equationEquation 20.2.353C4F71AC116EThe trace operator
equationEquation 20.2.366EEF71F49B54The trace operator
equationEquation 20.2.37E74D68200A33The trace operator
section20.2.38 Restricted volumes of line bundlesEDE927CBCEA7The trace operator
theoremTheorem 20.2.39230AF20F8E84The trace operator
equationEquation 20.2.4092546FA16294The trace operator
equationEquation 20.2.4197B6F891A18AThe trace operator
theoremTheorem 20.2.424F486DDB8BD0The trace operator
remarkRemark 20.2.43DB535DD3B5A9The trace operator
section20.2.44 Analytic Bertini theorems6B19CA7EC662The trace operator
theoremTheorem 20.2.457931695F46E8The trace operator
remarkRemark 20.2.466A24558044D5The trace operator
equationEquation 20.2.4762AF47325D23The trace operator
remarkRemark 20.2.48E131A2FB1C6AThe trace operator
corollaryCorollary 20.2.4950ABF7A3834BThe trace operator
corollaryCorollary 20.2.50B590E31078F3The trace operator
lemmaLemma 20.2.51CB547C0AF98FThe trace operator
corollaryCorollary 20.2.523AEE64F8D403The trace operator
propositionProposition 20.2.53A2BD7EA3B4B4The trace operator
lemmaLemma 20.2.54A172685DB2ADThe trace operator
corollaryCorollary 20.2.558F47DBEBF9D3The trace operator
lemmaLemma 20.2.56 (Reverse Bertini theorem)A7BAA368EEF9The trace operator
equationEquation 20.2.57133BE510F62FThe trace operator
equationEquation 20.2.58C42AE95D23AFThe trace operator
section20.1.1 The notion of b-divisors2C74B684280CThe theory of b-divisors
propositionProposition 20.1.27E1DC3BE93AFThe theory of b-divisors
equationEquation 20.1.373D0C455ED1DThe theory of b-divisors
subsection20.1.4 Nef b-divisors6590FB79F2E7The theory of b-divisors
definitionDefinition 20.1.5178D74F44C30The theory of b-divisors
definitionDefinition 20.1.6E6F6681E7170The theory of b-divisors
lemmaLemma 20.1.7F5B2C1F1E318The theory of b-divisors
equationEquation 20.1.83BFB2E87E034The theory of b-divisors
exampleExample 20.1.97AF528B0FB3BThe theory of b-divisors
definitionDefinition 20.1.10908967DFB37EThe theory of b-divisors
definitionDefinition 20.1.11FB2640F8F828The theory of b-divisors
definitionDefinition 20.1.12890427BD65D5The theory of b-divisors
subsection20.1.13 The b-divisors of currentsB2ED24A0AA8AThe theory of b-divisors
definitionDefinition 20.1.14165D43352903The theory of b-divisors
theoremTheorem 20.1.156EB1B4E1D222The theory of b-divisors
equationEquation 20.1.161D610D16E240The theory of b-divisors
equationEquation 20.1.1735EFEAF9683DThe theory of b-divisors
equationEquation 20.1.18725B7EF61286The theory of b-divisors
propositionProposition 20.1.1987D0CCEB6DF9The theory of b-divisors
equationEquation 20.1.2064E2A79320D9The theory of b-divisors
equationEquation 20.1.21808CB322D92DThe theory of b-divisors
equationEquation 20.1.22762B9BFFA0EDThe theory of b-divisors
lemmaLemma 20.1.23476687DD2394The theory of b-divisors
theoremTheorem 20.1.24EA1E72841C6DThe theory of b-divisors
equationEquation 20.1.2536BC6A8ED233The theory of b-divisors
equationEquation 20.1.266E5120AB2EE3The theory of b-divisors
corollaryCorollary 20.1.27362BC48145F6The theory of b-divisors
theoremTheorem 20.1.28C3EFB85C68F6The theory of b-divisors
equationEquation 20.1.2979F72595C6AEThe theory of b-divisors
equationEquation 20.1.30F7913C6F4559The theory of b-divisors
corollaryCorollary 20.1.311D8C494FD558The theory of b-divisors
definitionDefinition 20.1.32AB54064E3778The theory of b-divisors
corollaryCorollary 20.1.33F249270AD497The theory of b-divisors
corollaryCorollary 20.1.345108D6DF4DAEThe theory of b-divisors
corollaryCorollary 20.1.35DF4E1BF96D34The theory of b-divisors
corollaryCorollary 20.1.36BF517753D33BThe theory of b-divisors
corollaryCorollary 20.1.37926F05B89741The theory of b-divisors
section20.1.38 Properties of the intersection product066B0B2E68CAThe theory of b-divisors
definitionDefinition 20.1.39BFB26AC5F975The theory of b-divisors
lemmaLemma 20.1.40DFC2368D5436The theory of b-divisors
definitionDefinition 20.1.4160343D71D40AThe theory of b-divisors
equationEquation 20.1.4233927991E020The theory of b-divisors
propositionProposition 20.1.43AEC81016D4BCThe theory of b-divisors
lemmaLemma 20.1.44D57BEED0545EThe theory of b-divisors
propositionProposition 20.1.459D4AD4B277B4The theory of b-divisors
propositionProposition 20.1.462E8615B14EC8The theory of b-divisors
propositionProposition 20.1.47754B050BFA6CThe theory of b-divisors
equationEquation 20.1.487E0BA3EB7A39The theory of b-divisors
propositionProposition 20.1.49FE8A163BAA22The theory of b-divisors
remarkRemark 20.1.50D7547FA69C91The theory of b-divisors
section20.3.1 Restricted ideals and trace data65B836952506Restriction and intrinsic trace data
theoremTheorem 20.3.27CD874C73EC9Restriction and intrinsic trace data
equationEquation 20.3.366AFB7F7397BRestriction and intrinsic trace data
propositionProposition 20.3.489D22AB4CBD8Restriction and intrinsic trace data
section20.3.5 Comparison with ordinary restriction95B8C7040FB9Restriction and intrinsic trace data
propositionProposition 20.3.6FE94B20EED19Restriction and intrinsic trace data
subsection20.3.7 Logarithmic divisors with a small-curvature residual14C5248C16DARestriction and intrinsic trace data
propositionProposition 20.3.8C4231F8A37FDRestriction and intrinsic trace data
section20.3.9 Realization on a compact curve0A6FA0C6A351Restriction and intrinsic trace data
theoremTheorem 20.3.10879739B42EBBRestriction and intrinsic trace data
equationEquation 20.3.1129F47A0F8A5DRestriction and intrinsic trace data
equationEquation 20.3.1257C6BC849EC1Restriction and intrinsic trace data
propositionProposition 20.3.13B3BE3C08418ARestriction and intrinsic trace data
section20.3.14 The Poisson equation in the atomic trace typeE07C398EC98FRestriction and intrinsic trace data
theoremTheorem 20.3.15AFC6C3B6368CRestriction and intrinsic trace data
section20.3.16 Singularity distance on a curve7EDE08C90FE8Restriction and intrinsic trace data
propositionProposition 20.3.17F8A7A320265ARestriction and intrinsic trace data
equationEquation 20.3.18D980BA19141ERestriction and intrinsic trace data
subsection20.3.19 Total variation of the atomic dataFED68B5DF3EBRestriction and intrinsic trace data
lemmaLemma 20.3.20A374F6B4426ERestriction and intrinsic trace data
theoremTheorem 20.3.212432A5A2AE14Restriction and intrinsic trace data
corollaryCorollary 20.3.22D416A3AB0845Restriction and intrinsic trace data
section20.3.23 The trace envelope53884234AA76Restriction and intrinsic trace data
propositionProposition 20.3.24787C2A9E0C8DRestriction and intrinsic trace data
equationEquation 20.3.25AA3FC60BB1F2Restriction and intrinsic trace data
subsection20.3.26 Limits of restricted analytic envelopes9FDA3A326A26Restriction and intrinsic trace data
theoremTheorem 20.3.275FE15106F340Restriction and intrinsic trace data
equationEquation 20.3.285377E31C95CDRestriction and intrinsic trace data
subsection20.3.29 Envelopes on modificationsDB975E954111Restriction and intrinsic trace data
theoremTheorem 20.3.307EFA5C202804Restriction and intrinsic trace data
equationEquation 20.3.3113DF98C40E87Restriction and intrinsic trace data
subsection20.3.32 Exact realization on a Kähler submanifold4C8D52A64B1ARestriction and intrinsic trace data
theoremTheorem 20.3.33D1551EE7322BRestriction and intrinsic trace data
subsection20.3.34 Positive mass and prescribed equationsA7CED8F92D7BRestriction and intrinsic trace data
theoremTheorem 20.3.355CC5BF99E84FRestriction and intrinsic trace data
corollaryCorollary 20.3.36C5512844A274Restriction and intrinsic trace data
section20.3.37 Exact traces over a curve3F27B145B165Restriction and intrinsic trace data
equationEquation 20.3.38D1F9CCD69DB1Restriction and intrinsic trace data
theoremTheorem 20.3.39A30F4066C584Restriction and intrinsic trace data
equationEquation 20.3.4034E4C9363222Restriction and intrinsic trace data
section20.3.41 Exact traces on Hopf manifolds06493A31E440Restriction and intrinsic trace data
equationEquation 20.3.42974BDFDF0550Restriction and intrinsic trace data
subsection20.3.43 Divisors under a dominant mapA7CD7872B787Restriction and intrinsic trace data
lemmaLemma 20.3.44C53A2B3087ABRestriction and intrinsic trace data
equationEquation 20.3.4509D69C0F18B7Restriction and intrinsic trace data
subsection20.3.46 The full trace envelope286FEC8F6AC1Restriction and intrinsic trace data
equationEquation 20.3.471D55891047AARestriction and intrinsic trace data
theoremTheorem 20.3.4863EC1D93ABC1Restriction and intrinsic trace data
equationEquation 20.3.49E6C82B883427Restriction and intrinsic trace data
equationEquation 20.3.503B4C37A0F595Restriction and intrinsic trace data
subsection20.3.51 Mass and the ideal class8065AC6F9F0FRestriction and intrinsic trace data
propositionProposition 20.3.5213817C91E7B4Restriction and intrinsic trace data
equationEquation 20.3.53B5F41C0BB745Restriction and intrinsic trace data
exampleExample 20.3.54D8423B168573Restriction and intrinsic trace data
corollaryCorollary 20.3.55313F5BBD3487Restriction and intrinsic trace data
subsection20.3.56 Decreasing analytic modelsF128B6972881Restriction and intrinsic trace data
propositionProposition 20.3.57B0E6CF965FABRestriction and intrinsic trace data
section20.3.58 Product realizations and their measures291ACA57C265Restriction and intrinsic trace data
propositionProposition 20.3.59AA3908FEC2B4Restriction and intrinsic trace data
equationEquation 20.3.60D8F334251727Restriction and intrinsic trace data
equationEquation 20.3.61BB2C048CAB3BRestriction and intrinsic trace data
propositionProposition 20.3.6265113AF2CE2FRestriction and intrinsic trace data
section20.3.63 Ordinary trace volumes on a nonclosed ruled surface070063ADB7C2Restriction and intrinsic trace data
equationEquation 20.3.64691AF485D1C2Restriction and intrinsic trace data
lemmaLemma 20.3.6572230836D2EERestriction and intrinsic trace data
theoremTheorem 20.3.669EDE5892D63BRestriction and intrinsic trace data
equationEquation 20.3.67EEBF58093B80Restriction and intrinsic trace data
equationEquation 20.3.689099C8F4472BRestriction and intrinsic trace data
equationEquation 20.3.69F060F71204CBRestriction and intrinsic trace data
subsection20.3.70 Ambient ideal orders and ordinary type4746F2B8424ARestriction and intrinsic trace data
propositionProposition 20.3.71C33DAF9C7870Restriction and intrinsic trace data
subsection20.3.72 Pullback and strict representativesD36C5D62DF35Restriction and intrinsic trace data
propositionProposition 20.3.7355D866148BAFRestriction and intrinsic trace data
propositionProposition 20.3.74E91643D06F3ERestriction and intrinsic trace data
section20.4.1 Divisorial trace currents9A1D9C97FB7BDivisorial currents on compact modifications
theoremTheorem 20.4.2A67FA9DFB4BCDivisorial currents on compact modifications
equationEquation 20.4.3B7771456202CDivisorial currents on compact modifications
section20.4.4 Compatibility between models70BE0BB07401Divisorial currents on compact modifications
theoremTheorem 20.4.576E7A41B366ADivisorial currents on compact modifications
equationEquation 20.4.655CA1CD82506Divisorial currents on compact modifications
section20.4.7 The trace envelope as a common majorantB7F265C09335Divisorial currents on compact modifications
propositionProposition 20.4.8C592843AB1E3Divisorial currents on compact modifications
section20.4.9 Bott–Chern classes and closed currents9F22F2EB9CD1Divisorial currents on compact modifications
lemmaLemma 20.4.10C288B8AF0D4EDivisorial currents on compact modifications
definitionDefinition 20.4.116EC6FB71DCF4Divisorial currents on compact modifications
propositionProposition 20.4.12E2830D0CF2C8Divisorial currents on compact modifications
section20.4.13 The b-divisor class of a closed positive currentF45AA1F2A651Divisorial currents on compact modifications
lemmaLemma 20.4.14691315D9AA1CDivisorial currents on compact modifications
theoremTheorem 20.4.1525F4A4E3755DDivisorial currents on compact modifications
section20.4.16 Nef Cartier realizationsEB6B5025AAF9Divisorial currents on compact modifications
lemmaLemma 20.4.176370BDE30D1BDivisorial currents on compact modifications
theoremTheorem 20.4.180AD88DBD3706Divisorial currents on compact modifications
section20.4.19 Continuous representatives and pullbackD9E4FD0A9290Divisorial currents on compact modifications
lemmaLemma 20.4.20721E66DF8C88Divisorial currents on compact modifications
equationEquation 20.4.21641BFD39D2E5Divisorial currents on compact modifications
propositionProposition 20.4.22D7373D3639FADivisorial currents on compact modifications
equationEquation 20.4.23EBECC85902E6Divisorial currents on compact modifications
corollaryCorollary 20.4.24AC7947F570DBDivisorial currents on compact modifications
section20.4.25 Exceptional divisors between trace modelsF423B2B60EE2Divisorial currents on compact modifications
theoremTheorem 20.4.26400A4F39FD56Divisorial currents on compact modifications
equationEquation 20.4.27D5C95521E68EDivisorial currents on compact modifications
equationEquation 20.4.28398DC29813C5Divisorial currents on compact modifications
equationEquation 20.4.293A85C5091414Divisorial currents on compact modifications
corollaryCorollary 20.4.3074858EC68B34Divisorial currents on compact modifications
exampleExample 20.4.3174A0526E4030Divisorial currents on compact modifications

21. Toric and convex geometry

definitionDefinition 21.2.14A9F1082E9C2Closed convex functions and supporting slopes
equationEquation 21.2.2C59E34BB3C43Closed convex functions and supporting slopes
lemmaLemma 21.2.334526AA883B2Closed convex functions and supporting slopes
propositionProposition 21.2.472D7B299EAB8Closed convex functions and supporting slopes
equationEquation 21.2.57BAA42793D50Closed convex functions and supporting slopes
propositionProposition 21.2.6EAC7AAEC7279Closed convex functions and supporting slopes
subsection21.2.7 Invariant functions on tubes and complex toriEFCDD26ABB1EClosed convex functions and supporting slopes
propositionProposition 21.2.8909102168A21Closed convex functions and supporting slopes
corollaryCorollary 21.2.90C12F36D1EDDClosed convex functions and supporting slopes
definitionDefinition 21.3.1103D0D196EAFLegendre duality and convex envelopes
theoremTheorem 21.3.2011FA81B0000Legendre duality and convex envelopes
propositionProposition 21.3.353A9D470F1ACLegendre duality and convex envelopes
definitionDefinition 21.4.1E8746DA689A7Slope bodies and bounded growth
propositionProposition 21.4.2D6E7C0B06FADSlope bodies and bounded growth
lemmaLemma 21.4.39AEE54E3B1E1Slope bodies and bounded growth
propositionProposition 21.4.4374D75E96CF6Slope bodies and bounded growth
exampleExample 21.4.556EB803309C2Slope bodies and bounded growth
lemmaLemma 21.4.6F7AECD623994Slope bodies and bounded growth
theoremTheorem 21.5.1409656CC47D4The real Monge–Ampère measure
theoremTheorem 21.5.2F4072D9ED060The real Monge–Ampère measure
exampleExample 21.5.36BCC82ADB7C7The real Monge–Ampère measure
propositionProposition 21.5.45D8A7A7D734EThe real Monge–Ampère measure
equationEquation 21.5.5D38381F0AC43The real Monge–Ampère measure
lemmaLemma 21.6.1FCDAD9CEC8CECompact convex envelopes and full-mass limits
lemmaLemma 21.6.2F2FABB0A312FCompact convex envelopes and full-mass limits
definitionDefinition 21.7.1B45BB086BF76Lattice points and volume
propositionProposition 21.7.2EDD78BC61BD7Lattice points and volume
theoremTheorem 21.7.3831AEEB7084FLattice points and volume
corollaryCorollary 21.7.4CF4226A5034FLattice points and volume
definitionDefinition 21.8.1E11C7A4F96ACSmooth lattice polytopes and projective charts
lemmaLemma 21.8.241C54729DBFDSmooth lattice polytopes and projective charts
theoremTheorem 21.8.3E49F0370E203Smooth lattice polytopes and projective charts
propositionProposition 21.9.19FD0EEED0DBFFans, divisors and character sections
propositionProposition 21.9.27B4A7CC50CDCFans, divisors and character sections
propositionProposition 21.9.3C62B8871AA1CFans, divisors and character sections
lemmaLemma 21.9.400432B20A966Fans, divisors and character sections
theoremTheorem 21.9.569D3D97E7701Fans, divisors and character sections
theoremTheorem 21.10.1B471B994CE64The Guillemin reference metric
section21.12.1 Toric setupDD9C13434AA7Toric pluripotential theory on ample line bundles
equationEquation 21.12.2338B3FB48871Toric pluripotential theory on ample line bundles
equationEquation 21.12.3EA1385DD984DToric pluripotential theory on ample line bundles
exampleExample 21.12.44214559F0471Toric pluripotential theory on ample line bundles
captionFigure 21.12.5987E79283076Toric pluripotential theory on ample line bundles
section21.12.6 Toric plurisubharmonic functionsE5B138E4CBB0Toric pluripotential theory on ample line bundles
lemmaLemma 21.12.71F4ACCC03A78Toric pluripotential theory on ample line bundles
equationEquation 21.12.8836C60871D35Toric pluripotential theory on ample line bundles
equationEquation 21.12.9CC04BEE427C7Toric pluripotential theory on ample line bundles
equationEquation 21.12.10AA66EBA79840Toric pluripotential theory on ample line bundles
equationEquation 21.12.11C0E06297D5EFToric pluripotential theory on ample line bundles
exampleExample 21.12.12CD7BA5BA48AFToric pluripotential theory on ample line bundles
theoremTheorem 21.12.138B5B73657BF7Toric pluripotential theory on ample line bundles
equationEquation 21.12.140CA6FBB8B2BFToric pluripotential theory on ample line bundles
equationEquation 21.12.1540DB9807D05AToric pluripotential theory on ample line bundles
exampleExample 21.12.169BDBC1FA226AToric pluripotential theory on ample line bundles
captionFigure 21.12.171AC4B1EDD9A2Toric pluripotential theory on ample line bundles
captionFigure 21.12.18CC775DD29F71Toric pluripotential theory on ample line bundles
propositionProposition 21.12.19EEFA0F6B0CF6Toric pluripotential theory on ample line bundles
propositionProposition 21.12.20D60ADA8CEE9BToric pluripotential theory on ample line bundles
propositionProposition 21.12.21419A1D9F18A9Toric pluripotential theory on ample line bundles
exampleExample 21.12.222B1E84EAAE13Toric pluripotential theory on ample line bundles
propositionProposition 21.12.2373034E124A9AToric pluripotential theory on ample line bundles
propositionProposition 21.12.24725063C1ED05Toric pluripotential theory on ample line bundles
equationEquation 21.12.25C2BF3D2E88B4Toric pluripotential theory on ample line bundles
equationEquation 21.12.26E34E6BC222DCToric pluripotential theory on ample line bundles
definitionDefinition 21.12.27CC64AE504A19Toric pluripotential theory on ample line bundles
exampleExample 21.12.28E6289865C7A4Toric pluripotential theory on ample line bundles
propositionProposition 21.12.2911748980D2DFToric pluripotential theory on ample line bundles
equationEquation 21.12.30F8C4AC16B8F2Toric pluripotential theory on ample line bundles
theoremTheorem 21.12.31 (Yao)E109952155FBToric pluripotential theory on ample line bundles
equationEquation 21.12.32C40469D9226BToric pluripotential theory on ample line bundles
equationEquation 21.12.3339F3ED71C89DToric pluripotential theory on ample line bundles
equationEquation 21.12.34428CD3ABF6EFToric pluripotential theory on ample line bundles
corollaryCorollary 21.12.3505A4124B7E31Toric pluripotential theory on ample line bundles
exampleExample 21.12.3637CEF2EAED80Toric pluripotential theory on ample line bundles
propositionProposition 21.12.37852502204924Toric pluripotential theory on ample line bundles
propositionProposition 21.12.3855BFC442EA72Toric pluripotential theory on ample line bundles
propositionProposition 21.12.3975730375E037Toric pluripotential theory on ample line bundles
equationEquation 21.12.40F02AE1451B98Toric pluripotential theory on ample line bundles
equationEquation 21.12.413654D70C9AC4Toric pluripotential theory on ample line bundles
corollaryCorollary 21.12.42E80DDB8667C8Toric pluripotential theory on ample line bundles
definitionDefinition 21.12.430B726BDBFF06Toric pluripotential theory on ample line bundles
corollaryCorollary 21.12.4412D008DD3794Toric pluripotential theory on ample line bundles
section21.13.1 Toric setupE05D6993D0C8Toric pluripotential theory on big line bundles
equationEquation 21.13.22A3462C0F498Toric pluripotential theory on big line bundles
equationEquation 21.13.3077C9BE27980Toric pluripotential theory on big line bundles
equationEquation 21.13.488DD52C918D3Toric pluripotential theory on big line bundles
equationEquation 21.13.518C40EE2BAB3Toric pluripotential theory on big line bundles
equationEquation 21.13.60CCBC0846420Toric pluripotential theory on big line bundles
equationEquation 21.13.7498E343E626CToric pluripotential theory on big line bundles
propositionProposition 21.13.8111BC1BBBDE6Toric pluripotential theory on big line bundles
equationEquation 21.13.90AA0B0B933FCToric pluripotential theory on big line bundles
corollaryCorollary 21.13.100D3DB05576E8Toric pluripotential theory on big line bundles
equationEquation 21.13.117480F81CDDBDToric pluripotential theory on big line bundles
exampleExample 21.13.12C51FE704A3B3Toric pluripotential theory on big line bundles
captionFigure 21.13.1385FCE3A7A731Toric pluripotential theory on big line bundles
captionFigure 21.13.1416FDB2620E86Toric pluripotential theory on big line bundles
exampleExample 21.13.15B328D39711DFToric pluripotential theory on big line bundles
captionFigure 21.13.16C619293CCE8AToric pluripotential theory on big line bundles
exampleExample 21.13.17FF229F946E51Toric pluripotential theory on big line bundles
captionFigure 21.13.18EFB6C6617A59Toric pluripotential theory on big line bundles
exampleExample 21.13.199A122D0D1FDFToric pluripotential theory on big line bundles
captionFigure 21.13.20BFF7444AF5BEToric pluripotential theory on big line bundles
section21.13.21 Toric partial Okounkov bodiesE81A302A1530Toric pluripotential theory on big line bundles
lemmaLemma 21.13.22A11981674229Toric pluripotential theory on big line bundles
equationEquation 21.13.23CDCD55296D94Toric pluripotential theory on big line bundles
subsection21.13.24 Newton bodiesB09F60655D0AToric pluripotential theory on big line bundles
definitionDefinition 21.13.25D7E78CFDB281Toric pluripotential theory on big line bundles
equationEquation 21.13.26E09FCEF2675CToric pluripotential theory on big line bundles
definitionDefinition 21.13.2722C597E4DEB2Toric pluripotential theory on big line bundles
propositionProposition 21.13.2849A9A66B3A4DToric pluripotential theory on big line bundles
equationEquation 21.13.2904F5C34B36CAToric pluripotential theory on big line bundles
equationEquation 21.13.305FCC3FDF4C75Toric pluripotential theory on big line bundles
subsection21.13.31 Partial Okounkov bodies962BB5F87752Toric pluripotential theory on big line bundles
equationEquation 21.13.321EA0A3B6C7EAToric pluripotential theory on big line bundles
propositionProposition 21.13.330ED70BB2A7D0Toric pluripotential theory on big line bundles
equationEquation 21.13.3435302703CBCCToric pluripotential theory on big line bundles
exampleExample 21.13.3574A6D2D50F46Toric pluripotential theory on big line bundles
exampleExample 21.13.36BB6FCAD03597Toric pluripotential theory on big line bundles
captionFigure 21.13.3751BA09C15652Toric pluripotential theory on big line bundles
lemmaLemma 21.13.38FCDBEF4DBD54Toric pluripotential theory on big line bundles
theoremTheorem 21.13.398F3CCCDB83ECToric pluripotential theory on big line bundles
equationEquation 21.13.403FD1A27C837DToric pluripotential theory on big line bundles
equationEquation 21.13.4104CC3F6E758BToric pluripotential theory on big line bundles
corollaryCorollary 21.13.42B202075FB15EToric pluripotential theory on big line bundles
corollaryCorollary 21.13.43C521E79AA7F6Toric pluripotential theory on big line bundles
equationEquation 21.13.446C720B821C17Toric pluripotential theory on big line bundles
theoremTheorem 21.13.451293B87EF09FToric pluripotential theory on big line bundles
equationEquation 21.13.46188F0AD7B28CToric pluripotential theory on big line bundles
equationEquation 21.13.4782B62A741230Toric pluripotential theory on big line bundles
equationEquation 21.13.4883935CB0C416Toric pluripotential theory on big line bundles
equationEquation 21.13.496DBC736A41CFToric pluripotential theory on big line bundles
equationEquation 21.13.50F3EAEAF94235Toric pluripotential theory on big line bundles
equationEquation 21.13.5170B4F53FBB2EToric pluripotential theory on big line bundles
section21.13.52 The pluripotential theoryE0A57B09F819Toric pluripotential theory on big line bundles
theoremTheorem 21.13.5346AFD20390FFToric pluripotential theory on big line bundles
corollaryCorollary 21.13.541D4A688AAB93Toric pluripotential theory on big line bundles
propositionProposition 21.13.554E754992D548Toric pluripotential theory on big line bundles
propositionProposition 21.13.56A0B71330F2E6Toric pluripotential theory on big line bundles
propositionProposition 21.13.57BC781EB76285Toric pluripotential theory on big line bundles
propositionProposition 21.13.580728D8B152D3Toric pluripotential theory on big line bundles
equationEquation 21.13.59D5353E18A697Toric pluripotential theory on big line bundles
corollaryCorollary 21.13.60396E3B9D09BDToric pluripotential theory on big line bundles
corollaryCorollary 21.13.61E1B465B480ABToric pluripotential theory on big line bundles
propositionProposition 21.13.62D5C47A3AE82DToric pluripotential theory on big line bundles
equationEquation 21.13.63F494EF2554FBToric pluripotential theory on big line bundles
equationEquation 21.13.6471DDC2004810Toric pluripotential theory on big line bundles
equationEquation 21.13.65204454C17F1DToric pluripotential theory on big line bundles
corollaryCorollary 21.13.6603A225BFC03CToric pluripotential theory on big line bundles
equationEquation 21.13.67D3A1BA3F5322Toric pluripotential theory on big line bundles
corollaryCorollary 21.13.68551D529C92A2Toric pluripotential theory on big line bundles
equationEquation 21.13.698A0A9FDE2DF5Toric pluripotential theory on big line bundles
equationEquation 21.13.70C4D285074403Toric pluripotential theory on big line bundles
corollaryCorollary 21.13.7123F2FB292586Toric pluripotential theory on big line bundles
equationEquation 21.13.72F5DBB23C8310Toric pluripotential theory on big line bundles
corollaryCorollary 21.13.73AFC1919DC91AToric pluripotential theory on big line bundles
equationEquation 21.13.74FE80C413846EToric pluripotential theory on big line bundles
theoremTheorem 21.13.750BA5B8CFD15FToric pluripotential theory on big line bundles
captionFigure 21.13.769C7AF9D665D8Toric pluripotential theory on big line bundles
equationEquation 21.13.7750B6C6350780Toric pluripotential theory on big line bundles
equationEquation 21.13.7808543024F67DToric pluripotential theory on big line bundles
corollaryCorollary 21.13.7986EC82D6C082Toric pluripotential theory on big line bundles
equationEquation 21.13.80E330AEFE18FDToric pluripotential theory on big line bundles
corollaryCorollary 21.13.819B7387A93262Toric pluripotential theory on big line bundles
exampleExample 21.13.82A14927C0973AToric pluripotential theory on big line bundles
captionFigure 21.13.83F8805F2A5BD0Toric pluripotential theory on big line bundles
theoremTheorem 21.13.8438F3E4DFCD04Toric pluripotential theory on big line bundles
equationEquation 21.13.85862A2BCBD614Toric pluripotential theory on big line bundles
equationEquation 21.13.868E6671867292Toric pluripotential theory on big line bundles
section21.11.1 The notion of convex functionsFB8022E8BCA8Convex functions and convex bodies
definitionDefinition 21.11.23B5FF6A6C162Convex functions and convex bodies
definitionDefinition 21.11.34FA12A948A00Convex functions and convex bodies
lemmaLemma 21.11.447309C75E6C1Convex functions and convex bodies
exampleExample 21.11.51532A37DD27FConvex functions and convex bodies
exampleExample 21.11.68CBDC69B6C55Convex functions and convex bodies
definitionDefinition 21.11.70ECB6CF7B06DConvex functions and convex bodies
propositionProposition 21.11.868114EF2F29CConvex functions and convex bodies
definitionDefinition 21.11.98F179BE28950Convex functions and convex bodies
definitionDefinition 21.11.106561D15F1216Convex functions and convex bodies
definitionDefinition 21.11.114D25259F16ECConvex functions and convex bodies
propositionProposition 21.11.12B99B20CD6166Convex functions and convex bodies
lemmaLemma 21.11.132F4F518BD56DConvex functions and convex bodies
equationEquation 21.11.14752978EE01F6Convex functions and convex bodies
definitionDefinition 21.11.1561B746F5E884Convex functions and convex bodies
propositionProposition 21.11.16E013E1966E44Convex functions and convex bodies
theoremTheorem 21.11.177E0B71CF3B0EConvex functions and convex bodies
propositionProposition 21.11.18B6B15657CD64Convex functions and convex bodies
definitionDefinition 21.11.19CC63E6A637E8Convex functions and convex bodies
theoremTheorem 21.11.20A7E9B5A114CAConvex functions and convex bodies
section21.11.21 Legendre transform67B012704E18Convex functions and convex bodies
definitionDefinition 21.11.222E5E29E9A58BConvex functions and convex bodies
equationEquation 21.11.23E1AD8A1437E2Convex functions and convex bodies
theoremTheorem 21.11.24258227E5D497Convex functions and convex bodies
exampleExample 21.11.251A7C0B007516Convex functions and convex bodies
corollaryCorollary 21.11.268C4DFDD0ADE0Convex functions and convex bodies
equationEquation 21.11.27E918B6149A40Convex functions and convex bodies
equationEquation 21.11.283B0DC2BF53F0Convex functions and convex bodies
propositionProposition 21.11.29C3CC440747D6Convex functions and convex bodies
definitionDefinition 21.11.30D6A0CB1B8846Convex functions and convex bodies
equationEquation 21.11.31F6F028287E9DConvex functions and convex bodies
definitionDefinition 21.11.3208E83421373CConvex functions and convex bodies
theoremTheorem 21.11.33A57F525B7ABCConvex functions and convex bodies
propositionProposition 21.11.348B18F2B2DE3BConvex functions and convex bodies
equationEquation 21.11.35C0D69D19C109Convex functions and convex bodies
propositionProposition 21.11.367A76E56311F5Convex functions and convex bodies
propositionProposition 21.11.3736CA6A68994EConvex functions and convex bodies
propositionProposition 21.11.386FCC352452C3Convex functions and convex bodies
equationEquation 21.11.39D5ABA398BA7BConvex functions and convex bodies
section21.11.40 Classes of convex functions4F281A03F04DConvex functions and convex bodies
definitionDefinition 21.11.412AEBD4E3830AConvex functions and convex bodies
propositionProposition 21.11.42A63267E9EA69Convex functions and convex bodies
propositionProposition 21.11.43281F97D00D5DConvex functions and convex bodies
definitionDefinition 21.11.440E804C9E9F89Convex functions and convex bodies
propositionProposition 21.11.45C0908AA2E3B2Convex functions and convex bodies
propositionProposition 21.11.4693CDDD09D6B0Convex functions and convex bodies
lemmaLemma 21.11.4753E69530968EConvex functions and convex bodies
equationEquation 21.11.48B39039E9725DConvex functions and convex bodies
section21.11.49 Monge–Ampère measuresEAC3497DDCF4Convex functions and convex bodies
definitionDefinition 21.11.50FF5885207F9EConvex functions and convex bodies
equationEquation 21.11.5154CA724C5825Convex functions and convex bodies
propositionProposition 21.11.5284CD4F8DE81DConvex functions and convex bodies
propositionProposition 21.11.53F30A95FDA78EConvex functions and convex bodies
equationEquation 21.11.54492151B44AFAConvex functions and convex bodies
theoremTheorem 21.11.552DC9BC6F295AConvex functions and convex bodies
theoremTheorem 21.11.56ED65AA5C2E92Convex functions and convex bodies
lemmaLemma 21.11.578DE662F4EFE6Convex functions and convex bodies
theoremTheorem 21.11.5859E2455286B4Convex functions and convex bodies
section21.11.59 Several lemmata6F0D797F0564Convex functions and convex bodies
lemmaLemma 21.11.60B1B557E6282FConvex functions and convex bodies
equationEquation 21.11.61EF4C4E285442Convex functions and convex bodies
lemmaLemma 21.11.6281D915E5313DConvex functions and convex bodies
lemmaLemma 21.11.6366B643EAC79AConvex functions and convex bodies
section21.14.1 Weighted sections on the projective base10355FA4232CHopf fibrations and toric potentials
theoremTheorem 21.14.2DCED1F266D4FHopf fibrations and toric potentials
section21.14.3 Descended measures and equations544154BAA081Hopf fibrations and toric potentials
theoremTheorem 21.14.494243AEE3360Hopf fibrations and toric potentials
equationEquation 21.14.5D50C510DA3A9Hopf fibrations and toric potentials
section21.14.6 Logarithmic coordinates and the gradient simplex11FD5CD723A9Hopf fibrations and toric potentials
equationEquation 21.14.75DB0E5246A09Hopf fibrations and toric potentials
propositionProposition 21.14.8F23340CADBE0Hopf fibrations and toric potentials

22. Non-Archimedean pluripotential theory

section22.1.1 The definition of non-Archimedean metricsFB3B4C37DD45Non-Archimedean pluripotential theory
definitionDefinition 22.1.2657951EB240BNon-Archimedean pluripotential theory
equationEquation 22.1.3A99876729E5FNon-Archimedean pluripotential theory
remarkRemark 22.1.4C864B821AD8BNon-Archimedean pluripotential theory
propositionProposition 22.1.5B80F3474DB2FNon-Archimedean pluripotential theory
propositionProposition 22.1.62CB4FA8703B1Non-Archimedean pluripotential theory
definitionDefinition 22.1.7A92D6DFAC0CFNon-Archimedean pluripotential theory
propositionProposition 22.1.847A4DA9EBA71Non-Archimedean pluripotential theory
lemmaLemma 22.1.9587EEDE88F4FNon-Archimedean pluripotential theory
lemmaLemma 22.1.109A2678C2C223Non-Archimedean pluripotential theory
equationEquation 22.1.11717C364BF812Non-Archimedean pluripotential theory
equationEquation 22.1.1280067294986DNon-Archimedean pluripotential theory
exampleExample 22.1.13796F447D96B6Non-Archimedean pluripotential theory
definitionDefinition 22.1.14CCAAC9DA4E2FNon-Archimedean pluripotential theory
propositionProposition 22.1.153943AA7119C5Non-Archimedean pluripotential theory
equationEquation 22.1.1629D3530031C6Non-Archimedean pluripotential theory
propositionProposition 22.1.17817C1BB3E570Non-Archimedean pluripotential theory
equationEquation 22.1.18258B0F2A2F95Non-Archimedean pluripotential theory
corollaryCorollary 22.1.19347399E47277Non-Archimedean pluripotential theory
section22.1.20 Operations on non-Archimedean metricsC20FEECB51B7Non-Archimedean pluripotential theory
definitionDefinition 22.1.21DA2917901B54Non-Archimedean pluripotential theory
propositionProposition 22.1.228C5A487B9CCCNon-Archimedean pluripotential theory
propositionProposition 22.1.23F7D709A21AE1Non-Archimedean pluripotential theory
definitionDefinition 22.1.2493939D1E527CNon-Archimedean pluripotential theory
propositionProposition 22.1.2506DFF1DE1519Non-Archimedean pluripotential theory
propositionProposition 22.1.26E9E94041F68DNon-Archimedean pluripotential theory
definitionDefinition 22.1.27D1E32864F8F0Non-Archimedean pluripotential theory
propositionProposition 22.1.28ECFDC5EDD4F2Non-Archimedean pluripotential theory
propositionProposition 22.1.2987CB041BFCA3Non-Archimedean pluripotential theory
propositionProposition 22.1.3011099644479BNon-Archimedean pluripotential theory
definitionDefinition 22.1.31B819F1A85E09Non-Archimedean pluripotential theory
propositionProposition 22.1.329CACCE9D7690Non-Archimedean pluripotential theory
propositionProposition 22.1.3304004F65B932Non-Archimedean pluripotential theory
definitionDefinition 22.1.34C172DAF1B3E6Non-Archimedean pluripotential theory
equationEquation 22.1.35AB5F1151338CNon-Archimedean pluripotential theory
propositionProposition 22.1.3674CC0BF97B41Non-Archimedean pluripotential theory
propositionProposition 22.1.3721737B0303A5Non-Archimedean pluripotential theory
propositionProposition 22.1.38938848E420F9Non-Archimedean pluripotential theory
propositionProposition 22.1.3915E20104530BNon-Archimedean pluripotential theory
equationEquation 22.1.40AA5696D80091Non-Archimedean pluripotential theory
definitionDefinition 22.1.4165E2E07550F1Non-Archimedean pluripotential theory
equationEquation 22.1.42827FE14EA1A0Non-Archimedean pluripotential theory
equationEquation 22.1.43341067287939Non-Archimedean pluripotential theory
propositionProposition 22.1.4407B251D899EFNon-Archimedean pluripotential theory
propositionProposition 22.1.45C8734FD15163Non-Archimedean pluripotential theory
definitionDefinition 22.1.46EB1311598A32Non-Archimedean pluripotential theory
propositionProposition 22.1.4768FB6124EEB2Non-Archimedean pluripotential theory
propositionProposition 22.1.48EDFB2682E78ENon-Archimedean pluripotential theory
propositionProposition 22.1.491743F9AEA475Non-Archimedean pluripotential theory
definitionDefinition 22.1.50C0E37055D126Non-Archimedean pluripotential theory
equationEquation 22.1.516697E3349082Non-Archimedean pluripotential theory
equationEquation 22.1.52284D786E2B4ENon-Archimedean pluripotential theory
propositionProposition 22.1.53E7009D90BCC6Non-Archimedean pluripotential theory
section22.1.54 Duistermaat–Heckman measures3C0D12E80020Non-Archimedean pluripotential theory
definitionDefinition 22.1.559ADEFD2FF381Non-Archimedean pluripotential theory
theoremTheorem 22.1.560AD7F80A793DNon-Archimedean pluripotential theory
equationEquation 22.1.57A51020D58E6FNon-Archimedean pluripotential theory
equationEquation 22.1.5851031202809CNon-Archimedean pluripotential theory
equationEquation 22.1.59665B92171EA8Non-Archimedean pluripotential theory
propositionProposition 22.1.60492A145E9067Non-Archimedean pluripotential theory
equationEquation 22.1.61E10A583B2E40Non-Archimedean pluripotential theory
equationEquation 22.1.62636AFE1086C1Non-Archimedean pluripotential theory
remarkRemark 22.1.63CF750A31909ENon-Archimedean pluripotential theory
section22.1.64 Comparison with Boucksom–Jonsson's theoryB0164F7B261ENon-Archimedean pluripotential theory
subsection22.1.65 A brief recap of Boucksom–Jonsson's theory82D43892E8F6Non-Archimedean pluripotential theory
subsubsection22.1.66 Valuations618C84432CDANon-Archimedean pluripotential theory
definitionDefinition 22.1.67433FF564E9A3Non-Archimedean pluripotential theory
equationEquation 22.1.6818B5288E3A41Non-Archimedean pluripotential theory
equationEquation 22.1.693B8E87503101Non-Archimedean pluripotential theory
subsubsection22.1.70 Non-Archimedean plurisubharmonic functions3CE1C3E79114Non-Archimedean pluripotential theory
definitionDefinition 22.1.71EEADED50AAC3Non-Archimedean pluripotential theory
theoremTheorem 22.1.720DAB63894803Non-Archimedean pluripotential theory
propositionProposition 22.1.738E1FC46DBC87Non-Archimedean pluripotential theory
propositionProposition 22.1.7485809FFB162FNon-Archimedean pluripotential theory
theoremTheorem 22.1.756B2CF0C2744ANon-Archimedean pluripotential theory
conjectureConjecture 22.1.7695739E70234BNon-Archimedean pluripotential theory
subsection22.1.77 The analytifications5CF97BFE6BF8Non-Archimedean pluripotential theory
subsubsection22.1.78 The transcendental setting1630658F75F9Non-Archimedean pluripotential theory
definitionDefinition 22.1.79322D4668B967Non-Archimedean pluripotential theory
equationEquation 22.1.802CFF9C7C56DBNon-Archimedean pluripotential theory
equationEquation 22.1.81DFFDD4E75B4DNon-Archimedean pluripotential theory
definitionDefinition 22.1.82866728488224Non-Archimedean pluripotential theory
equationEquation 22.1.8345957E422015Non-Archimedean pluripotential theory
propositionProposition 22.1.84A87B7BF0C3D5Non-Archimedean pluripotential theory
equationEquation 22.1.85B5F5AD50A5ADNon-Archimedean pluripotential theory
definitionDefinition 22.1.8654C62141C0D9Non-Archimedean pluripotential theory
equationEquation 22.1.872B233D0F0F88Non-Archimedean pluripotential theory
remarkRemark 22.1.889911A537C86CNon-Archimedean pluripotential theory
lemmaLemma 22.1.896784060E951FNon-Archimedean pluripotential theory
equationEquation 22.1.90F46F1129051CNon-Archimedean pluripotential theory
subsubsection22.1.91 The algebraic settingF432DF631582Non-Archimedean pluripotential theory
lemmaLemma 22.1.92A8896460B129Non-Archimedean pluripotential theory
lemmaLemma 22.1.939434BE558E6ANon-Archimedean pluripotential theory
lemmaLemma 22.1.9476252E7717ACNon-Archimedean pluripotential theory
equationEquation 22.1.956126C03E6587Non-Archimedean pluripotential theory
lemmaLemma 22.1.96AF0C82DE24E9Non-Archimedean pluripotential theory
lemmaLemma 22.1.97E61430B531F3Non-Archimedean pluripotential theory
propositionProposition 22.1.986A232EAE791CNon-Archimedean pluripotential theory
equationEquation 22.1.999186BF0A5F55Non-Archimedean pluripotential theory
corollaryCorollary 22.1.1009D2A9B228E44Non-Archimedean pluripotential theory
subsection22.1.101 The comparison theoremF1438A60D3D4Non-Archimedean pluripotential theory
equationEquation 22.1.102E99D2AFA897ENon-Archimedean pluripotential theory
equationEquation 22.1.103DB2DA874FB8CNon-Archimedean pluripotential theory
equationEquation 22.1.104F161F9E1E0FANon-Archimedean pluripotential theory
lemmaLemma 22.1.105CB69CB164255Non-Archimedean pluripotential theory
equationEquation 22.1.10699B9B27CF451Non-Archimedean pluripotential theory
lemmaLemma 22.1.107AB6A7C1FD159Non-Archimedean pluripotential theory
equationEquation 22.1.108C18150DD4B94Non-Archimedean pluripotential theory
propositionProposition 22.1.1099F673331DE01Non-Archimedean pluripotential theory
equationEquation 22.1.110115AEB4414B4Non-Archimedean pluripotential theory
theoremTheorem 22.1.1119B811DBD89E3Non-Archimedean pluripotential theory
theoremTheorem 22.1.112CB40FF867A00Non-Archimedean pluripotential theory
theoremTheorem 22.1.113D613676A36CANon-Archimedean pluripotential theory
equationEquation 22.1.1140C743FB2F354Non-Archimedean pluripotential theory
equationEquation 22.1.1158B0003DFBC94Non-Archimedean pluripotential theory
lemmaLemma 22.1.1161861C279EAB7Non-Archimedean pluripotential theory

23. Hopf and locally conformally Kähler geometry

theoremTheorem 23.1.1 (Xia–Zhang)92B2FB434C79Unbounded mass on Hopf manifolds
section23.1.2 The Hopf metric and its massCFFF1A0C2E2EUnbounded mass on Hopf manifolds
equationEquation 23.1.325275E5CD565Unbounded mass on Hopf manifolds
propositionProposition 23.1.4DDEBFFAFCD1FUnbounded mass on Hopf manifolds
equationEquation 23.1.5C46F0B405DC2Unbounded mass on Hopf manifolds
section23.1.6 The radial–vertical block70014772CD9CUnbounded mass on Hopf manifolds
lemmaLemma 23.1.76674C9FC8075Unbounded mass on Hopf manifolds
equationEquation 23.1.8814E85A56944Unbounded mass on Hopf manifolds
section23.1.9 Heat potentials and the mixed derivative1CD2386648C8Unbounded mass on Hopf manifolds
equationEquation 23.1.10097EB0CA92F4Unbounded mass on Hopf manifolds
equationEquation 23.1.1127E901CC3829Unbounded mass on Hopf manifolds
lemmaLemma 23.1.1214E6CDA9AAC4Unbounded mass on Hopf manifolds
equationEquation 23.1.139CCEEAD6F62CUnbounded mass on Hopf manifolds
equationEquation 23.1.14D87F1ADF41BFUnbounded mass on Hopf manifolds
equationEquation 23.1.15E73DDDEDFD02Unbounded mass on Hopf manifolds
equationEquation 23.1.16464524C306E0Unbounded mass on Hopf manifolds
lemmaLemma 23.1.177C9989D90F66Unbounded mass on Hopf manifolds
equationEquation 23.1.188EBD296AAFDAUnbounded mass on Hopf manifolds
lemmaLemma 23.1.19AB7F46061184Unbounded mass on Hopf manifolds
equationEquation 23.1.20ED58C2434E12Unbounded mass on Hopf manifolds
section23.1.21 Variation of the heat time525E30DDFE7AUnbounded mass on Hopf manifolds
equationEquation 23.1.22F445D65CCB92Unbounded mass on Hopf manifolds
equationEquation 23.1.23C76E171B8E17Unbounded mass on Hopf manifolds
equationEquation 23.1.2403733983FD0FUnbounded mass on Hopf manifolds
lemmaLemma 23.1.253F868B74CEBDUnbounded mass on Hopf manifolds
equationEquation 23.1.26198FD1FE8553Unbounded mass on Hopf manifolds
equationEquation 23.1.27477FF03F5EC1Unbounded mass on Hopf manifolds
equationEquation 23.1.28A811248FC3F3Unbounded mass on Hopf manifolds
equationEquation 23.1.291BA0538C7E35Unbounded mass on Hopf manifolds
equationEquation 23.1.307C466FD3CC31Unbounded mass on Hopf manifolds
section23.1.31 A base correction and unbounded massD7D1053AF199Unbounded mass on Hopf manifolds
section23.1.32 Higher-dimensional Hopf manifolds28A754700E01Unbounded mass on Hopf manifolds
lemmaLemma 23.1.33264893FAE204Unbounded mass on Hopf manifolds
equationEquation 23.1.34D216F8E7C5AAUnbounded mass on Hopf manifolds
theoremTheorem 23.1.35D3E13E0C86B3Unbounded mass on Hopf manifolds
corollaryCorollary 23.1.3677FC30E16968Unbounded mass on Hopf manifolds
section23.1.37 Minimum mass and fiber averagingCE2794970ECDUnbounded mass on Hopf manifolds
propositionProposition 23.1.38E42306969D71Unbounded mass on Hopf manifolds
equationEquation 23.1.3918B65325175FUnbounded mass on Hopf manifolds
corollaryCorollary 23.1.4009DD8DF2EF76Unbounded mass on Hopf manifolds
section23.1.41 Sharp damping under unitary rotations5746F7AEAB40Unbounded mass on Hopf manifolds
subsection23.1.42 Positive fiber flux9880531015C7Unbounded mass on Hopf manifolds
equationEquation 23.1.43BDDF6CEA3CFEUnbounded mass on Hopf manifolds
lemmaLemma 23.1.44EF821157486EUnbounded mass on Hopf manifolds
equationEquation 23.1.4574322EB52E76Unbounded mass on Hopf manifolds
lemmaLemma 23.1.46712BD3EA719CUnbounded mass on Hopf manifolds
propositionProposition 23.1.47316CC905C469Unbounded mass on Hopf manifolds
equationEquation 23.1.481160D828BCBBUnbounded mass on Hopf manifolds
equationEquation 23.1.4929C345F5CC8DUnbounded mass on Hopf manifolds
subsection23.1.50 Heat concentration at large depth92674EE5B61BUnbounded mass on Hopf manifolds
lemmaLemma 23.1.5121E7B1647FA9Unbounded mass on Hopf manifolds
theoremTheorem 23.1.521394B3372728Unbounded mass on Hopf manifolds
equationEquation 23.1.5326883D1656D1Unbounded mass on Hopf manifolds
equationEquation 23.1.5483FE97565E82Unbounded mass on Hopf manifolds
corollaryCorollary 23.1.550ACED0B738A9Unbounded mass on Hopf manifolds
theoremTheorem 23.1.56F1398FE81D8DUnbounded mass on Hopf manifolds
equationEquation 23.1.57AC3F3934DA9FUnbounded mass on Hopf manifolds
equationEquation 23.1.58CF8AE8E2EB06Unbounded mass on Hopf manifolds
equationEquation 23.1.59C7F482FD7515Unbounded mass on Hopf manifolds
subsection23.1.60 Concentration at the critical weight4B61F80092C8Unbounded mass on Hopf manifolds
lemmaLemma 23.1.615F2A120794A6Unbounded mass on Hopf manifolds
theoremTheorem 23.1.628BCBC6CB58C6Unbounded mass on Hopf manifolds
corollaryCorollary 23.1.63E636296BB209Unbounded mass on Hopf manifolds
section23.1.64 Limits of shrinking smooth potentialsC99F8C57CD9AUnbounded mass on Hopf manifolds
equationEquation 23.1.65ABD574F21AEBUnbounded mass on Hopf manifolds
equationEquation 23.1.66823EC0E922FDUnbounded mass on Hopf manifolds
subsection23.1.67 Positive quadratic measures52E95F708FD4Unbounded mass on Hopf manifolds
lemmaLemma 23.1.681E196E409754Unbounded mass on Hopf manifolds
subsection23.1.69 Orbital mixtures4D8838BCFD12Unbounded mass on Hopf manifolds
propositionProposition 23.1.70A71DAFFE572EUnbounded mass on Hopf manifolds
subsection23.1.71 Fiber measures7802C5C9B69DUnbounded mass on Hopf manifolds
lemmaLemma 23.1.72D04CA3BB7868Unbounded mass on Hopf manifolds
lemmaLemma 23.1.731C279783821EUnbounded mass on Hopf manifolds
theoremTheorem 23.1.747C4D0FB8D775Unbounded mass on Hopf manifolds
subsection23.1.75 Changing the Hermitian metric510437B8C8BBUnbounded mass on Hopf manifolds
theoremTheorem 23.1.76C1374FAD9C60Unbounded mass on Hopf manifolds
equationEquation 23.1.77D98350F0404CUnbounded mass on Hopf manifolds
section23.2.1 The Lee form and the Kähler coveringE0F12265EE84Locally conformally Kähler geometry
definitionDefinition 23.2.248F6C713AFE3Locally conformally Kähler geometry
equationEquation 23.2.32E90E918DF55Locally conformally Kähler geometry
equationEquation 23.2.40F9BDB92C6B1Locally conformally Kähler geometry
section23.2.5 Gauduchon representatives and the Lee classA8F518936B0ALocally conformally Kähler geometry
equationEquation 23.2.629D45DC05342Locally conformally Kähler geometry
propositionProposition 23.2.711EBB7AD1165Locally conformally Kähler geometry
equationEquation 23.2.835851308DD2BLocally conformally Kähler geometry
equationEquation 23.2.9624B1D55AC6ELocally conformally Kähler geometry
corollaryCorollary 23.2.10F6950D5704C9Locally conformally Kähler geometry
propositionProposition 23.2.1126993B1C8C74Locally conformally Kähler geometry
conjectureConjecture 23.2.124EA8A1D43A8CLocally conformally Kähler geometry
subsection23.2.13 The least eigenvalue of Lee curvature942B6FCFDC1ALocally conformally Kähler geometry
propositionProposition 23.2.14E5F423FF7707Locally conformally Kähler geometry
theoremTheorem 23.2.15B8583A0E17A6Locally conformally Kähler geometry
exampleExample 23.2.16BBCE95D98A62Locally conformally Kähler geometry
exampleExample 23.2.17EB59EA55107FLocally conformally Kähler geometry
section23.2.18 Ordinary and twisted potentials435B9E1C507BLocally conformally Kähler geometry
equationEquation 23.2.19A1632B661AC9Locally conformally Kähler geometry
propositionProposition 23.2.20B365A2A75E69Locally conformally Kähler geometry
equationEquation 23.2.2148546D806E67Locally conformally Kähler geometry
equationEquation 23.2.22F2EE69C448EBLocally conformally Kähler geometry
subsection23.2.23 The Vaisman metric on the Hopf manifoldD1CB266917DALocally conformally Kähler geometry
propositionProposition 23.2.2432084430C8ADLocally conformally Kähler geometry
section23.2.25 Volume equations in Gauduchon gauge29CDAA9CF041Locally conformally Kähler geometry
equationEquation 23.2.26C2B05F8B6DCALocally conformally Kähler geometry
problemProblem 23.2.273B2DB14C0DFFLocally conformally Kähler geometry
subsection23.2.28 Invariant volumes on Vaisman manifoldsA3211952AB85Locally conformally Kähler geometry
theoremTheorem 23.2.2919781C771E9DLocally conformally Kähler geometry
equationEquation 23.2.309A2166571118Locally conformally Kähler geometry
equationEquation 23.2.31E3B61067B040Locally conformally Kähler geometry
propositionProposition 23.2.32FCE8FB3BF729Locally conformally Kähler geometry
equationEquation 23.2.331958C8BBF316Locally conformally Kähler geometry
equationEquation 23.2.348455EE64C6A0Locally conformally Kähler geometry
subsection23.2.35 Radial volumes on scalar Hopf manifoldsAB1C295D0BF1Locally conformally Kähler geometry
equationEquation 23.2.362622F07FBE04Locally conformally Kähler geometry
lemmaLemma 23.2.37AE38387ABD03Locally conformally Kähler geometry
equationEquation 23.2.380452243D80CFLocally conformally Kähler geometry
equationEquation 23.2.399A128F70476FLocally conformally Kähler geometry
lemmaLemma 23.2.401D02D96B9AF2Locally conformally Kähler geometry
theoremTheorem 23.2.41C4C3411B46C3Locally conformally Kähler geometry
equationEquation 23.2.4207070AFDA51CLocally conformally Kähler geometry
propositionProposition 23.2.43503628CC8BE5Locally conformally Kähler geometry
propositionProposition 23.2.44A642AD47FF61Locally conformally Kähler geometry
exampleExample 23.2.45FDF7812789EFLocally conformally Kähler geometry
section23.2.46 Chern–Ricci flatness and Vaisman rigidityF6350B804805Locally conformally Kähler geometry
theoremTheorem 23.2.47C4981F6B1F78Locally conformally Kähler geometry
equationEquation 23.2.48C15BADA6DBC4Locally conformally Kähler geometry
equationEquation 23.2.49B65427234DA6Locally conformally Kähler geometry
equationEquation 23.2.50900FA1D2D218Locally conformally Kähler geometry
equationEquation 23.2.511F4612E67B59Locally conformally Kähler geometry
subsection23.2.52 A flat conformal gauge that is not GauduchonE87B4D0DAD68Locally conformally Kähler geometry
exampleExample 23.2.53E37387C4675BLocally conformally Kähler geometry
conjectureConjecture 23.2.548DB45FD59BADLocally conformally Kähler geometry
section23.2.55 The Lee–Gauduchon cone25E90D86853ALocally conformally Kähler geometry
lemmaLemma 23.2.5670F067DD267FLocally conformally Kähler geometry
definitionDefinition 23.2.57FC269BC81E64Locally conformally Kähler geometry
equationEquation 23.2.58FB0122E5265CLocally conformally Kähler geometry
propositionProposition 23.2.59EBFA309383E5Locally conformally Kähler geometry
theoremTheorem 23.2.60DB27896B9A96Locally conformally Kähler geometry
subsection23.2.61 Bimeromorphic invariance5156B6A245EBLocally conformally Kähler geometry
lemmaLemma 23.2.62D4DABE9DA177Locally conformally Kähler geometry
lemmaLemma 23.2.63ED85C468C237Locally conformally Kähler geometry
theoremTheorem 23.2.64403FA017D33BLocally conformally Kähler geometry
subsection23.2.65 Balanced metrics and the Hopf cone197B18655594Locally conformally Kähler geometry
corollaryCorollary 23.2.66BF1CE00DF8A7Locally conformally Kähler geometry
section23.2.67 Strongly Gauduchon metrics5A635830FDEBLocally conformally Kähler geometry
definitionDefinition 23.2.68ADA9E5EAD64CLocally conformally Kähler geometry
propositionProposition 23.2.694CCB765ECCA2Locally conformally Kähler geometry
corollaryCorollary 23.2.7086E2247D209DLocally conformally Kähler geometry
equationEquation 23.2.7124F2C25D4A8FLocally conformally Kähler geometry
exampleExample 23.2.721EE117A3B0B0Locally conformally Kähler geometry

24. Problems in pluripotential theory

subsection24.2.1 Individual density thresholds33D327BAD4E0Threshold equations and singular limits
problemProblem 24.2.2FBE8C5BA5CA4Threshold equations and singular limits
subsection24.2.3 Dual attainment and exact equations23806B61F487Threshold equations and singular limits
subsection24.2.4 Regularity of singular solutions228EC8A6CA8DThreshold equations and singular limits
subsection24.2.5 Uniqueness and selectionDFEF338E1977Threshold equations and singular limits
subsection24.3.1 Collapse of normalized Gauduchon testsC56273368CE1Positive Gauduchon volume and bigness
propositionProposition 24.3.232F82EA89236Positive Gauduchon volume and bigness
equationEquation 24.3.3C672C5EDF4D5Positive Gauduchon volume and bigness
subsection24.4.1 Nef classes, bigness and the Fujiki class806079150C7AClosed positivity and geometric equations
equationEquation 24.4.2018A7DA7A817Closed positivity and geometric equations
problemProblem 24.4.31E16C9C72539Closed positivity and geometric equations
subsection24.4.4 Positive lower mass and concentrationE1772510D012Closed positivity and geometric equations
equationEquation 24.4.50EA28327A7B8Closed positivity and geometric equations
problemProblem 24.4.681116FD94515Closed positivity and geometric equations
subsection24.4.7 Primitive elliptic bundles148E1DFBEDF1Closed positivity and geometric equations
propositionProposition 24.4.88E5CAA6E5122Closed positivity and geometric equations
equationEquation 24.4.9A13181302F35Closed positivity and geometric equations
lemmaLemma 24.4.10987A41833787Closed positivity and geometric equations
propositionProposition 24.4.11E63ACDB15696Closed positivity and geometric equations
exampleExample 24.4.128D6FF393E593Closed positivity and geometric equations
propositionProposition 24.4.133D0A0BF3A375Closed positivity and geometric equations
problemProblem 24.4.14B341DA2B6EFEClosed positivity and geometric equations
subsection24.4.15 Null loci and the singularities of the Chern–Ricci flowF6F63D20E784Closed positivity and geometric equations
subsection24.4.16 Balanced and pluriclosed metricsB8BA04F9415AClosed positivity and geometric equations
lemmaLemma 24.4.172066C3EEC6F6Closed positivity and geometric equations
equationEquation 24.4.183EA53204CDA3Closed positivity and geometric equations
lemmaLemma 24.4.194865DD012C9CClosed positivity and geometric equations
theoremTheorem 24.4.20781DAEC1B4CEClosed positivity and geometric equations
equationEquation 24.4.21B755FCE3BC15Closed positivity and geometric equations
problemProblem 24.4.2258C2B4B7107EClosed positivity and geometric equations
subsection24.4.23 Pluriclosed metrics under deformation39D58BE832C0Closed positivity and geometric equations
lemmaLemma 24.4.2490818FEADA4EClosed positivity and geometric equations
equationEquation 24.4.25A0AA509F3862Closed positivity and geometric equations
equationEquation 24.4.26321E95CFB730Closed positivity and geometric equations
equationEquation 24.4.27D218CA2053F4Closed positivity and geometric equations
theoremTheorem 24.4.285C15639B8E53Closed positivity and geometric equations
subsection24.4.29 Morse inequalities and the choice of volumeB20929459DA6Closed positivity and geometric equations
equationEquation 24.4.30C29E513B11B7Closed positivity and geometric equations
problemProblem 24.4.31EF3C6D763307Closed positivity and geometric equations
subsection24.4.32 Oeljeklaus–Toma manifolds68A7CB3104FEClosed positivity and geometric equations
problemProblem 24.4.337E8975FF8CBBClosed positivity and geometric equations
problemProblem 24.4.34A65ADE22C9EDClosed positivity and geometric equations
problemProblem 24.6.1B93ADA1B6064Degeneration orders and finite-distance boundaries
conjectureConjecture 24.7.17C0A1CE5EC77Weak Gauduchon adjoints
subsection24.8.1 Weak convergence and singularity data163A8FDB4B95Topologies on singularity types
subsection24.8.2 Volume comparison and analytic approximationEB37AB0B2D63Topologies on singularity types
subsection24.8.3 Polar curvatureB245A62A943CTopologies on singularity types
problemProblem 24.8.435C633451730Topologies on singularity types
problemProblem 24.9.1D81AB6932636Trace realization and recovery
subsection24.9.2 Recovery of compatible divisor systems5A6D078648B3Trace realization and recovery
problemProblem 24.9.3E8E7FDD4A1BFTrace realization and recovery
problemProblem 24.10.1D2B2B955B9A6Sections, kernels and the dimension of growth
problemProblem 24.10.24813DF6B9029Sections, kernels and the dimension of growth
subsection24.10.3 Growing vanishing orders on curves5FCA86C5A588Sections, kernels and the dimension of growth
problemProblem 24.10.44C662988834FSections, kernels and the dimension of growth
problemProblem 24.11.1A63319B4B931Equations on singular spaces

25. Historical and bibliographical notes

section25.1.1 Measures and distributionsE6CD9442A846Historical and bibliographical notes
section25.1.2 Harmonic functions and boundary values16FFFBC63146Historical and bibliographical notes
section25.1.3 Subharmonic functions and polar sets7A117C94C896Historical and bibliographical notes
section25.1.4 Energy, capacity and exceptional setsF2A7FD4CCC54Historical and bibliographical notes
section25.1.5 Fine topology and balayage3BA364E0394CHistorical and bibliographical notes
section25.1.6 Fourier analysis and complex-line incidence342F8D0FF9AEHistorical and bibliographical notes
section25.1.7 Differential forms, currents and holomorphic divisorsCE9C93DDA94AHistorical and bibliographical notes
section25.1.8 Plurisubharmonic functions and invariant minimaE49C1A0D4682Historical and bibliographical notes
section25.1.9 Bounded products and nonlinear capacityE66D40EC5904Historical and bibliographical notes
section25.1.10 Pluripolar sets and the plurifine topology1AFE6A788B80Historical and bibliographical notes
section25.1.11 Weighted estimates and local extensionE796BF553819Historical and bibliographical notes
section25.1.12 Lelong numbers and multiplier ideals2BFBF0A17586Historical and bibliographical notes
section25.1.13 Analytic singularities and regularizationC1D97719C08EHistorical and bibliographical notes
section25.1.14 Global potentials and positivity lociF9B38BEB0BCFHistorical and bibliographical notes
section25.1.15 Non-pluripolar products0948B3754F28Historical and bibliographical notes
section25.1.16 Envelopes and model potentials3F57C7AFD15CHistorical and bibliographical notes
section25.1.17 Relative energy and integration by parts889877FC2044Historical and bibliographical notes
section25.1.18 Weak geodesics4C762779903DHistorical and bibliographical notes
section25.1.19 Extremal slopes and singularity cutoffs3EBE4FDCA90EHistorical and bibliographical notes
section25.1.20 Convex duality and real Monge–Ampère measures482A6169B3A6Historical and bibliographical notes
section25.1.21 Analytic spaces and modificationsF14EBE94C297Historical and bibliographical notes
section25.23.1 A brief historyC7AB6352651BComments
section25.23.2 Open problems8B52D455AC39Comments
conjectureConjecture 25.23.3F96A00F2C17AComments
problemProblem 25.23.46DBA7A1F707BComments
problemProblem 25.23.56BD8E77C1AAFComments
problemProblem 25.23.60C91C5C2794AComments
problemProblem 25.23.77FAD0FAF17B7Comments
problemProblem 25.23.8856DCCA91F34Comments
problemProblem 25.23.918561C2CB4F4Comments
subsection25.11.1 Singularity volumes and relative classesFFA645E73814Volumes under mass bounds
subsection25.11.2 Model topology and analytic approximation572621232871Volumes under mass bounds
subsection25.11.3 Curves and zero-volume limits9AB6A1306853Volumes under mass bounds
subsection25.12.1 Equations of prescribed ordinary type7C4DEB2F0F6BEquations with prescribed singularities
subsection25.12.2 General measures in semipositive backgrounds4D81B44E0A9BEquations with prescribed singularities
subsection25.12.3 Big backgrounds under mass boundsBCD8D94DC861Equations with prescribed singularities

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BibTeX citation

@misc{pome-entries,
  author = {{Pome contributors}},
  title = {All entries},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/entries/}
}

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