25.1 Historical and bibliographical notes`65E16F3944C7`

65E16F3944C7

These notes give the sources of the results in this book. For each chapter we indicate where the main results were first proved or where they can be found in the literature, compare our statements, proofs and normalizations with those of the sources, and point to alternative presentations. The sections follow the order of the book: each concerns one chapter, except that the notes on section 17.1 occupy three sections, and the last section concerns section 9.1. A few ideas recur throughout the book: a function is recovered from its distributional derivatives, exceptional sets are controlled by capacities, nonlinear measures are constructed by approximation with bounded potentials, and singular potentials are compared through their envelopes. These ideas have different origins, and their formulation has changed as the theory moved from domains in Euclidean space to compact complex manifolds.

When the book is compared with another source, the conventions of that source matter as much as its hypotheses. We use $\mathrm{dd}^{\mathrm{c}}=\mathrm i\partial \bar\partial /(2\pi )$; Lelong numbers are normalized so that $\log |z|^2$ has Lelong number $1$ at the origin, and multiplier ideals are defined with the weight $\mathrm e^{-\varphi }$. Several of the sources cited below use the operator $\mathrm{dd^ c}=\mathrm i\partial \bar\partial /\pi $, measure Lelong numbers against $\log |z|$ and use the weight $\mathrm e^{-2\varphi }$. A source with the last two conventions assigns to $\varphi =u/2$ the Lelong numbers and multiplier ideals that we assign to $u$. If the operator is multiplied by a constant $c>0$, then at fixed potentials a product of $p$ factors of the form $\mathrm{dd}^{\mathrm{c}}u$ is multiplied by $c^ p$. We record these conversions where they matter.

Theorem and page numbers refer to the cited version of a work. When this version is a preprint rather than the published article, whose numbering or scope may differ, we say so. Two existence theorems are used without proof, namely the resolution and principalization theorems (theorem 9.6.6, theorem 9.6.7); their source is given in subsection 25.1.21.

25.1.1 Measures and distributions`E6CD9442A846`

The representation of positive linear functionals by measures, theorem 2.3.3, is the Riesz representation theorem. For its formulation on locally compact, $\sigma $-compact spaces compare [Theorem 24, Tao09Radon]; on open subsets of Euclidean space our proof constructs the measure from a metric outer measure and identifies the functional by slicing functions into level sets. The regularity and density statements in proposition 2.2.2, proposition 2.3.5 correspond to [Theorem 21, Exercise 22 and Proposition 10, Tao09Radon]. For the measurability part of the outer-measure construction compare [Theorem 5, Tao10Products]. The assertions on product measures and the interchange of integrals in proposition 2.2.7 form the Tonelli–Fubini theorem for $\sigma $-finite measures; compare [Theorem 36 and its proof, Tao10Products], where the monotone class lemma (Lemma 35 there) is applied to the Boolean algebra of finite disjoint unions of rectangles. We instead apply Dynkin's $\pi $–$\lambda $ theorem, lemma 2.2.6, to the rectangles themselves.

For the local order bound, regularization by mollifiers and the fact that positive distributions are measures (definition 2.4.1, proposition 2.5.3, theorem 2.5.6) compare [Section 1, Exercises 4–5; Section 2, Definition 1, Exercise 15 and Lemma 2, Tao09Distributions]. That source writes the test function first and gives the space of distributions the conjugate complex structure. Our pairing $\langle u,\chi \rangle =\int u\chi \, \mathrm d x$ is complex linear in both variables, the convention used for currents later in the book; with this pairing the derivative of a distribution is defined with the same sign as in that source.

The selection theorem for sequences of measures, theorem 2.6.5, is proved by a diagonal argument over countably many test functions. The compactness theorem for nets, theorem 2.7.3, rests instead on Tychonoff's theorem for a product of intervals, which we prove with ultrafilters (lemma 2.7.2); this is where the axiom of choice enters. Subnets are Willard subnets throughout the book, and proposition 2.7.1 characterizes compactness in these terms. The Arzelà–Ascoli theorem, proposition 2.7.4, is proved in the usual way, by showing that an equicontinuous and pointwise bounded family is uniformly bounded and totally bounded for the uniform metric.

For the differentiation of integrals compare [Section 2, Theorems 32 and 36, Exercise 38 and Lemma 39, Tao10Differentiation]. In theorem 2.8.3 we use cubes instead of balls. The covering argument is the same: a finite disjoint subfamily of the cubes covers the original cubes after enlargement by the factor three, and the constant in the maximal inequality becomes $3^ d$. The Poincaré-type inequality lemma 2.8.4 follows by integrating the fundamental theorem of calculus along segments parallel to the coordinate axes.

For the Radon–Nikodym theorem compare [Corollary 1, following Theorem 2, Tao09RN]. The book uses only the case of a Borel measure that is dominated, at least locally, by a constant multiple of a Radon measure, lemma 2.8.1. We prove it by an $L^2$ argument on half-open dyadic cubes, so that no assumption on the measure of cube boundaries is needed. This lemma provides, for instance, the densities of the coefficient measures in lemma 7.14.3.

25.1.2 Harmonic functions and boundary values`16FFFBC63146`

Fundamental solutions, the regularity of distributional solutions of the Laplace equation, the Poisson formula and Harnack's inequality are treated in [Theorems 3.1.2–3.1.7, pp. 117–123, Hor07]. As there, the fundamental solution $E$ of proposition 3.1.13 satisfies $\Delta E=\delta _0$, whereas our Green functions are positive and satisfy $-\Delta G=\delta _ x$. Our proof of Weyl's lemma, theorem 3.1.11, uses radial averaging and convolution; Theorem 3.1.3 of that book contains in addition a statement on analytic continuation. The Poisson formula and Harnack's inequality (theorem 3.1.18, proposition 3.1.19) are stated with the explicit Poisson kernel and its two-sided bounds, and they include the case of dimension one.

Perron's method, barriers and the solution of the Dirichlet problem for continuous boundary data are developed in [Section 3.3, pp. 171–174, especially Theorem 3.3.3, Definition 3.3.4 and Theorem 3.3.5, Hor07]. In theorem 3.1.25, theorem 3.1.27 we work with continuous subsolutions. For the characterization of regular boundary points by barriers compare also [Theorem 3.4.17, pp. 219–220, Hor07]. Our local barrier criterion, proposition 3.1.30, uses a positive continuous supersolution near the given boundary point whose continuous extension to the nearby boundary vanishes only at that point; the theorem in Hörmander's book allows more general exceptional sets. The Green function of theorem 3.1.34 is constructed only on bounded regular domains; within this class the theorem also shows that Green functions increase with the domain and converge along increasing sequences of domains whose union is again bounded and regular.

25.1.3 Subharmonic functions and polar sets`7A117C94C896`

The characterizations of subharmonic functions by comparison with harmonic functions and by mean values, their local integrability, the Riesz decomposition and the distributional characterization are developed in [Theorem 3.2.3, Corollary 3.2.8 and Theorems 3.2.9–3.2.11, pp. 143–148, Hor07]; see theorem 3.4.5, proposition 3.4.8, theorem 3.4.14, theorem 3.4.12. Since a subharmonic function is the limit of its means at every point, the canonical representative of a distribution with positive Laplacian is determined by the distribution at every point, not only almost everywhere. Our Riesz measure is $\Delta u$, without a factor $1/(2\pi )$; thus a logarithmic pole in the plane has Riesz mass $2\pi $.

For the compactness theorem and Hartogs' lemma (theorem 3.4.19, theorem 3.4.20) compare [Theorems 3.2.12–3.2.13, pp. 149–151, Hor07]. Theorem 3.2.13 there also contains convergence statements in Sobolev spaces and against measures; the latter are discussed in the next section. Choquet's lemma and the regularized upper envelope (lemma 3.4.22, theorem 3.4.23) correspond to [Proposition 3.4.4, pp. 205–206, Hor07]; theorem 3.4.23 also asserts that the pointwise supremum of an arbitrary family is Lebesgue measurable. For the local definition of polar sets and the removable singularity theorem (definition 3.4.26, theorem 3.4.27) compare [Definition 3.4.1 and Theorem 3.4.3, pp. 203–205, Hor07]. Our proof works directly with the local subharmonic functions of the definition, and the connectedness of the complement of a closed polar set in a domain follows from the maximum principle.

25.1.4 Energy, capacity and exceptional sets`F2A7FD4CCC54`

The continuity principle and the restriction of a measure to compact sets on which its potential is continuous (theorem 4.1.10) follow the argument of [Theorems 3.4.6–3.4.7, pp. 207–208, Hor07]; the approximation lemma of Lusin type used there is lemma 4.1.9. The criterion for polarity of compact sets is [Theorem 3.4.5, pp. 206–208, Hor07]. Our relative capacity of a compact set is the total mass of the Riesz measure $\Delta u$ of its regularized relative extremal function $u$, and our Green functions are positive with $-\Delta G=\delta $; the numerical values of the capacity depend on these normalizations.

Strong subadditivity, outer regularity and capacitability are developed in [Section 3.4, pp. 209–216, especially Theorems 3.4.9 and 3.4.12 and Proposition 3.4.13, Hor07]. Choquet's capacitability theorem, theorem 4.1.23, which underlies the capacitability assertion of that section, is proved here by approximating continuous images of countable intersections of countable unions of compact sets from inside by compact sets. The convergence of integrals against measures with continuous potentials in lemma 4.1.25 is part of [Theorem 3.2.13, pp. 150–151, Hor07], and the fact that the exceptional sets of the limit theorems are polar, theorem 4.1.26, corresponds to [Theorem 3.4.14, pp. 216–217, Hor07]. This polarity strengthens the equality almost everywhere of an upper envelope and its regularization in theorem 3.4.23.

For the identity between Green energy and Dirichlet energy and for quasi-continuity compare [Propositions 3.4.19 and 3.4.21, pp. 221–223, Hor07]. In proposition 4.1.29 we work with the explicit, symmetric Green function of a ball. The positivity of the Newtonian energy and of the logarithmic energy (proposition 4.1.5) is proved by expressing the kernels as integrals of Gaussians. For the logarithmic kernel, positivity and the Cauchy–Schwarz inequality are proved, and used, only for signed measures of total mass zero. The equilibrium measure is obtained by minimizing the energy, and proposition 4.1.27 describes its potential (Frostman's theorem) with our normalization of the kernels.

The example [Example 6.1.3, XiaSampleBook] uses an atom-free measure on a polar Cantor set and refers to [p. 31, Car83] for the Cantor set. Such a measure $\mu $ is constructed explicitly in proposition 4.1.12: the Cantor set $K$ is built from a rapidly decreasing sequence of lengths, so that the subharmonic function $p(z)=\int _ K\log |z-a|^2\, \mathrm d\mu (a)$ equals $-\infty $ at every point of $K$. The remaining ingredients of that example, namely the potential on the projective line, its integrability and the computation of its mass in degree one, are proposition 13.4.10, corollary 6.9.3, example 13.4.11.

25.1.5 Fine topology and balayage`3BA364E0394C`

Fuglede attributes the classical fine topology to Cartan and the quasi-Lindelöf principle to Doob in [Introduction, p. 227, Fug71Fine]. His preliminaries [Section 1, pp. 228–230, Fug71Fine] are formulated in an axiomatic harmonic space satisfying the domination axiom and admitting a positive potential, and they assume the general duality theory of sweeping before fine connectedness is studied. In the Euclidean setting of this book the quasi-Lindelöf principle, theorem 4.2.22, follows directly from Choquet's lemma and the polarity of the exceptional set of an upper envelope. By example 4.2.8, the fine topology is not Lindelöf, so a countable subfamily can only be expected to cover the union of a family of finely open sets up to a polar set.

Reduction and balayage (theorem 4.2.14, theorem 4.2.16, theorem 4.2.18) are treated only on a ball, where the Green function is explicit and symmetric and Green energy is a Dirichlet integral by proposition 4.2.11. In this setting the swept measure is determined by an orthogonality relation for the energy, which also gives additivity, and the general duality of sweeping recalled by Fuglede is not required. Example 4.2.20 expresses the swept potential of a Dirac measure through harmonic measure, for a compact set whose complement in the ball is a regular domain. Proposition 4.2.25 is proved directly from the definitions; it shows that for $n\geq 2$ the classical fine topology of $\mathbb C^ n=\mathbb R^{2n}$ differs from the topology generated by the upper semicontinuous functions that are subharmonic or identically $-\infty $ on complex lines, which is the plurifine topology of section 5.47 (example 5.47.17).

25.1.6 Fourier analysis and complex-line incidence`342F8D0FF9AE`

For Fourier inversion on the Schwartz space and Plancherel's theorem (lemma 2.9.2, theorem 2.9.4) compare [Exercises 38–42, formula (11), Theorem 43 and Exercise 44, Tao09Fourier]. Our Fourier transform uses the character $\mathrm e^{-2\pi \mathrm i x\cdot \xi }$. Gaussian integration gives the exact constant in the Fourier expression of the Riesz energy, theorem 2.9.6; the kernel there is $|x-y|^{-s}$, without the normalization of a fundamental solution.

For the dyadic decomposition compare [Proposition 4.3 and proof, pp. 23–24, Tao06Fourier3]; for the weak type $(1,1)$ estimate and its consequences by interpolation and duality compare [Lemma 2.7, Corollaries 2.9–2.10 and proofs, pp. 7–9, Tao06Fourier4]. We treat only the Hessian multipliers $-\xi _ a\xi _ b/|\xi |^2$ (lemma 2.9.10), whose kernels are obtained by differentiating the heat kernel. Their weak type $(1,1)$ bound and their $L^ p$ bounds are lemma 2.9.11, theorem 2.9.12; the exponents $p>2$ are obtained by duality. The interpolation inequality lemma 2.9.15 and Morrey's inequality lemma 2.9.16 are used for metric Laplacians in section 16.10.

The star-shaped annular sets, the projections onto complex hyperplanes and the incidence argument follow [Lemmas 1.3–1.5, pp. 2–4, PSHD]; this manuscript carries no author's name and no publication data. The Borel selection required by the argument is lemma 2.9.21: the sets of high values on circles are closed and, by the submean inequality, have positive length, so a median with respect to arc length gives a Borel choice, and no general measurable selection theorem for compact-valued maps is needed. The unitarily invariant measures are constructed from the Gaussian measure on matrices, and volumes in projective space are controlled by an explicit estimate for projective balls instead of volume charts. In corollary 2.9.28 the function is assumed to be Borel; in the proof of theorem 5.8.4 this measurability is established before the corollary is applied.

25.1.7 Differential forms, currents and holomorphic divisors`CE9C93DDA94A`

For positive forms and currents, the characterization of weak positivity by restriction to complex subspaces and the fact that positive currents have order zero, compare [Chapter III, Section 1.A–B, pp. 129–134, especially (1.4), (1.6) and (1.14), DemBook]. In addition, proposition 5.3.2 shows that $\beta ^ p$ lies in the interior of the cone of strongly positive forms, which gives the trace bounds for positive currents; the proof averages over sign changes and permutations of the coordinates. Our currents are indexed by bidegree, whereas Demailly's subscripts there refer to bidimension. Pushforward and pullback of currents (proposition 5.6.1, proposition 5.6.2) correspond to [Chapter III, (1.17), p. 134, DemBook]; for the pushforward the map need only be proper on the support of the current.

The mixed Hodge–Riemann theorem of linear algebra, due to Timorin, is recorded in [Proposition 2.1, p. 4, DN2005mixedv2] (second arXiv version). Lemma 5.3.4 is its case of real $(1,1)$-forms, written with mixed discriminants of Hermitian matrices; we prove it by induction on principal minors and deformation to identity matrices, where the form is given by an explicit trace formula.

For local potentials of closed positive $(1,1)$-currents compare [Chapter III, Section 1.18 and (1.19), p. 135, DemBook]. Our proof of theorem 5.4.3 is different: we subtract the complex Hessian of the potential of the trace measure, and the remainder, which has trace zero and therefore harmonic coefficients, is $\mathrm{dd}^{\mathrm{c}}$-exact by the smooth local result lemma 5.4.2. In proposition 5.4.4 the composition with a holomorphic map is taken pointwise, for the canonical representative; it is proved by radial regularization.

Our proof of the Weierstrass preparation theorem, theorem 5.5.2, is the proof by contour integrals in [Chapter II, Section 2.A, pp. 79–80, Theorem (2.1), DemBook], where it is attributed to Siegel. The Poincaré–Lelong formula, theorem 5.5.6, is [Chapter III, (2.15), pp. 143–144, DemBook]. The expression $(\mathrm{i}/\pi )\partial \bar\partial \log |f|$ appearing there equals our $\mathrm{dd}^{\mathrm{c}}\log |f|^2$, while the operator $\mathrm{dd^ c}$ used subsequently in that book is twice ours. In our proof, slicing by coordinate lines and lemma 5.5.5 show that $\mathrm{dd}^{\mathrm{c}}\log |f|^2$ puts no mass on the singular locus of the divisor, so that it equals the current of integration over the regular part, weighted by multiplicities. Example 5.5.8 gives the corresponding identity in $L^1_{\mathrm{loc}}$ for quotients of holomorphic functions.

25.1.8 Plurisubharmonic functions and invariant minima`E49C1A0D4682`

The theorem that a function which is subharmonic or identically $-\infty $ on complex lines is automatically upper semicontinuous is proved in the unsigned manuscript [Theorem 1.1 and its proof, pp. 1–6, PSHD], and our proof of theorem 5.8.4 completes the argument given there. In that manuscript, Lemma 1.2, pp. 1–2, proves measurability by induction on the dimension, using means over bounded fibers; Proposition 2.1, pp. 4–5, identifies a finite function that is subharmonic on lines with the canonical representative; and the final proof, p. 6, uses truncation. The incidence results needed for this argument are proved in section 2.9, and proposition 5.4.4 describes the restrictions of the canonical representative to complex lines without using the theorem. We keep the usual definition of plurisubharmonic functions (definition 5.8.1), which requires upper semicontinuity; theorem 5.8.4, proved shortly afterwards, shows that this condition is redundant.

The distributional characterization, upper envelopes and compactness of plurisubharmonic functions (theorem 5.9.1, theorem 5.10.2, theorem 5.10.4) are deduced from the corresponding results for subharmonic functions in section 3.4, section 4.1. At this stage the set where an upper envelope differs from its regularization is only known to be real polar, that is, polar as a subset of $\mathbb R^{2n}$ in local charts; its pluripolarity is proved in section 5.47. The extension theorem (theorem 5.11.3) gives both local boundedness and extension across a relatively closed set that is locally $\mathcal H^{2n-3}$-null; to apply it to analytic sets of codimension at least two (corollary 9.3.18) one needs the local estimate for the Hausdorff measure of analytic sets in terms of their dimension.

For $L^ p$ integrability and convergence in $L^ p_{\mathrm{loc}}$ compare [Theorem 1.48, parts 1–2 and proof Steps 1–2, pp. 30–32, GZ17]. As there, we pass from discs to polydiscs (lemma 5.10.5, proposition 5.10.8, theorem 5.10.10), but we use the Riesz decomposition with a compactly supported measure instead of the Poisson–Jensen formula on the boundary. Since our Riesz measure is $\Delta v$, its logarithmic potential carries the factor $1/(2\pi )$. The bound (5.10.9) gives uniform estimates on smaller polydiscs, and convergence in $L^ p_{\mathrm{loc}}$ then follows from Hölder's inequality. The identity with the factor two in the truncation display of [p. 32, GZ17] is incorrect, as is also noted in an unofficial list of errata for that book.

The local exponential integrability estimates (lemma 5.10.11, proposition 5.10.12) are proved with the same compactly supported Riesz decomposition and Jensen's inequality (lemma 2.2.4), with an exponent that depends only on a local $L^1$ bound; a finite covering by charts then gives the uniform estimate for normalized quasi-plurisubharmonic functions on a compact manifold, corollary 10.3.3.

The partial Legendre transform is due to Kiselman [Kis78]. For his minimum principle and the regularization argument in its proof compare [Chapter I, Section 7.B, Theorem (7.5) and its proof, p. 57, DemBook]. In theorem 5.12.1 the existence of an invariant exhaustion is a hypothesis, and the smooth case is proved with a Schur complement. Such an exhaustion is constructed for product tubes over open convex sets in corollary 5.12.2; the case of general pseudoconvex tubes, corollary 6.3.6, uses in addition the plurisubharmonicity of $-\log $ of the boundary distance.

25.1.9 Bounded products and nonlinear capacity`E66D40EC5904`

The recursive definition of products of currents $\mathrm{dd}^{\mathrm{c}}u$ with $u$ locally bounded and plurisubharmonic, the Chern–Levine–Nirenberg estimates and continuity along decreasing sequences are developed in [Chapter III, Section 3, (3.1)–(3.11), pp. 144–150, DemBook]; see theorem 7.2.5, theorem 7.3.7. In our proof of decreasing continuity, the integration by parts formula lemma 7.3.3 for functions that coincide near the boundary of a ball is combined with an induction on the number of factors, which also gives the symmetry of the product at each stage. The weighted estimate theorem 7.4.1 concerns only the measures $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \beta ^{n-q}$; a version relative to a closed positive current, for one factor at a time, is lemma 13.4.1.

For continuity along increasing sequences, the behavior of products under maxima and the comparison principle (theorem 7.6.1, proposition 7.3.10, theorem 7.3.11) compare [Theorem 3.23, pp. 89–91, and Theorems 3.27 and 3.29, pp. 93–94, GZ17]. The domination principle corollary 7.3.12 is [Proposition 1.2.10, XiaSampleBook]; we prove it by a strictly plurisubharmonic quadratic perturbation and the comparison principle, applied on the given bounded open set itself.

The Monge–Ampère capacity, quasi-continuity and convergence in capacity (definition 7.4.2, theorem 7.5.2, theorem 7.5.8) appear in [Proposition 4.19, Theorem 4.20 and Lemma 4.21, pp. 110–112, and Theorem 4.26, pp. 114–115, GZ17]. Our capacity is defined with competitors taking values between zero and one, whereas the competitors in [Definition 4.16, p. 108, GZ17] take values between $-1$ and $0$; since adding a constant does not change Monge–Ampère measures, the two normalizations give the same capacity. Both Demailly's book and [GZ17] use the operator $\mathrm{dd^ c}=(\mathrm{i}/\pi )\partial \bar\partial $, twice ours, so their Monge–Ampère measures and capacities are $2^ n$ times ours.

The estimate of proposition 7.4.5 is attributed to Błocki in [Proposition 3.21, pp. 88–89, GZ17]. The power $M^{n+1}$ of the oscillation printed there should be $M^ n$: the proof uses $n$ integrations by parts, and the left-hand side is homogeneous of degree $n$ in $v$. We prove the estimate with $M^ n$, under the assumption that $u=h$ near the boundary of the ball, first for smooth functions and then, by regularization and corollary 7.3.8, for bounded ones. Corollary 7.6.2 shows that the set where an upper envelope differs from its regularization has capacity zero; this is used in section 5.47 to show that such sets are pluripolar.

The solution of the homogeneous Dirichlet problem on the ball, theorem 7.7.4, goes back to [BT76]. Averaging over automorphisms that move the center exactly to a prescribed point is carried out for polydiscs in [Section 2.3, Theorem 2.3.2 and Proposition 2.3.3, pp. 50–52, Blo02Pluripotential]; lemma 7.7.1 provides the corresponding automorphisms of the ball. Compare also [Theorems 5.13–5.14, pp. 143–149, GZ17]. In Step 3 of the proof of theorem 7.7.4, the determinant of the complex Hessian of the solution is shown to vanish almost everywhere by comparing the solution with its second-order Taylor polynomial at points where this expansion holds. For functions with bounded real Hessian, the expansion holds almost everywhere by proposition 2.8.5, which is proved with the differentiation of integrals and the Poincaré inequality on cubes. The homogeneous balayage of corollary 7.7.5, together with the comparison principle, shows that a locally bounded plurisubharmonic function is maximal if and only if its Monge–Ampère measure vanishes (theorem 7.7.7); in particular, maximality is a local property for locally bounded functions.

The characterization of the inequality $(\mathrm{dd}^{\mathrm{c}}u)^ n\geq f\eta ^ n$, where $\eta =\mathrm{dd}^{\mathrm{c}}|z|^2$, by upper test functions and the linearization of the determinant (proposition 7.8.3) are proved in [Proposition 1.5 and Lemma 1.7, pp. 5–7, EGZ10ViscosityV1], the first arXiv version of the later article by Eyssidieux, Guedj and Zeriahi. The passage from upper test functions to distributional inequalities, which the proof in that preprint quotes from Hörmander, is lemma 7.8.2. The trace matrices used there have determinant $n^{-n}$; multiplying them by $n$ gives our matrices of determinant one.

The inequality for mixed products of smooth functions and its extension to bounded functions whose Monge–Ampère measures have densities with respect to Lebesgue measure are stated in [Theorems 1.1–1.2, pp. 1–2, Din07MixedV1], the first arXiv version of Dinew's article; the extension is attributed there to Kołodziej and Błocki. We prove the case of continuous densities, corollary 7.8.4, by linearization and decreasing convolution, and the case of a common measure in corollary 7.12.6. At fixed potentials, the smooth measure displayed by Dinew is $(4\pi )^ n$ times ours.

The Dirichlet problem on a ball with continuous density, theorem 7.9.2, follows the argument of [Theorems 5.13–5.14 and Corollary 5.15, pp. 143–149, GZ17]. Since the automorphisms of lemma 7.7.1 move the center exactly to a prescribed point, the preliminary Lipschitz comparison with translations used there is not needed. Under such an automorphism $S$ the $n$-th root of the density transforms into $(f^{1/n}\circ S)|\det _{\mathbb C}DS|^{2/n}$; on p. 145 this factor is written first without the absolute value and later without the derivative. The symmetric error is quadratic, and the half-sum is taken only once; this corrects the formulas printed on pp. 145–146. The stability estimate (7.9.3) contains a term for the boundary data, which is omitted on p. 149, and its proof does not require monotone approximation of the boundary data and of the density. Densities are taken with respect to $\eta ^ n=(\mathrm{dd}^{\mathrm{c}}|z|^2)^ n$; at fixed potentials, the operator of [GZ17] gives $2^ n$ times our measures.

The extension to bounded densities (lemma 7.10.1, theorem 7.10.2) corresponds to [Theorem 5.17 and Lemma 5.18, pp. 150–151, GZ17]. In lemma 7.10.1 the comparison is made on strict sublevel sets, and only the shifted sublevel set is required to be relatively compact. In the proof of theorem 7.10.2 the approximating densities have an explicit common bound. Convergence in capacity of the approximating solutions is proved without assuming that the solution for a bounded measurable density is continuous inside the ball, by applying Hartogs' lemma on compact sets outside the exceptional sets of quasi-continuity. Approximating the $n$-th roots of the densities by convolution extends the stability estimate (7.9.3) to bounded densities.

The class $\mathcal E_0(B)$ of definition 7.11.1 is due to Cegrell: its definition on hyperconvex domains is recalled in [Section 3, p. 161, Ceg04], where the class is said to have been introduced in [Ceg98], and it is also recalled in [Section 2, p. 3, Din07MixedV1]. The moment condition used in Dinew's article to approximate a measure by its averages over cubes is stated in Theorem 2.2, pp. 4–5, whose conclusions on approximation and stability are quoted from other sources. Theorem 7.11.2 establishes the moment condition for measures dominated by the Monge–Ampère measure of a bounded plurisubharmonic function; it follows from Błocki's estimate proposition 7.4.5 by integration over level sets, after truncating the function with zero boundary values and passing to a smaller ball. The constants depend explicitly on the mass of the measure and on the oscillation of the subsolution, and the estimate also holds in dimension one.

By proposition 7.11.5, which combines this estimate with the local $L^ p$ bound proposition 5.10.8, the cube averages satisfy the moment condition uniformly. Applied to the solutions with zero boundary values, this gives uniform bounds for their local energy moments, and applied to the measure $\eta ^ n$, a constant multiple of Lebesgue measure, it gives compactness in $L^1$ along subsequences. At fixed potentials, Dinew's operator multiplies measures and local energy moments by $(4\pi )^ n$; because of the power of the mass on its right-hand side, both sides of (7.11.4) are multiplied by the same factor.

The convergence of the energy integrals of the approximating solutions is asserted in [Theorem 2.2(3), p. 5, Din07MixedV1], where it is quoted from Cegrell. Theorem 7.11.11 proves it for compactly supported measures dominated by the Monge–Ampère measure of a bounded plurisubharmonic function, along every subsequence of solutions that converges in $L^1$. One ingredient is the continuity of integrals against a fixed measure that does not charge sets of capacity zero, lemma 7.11.9.

Convergence in capacity, as a consequence of the convergence of the energy integrals, is quoted in [Theorem 2.3, p. 5, Din07MixedV1]. In theorem 7.11.24 we derive it from the comparison of capacities of strict sublevel sets (lemma 7.10.1) and the bound on second moments. By lemma 7.11.19, truncation preserves the total Monge–Ampère mass of a function in $\mathcal E_0(B)$; the lemma also bounds the change of the measure caused by truncation in terms of the energy, and shows that Monge–Ampère measures with bounded energy are uniformly absolutely continuous with respect to capacity.

The definition of the solution as an envelope of subsolutions is recalled in [Definition 2.4 and the following paragraph, p. 5, Din07MixedV1]. We solve the Dirichlet problem with zero boundary values, and identify the solution with this envelope, for compactly supported measures on a ball that are dominated by the Monge–Ampère measure of a bounded plurisubharmonic function (theorem 7.11.30).

The mixed energy inequalities used in the approximation argument are recorded in [Section 2, p. 4, Din07MixedV1] and attributed there to Cegrell. For their proof by integration by parts and the Cauchy–Schwarz inequality compare [Lemma 5.4 and Theorem 5.5, pp. 168–170, Ceg04]; Theorem 5.5 there has a fixed weight and exponents $1/n$. Theorem 7.11.33 is the version with $n+1$ functions, with an explicit but weaker constant. If $\mathrm{dd}^{\mathrm{c}}$ is multiplied by a constant $c>0$, both sides of (7.11.34) are multiplied by $c^ n$. Corollary 7.11.38 uses this bound to discard the sets where the functions are very negative before continuity in capacity is applied; the limit functions there are locally bounded.

Dinew's inequality for bounded functions and a common measure is [Theorem 3.3, p. 9, Din07MixedV1]. Its proof by approximation (Proposition 3.1 and Lemma 3.2 there, pp. 7–9) relies on results on the local Dirichlet problem, energies and continuity in capacity that are quoted from other sources. Theorem 7.12.1 allows $n$ different functions; its proof uses the same approximation by cube averages, with these inputs proved in the preceding sections. The inequality for mixed discriminants, lemma 7.8.6, follows from the reversed Cauchy–Schwarz inequality given by lemma 5.3.4, and linearization of the determinant then gives the case of continuous densities, corollary 7.8.8. In this bounded setting the common measure may charge pluripolar sets. The extension to unbounded functions in Cegrell's classes, [Theorem 3.4, Din07MixedV1], is not treated here.

The computation on contact sets in [Lemma 2.4.7, Step 1, XiaSampleBook] uses Stampacchia's theorem that weak derivatives vanish almost everywhere on level sets, but it also asserts that a function in $W^{2,1}$ has a classical real Hessian. A complete argument is given in [Lemma 3.1 and Theorem 3.2, pp. 4–8, DNT2019v1] (first arXiv version). There $C^{1,\overline1}$ means that the Laplacian with respect to a metric is bounded, not that the full real Hessian is bounded, and elliptic $W^{2,p}$ regularity is used.

In this book the vanishing of weak derivatives on level sets is lemma 7.13.1, proved with a smooth truncated chain rule, and functions with bounded Euclidean Laplacian have real $W^{2,2}$ derivatives by lemma 7.13.2. With these lemmas we prove proposition 7.13.3, proposition 7.13.4 for regular potentials and obstacles in $W^{2,1}_{\mathrm{loc}}$; the density on the contact set does not require the determinant of the obstacle to be integrable on the whole domain. The operator used in [DNT2019v1] is $\mathrm{dd^ c}=\mathrm i\partial \bar\partial $, which is $2\pi $ times ours.

25.1.10 Pluripolar sets and the plurifine topology`1AFE6A788B80`

Josefson proved that locally pluripolar subsets of $\mathbb C^ n$ are globally pluripolar [Jos78]; see theorem 5.47.10. The approach through capacity is due to Bedford and Taylor [BT82]; for relative extremal functions compare [Theorem 4.32, pp. 118–119, Theorem 4.34, pp. 119–121, and Theorem 4.40 and Corollary 4.41, pp. 124–125, GZ17]. The identities for the masses of relative extremal functions of compact and of open sets (proposition 5.47.6, proposition 5.47.7) are proved with the homogeneous balayage of corollary 7.7.5; together with approximation by decreasing open sets they give the characterization of pluripolar sets by capacity, theorem 5.47.8.

For the local base of plurifine neighborhoods in proposition 5.47.15 compare [arXiv version, Lemma 3.1, pp. 3–4, EMW06]. The main theorem of that paper, the local connectedness of the plurifine topology, is not treated here; its proof also uses finely subharmonic functions. The plurifine quasi-Lindelöf principle, theorem 5.47.20, follows from Choquet's lemma and the pluripolarity of the exceptional sets of upper envelopes (corollary 5.47.11). The plurifine locality of Bedford–Taylor products, theorem 5.47.22, is deduced from their behavior under maxima (proposition 7.3.10), up to exceptional sets that products of bounded functions do not charge. In subsection 5.47.24 the difference of two plurisubharmonic functions, and its quasi-extrema, are considered only outside their common pole set, where the difference is defined.

25.1.11 Weighted estimates and local extension`E796BF553819`

For unbounded operators between Hilbert spaces, their adjoints and complexes of such operators compare [Chapter VIII, Section 1, pp. 363–365, DemBook]; the facts used here are proved in lemma 6.2.3, theorem 6.2.4. Hörmander's estimate on pseudoconvex domains, with the method of three weights, is presented in [Lemma 7.2, Proposition 7.3 and Theorem 7.1, pp. 35–39, Blo14SCV], and our proof of theorem 6.4.7 follows this presentation; in the density lemma (lemma 6.4.2) we verify that multiplication by cutoff functions preserves the domain of the adjoint before smoothing. Theorem 6.4.9 allows a plurisubharmonic weight together with a smooth weight whose complex Hessian is positive semidefinite; it is proved by approximation with weights for which the right-hand sides are bounded pointwise.

For Richberg's regularization and the boundary distance of pseudoconvex domains (lemma 6.3.3, theorem 6.3.4) compare [Chapter I, (5.18)–(5.21), pp. 43–44, and (7.1)–(7.2), pp. 53–55, DemBook]. These results give the invariant exhaustion needed in corollary 6.3.6.

Our proof of the Ohsawa–Takegoshi extension theorem, theorem 6.5.4, follows Chen's method with a bounded twisting function, as presented in [Theorems 9.1–9.3 and Lemma 9.4, pp. 46–48, Blo14SCV]. The auxiliary weight of lemma 6.5.3 is $-\log (s+\log s)$, where the function $s=-\log (|\zeta |^2+\varepsilon ^2)$ is positive. The constant in theorem 6.5.4 depends only on the radius in the normal direction, and iteration of the theorem gives the estimate with prescribed point value, corollary 6.5.8.

25.1.12 Lelong numbers and multiplier ideals`2BFBF0A17586`

For the Jensen–Lelong formula compare [Chapter III, (6.1)–(6.11), pp. 162–166, DemBook]. The book uses only its form on Euclidean balls, theorem 6.7.6. For smooth functions the formula follows from the derivative of spherical means, proposition 3.1.5, and the monotonicity of the density from the convexity of circle means on complex lines through the center; regularization, with lower bounds for spherical means given by the Poisson kernel, gives the general case. Demailly's operator there is twice ours, and he measures growth against $\log |z|$; the density in (6.7.5) is written with our $\mathrm{dd}^{\mathrm{c}}$ and $\log |z|^2$, so that $\nu (\log |z|^2,0)=1$.

The coherence of multiplier ideals, theorem 6.8.9, and its proof by generation from a weighted Hilbert space of holomorphic functions are presented in [Proposition 5.7, pp. 36–37, DemAnalyticOnline]; page numbers refer to the manuscript on Demailly's web page. The weight there is $\mathrm e^{-2\varphi }$, which becomes our $\mathrm e^{-\varphi }$ after doubling the potential. For the strong Noetherian property and the Artin–Rees lemma used in the proof compare [Chapter II, (3.22), p. 90, and Chapter IX, (1.1)–(1.3), pp. 403–404, DemBook]. Hilbert's basis theorem, the Artin–Rees lemma and Krull's intersection theorem are proved in lemma 6.8.1. We prove the strong Noetherian property in lemma 6.8.4, and the uniform generation by elements of the Hilbert space, with bounded coefficients, in lemma 6.8.10, using the completeness of the Hilbert space.

Siu's theorem on the analyticity of the upper level sets of Lelong numbers, theorem 6.9.5, goes back to [Siu74]. The proof through the orders of vanishing of multiplier ideals is Demailly's method; compare [Corollary 14.3, p. 138, DemAnalyticOnline] and [Theorem 10.2 and Lemma 10.3, p. 42, Bou23Notes], the version of Boucksom's notes dated 11 September 2023. The proofs of the restriction formula and of subadditivity by products and the diagonal (theorem 6.10.1), and the proof of the pullback formula by the graph (corollary 6.10.5), are taken from [(15.1)–(15.3), pp. 158–159, DemAnalyticOnline]. In (6.10.2), the restriction of $\mathcal I(\varphi )$ to $Z$ means its image ideal in $\mathcal O_ Z$.

Strong openness, corollary 6.11.6, is due to Guan and Zhou [GZ15Strong]. Our proof of theorem 6.11.4, corollary 6.11.6 follows their argument by analytic curves and extension in Sections 3.1–3.2, pp. 609–611, of the published article, and its monotone version in Section 3.3, pp. 611–612. The curve is constructed in lemma 6.11.1 by blowing up an ideal, and the lower bound for the norms of extensions is lemma 6.11.2. In the induction the slices are taken in a fixed polydisc, so that the constant of the extension theorem does not depend on the slice or on the weight.

That a weight with zero Lelong number at a point has trivial multiplier ideals there, corollary 6.9.3, is a special case of Skoda's integrability theorem; compare [Lemma 5.6(a), pp. 35–36, DemAnalyticOnline]. Here it is deduced from the bounds for the orders of multiplier ideals in lemma 6.9.1, which rely on the extension with prescribed point value of corollary 6.5.8. In that source the potential $\varphi =u/2$ corresponds to our $u$, by the conversion described at the beginning of these notes.

For discrepancies, the change of variables formula and the valuative criterion for multiplier ideals compare [Proposition 10.12, Theorem 10.13 and Corollary 10.18, pp. 44–46, Bou23Notes]; Boucksom's conventions $\mathrm e^{-2\varphi }$ and $\log |z|$ are converted as above. We state the criterion, theorem 6.12.16, for a holomorphic germ, with one positive parameter $\varepsilon $ that works for all divisors over a neighborhood of the point. For weights of the form $c\log \sum _ j|g_ j|^2+O(1)$, the strict threshold on a log resolution in theorem 6.12.8 involves every component of the resolved divisor that meets the fiber over the point, including the strict transforms of hypersurfaces in $X$, not only the exceptional ones.

25.1.13 Analytic singularities and regularization`C1D97719C08E`

Approximation by Bergman kernels is Demailly's construction; compare [Proposition 3.1 and its proof, pp. 13–14, DemRegOnline] and [Theorem 14.2, pp. 136–138, DemAnalyticOnline]. Page numbers of the former refer to the manuscript on Demailly's web page. These sources use the weight $\mathrm e^{-2m\varphi }$ and the factor $(2m)^{-1}$, whereas theorem 6.15.4 uses $\mathrm e^{-m\varphi }$ and $m^{-1}$, in accordance with our normalization of Lelong numbers. The comparison of the Bergman kernel with finitely many generators and its form on a log resolution (lemma 6.15.3) are derived from the bounded generation of lemma 6.15.1(3), a consequence of lemma 6.8.10.

The computation for the logarithm of a sum of local Bergman kernels weighted by cutoff functions (lemma 6.17.1) is [Lemma 3.5, p. 15, DemRegOnline], and the gluing over overlapping balls in Lemma 3.6, pp. 16–17, of that paper is explained by the more general Lemma 4.6 and its proof, pp. 19–22. The comparison of the local kernels on overlaps and the gluing by maxima (lemma 6.16.2, lemma 6.16.1) follow the argument of Vu and Xia in the unpublished manuscript [Lemmas patching and regularization, VuXiaNakamaye]. Theorem 6.16.3 has exactly the two conclusions of the regularization lemma there, and its curvature constant depends only on the fixed geometry.

Our definition 6.14.12 asks for the local form with smooth remainder on every log resolution of the ideal, that is, on every composite of blowups along closed submanifolds that transforms it into the ideal of a divisor with simple normal crossings support (section 6.14). The definition in [Chapter on plurisubharmonic functions, definition of gentle analytic singularities, XiaSampleBook] asks for such a form on one Kähler modification. Since log resolutions exist and are Kähler (proposition 6.14.11), our requirement is the stronger one, and the approximations of theorem 6.17.6 satisfy it.

For the comparison by extension from the diagonal, the dyadic correction and the passage from the integrability of a gap to inclusions of ideals compare [arXiv version, Section 2.2, Steps 2–3, pp. 12–14, DPS01]. Theorem 2.2.1 there, proved on pp. 9–14 of the same version, concerns equisingular approximation, and Remark 2.2.2 distinguishes the case of logarithmic poles. Theorem 6.17.6 is a different statement: the approximations have the prescribed gentle analytic types $(2^{-j},\mathcal I(2^ j\varphi ))$, and their monotonicity and the inclusions of ideals between consecutive dyadic scales are derived from the kernels with cutoffs, the comparison on the diagonal and lemma 6.17.4. To compare the approximations, the growth estimate lemma 6.18.1, proved on finitely many local log resolutions, suffices, so that no division theorem is needed.

The ideal $\mathcal I_\infty $ of germs of bounded growth (definition 6.19.1) is used in [Chapter on plurisubharmonic functions, the lemma following the definition of $\mathcal I_\infty $ and Lemma 1.6.3, XiaSampleBook]. Its coherence for potentials with analytic singularities, theorem 6.19.2, is proved with a cone of fractional powers, finite normalization and finitely many homogeneous generators; the direct image under the finite map is handled by division by monic polynomials, so the general theorem on direct images under proper maps is not needed. Corollary 6.19.3 derives from the discrepancies on log resolutions the inclusion for large multiples that is needed to compare approximations; the sharper estimate in the source, with a loss depending on the dimension, is not proved here.

25.1.14 Global potentials and positivity loci`F9B38BEB0BCF`

For elliptic operators on compact manifolds and the Laplacian on forms compare [Chapter VI, Section 2, pp. 289–290, and Sections 3.1–3.2, pp. 291–294, DemBook]. The results on Sobolev spaces, compactness and regularity used here are proved directly in lemma 10.2.1, lemma 10.2.5, by Fourier truncation, difference quotients and duality. The proof of the $\mathrm{dd}^{\mathrm{c}}$-lemma, theorem 10.2.15, uses the scalar trace equation and the identity for the Hodge star of primitive $(1,1)$-forms; the factor $-1/(4\pi )$ in the trace identity (10.2.13) comes from our normalization of $\mathrm{dd}^{\mathrm{c}}$ and from the sign convention that makes the Riemannian Laplacian nonnegative.

The method of primitive representatives for the mixed Hodge–Riemann relations is developed in [Propositions 2.3–2.4 and proof of Theorem A, pp. 5–8, DN2005mixedv2]. For real $(1,1)$-classes, lemma 10.2.9, theorem 10.4.3 make a representative primitive at every point by solving a scalar equation in divergence form. The operator $\mathrm{dd^ c}=\mathrm{i}\partial \bar\partial /\pi $ of that source is twice ours; doubling its correcting function gives the same representative. Corollary 10.4.5 derives the lower bound of mixed intersection numbers of nef classes by the geometric mean of their volumes from the concavity of the logarithms of mixed intersection numbers, first for Kähler classes and then, by perturbation, for nef classes.

For positive cones, potentials with minimal singularities and the divisorial Zariski decomposition compare [Sections 2.1–2.3, pp. 47–49, Sections 2.7–2.8, pp. 51–53, and Section 3.1, pp. 54–55, Bou04]. These locators refer to the published article, whose title and numbering differ from those of the arXiv version of 2002. Siu's decomposition, theorem 10.4.8, including the uniqueness of the decomposition, is proved by subtracting divisors with the help of the uniform logarithmic growth of local potentials along a divisor (proposition 6.9.8). The proof of proposition 10.5.2 is the argument of Proposition 3.6 of the published article, by convex combination with a Kähler current. As in that article, minimal multiplicities are defined by means of currents $T\geq -\varepsilon \omega $; for big classes they coincide with the Lelong numbers of a potential with minimal singularities, and in general they are bounded above by these Lelong numbers.

That the non-Kähler locus of a big class is the pole set of a single Kähler current with analytic singularities is [Definition 3.16 and Theorem 3.17, pp. 59–60, Bou04]. Our proof of theorem 10.5.4 follows Boucksom's argument, based on the stationarity of increasing sequences of coherent ideals on a compact manifold and on taking maxima, with the approximation results of section 6.13. The theorem also contains the identity (10.5.5), with conventions that include classes that are not big or not pseudo-effective. The last part of Boucksom's Theorem 3.17, a criterion in terms of restrictions to arbitrary analytic subsets, is not treated here.

The conventions for psh metrics, written as quadratic forms, and for the Fubini–Study metrics defined by sections are those of [Chapter on plurisubharmonic functions, Definitions 1.8.3, 1.8.6 and 1.8.7, XiaSampleBook]. In proposition 10.6.9 the convex combinations have rational coefficients, and closure under these operations is proved by clearing denominators of the degrees and multiplying sections. The local weight of a metric in [Section 2.4, pp. 49–50, Bou04] is $\mathrm e^{-2\varphi }$; doubling the potential gives our weight $\mathrm e^{-\varphi }$ for quadratic metrics, and the curvature current is the same.

For the $L^2$ estimate for line bundles on Kähler manifolds, lemma 10.7.6, compare [Chapter VI, Section 6.1, pp. 304–306, and Chapter VII, Theorem 1.1, Theorem 1.2 and Corollary 1.3, pp. 329–330, DemBook]. We prove the commutator identity, the density of smooth forms for the graph norm and the solvability in Hilbert space, and we show that for forms of top degree the curvature term in the estimate does not depend on the auxiliary complete metric.

Complete metrics and weights with logarithmic poles are also used in [Theorem 5.1 and its proof, pp. 18–22, Hisamoto2012v2] (second arXiv version), where Theorem 5.1 assumes a smooth metric and Theorem 1.4 assumes projectivity. Theorem 10.7.9 concerns extension from a point of a compact Kähler manifold. It is proved by applying the gentle approximations of section 6.13 to the integrable error introduced by a fixed cutoff function. The weighted spaces of sections in lemma 10.7.11 provide the finite-dimensional Hilbert spaces and the curvature of the background used in [Proposition 1.8.2, XiaSampleBook].

For the twisted $L^2$ estimate and the treatment of the error from the initial gluing compare [Section 2, pp. 5–9, and Appendix, pp. 14–15, Cao2017v2]; page numbers refer to the second arXiv version, posted on 3 October 2017 and dated 4 October in the manuscript. Theorem 1.3 there treats a submanifold defined by a global section of a vector bundle, under assumptions on curvature and approximation. Theorem 10.8.18 applies to an arbitrary closed connected submanifold $Y$. The pole along $Y$ is given by the function $R_ Y$ of lemma 10.8.8, a sum of local squared normal distances weighted by squares of cutoff functions.

The estimate (10.8.19) is first proved for weights with gentle analytic singularities. The local gain from the logarithmic factor is lemma 10.8.6, the error of the initial gluing is controlled by strong openness and Hölder's inequality in lemma 10.8.11, and lemma 10.8.15 computes the limit over thin annular neighborhoods of $Y$ for gentle weights. Approximation by decreasing gentle weights then gives the estimate for the original metric, with the same constant. Corollary 10.8.22 recovers [Theorem 1.8.1, XiaSampleBook]; the threshold for the tensor power and the conversion between the volume forms, which involves factorials, are stated explicitly.

The extension of potentials from a submanifold, theorem 10.8.26, follows [arXiv version, Theorem 6.1 and its proof, Lemma 6.2, pp. 19–22, DRWNXZ]. That source uses the same operator $\mathrm{dd}^{\mathrm{c}}=\mathrm i\partial \bar\partial /(2\pi )$ (p. 5) and the same notion of analytic singularities, with rational coefficients and bounded remainder (Definition 2.2, p. 6). The lemma on poles along the submanifold quoted there is replaced by lemma 10.8.8. Strict positivity near the submanifold follows from an estimate for block matrices, and the local extensions are formed as logarithms of sums of exponentials (lemma 10.8.25); the exponent of the normal pole is taken small enough to absorb the error coming from the Hölder continuity of the projection. Since the comparison of the local extensions on overlaps holds also at common poles, the glued maximum involves a fixed set of indices near each point of the submanifold.

Josefson's theorem on compact complex manifolds, theorem 10.9.19, is due to Vu [arXiv version, Theorem 1.1 and its proof, pp. 8–9, Vu19]. The operator $\mathrm i\partial \bar\partial /\pi $ used there is twice ours. We follow Vu's argument, with a weighted sum of global poles and two extremal limits, but replace his comparison of capacities in exponential form by the estimate (10.9.17). Positivity of the mass is needed only for the one bounded limit function, not as a uniform lower bound over all bounded potentials.

For products with a background form that is not closed and the modified comparison principle compare [Section 1, pp. 3–4, and Theorem 2.3 and its proof, pp. 5–6, KN2015v2], the second arXiv version of the article by Kołodziej and Nguyen, whose operator is again twice ours. The one-factor inequality on which the modified comparison principle rests is lemma 10.9.4; we prove it for smooth approximations with a convex cutoff and a partition of unity, and pass to the limit by the continuity of Bedford–Taylor products with polynomial weights (corollary 7.3.8) and uniform approximation of the weights by polynomials. The modified comparison principle is proved here only with the fixed margin $\omega _ v\geq \frac12\omega $, proposition 10.9.10.

In proposition 10.9.16 the background form is frozen on small balls, so that the balayage of ordinary plurisubharmonic functions applies. In this way, the proof that the extremal function puts no Monge–Ampère mass outside the compact set avoids solving the homogeneous Dirichlet problem or smooth Monge–Ampère equations on Hermitian manifolds.

25.1.15 Non-pluripolar products`0948B3754F28`

The definition of non-pluripolar products by truncation at finite levels, locally and on compact Kähler manifolds, is developed in [Section 1.2, Proposition 1.4, Lemma 1.5 and Proposition 1.6, pp. 203–207, BEGZ10]. Multilinearity is proved in lemma 13.2.2, proposition 13.2.6 by truncating all factors at a common level. In lemma 13.2.2 the convergence of the restricted products is stated for Borel sets $E_ k$ such that $\mathbf1_{E_ k}\to 1$ almost everywhere for the trace measure of the product; this holds if the $E_ k$ increase and their union has pluripolar complement, whereas the condition on the union alone does not suffice. In example 13.2.9 the computation of the Hessian shows that the function is plurisubharmonic with the sign opposite to the one displayed in [Example 1.3, pp. 204–205, BEGZ10].

That non-pluripolar products are closed, theorem 13.3.1, is [Theorem 1.8 and Lemma 1.9, pp. 207–210, BEGZ10]. The gradient $\mathrm dv\wedge \Theta $ of the product of a bounded plurisubharmonic function $v$ and a closed positive current $\Theta $, and the corresponding chain rule, are established in lemma 7.14.3 by decreasing approximation; the proof of theorem 13.3.1 then estimates the error made by the cutoff functions. The gradient energy $\mathrm dv\wedge \mathrm d^{\mathrm c}v\wedge \Theta $, with $\mathrm d^{\mathrm c}$ as in (5.2.10), is used only as a positive measure, and no closedness property of it is needed. On a compact Kähler manifold, Stokes' formula for bounded potentials (proposition 13.5.1) and the mass bound with shifted backgrounds (theorem 13.5.4) are proved before global truncations are used.

The lower bound for mixed products in terms of a common measure is [Proposition 1.11 and its proof, p. 211, BEGZ10]; it applies Dinew's inequality for bounded functions before passing to the restrictions at finite levels. Proposition 13.5.8 is the statement for $n$ currents, deduced from theorem 7.12.1; the case of two currents, corollary 13.5.10, follows by repeating factors. Since non-pluripolar products put no mass on pluripolar sets, the domination hypotheses force the densities to vanish almost everywhere for the common measure on the union of the local pole sets; this allows us to remove the restrictions to finite levels for an arbitrary measure.

The compatibility of non-pluripolar products with Bedford–Taylor products of bounded factors, proposition 13.4.3, is [Proposition 2.2.1(iii), XiaSampleBook]. Our proof uses the fact that Bedford–Taylor products with a current that does not charge pluripolar sets again do not charge them, lemma 13.4.2, which follows from the Chern–Levine–Nirenberg estimate relative to a current, lemma 13.4.1; the products are then identified with the help of bounded plurisubharmonic cutoff functions.

For the identity in degree one, proposition 13.4.7, compare [Proposition 3.3, pp. 3–4, ElKadiri2023v3] (third arXiv version). That proposition concerns the domain of definition of the Monge–Ampère operator and refers to an earlier theorem for the equality. We prove the identity first in one variable, by approximating the Riesz measure from below by measures with continuous potentials, and then in several variables by integration over the fibers of coordinate projections. Strong positivity (proposition 13.2.5) passes from products of smooth forms (proposition 5.3.2) to Bedford–Taylor products and then, by lemma 13.2.1, to their restrictions to Borel sets.

The fiber integration formula and its application to mixed products on compact manifolds follow [Chapter on non-pluripolar products, Lemma 2.1.1 and Corollary 2.2.1, XiaSampleBook]. The measurability of the fiber measures that is needed to integrate indicator functions of level sets is proved in lemma 13.4.4, the formula for Bedford–Taylor products of bounded functions is lemma 13.4.5, and corollary 13.6.1 passes to the limit along the increasing restricted products.

The comparison of masses in lemma 13.8.1, for potentials that are locally bounded outside the closed pole set of a quasi-plurisubharmonic function, follows the argument with compact supports in [Theorem 1.16 and its proof, pp. 217–218, BEGZ10]. The auxiliary projective construction is due to Witt Nyström [arXiv version, Sections 3–4, pp. 4–9, WN19]. The projection onto the simplex and the integration over angles are carried out in lemma 13.8.2. With the exterior powers used in this book, which are not divided by factorials, the formula of that lemma contains the factor $\binom {n+N}{n}$, and the Hessian contains $2\sum _ i\mathrm dx_ i\wedge \mathrm d^{\mathrm c}x_ i$. Both factors follow by direct differentiation and expansion, and they are missing from the calculation displayed in the preprint; the common positive binomial factor cancels in the proof of monotonicity. Theorem 13.8.7 handles classes that need not be big by adding $\varepsilon \omega $, so it does not depend on the criterion for bigness by positive mass, theorem 17.5.37.

The invariance of mixed masses under bounded perturbations, the lower bound for limits in capacity and the monotonicity of mixed masses are proved in [Proposition 2.1, Theorem 2.3 and Theorem 2.4, pp. 2054–2057, DDNL18mono]. Our theorem 13.7.5 uses the local continuity of weighted products under convergence in capacity and truncations at fixed levels, and corollary 13.8.8 obtains invariance from theorem 13.8.7 by polarization and then monotonicity by decreasing maxima. Positive mixed intersection numbers and full mass are introduced in [Definition 1.17, Proposition 1.20 and Definition 1.21, pp. 218–222, BEGZ10]; proposition 13.9.2 gives the bound for these masses that is used later. The comparison and domination principles, proposition 13.9.5, corollary 13.9.6, follow [arXiv version, Corollaries 4.1–4.2, pp. 8–9, WN19]; they keep the comparison hypotheses on singularities, and pointwise domination requires a potential with minimal singularities, as stated there.

The version with varying weights, corollary 13.7.9, is [arXiv version, Theorem 2.6 and its proof, pp. 10–12, DDNLsurv], the reference given in [Theorem 2.4.2, XiaSampleBook]. We keep the argument with a cutoff at a finite level and complementary weights, but work locally in every mixed degree before gluing on the compact manifold; the limit is taken with the theorem for bounded weights proved earlier. The survey uses $\mathrm d^{\mathrm c}=\mathrm i(\bar\partial -\partial )/(2\pi )$, so its operator $\mathrm d\mathrm d^{\mathrm c}$ is twice ours. At fixed potentials and with doubled backgrounds, each product of $p$ factors is $2^ p$ times ours, and this common factor cancels in the inequalities for lower limits and for masses. Proposition 13.7.8 establishes the statement of [Proposition 2.3.1, XiaSampleBook] for arbitrary monotone nets; the equivalence of the two notions of convergence in capacity is asserted for plurisubharmonic functions, not for arbitrary Borel functions with values in $[-\infty ,\infty ]$.

25.1.16 Envelopes and model potentials`3F57C7AFD15C`

Envelopes with obstacles, rooftop envelopes and envelopes of singularity types are used throughout [XiaSampleBook]. The argument for lower semicontinuous obstacles and the two reductions for quasi-continuous obstacles are in [Lemma 2.4.3 and Theorem 2.4.4, XiaSampleBook]. We follow them in lemma 16.13.3, theorem 16.13.4, using the local homogeneous Dirichlet problem and the capacity theory of section 7.1. Since potentials with minimal singularities are locally bounded outside the pole set of a reference potential with analytic singularities, the argument can be carried out in charts where they are bounded; it uses no differentiability of the envelope.

The concentration theorem of that source also allows pseudo-effective classes and quasi-continuous obstacles that are not bounded above. Obstacles that are not bounded above are already covered by theorem 16.13.4, because capping the obstacle strictly above the envelope changes neither the envelope nor its contact set. Pseudo-effective classes are treated in corollary 17.5.38, after the criterion for bigness: if the class is not big, every non-pluripolar measure of top degree vanishes.

For functions whose Laplacian with respect to a metric is bounded, and for the reduction to regular contact, compare [p. 4 and Lemma 3.1, pp. 4–5, DNT2019v1]. Such functions are locally in $W^{2,2}$ by lemma 16.14.4, which is derived from the elliptic theory of lemma 10.2.5. Proposition 16.14.6 applies the vanishing of weak derivatives on level sets to obstacles in $W^{2,1}_{\mathrm{loc}}$, for instance continuous obstacles with bounded metric Laplacian; the computation on the contact set needs only this local regularity. The stronger elliptic $W^{2,p}$ estimate quoted in the source is proved separately in proposition 16.14.9, by freezing the principal coefficients and testing the equation against a regularization of $u|u|^{p-2}$, and corollary 16.14.11 then shows that the signed determinant of such obstacles is integrable on all of $X$.

Approximation of envelopes by Monge–Ampère equations with an exponential penalty, and the limit as the parameter tends to infinity, are due to Berman; the equations and estimates are presented in [equations (2.1)–(2.2), pp. 282–283, Tos18], where smooth solvability is referred to Aubin and Yau. We prove solvability for positive parameters in theorem 16.15.20 by the continuity method, with a contraction argument in Sobolev spaces and a priori estimates along the path. The third-order estimate, lemma 16.15.13, bounds the difference of two Chern connections. Its computation follows the tensor calculation in [Proposition 2.8 and Theorem 2.9, pp. 24–26, SongWeinkove2012v1], the first arXiv version of Song and Weinkove's lecture notes, where such estimates are traced back to a calculation of Calabi and the explicit formula in the parabolic case is attributed to Phong, Šešum and Sturm. Our calculation is elliptic: the penalized equation bounds the Ricci tensor and its covariant derivative in terms of the difference of connections.

The constants in the uniform trace bound and in the convergence to the envelope (lemma 16.15.7, proposition 16.15.21) depend only on the fixed geometry, on a bound for the obstacle $f$ and on an upper bound for the trace $\Lambda \mathrm{dd}^{\mathrm{c}}f=-Lf/(4\pi )$ of its complex Hessian, that is, on a lower bound for its metric Laplacian $Lf$. Tosatti writes forms with $\mathrm i\partial \bar\partial $; in the notation of section 16.15, the coefficient matrices $h$ and $k$ represent $2\pi \omega $ and $2\pi \kappa $, so that $g_{i\bar j}=k_{i\bar j}+a_{i\bar j}$ is the coefficient matrix of $2\pi (\kappa +\mathrm{dd}^{\mathrm{c}}a)$.

That the envelope of the minimum of finitely many functions with bounded metric Laplacian again has bounded metric Laplacian is [Theorem 2.5(ii), p. 5, DarvasRubinstein2014v1] (first arXiv version). We prove it in theorem 16.15.24 by smoothing the finite minima and applying the uniform trace estimate above; in the trace of the complex Hessian of the smoothed minimum, the variance of the gradients enters with a negative sign, so the upper bound for this trace is preserved. Convolution in charts and the elliptic estimates for all finite $p$ (proposition 16.14.9) control the commutator of the convolution with the operator $\Lambda \mathrm{dd}^{\mathrm{c}}$, whose coefficients vary.

The estimate for the full real Hessian (lemma 16.16.1, theorem 16.16.7) follows Tosatti's argument with penalized equations, [pp. 283–287, Tos18]. For the absorption of the negative term with third derivatives, Tosatti refers to the calculation of Chu, Tosatti and Weinkove, which is given in full in [arXiv version, Section 2, Lemmas 2.1–2.2, pp. 4–10, CTW17]. In this book the formulas for derivatives of eigenvalues and every absorption step are proved directly. The perturbation by a projection subtracts one from the eigenvalues other than the largest one; keeping track of this shift gives the bound $2\lambda +3$ in the comparison of positive forms. The real metric associated with $2\pi \kappa $ is $2\operatorname{Re}(k_{i\bar j}\, \mathrm dz_ i\otimes \mathrm d\bar z_ j)$, so that its Hessian trace is $2D_ ka$. Since the estimate uses, besides bounds for the obstacle and its gradient, only an upper bound for its real Hessian, the same proof applies to minima of finitely many smooth functions, and then to rooftop envelopes of finitely many $C^{1,1}$ functions.

The regularity of envelopes in big classes on the ample locus is [Theorem 1.1, pp. 1347–1349, DNT23]; this locator refers to the statement in the published article. The complete proof and the extension to finitely many obstacles are given in [arXiv version, Lemmas 3.1, 3.3–3.5, Section 4 and Theorem 4.2, pp. 3–19, DNT23]. The smoothing by the flow of a Chern connection goes back to [Proposition 2.9, pp. 5–6, Proposition 3.8, pp. 8–12, and Theorem 4.1, pp. 12–16, DemFlowOnline], a twenty-page manuscript cited with its own page numbers. In lemma 16.17.1 we carry out the computation with jets of a fixed finite order instead of an extension to infinite order, and compute the integrals of the weights directly. Demailly's normalization $\mathrm i\partial \bar\partial /\pi $ is twice our $\mathrm{dd}^{\mathrm{c}}$, while the auxiliary operator in our kernel computation is $\mathcal D=2\pi \mathrm{dd}^{\mathrm{c}}$.

Theorem 16.19.1 follows the two successive smoothing estimates of Di Nezza and Trapani and their use of high powers of the equations of the divisors on a resolution. Local holomorphic changes of the radius give the required uniform estimates without introducing the total space of the line bundle used by them, and lemma 16.18.1 handles the minimum over varying radii at the upper end point. We keep a fixed quadratic correction in the smoothing, keep the smooth background separate from the nonsmooth obstacle, and recover the varying metric Laplacian from the quadratic error by an expansion of the adjoint kernel. The final local estimate in real Sobolev spaces follows from the Euclidean Hessian multipliers of section 2.9.

The formula for the Monge–Ampère measure on the contact set for arbitrary singular potentials is [Theorem 3.2 and its proof, pp. 5–8, DNT2019v1], where the method is credited to earlier work of Witt Nyström. In theorem 16.20.2 we simplify the passage to the limit in the weights at levels: the limiting inequality is restricted to the contact set, where the positive weight cancels, so that a single truncation level and a single positive parameter suffice. For a general obstacle, whose metric Laplacian need not control its full Hessian, we keep the regular intermediate envelope.

Corollary 16.20.4 gives the formula of the source for the density on the contact set of an ordinary obstacle in big classes. Corollary 17.5.39, proved after the criterion for bigness, also covers pseudo-effective classes that are not big, in which all non-pluripolar measures of top degree vanish. All measures are taken with our normalization; with the operator $\mathrm i\partial \bar\partial $ of the source, products of $n$ factors, pure or mixed, are $(2\pi )^ n$ times ours.

The argument showing that envelopes of singularity types preserve mixed masses is in [Section 3B, p. 2060, DDNL18mono]; the continuity along increasing sequences that it requires is provided by corollary 13.8.9, proposition 16.21.2. The bounds for the Monge–Ampère measures of rooftop envelopes and of envelopes of singularity types are [Lemma 3.7 and Theorem 3.8, pp. 2060–2061, DDNL18mono]. In lemma 16.13.1, corollary 16.13.5, proposition 16.21.3 they are derived from convergence within a bounded singularity type and from the theorem on contact sets for obstacles. Here a measure is carried by the contact set if the complement of this Borel set has measure zero, which says nothing about the closed support of the measure.

The extrapolation argument and the domination principle (theorem 16.22.4, theorem 16.22.10) follow [Lemma 2.4.4 and Theorem 2.4.5, XiaSampleBook]. In lemma 16.22.1, a fixed point that preserves the mass is obtained as the regularized limit of an increasing sequence that almost maximizes an integral height at every step; this replaces the transfinite iteration in the proof in the source. For theorem 16.22.11, proposition 16.22.14 compare [Theorems 3.12 and 3.14, pp. 2063–2064, DDNL18mono]. There the domination principle is derived from the construction with prescribed densities in Corollary 3.9, p. 2062, whereas here, in a big class, it follows from extrapolation and the concentration on contact sets. The example with a $\log \log $ singularity, example 16.22.12, follows [Example 3.1.1, XiaSampleBook]; its mass is computed with the identity in degree one, proposition 13.4.7.

The criterion for the existence of rooftop envelopes, the substitution of potentials of full mass and the mass inequality (proposition 16.23.1, proposition 16.23.3, theorem 16.23.5) follow [Proposition 3.1.5, Lemma 3.1.2 and Theorem 3.1.3, XiaSampleBook]. The bound for upper limits needed there is isolated in lemma 16.23.4; it does not claim weak convergence of the measures along a decreasing sequence, which may lose mass.

Envelopes with an obstacle and a constraint on the singularity type (definition 16.25.1, proposition 16.25.2, proposition 16.25.3) extend [Definition 3.1.4, XiaSampleBook]. The bound on the contact set for bounded quasi-continuous obstacles follows from the theorems on relative envelopes and on concentration for obstacles; it does not identify a smooth density. Proposition 16.25.4 is [Proposition 3.1.15, XiaSampleBook], with an explicit defect term for potentials that do not have full mass. The relative capacity of definition 16.25.5, defined with competitors between $\phi -1$ and $\phi $, reduces to [Definition 2.3.1, XiaSampleBook] when $\phi $ has minimal singularities. Theorem 16.26.1 gives a complete proof of [Theorem 6.1.1, XiaSampleBook]; in big classes it provides the margin for rooftop envelopes used in [Theorem 3.1.1, VuXiaNakamaye].

Our definition of model potentials keeps the condition of positive mass from [Definition 3.1.3, XiaSampleBook]. For the constraints by multiplier ideals used in the chapter on envelopes of that book, proposition 16.24.1 combines the uniform valuative criterion theorem 6.12.16 with lemma 16.24.2, the formula $\nu (v,E)=\inf _ j\nu (v_ j,E)$ for the Lelong numbers along divisors of an upper envelope $v=(\sup _ jv_ j)^*$. In the proof of this formula, a local pullback argument shows that these Lelong numbers are preserved under the countable selection of Choquet's lemma. The proposition thus provides the closure property used in [Propositions 3.2.6 and 3.2.9, XiaSampleBook]; the construction itself and its algebraic consequences are not treated here.

The comparison of capacities in section 16.27 follows Lu [arXiv version, Lemmas 3.1–3.4 and Theorem 3.5, pp. 8–13, Lu21]. Integration by parts for bounded potentials on a compact manifold, lemma 13.5.2, is derived from the calculus of gradients in section 7.14, and, together with approximation by smooth bounded potentials, it gives the weighted estimate lemma 16.27.2. The envelopes with increasing penalties of lemma 16.27.5 provide the concentration on compact sets and the two growth bounds, which replace the properties of relative extremal functions quoted in the preprint. The comparison then proceeds in three steps: powers of model potentials in Kähler classes (proposition 16.27.12), comparison downwards for potentials of full mass (lemma 16.27.15), and comparison in both directions (theorem 16.27.17). Finally, proposition 16.27.18 compares, by a local extension construction, these capacities with the local capacities of section 7.1; in particular, local convergence in capacity implies convergence in each relative capacity.

25.1.17 Relative energy and integration by parts`889877FC2044`

The identity for differences of bounded type, lemma 17.2.3, follows [arXiv version, Lemma 4.1, pp. 13–15, Lu21]. Its proof reduces to bounded potentials by truncation, first for a strictly positive background; an expansion in polynomials then removes this restriction. The general integration by parts formula, theorem 17.2.5, follows the argument by differentiation and polarization in [arXiv version, proof of Theorem 1.2, pp. 15–16, Lu21]. In his introduction, pp. 1–3, Lu credits the earlier integration by parts formula to Mingchen Xia. The general formula is stated in the introduction of [Theorem 1.1 and discussion, pp. 1–2, Xia2019v1] (first arXiv version), which also describes a proof by a projective construction; the proof followed here is Lu's later argument with capacities.

Relative energy classes and the relative energy are defined in [Definitions 3.1.5–3.1.6, XiaSampleBook], and the cocycle formula for potentials of bounded type, the bounds by tangents, concavity and continuity along decreasing sequences are [Lemmas 3.1.3–3.1.5 and Proposition 3.1.16, XiaSampleBook]. In proposition 17.3.4 the first variation of the energy along a segment is computed directly from the integral formula for the energy and the integration by parts theorem, and the cocycle formula follows by integrating it. In proposition 17.3.12, proposition 17.3.13 the energy is extended by means of the canonical truncations $\max \{ u,\phi -k\} $, and continuity is proved for decreasing nets, not only for sequences, the limit being the pointwise limit.

For theorem 17.3.14, lemma 17.3.18, proposition 17.3.19 compare [Proposition 3.1.17, Lemma 3.1.6, Corollary 3.1.6 and Proposition 3.1.18, XiaSampleBook]. In our proof, a uniform bound at finite levels for the integral of a potential against its own Monge–Ampère measure shows that finite energy by itself implies full mass.

Continuity of the energy along increasing nets, proposition 17.4.1, follows [Proposition 3.1.20, XiaSampleBook]; the uniform bound for the tails is (17.4.2). The proof of [Proposition 3.1.21, XiaSampleBook] uses an exponential Monge–Ampère equation; in proposition 17.4.7 we instead integrate the bound for the measures of rooftop envelopes on contact sets, corollary 16.13.5. This closure of finite energy under rooftop envelopes guarantees that the slices of Perron segments have finite energy. The closure under upper envelopes of families and the envelopes of mutual singularities follow [Propositions 3.1.23–3.1.24, XiaSampleBook].

The derivative of the energy along affine segments, proposition 17.4.9, also appears in [Lemma 3.1.11, XiaSampleBook]. Here it follows from the cocycle formula, which is polynomial in the parameter. The family of smooth potentials on $\mathbb P^1$ in example 17.4.11 shows that convergence in $L^1$ does not imply convergence of the energy.

The characterization of solutions as maximizers of a functional and the derivative of the energy of projections (theorem 17.5.5, proposition 17.5.4) follow the variational approach of [Theorem 4.1 and Lemma 4.2, pp. 207–209, BBGZ13], whose normalization of the operator agrees with ours. We prove the derivative of the energy of projections directly, with a model potential as reference, from the theorem on contact sets and the bounds of the energy by tangents.

For Monge–Ampère equations relative to a model potential, with prescribed measure and with an exponential factor, compare [arXiv version, Theorem 4.3, p. 20; Theorem 5.3 and Lemma 5.4, pp. 24–25, DDNL19log]. That source uses $\mathrm{i}\partial \bar\partial $; its potential corresponding to our $u$ is $u/(2\pi )$, and its exponential parameter is $2\pi \lambda $. We solve the prescribed-measure equation with bounded densities (theorem 17.5.6) and the exponential equation for every finite non-pluripolar measure (theorem 17.5.13) by compactness in the variational approach. The relative comparison and domination principles give proposition 17.5.7 for every finite non-pluripolar measure, when the reference has positive mass.

For the continuity of integrals against a fixed non-pluripolar measure compare [Theorem 3.10, pp. 203–205, BBGZ13]. Lemma 17.5.11 proves the case of uniformly bounded functions. The upper limit is controlled outside a pluripolar set; for the lower limit we use convex combinations of the tails with almost minimal $L^2$ norm for the given measure, which form a Cauchy sequence by the parallelogram identity. Applied to exponentials, the lemma allows the same variational construction for arbitrary finite non-pluripolar measures. The lower bound for limits in lemma 17.5.14 follows [arXiv version, Lemma 2.8, pp. 12–13, DDNL19log]; a fixed lower measure is restricted before weak limits are taken. The existence of solutions with prescribed measure, theorem 17.5.15, is obtained instead by letting the exponential parameter tend to zero; for the existence statement compare Theorem 4.7, p. 23, of that preprint.

Uniqueness for the relative equations is proved in [Theorem 4.29 and its proof, pp. 2079–2080, DDNL18mono] by adapting the argument of Kołodziej and Dinew based on the concentration of mass; in that section the model reference has positive mass and small unbounded locus. Our theorem 17.5.17 allows an arbitrary model reference of positive mass. It follows the same strategy, using the existence theorem, the comparison and domination principles for full mass and the lower bound for products with respect to a common measure (corollary 13.5.11); the domination principle turns inequalities almost everywhere for the common measure into pointwise inequalities.

The iteration with capacities in theorem 17.5.20, corollary 17.5.26 follows [arXiv version, Theorem 3.3 and its proof, pp. 14–15, DDNL19log]. Here the measure is assumed to be dominated by the power $1+\delta $ of the capacity, and an iteration with geometrically decreasing steps shows directly that the capacities of the relevant sublevel sets vanish beyond a finite level. The uniform bound for moments is corollary 10.3.2.

The estimates of volume and of measures with $L^ p$ densities by capacity (proposition 16.27.8, corollary 16.27.10) follow [Proposition 4.30, pp. 2081–2082, DDNL18mono], which assumes a small unbounded locus and uses uniform Skoda integrability; the formulation for arbitrary model potentials is [arXiv version, Lemma 3.9 and Proposition 3.10, p. 18, DDNL19log]. Here the prerequisites are the uniform exponential bound corollary 10.3.3 and the penalized envelopes of lemma 16.27.5. At fixed potentials, the background $2\pi \theta $ of the source gives measures and capacities that are $(2\pi )^ n$ times ours, while their ratios to the mass of the reference are unchanged.

The criterion that a class containing a current of positive non-pluripolar mass is big, which is used in [Proposition 2.4.1, XiaSampleBook], is [Proposition 1.22, p. 222, BEGZ10]. There the criterion is attributed to Boucksom; in the published article [Section 4.3, Theorem 4.7, pp. 1057–1061, Bou02b] the numbering differs from that citation, and the volume is defined through the determinant of the absolutely continuous part. Our proof of theorem 17.5.37 uses instead the Monge–Ampère equations of section 17.5, solved in the big classes obtained by adding $\varepsilon \{ \omega \} $ to the given class, the mixed trace estimate lemma 17.5.33 and Lamari's criterion. The proof also shows that a pseudo-effective class is big if and only if its volume is positive, without using an identity between the determinant volume and the non-pluripolar volume.

Lamari's duality criterion, lemma 17.5.28, is [Lemma 3.3 and its proof, pp. 275–276, Lamari1999]; the cohomological statement on annihilators is on p. 267. We prove the Kähler case by extending a positive linear functional with the Hahn–Banach argument, together with the scalar Poisson equation and the identity for the trace and the Hodge star proved earlier.

The pointwise trace estimate for smooth forms and the integrated Cauchy–Schwarz argument are [Lemmas 3.1–3.2 and their proofs, pp. 4–6, Popovici2014v1] (first arXiv version). Lemma 17.5.33 states the estimate for an arbitrary strictly positive form $\tau $ of bidegree $(n-1,n-1)$, and extends it, by local decreasing convolution and linearization of the determinant, to currents whose local potentials are locally bounded outside a closed set of Lebesgue measure zero; the first factor on its left-hand side is the full mass of $S\wedge \tau $, including any mass on the poles. The preprint writes $\mathrm i\partial \bar\partial $ for potentials, whereas our proof uses the normalization of this book throughout.

The inequality between mixed masses and the geometric mean of the masses, with its consequences for the concavity of roots and logarithms of masses, is [arXiv version, Theorem 6.1 and its proof, p. 26, DDNL19log]; it is the result referred to in [Theorem 2.4.1, XiaSampleBook]. Theorem 17.5.41 follows the proof in the source by relative Monge–Ampère equations. Because this proof uses the criterion for bigness, the theorem comes after that criterion; the case of vanishing masses is treated separately. At fixed potentials, the operator $\mathrm i\partial \bar\partial $ of the source multiplies all measures of top degree by $(2\pi )^ n$; by homogeneity this factor cancels in the inequality.

The identities for envelopes of arbitrary mass in theorem 17.6.5 are [Theorem 1.2, p. 4, PSWZ26], an arXiv preprint; they include potentials of zero mass, to which theorem 16.22.11 does not apply. Their proof there (Section 5, Theorem 5.1, pp. 37–42) uses a uniform bound for normalized tails and the recovery of volume for perturbations of positive mass; proposition 17.6.1 follows this approach. Monotonicity gives the limit directly, and the solutions with constant density that are needed are provided by theorem 17.5.6. Applying the limit to each member of the family $\mathcal F_ u$ that defines the ceiling proves the identity $(\sup _{v\in \mathcal F_ u}v)^*=P_\theta [u]$, without a separate formula for the regularization of the ceiling and without a second family of perturbations.

The source uses the operator $\mathrm{dd^ c}=\mathrm i\partial \bar\partial /\pi $ (p. 7). At fixed potentials, its backgrounds $2\theta $ and $2\omega $ give measures of top degree that are $2^ n$ times ours, while the normalized tail ratios and the constant densities of the auxiliary equations are unchanged.

25.1.18 Weak geodesics`4C762779903D`

The definition of subgeodesics on a strip and their convexity and boundary behavior (definition 17.7.1, proposition 17.7.4) follow [Definition 4.1.1 and Proposition 4.1.2, XiaSampleBook], and lemma 17.7.6, lemma 17.7.5, theorem 17.7.8 follow the arguments of [Proposition 4.1.3, Lemma 4.1.1 and Proposition 4.2.1, XiaSampleBook]. Rooftop envelopes and the identity for envelopes of mutual singularities show that the Perron segment is not identically $-\infty $ and has the prescribed endpoints. The formula of proposition 17.7.10, which expresses the Perron segment through rooftop envelopes and a Legendre transform, follows by taking supporting affine functions of the convex function of time on each fiber where it is finite; the regularization in the space variable is retained.

That the relative energy is affine along weak geodesics, theorem 17.8.3, follows the argument of [Proposition 4.3.3, XiaSampleBook] on a compact extension of the strip. The proof uses the fiber integration formula corollary 13.6.1 and integration by parts for bounded differences, theorem 17.2.5. The vanishing of the non-pluripolar Monge–Ampère measure of a maximal plurisubharmonic function, bounded or not, is lemma 17.8.1; it is proved with the homogeneous balayages of the bounded truncations. The maximality of the complexified Perron segment, lemma 17.8.2, is proved with its envelope under rotations of the annulus, or under imaginary translations of the strip. Approximating the endpoints by their decreasing truncations then extends the identity from potentials of bounded type to potentials of finite energy.

25.1.19 Extremal slopes and singularity cutoffs`3EBE4FDCA90E`

The identity for the supremum of the difference along a weak geodesic is [Proposition 4.2.2, XiaSampleBook], in the chapter on rays; for the infimum, [Proposition 4.2.3, XiaSampleBook] refers to an analogous argument. Proposition 17.9.1 gives this argument by reversing the segment and using canonical representatives; it also identifies the essential extrema with the extrema over the set where the potentials are finite.

The assertion on bounded shifts that follows [Example 4.3.1, XiaSampleBook] is justified there by comparing long segments with truncated endpoints. Proposition 17.9.2 makes this comparison quantitative: a bound $C$ at the terminal endpoint contributes at most $Cs/T$ at time $s$ on a segment of length $T$. Passing to the regularized limits as $T\to \infty $ gives invariance under bounded perturbations and monotonicity with respect to the preorder $\preceq $ of singularities. The construction of the associated rays and the formula for their energy in terms of mixed masses are not treated here.

25.1.20 Convex duality and real Monge–Ampère measures`482A6169B3A6`

For the separation of convex sets and for the local regularity and subgradients of convex functions (lemma 21.2.3, proposition 21.2.4) compare [Theorem 2.1.10 and Corollaries 2.1.11–2.1.13, pp. 44–45; Theorem 2.1.22, pp. 55–56, Hor07]. Closed convex functions and Legendre conjugation are treated in [Propositions 2.2.1–2.2.2 and Theorem 2.2.4, pp. 66–68; Theorem 2.2.5, pp. 68–69; Theorem 2.2.8, p. 70, Hor07]. The corresponding duality statements in this book are theorem 21.3.2, proposition 21.3.3; in the formula for a sum in which the infimum is attained, both functions are assumed to be finite on the whole space.

The classes of convex functions, the formulas involving slope bodies and the real Monge–Ampère measures follow the appendix on convex analysis of [XiaSampleBook]; as there, the factor $n!$ is kept, with Lebesgue measure normalized by the lattice. In theorem 21.5.1, measurability with respect to the completed Lebesgue measure and countable additivity are proved using the fact that the inverse of the subgradient map is single-valued almost everywhere. Theorem 21.5.2 concerns vague convergence, and example 21.5.3 shows that narrow convergence requires an additional condition that prevents mass from escaping to infinity. The mixed measures of proposition 21.5.4 are obtained from the positivity of matrices, convolution and polarization. We keep the convention of the source for full mass when a slope body has empty interior, and example 21.4.5 shows that the hypotheses on slope bodies in the formula for sums cannot be omitted.

The lemma on the finiteness of rotation-invariant plurisubharmonic functions, [Lemma 1.2.1, XiaSampleBook], is proved there by the logarithmic convexity of circle means, and it provides the finiteness step in [Lemma 5.2.1, XiaSampleBook]. Proposition 21.2.8, Corollary 21.2.9 give a direct proof, by smoothing and the fact that the complex Hessian of a function invariant under imaginary translations is one quarter of its real Hessian; this proof does not use Kiselman's minimum principle.

For the conjugate of the maximum of finitely many convex functions, the convex appendix of [XiaSampleBook] uses [Theorem 16.5, Roc70]. Lemma 21.6.1 proves the case of functions whose domains lie in a compact set, which is the case of the finite suprema in the chapter on ample toric varieties, [Proposition 5.2.4, XiaSampleBook]; compactness takes care of points whose weights tend to zero. The computation of the energy in [Proposition 5.2.9, Step 1, XiaSampleBook] passes to the limit in integrals of a bounded function on a noncompact space against real Monge–Ampère measures; lemma 21.6.2 justifies this passage to the limit by a cutoff and the convergence of total masses.

The asymptotic count of lattice points used in [Section 5.2, XiaSampleBook] follows from the case of saturated cones in [Theorem 1.15, p. 939, KK12], where Kaveh and Khovanskii reduce the assertion to Riemann sums; their introduction states the more general theorem on the growth of semigroups on p. 929. Theorem 21.7.3 gives a complete proof for convex bodies by means of fundamental cells, allowing arbitrary translations, bodies of smaller dimension and sublinear errors at the boundary. The continuity of volume needed in its proof is proposition 21.7.2, and corollary 21.7.4 isolates the two-sided bound used in the toric computation of dimensions.

The chapter on ample toric varieties in [XiaSampleBook] relies on the theory of toric varieties developed in [CLS11]. Here the smooth projective toric variety of a Delzant lattice polytope is constructed directly from its characters in theorem 21.8.3; its fan and its boundary divisors are identified in proposition 21.9.1, proposition 21.9.2, and the basis of global sections given by characters, theorem 21.9.5, is obtained from Laurent expansions in the charts at the vertices. No general classification of toric varieties is needed.

The potential of theorem 21.10.1, built from the function $G_0=\sum _\rho \ell _\rho \log \ell _\rho $, is Guillemin's reference potential. For the Legendre computation compare [Proposition 2, pp. 7–8; Theorem 1, pp. 9–10, CDG03v1] (first arXiv version). That source uses $G=\frac12\sum _\rho \ell _\rho \log \ell _\rho $ and holomorphic logarithmic coordinates $y+\mathrm i\theta $; with $x=2y=\log |w|^2$ we have $G_0=2G$ and $F_0(x)=2F(x/2)$, up to an additive constant. Its convention $\mathrm d^{\mathrm c}f=-\mathrm df\circ J$ gives $\mathrm d\mathrm d^{\mathrm c}=2\mathrm i\partial \bar\partial $; with the curvature convention of this book the form is divided by $2\pi $. Smooth extension and strict positivity in the monomial charts are proved by the inverse function theorem at every face, including the vertices.

25.1.21 Analytic spaces and modifications`F14EBE94C297`

The Weierstrass division theorem and the Noetherian property of the ring of germs of holomorphic functions (proposition 9.2.2, proposition 9.2.4) follow [Chapter II, (2.3)–(2.7), pp. 80–81, DemBook]. For coherent sheaves and Oka's coherence theorem (lemma 9.2.6, theorem 9.2.7) compare [Chapter II, (3.8)–(3.22), pp. 87–90, DemBook]. Local irreducible components, local parametrization, the Nullstellensatz, dimension and Cartan's theorem on the coherence of the ideal sheaf of an analytic set are treated in [Chapter II, (4.1)–(4.31), pp. 91–100, DemBook]. The algebraic facts on finite field extensions and discriminants used in these constructions are proved in lemma 9.3.3 by linear algebra over the field of fractions, with matrices of multiplication and the trace form, and they are applied in theorem 9.3.4. In our proof of Cartan's theorem, theorem 9.3.10, a single denominator works at all nearby stalks, which is more than finite generation of each stalk.

The Nullstellensatz and the strict decrease of dimension for proper analytic subgerms of an irreducible germ (corollary 9.3.8, proposition 9.3.9) are deduced from the same local parametrization: division by the discriminant identifies each prime ideal with the ideal of its zero set, and the connectedness of the generic sheets gives the decrease of dimension. The proof of the Nullstellensatz includes the description of the radical as an intersection of prime ideals.

The support theorem for closed currents of order zero, theorem 9.3.20, is the case of bidegree $(1,1)$ of [Chapter III, (2.10)–(2.14), pp. 141–143, DemBook]. Our proof uses smooth normal coordinates, convolution in tangential directions and induction over the singular loci of the analytic set; the theorem is applied in lemma 10.5.7. In the proof of proposition 10.5.8, the pullback is corrected with the pushforward of a Kähler form, which gives the required strict positivity directly.

For universal denominators of weakly holomorphic functions and for normalization (proposition 9.4.3, theorem 9.4.6) compare [Chapter II, (7.1)–(7.15), pp. 110–116, DemBook]. The cusp and the pair of crossing lines in example 9.4.10 illustrate the difference between being normal and being unibranch. The Remmert–Stein extension theorem and Remmert's proper mapping theorem are proved together in [Chapter II, (8.7)–(8.8), pp. 118–121, DemBook]. Our proof of theorem 9.7.1, theorem 9.7.2 also treats them together, but it uses the resolution of singularities, theorem 9.6.6, to reduce to proper maps defined on manifolds, and it proves the statements on finite maps and on the dimension of hypersurfaces separately.

The resolution and principalization theorems (theorem 9.6.6, theorem 9.6.7) are the analytic versions of [Theorems 1.1 and 1.3, pp. 611 and 614; Definitions 2.1–2.2, pp. 618–619, BM08]; that the results apply to complex analytic spaces is stated on p. 611 there. The relative principalization theorem, theorem 9.7.5, follows the lemma on relative principalization in the unpublished manuscript [Lemma relative-principalization, VuXiaNakamaye]. At each step of this construction, the centers in the ambient manifold divide the ideal by exactly one exceptional factor, as required for the controlled transform of the restricted ideal. In our statement the submanifold is closed and the ambient manifold is connected, so that the global blowups and the identities between ideals hold as stated.

Theorem 9.8.1 states that a proper bimeromorphic map has connected fibers over the points at which the germ of the target is irreducible, in particular over unibranch spaces. Its proof follows the argument with complements of branches in [Appendix on unibranch spaces, XiaSampleBook]; properness is used to place the inverse image of a small neighborhood inside neighborhoods of the pieces of the fiber. The local upper bound near analytic subsets of codimension at least two in a reduced space, proposition 9.9.4, is proved by slicing with curves over a finite projection and the maximum principle for subharmonic functions. It complements the extension theorem on manifolds, corollary 9.3.18, which follows from the criterion in terms of Hausdorff measure.

The corollary on the realization of divisors, in the appendix of [XiaSampleBook] on unibranch spaces, underlies the comparison of multiplier ideals with Lelong numbers along divisors in Part I of that book. Theorem 9.8.6 proves this corollary by principalizing the successive centers and decreasing the order of the Jacobian along a fixed divisor. The definition of plurisubharmonic functions on reduced spaces by local ambient extension, definition 9.9.1, suffices for pullback by holomorphic maps, for the maximum principle on compact fibers and for the upper bound obtained by slicing with curves. On the manifolds used in Part I of that book, the descent of quasi-plurisubharmonic functions under modifications, proposition 10.3.9, is proved directly by extension across an analytic set of codimension at least two.

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@misc{pome-sections-historical-and-bibliographical-notes,
  author = {{Pome contributors}},
  title = {25.1 Historical and bibliographical notes},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/sections/historical-and-bibliographical-notes/}
}

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25.1 Historical and bibliographical notes65E16F3944C7

25.1.1 Measures and distributionsE6CD9442A846

25.1.2 Harmonic functions and boundary values16FFFBC63146

25.1.3 Subharmonic functions and polar sets7A117C94C896

25.1.4 Energy, capacity and exceptional setsF2A7FD4CCC54

25.1.5 Fine topology and balayage3BA364E0394C

25.1.6 Fourier analysis and complex-line incidence342F8D0FF9AE

25.1.7 Differential forms, currents and holomorphic divisorsCE9C93DDA94A

25.1.8 Plurisubharmonic functions and invariant minimaE49C1A0D4682

25.1.9 Bounded products and nonlinear capacityE66D40EC5904

25.1.10 Pluripolar sets and the plurifine topology1AFE6A788B80

25.1.11 Weighted estimates and local extensionE796BF553819

25.1.12 Lelong numbers and multiplier ideals2BFBF0A17586

25.1.13 Analytic singularities and regularizationC1D97719C08E

25.1.14 Global potentials and positivity lociF9B38BEB0BCF

25.1.15 Non-pluripolar products0948B3754F28

25.1.16 Envelopes and model potentials3F57C7AFD15C

25.1.17 Relative energy and integration by parts889877FC2044

25.1.18 Weak geodesics4C762779903D

25.1.19 Extremal slopes and singularity cutoffs3EBE4FDCA90E

25.1.20 Convex duality and real Monge–Ampère measures482A6169B3A6

25.1.21 Analytic spaces and modificationsF14EBE94C297