Chapter 12

Gauduchon variational volumeA93AA52F4C7A

Sections and subsections

1. 12.1 A Gauduchon variational volume881C26425D7A
2. 12.2 The quotient and the smooth equation43BF29E33E59
3. 12.3 The equality case of concavity and the first variation484879715C58
4. 12.4 Second variation on background classes8A687EAB6B68
5. 12.5 Recovering a positive class from its Gauduchon derivative9141861B4F50
6. 12.6 The Hessian metric and its completion466CEDF9AEBA
   1. 12.6.2 Product and blowup examples5429610F6E7A
   2. 12.6.4 Invariant backgrounds on compact Lie-group quotients8D8E5D56DAEF
   3. 12.6.21 Masses and reference densities in dimension threeD70657252686
   4. 12.6.31 Warped product cones and flat completionC0584AC307C7
   5. 12.6.38 Lengths of the two ends96431078A25B
   6. 12.6.42 Current cones and endpoint equations9FC10688E696
   7. 12.6.46 Limits of the base potentialsFDA6B324FD1B
   8. 12.6.48 Traces of minimizing metrics9C8B56E18A60
   9. 12.6.52 Projective and torus examplesDDBDE1BD78C7
7. 12.7 A compact dual problem over Gauduchon currents54ADA5D399DA
8. 12.8 Supporting probabilities represented by maximizing currents7C2BB0FDDDF5
9. 12.9 Integrability of Gauduchon determinant measuresD02094988A03
   1. 12.9.4 Uniform bounds near a smooth densityA0340E674DC4
   2. 12.9.11 Uniformly controlled complex Hessians5603574CC3A6
10. 12.10 Integrable densities in the compact Gauduchon problem5CB8CFC32A19
    1. 12.10.4 The tail condition and perturbations of the density5210ECD5227F
    2. 12.10.7 Supporting probabilities for an integrable density516A524E7A2A
    3. 12.10.10 Trace inequalities at the subsolution threshold07A010666AF5
    4. 12.10.16 A smooth maximizing form with an analytic degeneracy set8D604EFF48DF
    5. 12.10.19 The limit of the exponential equations604052B5061F
    6. 12.10.22 Envelope criteria and extension across the degeneracy setDC394A61A477
    7. 12.10.25 A globally smooth root of rank at least two8EACEED9426F
    8. 12.10.28 Two attained instances244D24A4396B
    9. 12.10.32 A full-mass point pole without a smooth maximizing testB4100C32D89A
    10. 12.10.36 Quantitative convergence of the currents5B8A428F9007
    11. 12.10.38 The classes of the divisorial components622292D98711
    12. 12.10.40 A branched projective exampleBE9EC228849D
    13. 12.10.42 A moment condition on maximizing tests47D5B2CA6B21
11. 12.11 Nef limits and nonattainmentEA449D56F8F7
12. 12.12 Subtraction with a closed nef core75F31FAA4616
13. 12.13 Subtraction on warped productsEA9388E29ABC
14. 12.14 Positive volume without bignessE4AB7B03C20D
    1. 12.14.9 Volume tests and threshold testsB51E860FD3F7
15. 12.15 Varying the probability measure8B161DDAE77F
    1. 12.15.3 Upper and lower masses53B6AFE7C911
16. 12.16 Curvature conditions for mass extremaA57EC3129CC8
17. 12.17 Curvature pairs and positive lower massC18EABD7D630
    1. 12.17.1 The bounded cubic identityEDD3B0732B25
    2. 12.17.6 The curvature criterionC72702069454
    3. 12.17.10 Negative-line-bundle quotients49FAED65F993
    4. 12.17.13 A Calabi–Eckmann family without LCK metrics51CD5F251745
18. 12.18 Geometric bounds and determinant concentrationAA652CC345D8
    1. 12.18.5 Intrinsic determinant measures and the complete profile5613B7E2253B
    2. 12.18.12 Fixed determinant data with degenerating ellipticity95D23527B325
    3. 12.18.14 The same obstruction for nonclosed pluriclosed metrics10AA5741C871
    4. 12.18.16 Concentration at exact maximizers for varying rootsA4F04193B5B9
19. 12.19 Vanishing lower mass108F40B2DFAA
    1. 12.19.2 The elliptic quotient1DD06E8B2862
    2. 12.19.6 Averaging the trace and compact curves57556869C08A
    3. 12.19.12 The volume of the positive root92006A543CD2
    4. 12.19.14 Consequences for dimension and equations8E3D46779F5F
    5. 12.19.17 A balanced metric and a finite upper massDEA97C6D08EB
20. 12.20 Orders of degeneration at the nef boundary3567ADAA7365
21. 12.21 Elliptic surface volumes and order spectraE4B29689FB79
22. 12.22 Comparison under an arbitrary modification9BED3D691F82
23. 12.23 Strict changes under one blowupD0D47B7389D7
24. 12.24 Positive-current subtraction and divisorial residuals57C0ECEB4F4C
25. 12.25 Integer orders in fixed projective conesE6D8A97F4D2C
    1. 12.25.4 Linear blowups and the fixed generators6EBDFF04B7CA
    2. 12.25.10 Equality of orders under modificationsA34399DFCD46
    3. 12.25.12 Subtracting effective divisorsA528EDFC5F96
26. 12.26 A maximum-rank comparison criterion493311C8FB2F
27. 12.27 Flat anticanonical curves0E9176045572
28. 12.28 Flat divisors and Hartogs extension25D1457FC429
29. 12.29 Punctured line bundles and elliptic quotients76F771FE4DA3
30. 12.30 Cotangent torsors on split elliptic ruled surfacesE4B698D2AEE3
31. 12.31 Symmetric tensors on split elliptic ruled surfaces7CF6C47CAA92
32. 12.32 Section algebras of cotangent extensions72E247F3A7CA
33. 12.33 Cotangent extensions with zero ruling componentC9CE007F34A7
34. 12.34 Current tests for smooth positivity509A5021C77D
    1. 12.34.3 Maximal time of the Chern–Ricci flow355CACEF1F3C
    2. 12.34.8 A positive-volume endpoint that is not big382AAED0F384
    3. 12.34.13 Twisting and distance boundsFFD5AB006AEF
35. 12.35 Divisorial constraints and residual volume269CE892EA5B
    1. 12.35.6 Varying divisorial coefficients2AD000453E7C
    2. 12.35.8 A ruled residual and its subtraction costE45BF47B8568
36. 12.36 A divisorial threshold on the Hopf threefold7F944D8C8D92