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Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics (2112.04290v2)

Published 8 Dec 2021 in math.AG, math.CV, and math.DG

Abstract: Let $X$ be a smooth projective variety. We construct partial Okounkov bodies associated to Hermitian pseudo-effective line bundles $(L,\phi)$ on $X$. We show that partial Okounkov bodies are universal invariants of the singularity of $\phi$. As an application, we generalize the theorem of Boucksom--Chen and construct Duistermaat--Heckman measures associated with finite energy metrics on the Berkovich analytification of an ample line bundle.

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References (57)
  1. “Geometry of higher rank valuations”, 2022 arXiv:2208.06237 [math.AG]
  2. “Growth of balls of holomorphic sections and energy at equilibrium” In Invent. Math. 181.2, 2010, pp. 337–394 DOI: 10.1007/s00222-010-0248-9
  3. Robert J. Berman and Bo Berndtsson “Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties” In Ann. Fac. Sci. Toulouse Math. (6) 22.4, 2013, pp. 649–711 DOI: 10.5802/afst.1386
  4. “Chern–Weil and Hilbert–Samuel formulae for singular hermitian line bundles”, 2021 arXiv:2112.09007 [math.AG]
  5. Robert J. Berman, Sébastien Boucksom and Mattias Jonsson “A variational approach to the Yau-Tian-Donaldson conjecture” In J. Amer. Math. Soc. 34.3, 2021, pp. 605–652 DOI: 10.1090/jams/964
  6. Robert Berman, Sébastien Boucksom and David Witt Nyström “Fekete points and convergence towards equilibrium measures on complex manifolds” In Acta Math. 207.1, 2011, pp. 1–27 DOI: 10.1007/s11511-011-0067-x
  7. “Okounkov bodies of filtered linear series” In Compos. Math. 147.4, 2011, pp. 1205–1229 DOI: 10.1112/S0010437X11005355
  8. “The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension” In J. Algebraic Geom. 22.2, 2013, pp. 201–248 DOI: 10.1090/S1056-3911-2012-00574-8
  9. “Monge-Ampère equations in big cohomology classes” In Acta Math. 205.2, 2010, pp. 199–262 DOI: 10.1007/s11511-010-0054-7
  10. Vladimir G. Berkovich “Spectral theory and analytic geometry over non-Archimedean fields” 33, Mathematical Surveys and Monographs American Mathematical Society, Providence, RI, 1990, pp. x+169 DOI: 10.1090/surv/033
  11. S. Boucksom, C. Favre and M. Jonsson “Valuations and plurisubharmonic singularities” In Publications of the Research Institute for Mathematical Sciences 44.2, 2008, pp. 449–494
  12. Sébastien Boucksom, Charles Favre and Mattias Jonsson “Differentiability of volumes of divisors and a problem of Teissier” In J. Algebraic Geom. 18.2, 2009, pp. 279–308 DOI: 10.1090/S1056-3911-08-00490-6
  13. José Ignacio Burgos Gil, Patrice Philippon and Martín Sombra “Arithmetic geometry of toric varieties. Metrics, measures and heights” In Astérisque, 2014, pp. vi+222
  14. Sébastien Boucksom, Tomoyuki Hisamoto and Mattias Jonsson “Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs” In Ann. Inst. Fourier (Grenoble) 67.2, 2017, pp. 743–841 URL: http://aif.cedram.org/item?id=AIF_2017__67_2_743_0
  15. “Global pluripotential theory over a trivially valued field” In Ann. Fac. Sci. Toulouse Math. (6) 31.3, 2022, pp. 647–836 DOI: 10.5802/afst.170
  16. S. Boucksom “Cônes positifs des variétés complexes compactes”, 2002
  17. S. Boucksom “Higher dimensional Zariski decompositions”, 2002 arXiv:0204336 [math.AG]
  18. S. Boucksom “Singularities of plurisubharmonic functions and multiplier ideals”, http://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf, 2017
  19. Junyan Cao “Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds” In Compos. Math. 150.11, 2014, pp. 1869–1902 DOI: 10.1112/S0010437X14007398
  20. “On the constant scalar curvature Kähler metrics, general automorphism group”, 2018 arXiv:1801.05907 [math.DG]
  21. “Newton-Okounkov bodies sprouting on the valuative tree” In Rend. Circ. Mat. Palermo (2) 66.2, 2017, pp. 161–194 DOI: 10.1007/s12215-016-0285-3
  22. “Asymptotic base loci via Okounkov bodies” In Adv. Math. 323, 2018, pp. 784–810 DOI: 10.1016/j.aim.2017.11.007
  23. “Okounkov bodies associated to pseudoeffective divisors” In J. Lond. Math. Soc. (2) 97.2, 2018, pp. 170–195 DOI: 10.1112/jlms.12107
  24. Tamás Darvas “Geometric pluripotential theory on Kähler manifolds” In Advances in complex geometry 735, Contemp. Math. Amer. Math. Soc., [Providence], RI, 2019, pp. 1–104 DOI: 10.1090/conm/735/14822
  25. Tamás Darvas, Eleonora Di Nezza and Chinh H. Lu “L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT metric geometry of big cohomology classes” In Ann. Inst. Fourier (Grenoble) 68.7, 2018, pp. 3053–3086 URL: http://aif.cedram.org/item?id=AIF_2018__68_7_3053_0
  26. Tamás Darvas, Eleonora Di Nezza and Chinh H. Lu “Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity” In Anal. PDE 11.8, 2018, pp. 2049–2087 DOI: 10.2140/apde.2018.11.2049
  27. Tamás Darvas, Eleonora Di Nezza and Chinh H. Lu “On the singularity type of full mass currents in big cohomology classes” In Compos. Math. 154.2, 2018, pp. 380–409 DOI: 10.1112/S0010437X1700759X
  28. Tamás Darvas, Eleonora Di Nezza and Chinh H. Lu “Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity” In Math. Ann. 379.1-2, 2021, pp. 95–132 DOI: 10.1007/s00208-019-01936-y
  29. Tamás Darvas, Eleonora Di Nezza and Hoang-Chinh Lu “The metric geometry of singularity types” In J. Reine Angew. Math. 771, 2021, pp. 137–170 DOI: 10.1515/crelle-2020-0019
  30. Jean-Pierre Demailly “Analytic methods in algebraic geometry” 1, Surveys of Modern Mathematics International Press, Somerville, MA; Higher Education Press, Beijing, 2012, pp. viii+231
  31. Jean-Pierre Demailly “On the cohomology of pseudoeffective line bundles” In Complex geometry and dynamics 10, Abel Symp. Springer, Cham, 2015, pp. 51–99
  32. Ya Deng “Transcendental Morse inequality and generalized Okounkov bodies” In Algebr. Geom. 4.2, 2017, pp. 177–202 DOI: 10.14231/AG-2017-009
  33. Tamás Darvas and Chinh H. Lu “Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry” In Geom. Topol. 24.4, 2020, pp. 1907–1967 DOI: 10.2140/gt.2020.24.1907
  34. Jean-Pierre Demailly, Thomas Peternell and Michael Schneider “Pseudo-effective line bundles on compact Kähler manifolds” In Internat. J. Math. 12.6, 2001, pp. 689–741 DOI: 10.1142/S0129167X01000861
  35. “Transcendental Okounkov bodies”, 2023 arXiv:2309.07584 [math.DG]
  36. “The volume of pseudoeffective line bundles and partial equilibrium” In Geometry & Topology, 2021 arXiv:2112.03827 [math.DG]
  37. “The closures of test configurations and algebraic singularity types” In Adv. Math. 397, 2022, pp. Paper No. 108198\bibrangessep56 DOI: 10.1016/j.aim.2022.108198
  38. “Foundations of rigid geometry. I”, EMS Monographs in Mathematics European Mathematical Society (EMS), Zürich, 2018, pp. xxxiv+829
  39. “Degenerate Complex Monge–Ampère Equations”, 2017
  40. Eiji Inoue “Entropies in μ𝜇\muitalic_μ-framework of canonical metrics and K-stability, II – Non-archimedean aspect: non-archimedean μ𝜇\muitalic_μ-entropy and μ𝜇\muitalic_μK-semistability”, 2022 arXiv:2202.12168 [math.AG]
  41. Shin-Yao Jow “Okounkov bodies and restricted volumes along very general curves” In Adv. Math. 223.4, 2010, pp. 1356–1371 DOI: 10.1016/j.aim.2009.09.015
  42. Kiumars Kaveh “Note on cohomology rings of spherical varieties and volume polynomial” In J. Lie Theory 21.2, 2011, pp. 263–283
  43. A.G. Khovanskiĭ “The Newton polytope, the Hilbert polynomial and sums of finite sets” In Funktsional. Anal. i Prilozhen. 26.4, 1992, pp. 57–63\bibrangessep96 DOI: 10.1007/BF01075048
  44. “Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory” In Ann. of Math. (2) 176.2, 2012, pp. 925–978 DOI: 10.4007/annals.2012.176.2.5
  45. “Convex bodies associated to linear series” In Ann. Sci. Éc. Norm. Supér. (4) 42.5, 2009, pp. 783–835 DOI: 10.24033/asens.2109
  46. “Holomorphic Morse inequalities and Bergman kernels” 254, Progress in Mathematics Birkhäuser Verlag, Basel, 2007, pp. xiv+422
  47. Andrei Okounkov “Why would multiplicities be log-concave?” In The orbit method in geometry and physics (Marseille, 2000) 213, Progr. Math. Birkhäuser Boston, Boston, MA, 2003, pp. 329–347
  48. Andrei Okounkov “Brunn-Minkowski inequality for multiplicities” In Invent. Math. 125.3, 1996, pp. 405–411 DOI: 10.1007/s002220050081
  49. Julius Ross and David Witt Nyström “Analytic test configurations and geodesic rays” In J. Symplectic Geom. 12.1, 2014, pp. 125–169 DOI: 10.4310/JSG.2014.v12.n1.a5
  50. R. Schneider “Convex bodies: the Brunn–Minkowski theory” Cambridge university press, 2014
  51. David Witt Nyström “Test configurations and Okounkov bodies” In Compos. Math. 148.6, 2012, pp. 1736–1756 DOI: 10.1112/S0010437X12000358
  52. David Witt Nyström “Transforming metrics on a line bundle to the Okounkov body” In Ann. Sci. Éc. Norm. Supér. (4) 47.6, 2014, pp. 1111–1161 DOI: 10.24033/asens.2235
  53. David Witt Nyström “Monotonicity of non-pluripolar Monge-Ampère masses” In Indiana Univ. Math. J. 68.2, 2019, pp. 579–591 DOI: 10.1512/iumj.2019.68.7630
  54. Mingchen Xia “Non-pluripolar products on vector bundles and Chern–Weil formulae on mixed Shimura varieties”, 2022 arXiv:2210.15342 [math.AG]
  55. Mingchen Xia “Mabuchi geometry of big cohomology classes” In J. Reine Angew. Math. 798, 2023, pp. 261–292 DOI: 10.1515/crelle-2023-0019
  56. Mingchen Xia “Pluripotential-theoretic stability thresholds” In Int. Math. Res. Not. IMRN, 2023, pp. 12324–12382 DOI: 10.1093/imrn/rnac186
  57. V.P. Zaharjuta “Transfinite diameter, Čebyšev constants and capacity for a compactum in Cnsuperscript𝐶𝑛C^{n}italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT” In Mat. Sb. (N.S.) 96(138), 1975, pp. 374–389\bibrangessep503
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