Abstract
The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays.
Acknowledgements
I would like to thank Rémi Reboulet and Lars Martin Sektnan for their invitation to the University of Gothenburg, in particular, Rémi who drew my attention to the problem of this article during my visit and shared some of his ideas. I also thank Sébastien Boucksom for many enlightening discussions on non-Archimedean pluripotential theory and related fields, and the anonymous referee for a careful reading. Finally, I would like to acknowledge the support of CNRS and École polytechnique.
References
[1] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1–2, 1–40. 10.1007/BF02392348Search in Google Scholar
[2] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren Math. Wiss. 223, Springer, Berlin 1976. 10.1007/978-3-642-66451-9Search in Google Scholar
[3] R. J. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc. 34 (2021), no. 3, 605–652. 10.1090/jams/964Search in Google Scholar
[4] B. Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math. 200 (2015), no. 1, 149–200. 10.1007/s00222-014-0532-1Search in Google Scholar
[5] B. Berndtsson, Probability measures associated to geodesics in the space of Kähler metrics, Algebraic and analytic microlocal analysis, Springer Proc. Math. Stat. 269, Springer, Cham (2018), 395–419. 10.1007/978-3-030-01588-6_6Search in Google Scholar
[6] R. Bhatia, Positive definite matrices, Princeton Ser. Appl. Math., Princeton University, Princeton 2007. Search in Google Scholar
[7] J.-M. Bismut and E. Vasserot, The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), no. 2, 355–367. 10.1007/BF01217912Search in Google Scholar
[8] S. Boucksom, Variational and non-archimedean aspects of the Yau–Tian–Donaldson conjecture, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Scientific, Hackensack (2018), 591–617. 10.1142/9789813272880_0069Search in Google Scholar
[9] S. Boucksom and D. Eriksson, Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, Adv. Math. 378 (2021), Article ID 107501. 10.1016/j.aim.2020.107501Search in Google Scholar
[10] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841. 10.5802/aif.3096Search in Google Scholar
[11] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability and asymptotics of energy functionals in Kähler geometry, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 9, 2905–2944. 10.4171/jems/894Search in Google Scholar
[12] S. Boucksom and M. Jonsson, Global pluripotential theory over a trivially valued field, Ann. Fac. Sci. Toulouse Math. (6) 31 (2022), no. 3, 647–836. 10.5802/afst.1705Search in Google Scholar
[13] S. Boucksom and M. Jonsson, A non-Archimedean approach to K-stability, II: Divisorial stability and openness, J. reine angew. Math. 805 (2023), 1–53. 10.1515/crelle-2023-0062Search in Google Scholar
[14] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Ann. of Math. Stud. 99, Princeton University, Princeton 1981. 10.1515/9781400881444Search in Google Scholar
[15] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999. 10.1007/978-3-662-12494-9Search in Google Scholar
[16] E. Calabi and X. X. Chen, The space of Kähler metrics. II, J. Differential Geom. 61 (2002), no. 2, 173–193. 10.4310/jdg/1090351383Search in Google Scholar
[17] D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables, Trends Math., Birkhäuser, Boston (1999), 1–23. 10.1007/978-1-4612-2166-1_1Search in Google Scholar
[18] H. Chen and C. Maclean, Distribution of logarithmic spectra of the equilibrium energy, Manuscripta Math. 146 (2015), no. 3–4, 365–394. 10.1007/s00229-014-0712-8Search in Google Scholar
[19] X. Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), no. 2, 189–234. 10.4310/jdg/1090347643Search in Google Scholar
[20] X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics, general automorphism group, preprint (2018), https://arxiv.org/abs/1801.05907. Search in Google Scholar
[21] X. Chen and Y. Tang, Test configuration and geodesic rays, Géométrie différentielle, physique mathématique, mathématiques et société. I, Astérisque 321, Société Mathématique de France, Paris (2008), 139–167, Search in Google Scholar
[22]
J. Chu, V. Tosatti and B. Weinkove,
[23] X. Dai, K. Liu and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41. 10.4310/jdg/1143593124Search in Google Scholar
[24] T. Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015), 182–219. 10.1016/j.aim.2015.08.005Search in Google Scholar
[25] T. Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017), no. 5, 1275–1313. 10.1353/ajm.2017.0032Search in Google Scholar
[26]
T. Darvas,
Weak geodesic rays in the space of Kähler potentials and the class
[27] T. Darvas and C. H. Lu, Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry, Geom. Topol. 24 (2020), no. 4, 1907–1967. 10.2140/gt.2020.24.1907Search in Google Scholar
[28] T. Darvas, C. H. Lu and Y. A. Rubinstein, Quantization in geometric pluripotential theory, Comm. Pure Appl. Math. 73 (2020), no. 5, 1100–1138. 10.1002/cpa.21857Search in Google Scholar
[29] T. Darvas and M. Xia, The closures of test configurations and algebraic singularity types, Adv. Math. 397 (2022), Paper No. 108198. 10.1016/j.aim.2022.108198Search in Google Scholar
[30]
J.-P. Demailly,
Estimations
[31] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. Search in Google Scholar
[32] J.-P. Demailly, Complex Analytic and Differential Geometry, Université de Grenoble, Grenoble 2012. Search in Google Scholar
[33] J.-P. Demailly, Extension of holomorphic functions defined on non reduced analytic subvarieties, The legacy of Bernhard Riemann after one hundred and fifty years. Vol. I, Adv. Lect. Math. (ALM) 35, International Press, Somerville (2016), 191–222. Search in Google Scholar
[34] E. Di Nezza and C. H. Lu, Geodesic distance and Monge–Ampère measures on contact sets, Anal. Math. 48 (2022), no. 2, 451–488. 10.1007/s10476-022-0159-1Search in Google Scholar
[35] S. Diverio, Segre forms and Kobayashi–Lübke inequality, Math. Z. 283 (2016), no. 3–4, 1033–1047. 10.1007/s00209-016-1632-ySearch in Google Scholar
[36] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2 196, American Mathematical Society, Providence (1999), 13–33. 10.1090/trans2/196/02Search in Google Scholar
[37] D. Eisenbud, Commutative algebra, Grad. Texts in Math. 150, Springer, New York 1995. 10.1007/978-1-4612-5350-1Search in Google Scholar
[38] S. Finski, Semiclassical Ohsawa–Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel, J. Differential Geom. 128 (2024), no. 2, 639–721. 10.4310/jdg/1727712891Search in Google Scholar
[39] S. Finski, On the metric structure of section ring, preprint (2022), https://arxiv.org/abs/2209.03853. Search in Google Scholar
[40] S. Finski, Submultiplicative norms and filtrations on section rings, preprint (2022), https://arxiv.org/abs/2210.03039. Search in Google Scholar
[41] S. Finski, The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds, J. Math. Pures Appl. (9) 189 (2024), Article ID 103586. 10.1016/j.matpur.2024.06.001Search in Google Scholar
[42] V. Guedj and A. Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607–639. 10.1007/BF02922247Search in Google Scholar
[43] V. Guedj and A. Zeriahi, Degenerate complex Monge–Ampère equations, EMS Tracts Math. 26, European Mathematical Society, Zürich 2017. 10.4171/167Search in Google Scholar
[44] R. Hartshorne, Ample vector bundles, Publ. Math. Inst. Hautes Études Sci. 29 (1966), 63–94. 10.1007/BF02684806Search in Google Scholar
[45] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
[46] T. Hisamoto, On the li reine angmit of spectral measures associated to a test configuration of a polarized Kähler manifold, J. reine angew. Math. 713 (2016), 129–148. 10.1515/crelle-2014-0021Search in Google Scholar
[47] G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math. 339, Springer, Berlin 1973. 10.1007/BFb0070318Search in Google Scholar
[48] S. Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153–166. 10.1017/S0027763000016615Search in Google Scholar
[49] J. Kollár, Lectures on resolution of singularities, Ann. of Math. Stud. 166, Princeton University, Princeton 2007. Search in Google Scholar
[50] C. Li, Geodesic rays and stability in the cscK problem, Ann. Sci. Éc. Norm. Supér. (4) 55 (2022), no. 6, 1529–1574. 10.24033/asens.2523Search in Google Scholar
[51] C. Li and C. Xu, Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180 (2014), no. 1, 197–232. 10.4007/annals.2014.180.1.4Search in Google Scholar
[52] X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progr. Math. 254, Birkhäuser, Basel 2007. Search in Google Scholar
[53] X. Ma and G. Marinescu, Toeplitz operators on symplectic manifolds, J. Geom. Anal. 18 (2008), no. 2, 565–611. 10.1007/s12220-008-9022-2Search in Google Scholar
[54] T. Mabuchi, Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227–252. Search in Google Scholar
[55]
T. Ohsawa,
On the extension of
[56] S. T. Paul and G. Tian, CM stability and the generalized Futaki invariant II, From probability to geometry II. Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque 328, Société Mathématique de France, Paris (2009), 339–354. Search in Google Scholar
[57] D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the Knudsen–Mumford expansion, J. Differential Geom. 78 (2008), no. 3, 475–496. 10.4310/jdg/1207834553Search in Google Scholar
[58] D. H. Phong and J. Sturm, Test configurations for K-stability and geodesic rays, J. Symplectic Geom. 5 (2007), no. 2, 221–247. 10.4310/JSG.2007.v5.n2.a3Search in Google Scholar
[59] D. H. Phong and J. Sturm, Regularity of geodesic rays and Monge–Ampère equations, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3637–3650. 10.1090/S0002-9939-10-10371-2Search in Google Scholar
[60] D. H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge–Ampere equations, Comm. Anal. Geom. 18 (2010), no. 1, 145–170. 10.4310/CAG.2010.v18.n1.a6Search in Google Scholar
[61] G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Math. 94, Cambridge University, Cambridge 1999. Search in Google Scholar
[62] R. Reboulet, Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy, J. reine angew. Math. 793 (2022), 59–103. 10.1515/crelle-2022-0059Search in Google Scholar
[63] R. Reboulet, The space of finite-energy metrics over a degeneration of complex manifolds, J. Éc. polytech. Math. 10 (2023), 659–701. 10.5802/jep.229Search in Google Scholar
[64] D. Rees, Lectures on the asymptotic theory of ideals, London Math. Soc. Lecture Note Ser. 113, Cambridge University, Cambridge 1988. 10.1017/CBO9780511525957Search in Google Scholar
[65] J. Ross and D. Witt Nyström, Analytic test configurations and geodesic rays, J. Symplectic Geom. 12 (2014), no. 1, 125–169. 10.4310/JSG.2014.v12.n1.a5Search in Google Scholar
[66] S. Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495–550. 10.2307/2374768Search in Google Scholar
[67] J. Song and S. Zelditch, Test configurations, large deviations and geodesic rays on toric varieties, Adv. Math. 229 (2012), no. 4, 2338–2378. 10.1016/j.aim.2011.12.025Search in Google Scholar
[68] G. Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015), no. 1–2, 451–484. 10.1007/s00208-014-1126-3Search in Google Scholar
[69] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99–130. 10.4310/jdg/1214445039Search in Google Scholar
[70] L. Wang and S. Zhou, Bergman kernels on degenerations, preprint (2023), https://arxiv.org/abs/2304.01943. Search in Google Scholar
[71] D. Witt Nyström, Test configurations and Okounkov bodies, Compos. Math. 148 (2012), no. 6, 1736–1756. 10.1112/S0010437X12000358Search in Google Scholar
[72] S. Zelditch, Szegő kernels and a theorem of Tian, Int. Math. Res. Not. IMRN 1998 (1998), no. 6, 317–331. 10.1155/S107379289800021XSearch in Google Scholar
[73] K. Zhang, Valuative invariants with higher moments, J. Geom. Anal. 32 (2022), no. 1, Paper No. 10. 10.1007/s12220-021-00827-6Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function
Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function