Diameter of Kähler currents
-
Vincent Guedj
and
Ahmed Zeriahi
Abstract
We establish upper bounds on the diameter of compact Kähler manifolds endowed with Kähler metrics whose volume form satisfies an Orlicz integrability condition. Our results extend previous estimates due to Fu–Guo–Song, Y. Li, and Guo–Phong–Song–Sturm. In particular, they do not involve any constraint on the vanishing of the volume form. Moreover, we show that singular Kähler–Einstein currents have finite diameter, provided that their local potentials are Hölder continuous.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-11-LABX-0040
Funding statement: This work has benefited from state aid managed by the ANR-11-LABX-0040, in connection with the research project HERMETIC, as well as by the ANR projects KARMAPOLIS and PARAPLUI and the Institut Universitaire de France.
References
[1] R. Berman and J.-P. Demailly, Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology, Progr. Math. 296, Birkhäuser/Springer, New York (2012), 39–66. 10.1007/978-0-8176-8277-4_3Search in Google Scholar
[2] R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj and A. Zeriahi, Kähler-Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. reine angew. Math. 751 (2019), 27–89. 10.1515/crelle-2016-0033Search in Google Scholar
[3] B. Berndtsson, The openness conjecture and complex Brunn–Minkowski inequalities, Complex geometry and dynamics, Abel Symp. 10, Springer, Cham (2015), 29–44. 10.1007/978-3-319-20337-9_2Search in Google Scholar
[4] F. Campana, H. Guenancia and M. Păun, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 6, 879–916. 10.24033/asens.2205Search in Google Scholar
[5] S.-K. Chiu and G. Székelyhidi, Higher regularity for singular Kähler–Einstein metrics, Duke Math. J. 172 (2023), no. 18, 3521–3558. 10.1215/00127094-2022-0107Search in Google Scholar
[6] D. Coman and V. Guedj, Quasiplurisubharmonic Green functions, J. Math. Pures Appl. (9) 92 (2009), no. 5, 456–475. 10.1016/j.matpur.2009.05.010Search in Google Scholar
[7] J.-P. Demailly, S. Dinew, V. Guedj, H. H. Pham, S. Kołodziej and A. Zeriahi, Hölder continuous solutions to Monge–Ampère equations, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 4, 619–647. 10.4171/jems/442Search in Google Scholar
[8] J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), no. 4, 525–556. 10.1016/S0012-9593(01)01069-2Search in Google Scholar
[9] J.-P. Demailly, T. Peternell and M. Schneider, Kähler manifolds with numerically effective Ricci class, Compos. Math. 89 (1993), no. 2, 217–240. Search in Google Scholar
[10] E. Di Nezza, V. Guedj and H. Guenancia, Families of singular Kähler–Einstein metrics, J. Eur. Math. Soc. (JEMS) 25 (2023), no. 7, 2697–2762. 10.4171/jems/1249Search in Google Scholar
[11] E. Di Nezza and C. H. Lu, Generalized Monge–Ampère capacities, Int. Math. Res. Not. IMRN 2015 (2015), no. 16, 7287–7322. 10.1093/imrn/rnu166Search in Google Scholar
[12] E. Di Nezza and S. Trapani, The regularity of envelopes, preprint (2021), https://arxiv.org/abs/2110.14314; to appear in Ann. Sci. Éc. Norm. Supér. (4). Search in Google Scholar
[13] P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009), no. 3, 607–639. 10.1090/S0894-0347-09-00629-8Search in Google Scholar
[14] X. Fu, B. Guo and J. Song, Geometric estimates for complex Monge–Ampère equations, J. reine angew. Math. 765 (2020), 69–99. 10.1515/crelle-2019-0020Search in Google Scholar
[15] P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math., John Wiley & Sons, New York 1978. Search in Google Scholar
[16] Q. Guan and X. Zhou, A proof of Demailly’s strong openness conjecture, Ann. of Math. (2) 182 (2015), no. 2, 605–616. 10.4007/annals.2015.182.2.5Search in Google Scholar
[17] V. Guedj, H. Guenancia and A. Zeriahi, Continuity of singular Kähler–Einstein potentials, Int. Math. Res. Not. IMRN 2023 (2023), no. 2, 1355–1377. 10.1093/imrn/rnab294Search in Google Scholar
[18] 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
[19] B. Guo, D. H. Phong, J. Song and J. Sturm, Sobolev inequalities on Kähler spaces, preprint (2023), https://arxiv.org/abs/2311.00221. Search in Google Scholar
[20] B. Guo, D. H. Phong, J. Song and J. Sturm, Diameter estimates in Kähler geometry, Comm. Pure Appl. Math. 77 (2024), no. 8, 3520–3556. 10.1002/cpa.22196Search in Google Scholar
[21]
B. Guo, D. H. Phong and F. Tong,
On
[22] B. Guo, D. H. Phong, F. Tong and C. Wang, On the modulus of continuity of solutions to complex Monge–Ampère equations, preprint (2021), https://arxiv.org/abs/2112.02354. Search in Google Scholar
[23] B. Guo and J. Song, Local noncollapsing for complex Monge–Ampère equations, J. reine angew. Math. 793 (2022), 225–238. 10.1515/crelle-2022-0069Search in Google Scholar
[24] H.-J. Hein and S. Sun, Calabi–Yau manifolds with isolated conical singularities, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 73–130. 10.1007/s10240-017-0092-1Search in Google Scholar
[25] R. Kobayashi, Kähler–Einstein metric on an open algebraic manifold, Osaka J. Math. 21 (1984), no. 2, 399–418. Search in Google Scholar
[26] S. Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998), no. 1, 69–117. 10.1007/BF02392879Search in Google Scholar
[27]
S. Kołodziej,
Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in
[28] J. Kovats, Dini–Campanato spaces and applications to nonlinear elliptic equations, Electron. J. Differential Equations 1999 (1999), Paper No. 37. Search in Google Scholar
[29] K. Kurdyka and P. Orro, Distance géodésique sur un sous-analytique, Rev. Mat. Univ. Complut. Madrid 10 (1997), 173–182. 10.5209/rev_REMA.1997.v10.17357Search in Google Scholar
[30] C. Li, Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds, J. reine angew. Math. 733 (2017), 55–85. 10.1515/crelle-2014-0156Search in Google Scholar
[31] Y. Li, On collapsing Calabi–Yau fibrations, J. Differential Geom. 117 (2021), no. 3, 451–483. 10.4310/jdg/1615487004Search in Google Scholar
[32] S. Lojasiewicz, Ensemble semi-analytiques, preprint IHES (1965). Search in Google Scholar
[33] M. Paun, On the Albanese map of compact Kähler manifolds with numerically effective Ricci curvature, Comm. Anal. Geom. 9 (2001), no. 1, 35–60. 10.4310/CAG.2001.v9.n1.a2Search in Google Scholar
[34] X. Rong and Y. Zhang, Continuity of extremal transitions and flops for Calabi–Yau manifolds, J. Differential Geom. 89 (2011), no. 2, 233–269. 10.4310/jdg/1324477411Search in Google Scholar
[35] Y. A. Rubinstein, Smooth and singular Kähler–Einstein metrics, Geometric and spectral analysis, Contemp. Math. 630, American Mathematical Society, Providence (2014), 45–138. 10.1090/conm/630/12665Search in Google Scholar
[36] G. Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. 10.1007/s002220050176Search in Google Scholar
[37] G. Tian and S.-T. Yau, Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory, Adv. Ser. Math. Phys. 1, World Scientific, Singapore (1987), 574–628. 10.1142/9789812798411_0028Search in Google Scholar
[38] V. Tosatti, Limits of Calabi–Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 4, 755–776. 10.4171/jems/165Search in Google Scholar
[39] A. Zeriahi, Remarks on the modulus of continuity of subharmonic functions, preprint (2020), https://arxiv.org/abs/2007.08399. Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Covariant projective representations of Hilbert–Lie groups
- Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity
- A Lindemann–Weierstrass theorem for 𝐸-functions
- Topology and dynamics of compact plane waves
- Diameter of Kähler currents
- CscK metrics near the canonical class
- On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
- The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound
- Hodge–Tate stacks and non-abelian 𝑝-adic Hodge theory of v-perfect complexes on rigid spaces
Articles in the same Issue
- Frontmatter
- Covariant projective representations of Hilbert–Lie groups
- Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity
- A Lindemann–Weierstrass theorem for 𝐸-functions
- Topology and dynamics of compact plane waves
- Diameter of Kähler currents
- CscK metrics near the canonical class
- On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
- The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound
- Hodge–Tate stacks and non-abelian 𝑝-adic Hodge theory of v-perfect complexes on rigid spaces
