[Convergence hybride de mesures de Kähler–Einstein]
Nous calculons la limite hybride (au sens de Boucksom–Jonsson) de la famille des formes volumes de Kähler–Einstein sur une dégénérescence de variétés canoniquement polarisées. La mesure limite est une somme pondérée de masses de Dirac en des valuations divisorielles, déterminées par la limite algébro-géométrique naturelle de la famille. Nous formulons aussi des remarques sur l’opérateur de Monge–Ampère non-archimédien et la continuité hybride des potentiels de Kähler–Einstein dans ce contexte.
We compute the hybrid limit (in the sense of Boucksom–Jonsson) of the family of Kähler–Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make some remarks on the non-archimedean Monge–Ampère operator and hybrid continuity of Kähler–Einstein potentials in this context.
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DOI : 10.5802/aif.3455
Keywords: Kähler–Einstein manifolds, degenerations, Berkovich spaces
Mot clés : Variétés de Kähler–Einstein, dégénérescences, espaces de Berkovich
Pille-Schneider, Léonard 1
CC-BY-ND 4.0
@article{AIF_2022__72_2_587_0,
author = {Pille-Schneider, L\'eonard},
title = {Hybrid convergence of {K\"ahler{\textendash}Einstein} measures},
journal = {Annales de l'Institut Fourier},
pages = {587--615},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {72},
number = {2},
year = {2022},
doi = {10.5802/aif.3455},
zbl = {07554664},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3455/}
}
TY - JOUR AU - Pille-Schneider, Léonard TI - Hybrid convergence of Kähler–Einstein measures JO - Annales de l'Institut Fourier PY - 2022 SP - 587 EP - 615 VL - 72 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3455/ DO - 10.5802/aif.3455 LA - en ID - AIF_2022__72_2_587_0 ER -
%0 Journal Article %A Pille-Schneider, Léonard %T Hybrid convergence of Kähler–Einstein measures %J Annales de l'Institut Fourier %D 2022 %P 587-615 %V 72 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3455/ %R 10.5802/aif.3455 %G en %F AIF_2022__72_2_587_0
Pille-Schneider, Léonard. Hybrid convergence of Kähler–Einstein measures. Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 587-615. doi : 10.5802/aif.3455. https://aif.centre-mersenne.org/articles/10.5802/aif.3455/
[1] Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta, Res. Math. Sci., Volume 2 (2015) no. 1, 7 | DOI | MR | Zbl
[2] Equations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978), pp. 63-95 | Zbl
[3] Smooth -adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl
[4] Kähler–Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal., Volume 24 (2014) no. 6, pp. 1683-1730 | DOI | Zbl
[5] Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl
[6] Uniqueness of K-polystable degenerations of Fano varieties, Ann. Math., Volume 190 (2019) no. 2, pp. 609-656 | MR | Zbl
[7] Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., Volume 44 (2008) no. 2, pp. 449-494 | DOI | MR | Zbl
[8] Solution to a non-archimedean Monge–Ampère equation, J. Am. Math. Soc., Volume 28 (2015) no. 3, pp. 617-667 | DOI | Zbl
[9] Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | DOI | Numdam | MR | Zbl
[10] Differentiability of non-archimedean volumes and non-archimedean Monge–Ampère equations (with an appendix by Robert Lazarsfeld) (2019) (https://arxiv.org/abs/1608.01919)
[11] Formes différentielles réelles et courants sur les espaces de Berkovich (2012) (https://arxiv.org/abs/1204.6277)
[12] Kähler–Einstein metrics and stability, Int. Math. Res. Not., Volume 2014 (2014) no. 8, pp. 2119-2125 | DOI | Zbl
[13] Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014) no. 1, pp. 63-106 | DOI | Zbl
[14] Singular Kähler–Einstein metrics, J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639 | DOI | Zbl
[15] Degeneration of endomorphisms of the complex projective space in the hybrid space (2017) (https://arxiv.org/abs/1611.08490)
[16] The Moduli of Extremal Compact Kähler Manifolds and generalized Weil–Petersson metrics, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 1, pp. 101-183 | DOI | Zbl
[17] Kähler–Einstein metrics and integral invariants, Lecture Notes in Mathematics, 1314, Springer, 1988 | DOI | MR | Zbl
[18] Toroidal Embeddings. I, Lecture Notes in Mathematics, 339, Springer, 1973 | DOI | Zbl
[19] Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013 (with the collaboration of Sándor Kovács) | DOI | Zbl
[20] Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 | DOI | Zbl
[21] Semi-stable extensions over 1-dimensional bases, Acta Math. Sin., Volume 34 (2018) no. 1, pp. 103-113 | DOI | MR | Zbl
[22] The dual complex of Calabi–Yau pairs, Invent. Math., Volume 205 (2016) no. 3, pp. 527-557 | DOI | MR | Zbl
[23] The complex Monge–Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | Zbl
[24] Affine structures and non-archimedean analytic spaces, The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003 (Progress in Mathematics), Volume 244, Birkhäuser, 2006, pp. 312-385 | MR | Zbl
[25] Weight functions on non-archimedean analytic spaces and the Kontsevich–Soibelman skeleton, Algebr. Geom., Volume 2 (2015) no. 3, pp. 365-404 | DOI | MR | Zbl
[26] The essential skeleton of a degeneration of algebraic varieties, Am. J. Math., Volume 138 (2016) no. 6, pp. 1645-1667 | DOI | MR | Zbl
[27] Collapsing K3 surfaces, tropical geometry and moduli compactifications of Satake, Morgan–Shalen type (2018) (https://arxiv.org/abs/1810.07685)
[28] On representations and compactifications of symmetric Riemannian spaces, Ann. Math., Volume 71 (1960), pp. 77-110 | DOI | MR | Zbl
[29] Riemannian geometry of Kahler–Einstein currents III: compactness of Kahler–Einstein manifolds of negative scalar curvature (2020) (https://arxiv.org/abs/2003.04709)
[30] Degeneration of Kähler–Einstein manifolds of negative scalar curvature (2017) (https://arxiv.org/abs/1706.01518)
[31] Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscr. Math., Volume 123 (2007) no. 4, pp. 381-451 | DOI | MR | Zbl
[32] Degeneration of Kähler–Einstein manifolds. I, Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990 (Greene, Robert, ed.) (Proceedings of Symposia in Pure Mathematics), Volume 54, American Mathematical Society, 1993, pp. 595-609 | Zbl
[33] Families of Calabi-Yau manifolds and canonical singularities, Int. Math. Res. Not., Volume 2015 (2015) no. 20, pp. 10586-10594 | DOI | MR | Zbl
[34] On the Ricci curvature of a compact Kähler manifold and complex Monge–Ampère equation I, Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411 | Zbl
[35] Collapsing of negative Kähler–Einstein metrics, Math. Res. Lett., Volume 22 (2015) no. 6, pp. 1843-11869 | DOI | MR | Zbl
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