Hybrid convergence of Kähler–Einstein measures
Annales de l'Institut Fourier, Volume 72 (2022) no. 2, pp. 587-615.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3455
Classification: 32Q20,  32P05,  14T05
Keywords: Kähler–Einstein manifolds, degenerations, Berkovich spaces
Pille-Schneider, Léonard 1

1 Département de mathématiques et applications Ecole Normale Supérieure 45 rue d’Ulm 75005 Paris
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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
TI  - Hybrid convergence of Kähler–Einstein measures
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 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/
UR  - https://zbmath.org/?q=an%3A07554664
UR  - https://doi.org/10.5802/aif.3455
DO  - 10.5802/aif.3455
LA  - en
ID  - AIF_2022__72_2_587_0
ER  - 
%0 Journal Article
%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://doi.org/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, Volume 72 (2022) no. 2, pp. 587-615. doi : 10.5802/aif.3455. https://aif.centre-mersenne.org/articles/10.5802/aif.3455/

[1] Amini, Omid; Baker, Matthew; Brugallé, Erwan; Rabinoff, Joseph Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta, Res. Math. Sci., Volume 2 (2015) no. 1, 7 | DOI | MR | Zbl

[2] Aubin, Thierry Equations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978), pp. 63-95 | Zbl

[3] Berkovich, Vladimir G. Smooth p-adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl

[4] Berman, Robert J.; Guenancia, Henri Kähler–Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal., Volume 24 (2014) no. 6, pp. 1683-1730 | DOI | Zbl

[5] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James 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] Blum, Harold; Xu, Chenyang Uniqueness of K-polystable degenerations of Fano varieties, Ann. Math., Volume 190 (2019) no. 2, pp. 609-656 | MR | Zbl

[7] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., Volume 44 (2008) no. 2, pp. 449-494 | DOI | MR | Zbl

[8] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Solution to a non-archimedean Monge–Ampère equation, J. Am. Math. Soc., Volume 28 (2015) no. 3, pp. 617-667 | DOI | Zbl

[9] Boucksom, Sébastien; Jonsson, Mattias 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] Burgos Gil, José; Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent 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] Chambert-Loir, Antoine; Ducros, Antoine Formes différentielles réelles et courants sur les espaces de Berkovich (2012) (https://arxiv.org/abs/1204.6277)

[12] Chen, Xiuxiong; Donaldson, Simon K.; Sun, Song Kähler–Einstein metrics and stability, Int. Math. Res. Not., Volume 2014 (2014) no. 8, pp. 2119-2125 | DOI | Zbl

[13] Donaldson, Simon K.; Sun, Song Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014) no. 1, pp. 63-106 | DOI | Zbl

[14] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Singular Kähler–Einstein metrics, J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639 | DOI | Zbl

[15] Favre, Charles Degeneration of endomorphisms of the complex projective space in the hybrid space (2017) (https://arxiv.org/abs/1611.08490)

[16] Fujiki, Akira; Schumacher, Georg 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] Futaki, Akito Kähler–Einstein metrics and integral invariants, Lecture Notes in Mathematics, 1314, Springer, 1988 | DOI | MR | Zbl

[18] Kempf, George R.; Knudsen, Finn F.; Mumford, David B.; Saint-Donat, Bernard Toroidal Embeddings. I, Lecture Notes in Mathematics, 339, Springer, 1973 | DOI | Zbl

[19] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013 (with the collaboration of Sándor Kovács) | DOI | Zbl

[20] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 | DOI | Zbl

[21] Kollár, János; Nicaise, Johannes; Xu, Chen Yang Semi-stable extensions over 1-dimensional bases, Acta Math. Sin., Volume 34 (2018) no. 1, pp. 103-113 | DOI | MR | Zbl

[22] Kollár, János; Xu, Chenyang The dual complex of Calabi–Yau pairs, Invent. Math., Volume 205 (2016) no. 3, pp. 527-557 | DOI | MR | Zbl

[23] Kołodziej, Sławomir The complex Monge–Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | Zbl

[24] Kontsevich, Maxim; Soibelman, Yan 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] Musţăta, Mircea; Nicaise, Johannes 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] Nicaise, Johannes; Xu, Chenyang The essential skeleton of a degeneration of algebraic varieties, Am. J. Math., Volume 138 (2016) no. 6, pp. 1645-1667 | DOI | MR | Zbl

[27] Odaka, Yuji; Oshima, Yoshiki Collapsing K3 surfaces, tropical geometry and moduli compactifications of Satake, Morgan–Shalen type (2018) (https://arxiv.org/abs/1810.07685)

[28] Satake, Ichirô On representations and compactifications of symmetric Riemannian spaces, Ann. Math., Volume 71 (1960), pp. 77-110 | DOI | MR | Zbl

[29] Song, Jian; Sturm, Jacob; Wang, Xiaowei Riemannian geometry of Kahler–Einstein currents III: compactness of Kahler–Einstein manifolds of negative scalar curvature (2020) (https://arxiv.org/abs/2003.04709)

[30] Song, Jong Degeneration of Kähler–Einstein manifolds of negative scalar curvature (2017) (https://arxiv.org/abs/1706.01518)

[31] Thuillier, Amaury 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] Tian, Gang 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] Tosatti, Valentino Families of Calabi-Yau manifolds and canonical singularities, Int. Math. Res. Not., Volume 2015 (2015) no. 20, pp. 10586-10594 | DOI | MR | Zbl

[34] Yau, Shing-Tung 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] Zhang, Yuguang Collapsing of negative Kähler–Einstein metrics, Math. Res. Lett., Volume 22 (2015) no. 6, pp. 1843-11869 | DOI | MR | Zbl

Cited by Sources: