Miyaoka–Yau inequalities and the topological characterization of certain klt varieties

Daniel Greb; Stefan Kebekus; Thomas Peternell

doi:10.5802/crmath.580

Article de recherche - Géométrie algébrique

Miyaoka–Yau inequalities and the topological characterization of certain klt varieties
[Inégalités de Miyaoka–Yau et caractérisation topologique de certaines variétés klt]

Daniel Greb ¹ ; Stefan Kebekus ² ; Thomas Peternell ³

¹ Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen, 45117 Essen, Germany
² Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany
³ Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 141-157.

Résumé
Abstract

Les quotients de boules, les variétés hyperelliptiques et les espaces projectifs sont caractérisés par leurs classes de Chern, comme les variétés pour lesquelles l’inégalité de Miyaoka–Yau devient une égalité. Les quotients de boules, les variétés abéliennes et les espaces projectifs sont aussi caractérisés topologiquement : si une variété projective complexe $X$ est homéomorphe à une variété de ce type, alors $X$ est elle-même de ce type. Dans cet article, des résultats similaires sont établis pour les variétés projectives avec des singularités klt qui sont homéomorphes à des quotients de boules singulières, à des quotients de variétés abéliennes, ou à des espaces projectifs.

Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka–Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projective manifold $X$ is homeomorphic to a variety of this type, then $X$ is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces.

Reçu le : 2023-06-16
Accepté le : 2023-10-13
Publié le : 2024-06-06

DOI : 10.5802/crmath.580

Classification : 32Q30, 32Q26, 14E20, 14E30
Keywords: Miyaoka–Yau inequality, klt singularities, uniformisation, homeomorphisms
Mot clés : Inégalité de Miyaoka–Yau, singularités klt, uniformisation, homéomorphismes

Affiliations des auteurs :

Daniel Greb ¹ ; Stefan Kebekus ² ; Thomas Peternell ³

¹ Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen, 45117 Essen, Germany
² Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany
³ Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Daniel Greb and Stefan Kebekus and Thomas Peternell},
     title = {Miyaoka{\textendash}Yau inequalities and the topological characterization of certain klt varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {141--157},
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     year = {2024},
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     language = {en},
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Daniel Greb; Stefan Kebekus; Thomas Peternell. Miyaoka–Yau inequalities and the topological characterization of certain klt varieties. Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 141-157. doi : 10.5802/crmath.580. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.580/

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