Equality in the Miyaoka–Yau inequality and uniformization of non-positively curved klt pairs

Benoît Claudon; Patrick Graf; Henri Guenancia

doi:10.5802/crmath.599

Article de recherche - Géométrie algébrique

Equality in the Miyaoka–Yau inequality and uniformization of non-positively curved klt pairs
[Cas d’égalité de l’inégalité de Miyaoka–Yau et uniformisation des paires klt à courbure négative]

Benoît Claudon ¹ ; Patrick Graf ² ; Henri Guenancia ³

¹ Univ Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France et Institut Universitaire de France
² Lehrstuhl für Mathematik I, Universität Bayreuth, 95440 Bayreuth, Germany
³ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 55-81.

Résumé
Abstract

Soit $(X, Δ)$ une paire klt compacte kählérienne pour laquelle $K_{X} + Δ$ est ample ou numériquement trivial, et $Δ$ à coefficients standard. Nous démontrons que, si l’inégalité de Miyaoka–Yau orbifold pour $(X, Δ)$ est une égalité, alors le revêtement universel orbifold de la paire est soit la boule (cas ample), soit l’espace affine (cas numériquement trivial).

Let $(X, Δ)$ be a compact Kähler klt pair, where $K_{X} + Δ$ is ample or numerically trivial, and $Δ$ has standard coefficients. We show that if equality holds in the orbifold Miyaoka–Yau inequality for $(X, Δ)$ , then its orbifold universal cover is either the unit ball (ample case) or the affine space (numerically trivial case).

Reçu le : 2023-05-09
Accepté le : 2023-12-06
Publié le : 2024-06-06

DOI : 10.5802/crmath.599

Classification : 32J27, 14J60
Keywords: Miyaoka–Yau inequality, orbifold uniformization, klt pairs
Mot clés : inégalité de Miyaoka–Yau, uniformisation orbifold, paires klt

Affiliations des auteurs :

Benoît Claudon ¹ ; Patrick Graf ² ; Henri Guenancia ³

¹ Univ Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France et Institut Universitaire de France
² Lehrstuhl für Mathematik I, Universität Bayreuth, 95440 Bayreuth, Germany
³ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Equality in the {Miyaoka{\textendash}Yau} inequality and uniformization of non-positively curved klt pairs},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {55--81},
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Benoît Claudon; Patrick Graf; Henri Guenancia. Equality in the Miyaoka–Yau inequality and uniformization of non-positively curved klt pairs. Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 55-81. doi : 10.5802/crmath.599. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.599/

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