Finite quotients of abelian varieties with a Calabi-Yau resolution

Cécile Gachet

doi:10.5802/jep.277

Finite quotients of abelian varieties with a Calabi-Yau resolution
[Quotients finis de variétés abéliennes admettant une résolution de Calabi-Yau]

Cécile Gachet ¹

¹ Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNRS UMR 7351, Parc Valrose 06108 Nice Cedex 2, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1219-1286.

Résumé
Abstract

Soit $A$ une variété abélienne et $G$ un groupe fini agissant librement en codimension $2$ par automorphismes sur $A$ . On s’intéresse ici aux conditions d’existence d’une résolution du quotient singulier $A / G$ qui soit une variété de Calabi-Yau. Tandis qu’en dimension $3$ , deux exemples de quotients admettant une telle résolution ont été construits par Oguiso dans un article de 1994, on montre ici qu’aucun quotient de la sorte n’existe en dimension $4$ . En dimension quelconque, on classifie les variétés abéliennes $A$ susceptibles d’admettre des quotients de la sorte, à isogénie près.

Let $A$ be an abelian variety, and $G \subset Aut (A)$ a finite group acting freely in codimension two. We discuss whether the singular quotient $A / G$ admits a resolution that is a Calabi-Yau manifold. While Oguiso constructed two examples in dimension 3, we show that there are none in dimension 4. We also classify up to isogeny the possible abelian varieties $A$ in arbitrary dimension.

Reçu le : 2023-01-24
Accepté le : 2024-10-04
Publié le : 2024-10-15

MR Zbl

DOI : 10.5802/jep.277

Classification : 14J32, 14K22, 14E15, 14L30
Keywords: Crepant resolution, abelian variety with complex multiplication, Calabi-Yau manifold
Mots-clés : Résolution crépante, variété abélienne à multiplication complexe, variété de Calabi-Yau

Affiliations des auteurs :

Cécile Gachet ¹

¹ Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNRS UMR 7351, Parc Valrose 06108 Nice Cedex 2, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Finite quotients of abelian varieties with {a~Calabi-Yau} resolution},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1219--1286},
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Cécile Gachet. Finite quotients of abelian varieties with a Calabi-Yau resolution. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1219-1286. doi : 10.5802/jep.277. https://jep.centre-mersenne.org/articles/10.5802/jep.277/

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