Finite quotients of abelian varieties with a Calabi-Yau resolution

Gachet, Cécile

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

Let $A$ be an abelian variety, and $G\subset \mathrm{Aut}\left(A\right)$ 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.

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.

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

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