Finite quotients of abelian varieties with a Calabi-Yau resolution

TL;DR

This paper investigates when quotients of abelian varieties by finite groups admit Calabi-Yau resolutions, providing classifications and showing the non-existence of such resolutions in dimension four.

Contribution

It classifies abelian varieties with free group actions in codimension two and proves the absence of Calabi-Yau resolutions in dimension four.

Findings

No Calabi-Yau resolutions exist in dimension four.

Classified abelian varieties admitting free actions in arbitrary dimensions.

Constructed two examples of Calabi-Yau resolutions in dimension three.

Abstract

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.