Smooth quotients of abelian varieties by finite groups

TL;DR

This paper classifies smooth quotients of abelian varieties by finite groups, showing they are products of elliptic curves or projective spaces under certain conditions.

Contribution

It provides a complete classification of smooth quotients of abelian varieties by finite groups fixing the origin, including special cases with irreducible actions.

Findings

When the group acts irreducibly, the abelian variety is a product of elliptic curves and the quotient is projective space.

In the general case with zero fixed points, the quotient is a product of projective spaces.

The classification covers all smooth quotients under the given conditions.

Abstract

We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is irreducible, we prove that $A$ is isomorphic to the self-product of an elliptic curve and $A/G\simeq \mathbb P^n$. In the general case, assuming $\dim(A^G)=0$, we prove that $A/G$ is isomorphic to a direct product of projective spaces.