Q-abelian and $\mathbb Q$-Fano finite quotients of abelian varieties

TL;DR

This paper characterizes finite quotients of abelian varieties, called fqav, focusing on Q-abelian and $ ext{Q}$-Fano types, and shows they serve as fundamental building blocks for all fqav's.

Contribution

It provides a characterization of Q-abelian and $ ext{Q}$-Fano fqav's via endomorphisms and describes their structure through finite quasiétale covers.

Findings

Q-abelian varieties are characterized by quasiétale polarized endomorphisms.

Every fqav has a finite quasiétale cover by a product of an abelian variety and a $ ext{Q}$-Fano fqav.

Q-Fano fqav's and Q-abelian varieties are fundamental building blocks of fqav's.

Abstract

We study finite quotients of abelian varieties (fqav for short) i.e. quotients of abelian varieties by finite groups. We show that Q-abelian varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors) are characterized by the existence of quasi\'etale polarized (or int-amplified) endomorphisms. We show that every fqav has a finite quasi\'etale cover by the product of an abelian variety and a $\mathbb Q$-Fano fqav. Using such coverings, we give a characterization of $\mathbb Q$-Fano fqav's, and show that $\mathbb Q$-Fano fqav's and Q-abelian varieties are ``building blocks'' of general fqav's.