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

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 quasiétale polarized (or int-amplified) endomorphisms. We show that every fqav has a finite quasiétale 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.

18 pages. Comments welcome