The Albanese morphism for hyperelliptic varieties

TL;DR

This paper explicitly describes the Albanese morphism of hyperelliptic varieties, detailing the Albanese variety and fibers, and explores implications for the derived category in specific cases.

Contribution

It provides an explicit description of the Albanese morphism for hyperelliptic varieties and analyzes the nature of its fibers and derived category indecomposability.

Findings

Fibers are abelian or hyperelliptic varieties.

Explicit description of the Albanese morphism in terms of $A$ and $G$.

Derived category of $X$ can be indecomposable in certain cases.

Abstract

We explicitly describe the Albanese morphism of a hyperelliptic variety, i.e., the quotient $X$ of an abelian variety $A$ by a finite group $G$ acting freely and not only by translations, by giving a description of the Albanese variety and the Albanese fibers in terms of $A$ and $G$. In particular, the fibers are themselves abelian or hyperelliptic varieties, and we investigate which can occur in explicit examples. As an application we show that the derived category of $X$ is indecomposable in certain cases.