Virtually abelian K\"ahler and projective groups

TL;DR

This paper characterizes virtually abelian groups that can serve as fundamental groups of compact K"ahler and projective varieties, establishing their equivalence and linking the K"ahler condition to integral symplectic representations.

Contribution

It provides a complete characterization of virtually abelian K"ahler groups, showing they are exactly the virtually abelian projective groups, and relates the K"ahler condition to symplectic representations.

Findings

Virtually abelian K"ahler groups are exactly the virtually abelian projective groups.

The K"ahler condition for these groups can be described via integral symplectic representations.

A virtually abelian group is K"ahler if and only if it is projective.

Abstract

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular, this allows to describe the K\"ahler condition for such groups in terms of integral symplectic representations.