Let $V$ be a smooth algebraic variety defined over complex numbers. Suppose that $G$ admits a free discrete action on $V$ so that $V/G$ is compact and Kahler (or algebraic). Is it ture that $G$ is virtually abelian (i.e., contains an abelian subgroup of finite index)?
If we don't ask $V/G$ to be algebraic (Kahler) there are lots of counterexamples, we can take a complex Lie group and quotient it by a co-compact lattice. But if $V/G$ is algebraic the only example that I can imagine is when $V\cong M\times \mathbb C^n$, where M is compact and $G$ preserves the product. Are there other examples? This question is related to Is the complex moduli of Quintic Calabi-Yau toric?
ADDED. Note that a compact complex nilmanifold, i.e. a complex manifold with a homogenious action of a nilpotent group is Kahler iff it is a trous, http://www.jstor.org/pss/2047375 . So a quotient of a complex nilpotent group by a co-compact subgroup never provides a counter-example.