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 
http://mathoverflow.net/questions/36388/is-the-complex-moduli-of-quintic-calabi-yau-toric