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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.

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I am not sure that this is correct, but have you tried upper triangular matrices over the complex numbers, divided out by the action of the upper triangular matrices with entries in $\mathbb{Z}[i]$? –  damiano Aug 23 '10 at 15:57
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Damiano: Your construction (for matrices of size at least $3\times 3$) gives a non-Kähler variety (it has non-closed 1-forms). –  Torsten Ekedahl Aug 23 '10 at 16:15
    
@Torsten: thanks for the answer! I was indeed stuck on proving that the quotient was Kaehler. I still think that "nilpotent" might be a better fit than "virtually abelian"... –  damiano Aug 23 '10 at 16:17
    
My earlier comment was wrong so I deleted it.There are examples of compact kahler manifolds with nilpotent fundamental groups due to Campana and Carlson-Toledo.Their universal cover maybe an algebraic variety. –  Mohan Ramachandran Aug 23 '10 at 22:06
    
How your question is related to mathoverflow.net/questions/36388/…? –  Mohammad F. Tehrani Sep 1 '10 at 2:14

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up vote 5 down vote accepted

The following article http://arxiv.org/abs/1102.2762 of BENOÎT CLAUDON, ANDREAS HÖRING, AND JÁNOS KOLLÁR answers positively the question, provided $V/G$ is projective, $V$ is quasiprojective and $\pi_1(V)=0$. Moreover, assuming abundance conjecture, they prove under the above conditions that $V$ is biholomophric to the product of $\mathbb C^n$ with a simply-connected variety.

Added. Interestingly, if we ask that $V/G$ is merely Kahler, the question seems to be open even for $V=\mathbb C^n$ for $n\ge 4$. It is related to (and follows immediately from) "Iitaka's conjecture" predicting that any such compact Kahler quotient $\mathbb C^n/G$ has a finite cover bi-holomorphic to a torus. Iitaka conjecture is discussed in the article "UNIFORMISATION IN DIMENSION FOUR: TOWARDS A CONJECTURE OF IITAKA" of Horing, Peternell, and Radloff : http://arxiv.org/abs/1103.5392

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