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.

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