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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.