Algebraic varieties with quasi-projective universal cover (Q2838634)

In the early 70's, Shafarevich conjectured that the universal cover \(\widetilde{X}\) of a smooth projective variety \(X\) over \(\mathbb{C}\) is holomorphically convex, that is, there is a proper holomorphic map \(f: \widetilde{X} \to Y\), such that \(f_* \mathcal{O}_{\widetilde{X}} \cong \mathcal{O}_Y\) and \(Y\) is Stein (Stein manifolds loosely speaking, but not completely precisely, correspond to affine varieties in the complex analytic setting). In this paper the authors prove that if \(X\) is a normal projective variety over \(\mathbb{C}\), then assuming the abundance conjecture the following are equivalentNEWLINENEWLINE(a) the universal cover \(\widetilde{X}\) of \(X\) is biholomorphic to a quasi-projective variety,NEWLINENEWLINE(b) \(\widetilde{X}\) is biholomorphic to \(F \times \mathbb{C}^m\), where \(F\) is a projective simply connected variety andNEWLINENEWLINE(c) there is a finite étale cover \(X' \to X\) that is a fiber bundle over an Abelian variety with simply connected fibers.NEWLINENEWLINEThis statement can be viewed both as a (conditional) positive answer to Shafarevich's conjecture in a special case, and also as a characterization of when \(\widetilde{X}\) is biholomorphic to a quasi-projective variety. Further note, that in fact only a special case of the abundance conjecture is assumed: if \(X\) is a smooth projective variety over \(\mathbb{C}\) such that \(K_X\) is nef, then \(K_X\) is semi-ample.NEWLINENEWLINEThe proof consists of two main steps. First, it is shown that the fundmental group of \(X\) is almost abelian, that is, it contains a subgroup of finite index, which is abelian. To this end the authors consider a minimal dimensional quasi-projective variety \(Y\) that has an infinite étale quasi-projective \(\Gamma\)-Galois cover \(\widetilde{Y}\) with \(\Gamma\) not almost abelian and study the effect of running the Minimal Model Program on \(Y\). The main idea is that this yields smaller dimensional examples as above, which is a contradiction. The fundamental group of \(X\) being almost abelian yields a finite Galois étale cover \(X' \to X\) with the fundamental group of \(X'\) being free, abelian. The second, more straightforward, part deals with showing that the Albanese morphism of \(X'\) is a locally trivial holomorphic fiber bundle. The assumption on the abundance conjecture is only used in the first part.