A note on the covering of projective surfaces (Q1024712)

Let \(X\) be a complex space with topology that has a countable base of open sets and such that there is a continuous plurisubharmonic proper function \(\varphi:X \to [0, \infty).\) Let \((Y, \pi)\) be a couple that consists of a complex space \(Y\) and a holomorphic map \(\pi: Y \to X\) having only discrete fibers. The main result of this paper is Theorem 1.1. If \(X\) carries a positive holomorphic line bundle and \(Y\) is irreducible and every holomorphic line bundle on \(Y\) is trivial, then \(Y\) is Stein and \(H^2 (Y, \mathbb Z) = 0.\) As a corollary of this fact the authors obtain the following: Let \(Y\) be a holomorphic covering of a projective surface \(X.\) If \(Y\) is irreducible and every holomorphic line bundle on \(Y\) is trivial, then \(Y\) is Stein and \(H^2 (Y, \mathbb Z) = 0.\) (Corollary 1.2)