Covering of a holomorphically convex manifold carrying a positive line bundle

Abstract

A classical theorem of Siegel [S] asserts that a bounded domain in C covering a compact complex manifold is a domain of holomorphy. In [Wa] Watanabe showed that if a complex manifold D covering a projective manifold M and satisfies H1(D,O∗) = 0, then D is a Stein manifold with H(D,Z) = 0, where O∗ is the sheaf of germs of nowhere-vanishing holomorphic functions and Z is the additive group of integers. The purpose of this paper is to study the case where the base of covering is a holomorphically convex complex manifold carrying a positive holomorphic line bundle. Recall that a complex manifold M is holomorphically convex if, for every infinite subset S of M without limit points, there is a holomorphic function f on M which is unbounded on S. Our main result is the following: