Weakly 1-Complete Manifold and Levi Problem

Abstract

Let X be a paracompact complex manifold of dimension «J>1. We call X a weakly 1-complete manifold if there exists a C°° plurisubharmonic function (? on X such that for every ceR (real number), <p~1((— oo? c)} is relatively compact in X. cp is called an exhaustion function of X. It is well known that holomorphically convex manifolds are weakly 1-complete. The converse is not true in general (cf. [1]), so one is led to the problem of seeking natural additional conditions which make weakly 1-complete manifolds holomorphically convex. The content of this article is divided into two parts; Section 1 is devoted to prove some properties of weakly 1-complete manifolds which have a nonconstant holomorphic function. In Section 2, first we present an application of the Nakano's vanishing theorem to the Levi problem on projective spaces and hyperquadrics. These results are not new (cf. [3]) but the method will be of some interest. Next, combining the Nakano's vanishing theorem with the result in Section 1 and a well known theorem (due to Bonnet) of differential geometry we obtain the following.