On $H^{p,q}(X,E)$ of Weakly 1-Complete Manifolds

Abstract

This note is a continuation of the author's previous work [4]. Let X be a complex manifold of dimension n and n: B— >X a holomorphic line bundle. We want to know the <?-th cohomology group of X with coefficients in the sheaf of germs of Bvalued holomorphic ^-forms on X. We denote it by H'(X,E). X is called weakly 1complete if there exists an exhausting C plurisubharmonic function <p on X. <p is called an exhaustion function. In [4] , we considered the case that B has a metric along the fibers whose curvature form is positive outside a compact subset K of X, and obtained the following theorems.