Vanishing Theorems on Complete K Ahler Manifolds and Their Applications

Semi-positive line bundles over compact Kahler manifolds have been the focus of studies for decades. Among them, there are several straddling vanishing theorems such as the Kodaira-Nakano Vanishing Theorem, Vesentini-Gigante-Girbau Vanishing Theorems and KawamataViehweg Vanishing Theorem. As a corollary of the above mentioned vanishing theorems one can easily show that a line bundle over compact Kahler manifolds with negative degree has no non-trivial holomorphic sections. The high cohomology vanishing theorems for non-compact complex manifolds were also studied by several authors. Among them, there are the Nakano's vanishing theorem for Nakano-positive vector bundle over weakly 1-complete manifolds, and Andreotti-Vesentini's vanishing theorem for the q-complete manifolds. In the case whereM is a non-compact manifold there are also many works on the nite dimensionness of cohomology group. One of these results proved by N. Mok in [16] gave the nite dimensional estimate for the space of L2-sections in the case where M is a complete noncompact Kahler manifold with nite volume. In this paper we rst show some vanishing theorems for the L2sections of the holomorphic vector bundles over complete nonparabolic Kahler manifolds. By applying the vanishing results and the L2-estimate of @ of Andreotti-Vesentini, we show, among other things, that if M is a non-parabolic Kahler manifold with nonnegative Ricci curvature and