The L^2 dbar method, weak Lefschetz theorems, and the topology of Kahler manifolds

Abstract

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the dbar-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth projective varieties of positive dimension and f is a holomorphic immersion of Y into X with ample normal bundle, then the image of the fundamental group of Y in that of X is of finite index. This result is obtained as a consequence of a direct generalization of Nori's theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in complex Euclidean space (of dimension at least 3) by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold X of dimension at least 3 has a strongly pseudoconvex end E and the Ricci curvature of X is bounded above by a negative constant, then, away from E, X has finite volume.