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 dbarmethod, 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.
 Publication:

arXiv eprints
 Pub Date:
 December 1997
 DOI:
 10.48550/arXiv.alggeom/9712030
 arXiv:
 arXiv:alggeom/9712030
 Bibcode:
 1997alg.geom.12030N
 Keywords:

 Algebraic Geometry;
 Mathematics  Algebraic Geometry;
 14E20
 EPrint:
 30 pages