Weak Lefschetz Theorems

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the ¯ ∂-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 : Y − → X is a holomorphic immersion with ample normal bundle, then the image of π 1 (Y) in π 1 (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 C ⋉ (n ≥ 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 n ≥ 3 has a strongly pseudoconvex end E and Ricci (X) ≤ −C for some positive constant C, then, away from E, X has finite volume. 0. Introduction In [No], Nori studied the fundamental group of complements of nodal curves with ample normal bundle in smooth projective surfaces. The main tool was the following weak Lefschetz theorem: Theorem (Nori). Suppose Φ : U − →X is a local biholomorphism from a connected complex manifold U into a connected smooth projective variety X of dimension at least 2 and U contains a connected effective divisor Y with compact support and ample normal bundle. Then, for every Zariski open subset Z of X, the image of π 1 (Φ −1 (Z)) in π 1 (Z) is of finite index.