The L ∂̄-method, Weak Lefschetz Theorems, and the Topology of Kähler Manifolds

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 π1(Φ −1(Z)) in π1(Z) is of finite index.