The $L^2$ $\bar \partial$-method, weak Lefschetz theorems, and the topology of Kähler manifolds

Abstract:

A new approach to Nori’s weak Lefschetz theorem is described. The new approach, which involves the $\bar \partial$-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 $f : Y \rightarrow X$ is a holomorphic immersion with ample normal bundle, then the image of $\pi _1(Y)$ in $\pi _1(X)$ 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 $\Bbb C ^n$ ($n\geq 3$) 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 $n\geq 3$ has a strongly pseudoconvex end $E$ and $\text {Ricci} (X) \leq -C$ for some positive constant $C$, then, away from $E$, $X$ has finite volume.