Topology of Kähler Manifolds with Weakly Pseudoconvex Boundary

Abstract

We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge 2$ boundary components (possibly $l=\infty$), then it has the first Betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge 1$ pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of $K$ is at least $l$. In either case, any boundary component has a nonvanishing first Betti number. If $K$ has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of $K$ is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.