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≥2 boundary components (possibly l=∞), 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≥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.