Topology of K\"ahler manifolds with weakly pseudoconvex boundary

TL;DR

This paper investigates the topology of non-compact Kähler manifolds with weakly pseudoconvex boundaries, establishing lower bounds on their first Betti number based on boundary components and end types, with applications to ALE and 4-manifolds.

Contribution

It extends previous results on Kähler manifolds with boundary by providing new topological bounds and simpler proofs, especially for ALE and 4-manifolds.

Findings

First Betti number is at least l-1 for manifolds with l boundary components.

Boundary components have non-vanishing first Betti number.

Results apply to Kähler ALE and 4-manifolds, extending prior work.

Abstract

We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge1$ pseudoconvex boundary components and at least one non-parabolic end, the first betti number of $K$ is at least $l$. In either case, any boundary component has non-vanishing first betti number. If $K$ has one pseudoconvex boundary component with vanishing first betti number, the first betti number of $K$ is also zero. Especially significant are applications to Kahler ALE manifolds, and to Kahler 4-manifolds. This significantly extends prior results in this direction (eg. Kohn-Rossi), and uses substantially simpler methods.