Pseudoconvex submanifolds in Kahler 4-manifolds

Abstract

On Kahler 4-manifolds, not necessarily compact or of finite topological type, we obtain relationships between the fundamental group of compact embedded Levi-flat or pseudoconvex submanifold and the fundamental group of the ambient manifold $M^4$. When a Levi-flat submanifold $V^3$ has finite fundamental group then $\pi_1(M^4)=\iota_*\pi_1(V^3)$; when a non-separating pseudoconvex submanifold $V^3$ has finite fundamental group, then $\pi_1(M^4)=\iota_*\pi_1(V^3)\rtimes\mathbb{Z}$. As applications, if a Kahler manifold (compact or not) has an embedded holomorphic $\mathbb{P}^1$ of positive self-intersection, it must intersect all other holomorphic $P^1$ of non-negative self-intersection, the fundamental group of $M^4$ is trivial, and no ALE or ALF ends exist. If a Levi-flat submanifold and an embedded holomorphic $\mathbb{P}^1$ of positive self-intersection both exist, they intersect. The total number of ALE plus ALF ends is zero or one regardless of what other kinds of ends exist. We provide examples, such as a 2-ended scalar-flat Kahler metric conformal to the Taub-NUT.