Pseudoconvex submanifolds in Kahler 4-manifolds

TL;DR

This paper explores the relationship between the fundamental groups of pseudoconvex submanifolds and their ambient Kahler 4-manifolds, revealing topological constraints and implications for the structure and ends of such manifolds.

Contribution

It establishes new fundamental group relations for Levi-flat and pseudoconvex submanifolds in Kahler 4-manifolds, with applications to the existence and intersection properties of holomorphic spheres.

Findings

Finite fundamental group of Levi-flat submanifolds implies the ambient group is the image of the submanifold's group.

Presence of a holomorphic ^1 with positive self-intersection constrains intersections and fundamental groups.

Total number of ALE or ALF ends is at most one, with examples including scalar-flat Kahler metrics conformal to Taub-NUT.

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.