2022/08/01 by Brian Weber, Weber, Brian
Mathematics · #Geometry and complex manifolds #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2208.00975
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 M4. When a Levi-flat submanifold V3 has finite fundamental group then π1(M4)=ι_*π1(V3); when a non-separating pseudoconvex submanifold V3 has finite fundamental group, then π1(M4)=ι_*π1(V3)\rtimesℤ. As applications, if a Kahler manifold (compact or not) has an embedded holomorphic ℙ1 of positive self-intersection, it must intersect all other holomorphic P1 of non-negative self-intersection, the fundamental group of M4 is trivial, and no ALE or ALF ends exist. If a Levi-flat submanifold and an embedded holomorphic ℙ1 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.