On the compactification of concave ends

TL;DR

This paper proves that the concave end of a complex 2-manifold with a proper strictly plurisubharmonic function can be compactified precisely when its first cohomology group is Hausdorff.

Contribution

It establishes a necessary and sufficient condition for the compactification of the concave end of certain complex manifolds based on cohomological properties.

Findings

Concave ends can be compactified if and only if the first cohomology is Hausdorff.

Provides a characterization linking geometric compactification to topological cohomology.

Advances understanding of the structure of complex manifolds with specific plurisubharmonic functions.

Abstract

Let X be a complex manifold of dimension 2, which admits a strictly plurisubharmonic function r which is proper as a function with values in the intervall ]inf r, sup r[. We prove that the concave end of X can be compactified, if and only if, the first cohomology of X is Hausdorff.