Weakly complete complex surfaces

TL;DR

This paper classifies weakly complete complex surfaces with real analytic plurisubharmonic exhaustion functions, identifying them as modifications of Stein spaces, fibrations over complex curves, or surfaces with Levi-flat hypersurfaces foliated by dense complex leaves.

Contribution

It provides a complete classification of such surfaces, introducing the concept of Grauert type surfaces and analyzing their geometric structures.

Findings

Surfaces of Grauert type have Levi-flat hypersurfaces foliated by dense complex leaves.

Not all weakly complete surfaces admit a real analytic plurisubharmonic exhaustion function.

Proper pluriharmonic functions characterize certain Levi-flat hypersurfaces in these surfaces.

Abstract

A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. An example of Brunella shows that not every weakly complete surface can be endowed with a real analytic plurisubharmonic exhaustion function. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the minimal singular set, i.e the set where every plurisubharmonic exhaustion function has a degenerate Levi form.