Weakly complete domains in Grauert type surfaces

TL;DR

This paper investigates the structure of weakly complete subdomains within Grauert type surfaces, showing they are either modifications of Stein spaces or the surfaces themselves, with applications to Hopf surfaces.

Contribution

It characterizes weakly complete subdomains of Grauert type surfaces, revealing their possible structures and extending understanding of their geometry.

Findings

Weakly complete subdomains are modifications of Stein spaces or Grauert type surfaces.

Results apply specifically to the case of Hopf surfaces.

Provides a classification of such subdomains based on their geometric properties.

Abstract

The aim of this short note is to investigate the geometry of weakly complete subdomains of Grauert type surfaces, i.e. open connected sets D, sitting inside a Grauert type surface X, which admit a smooth plurisubharmonic exhaustion function. We prove that they are either modifications of Stein spaces or Grauert type surfaces themselves and we apply these results to the special case of Hopf surfaces.