Non algebraicity of universal cover of K\"ahler surfaces

TL;DR

This paper investigates compact K"ahler surfaces with universal covers that are quasi-projective or quasi-K"ahler, establishing that if the universal cover is affine, the surface must be a quotient of a torus.

Contribution

It proves that compact K"ahler surfaces with affine universal covers are necessarily quotients of tori, revealing a new geometric classification under this condition.

Findings

Surfaces with affine universal cover are quotients of tori.

Universal cover being quasi-projective implies specific structure.

Characterization of K"ahler surfaces based on their universal cover.

Abstract

In this short note, we study compact K\"ahler surfaces whose universal cover can be realized as a quasi-projective (or quasi-K\"ahler) surface. In particular, we show that such a surface is a quotient of a torus if the universal cover is affine.