Threefolds with quasi-projective universal cover

TL;DR

This paper investigates compact Kähler threefolds with infinite fundamental groups whose universal covers can be compactified, revealing that such conditions strongly restrict the geometry, leading to a product structure involving affine space.

Contribution

It demonstrates that projective threefolds with infinite fundamental groups and quasi-projective universal covers are isomorphic to a product of an affine space with a simply connected manifold.

Findings

Universal cover can be expressed as a product involving affine space.

Strong restrictions on the geometry of threefolds with such universal covers.

Conditions imply the universal cover is quasi-projective and has a specific product structure.

Abstract

We study compact K\"ahler threefolds X with infinite fundamental group whose universal cover can be compactified. Combining techniques from $L^2$ -theory, Campana's geometric orbifolds and the minimal model program we show that this condition imposes strong restrictions on the geometry of X. In particular we prove that if a projective threefold with infinite fundamental group has a quasi-projective universal cover, the latter is then isomorphic to the product of an affine space with a simply connected manifold.