Numerical characterization of complex torus quotients

TL;DR

This paper characterizes quotients of complex tori by finite groups using numerical conditions on Chern classes, extending previous results to broader settings and including a Bogomolov--Gieseker inequality for singular spaces.

Contribution

It generalizes the characterization of complex torus quotients to non-projective cases and develops a Bogomolov--Gieseker inequality for stable sheaves on singular spaces.

Findings

Numerical vanishing condition characterizes complex torus quotients.

Extension of previous results to non-projective and higher-dimensional cases.

A version of the Bogomolov--Gieseker inequality for singular spaces.

Abstract

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb--Kebekus--Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov--Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.