Numerical characterization of complex torus quotients

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.

Equality in the Miyaoka–Yau inequality and uniformization of non-positively curved klt pairs

“…The result follows from standard arguments (see e.g. [15,Prop. 4.4] and references therein) once one has proved that the formation of c 2 (X ) • a is invariant under parallel transport along a locally trivial deformation, which we now prove.…”

“…The exact same arguments as in [15,Prop. 3.11] using orbifold forms instead of usual forms shows that the latter quantity is non-negative, and if it is zero, then we have c 2 (X ) • γ n−2 = 0 for any Kähler class γ on X .…”

“…This implies that T X is stable with respect to π * β, hence T X is stable with respect to π * β + ε β for ε > 0 small enough, cf e.g. [15,Prop. 3.4].…”

“…We will derive the general Kähler case from the projective one using a deformation argument, as in [15,Prop. 4.4].…”

“…The question of uniformizing spaces (as opposed to pairs) in the cases (I) and (II) has been considered in the framework of klt singularities. To quote a few relevant papers: [15,24,27,28,29,30,38]. This article grew out of an attempt to understand the general situation with an orbifold structure in codimension one.…”