Greb–Kebekus conjecture on orbifold Chern classes and torus quotients

Let $X$ be a compact complex space of dimension $n$ with klt singularities. A finite group action on a complex torus is free in codimension one if its quotient map is unramified outside a subset of codimension at least two. Let $\mathrm{\tilde c}_{2}(X)$ denote the orbifold second Chern class. Then the following are equivalent:

Greb–Kebekus conjecture.

1. $\mathrm{c}_{1}(X)=0\in \HH2.X.{\mathbb{R}}.$, and there exists a Kähler class $\alpha\in \HH2.X.{\mathbb{R}}.$ such that

$$\mathrm{\tilde c}_{2}(X)\cdot \alpha^{n-2}=0.$$

2. There exists a complex $n$-torus $T$ and a holomorphic action of a finite group $G$ on $T$, free in codimension one, such that

$$X\cong T/G.$$

This characterizes torus quotients using the vanishing of the orbifold second Chern class; the conjecture was formulated and proved in dimension three, while the general case remains open.