Semispecial tensors and quotients of the polydisc

TL;DR

The paper characterizes certain complex-projective varieties with klt singularities and ample canonical divisor as quotients of the polydisc using semispecial tensors, extending previous results to singular spaces.

Contribution

It generalizes a characterization of polydisc quotients to singular varieties by introducing semispecial tensors with reduced hypersurfaces.

Findings

X is a quotient of the polydisc if and only if it admits a semispecial tensor with reduced hypersurface.

Established the Bochner principle for holomorphic tensors on klt spaces in the negative Kähler–Einstein case.

Extended Catanese and Di Scala's result to singular spaces.

Abstract

Let $X$ be a complex-projective variety with klt singularities and ample canonical divisor. We prove that $X$ is a quotient of the polydisc by a group acting properly discontinuously and freely in codimension one if and only if $X$ admits a semispecial tensor with reduced hypersurface. This extends a result of Catanese and Di Scala to singular spaces, and answers a question raised by these authors. As a key step in the proof, we establish the Bochner principle for holomorphic tensors on klt spaces in the negative K\"{a}hler--Einstein case.