Projectively flat KLT varieties

TL;DR

This paper characterizes certain projective varieties with klt singularities that have a projectively flat cotangent sheaf, showing that torus quotients are unique under specific stability and Chern class conditions.

Contribution

It generalizes previous work by Jahnke-Radloff, establishing that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes, and extends results to nef cases.

Findings

Torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes.

Analogous results hold for varieties with nef normalized cotangent sheaves.

Provides a classification in the context of uniformisation problems.

Abstract

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.