Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor

TL;DR

This paper establishes a criterion linking projective flatness of sheaves on klt spaces to fundamental group representations, and applies it to characterize certain quotients of projective spaces and Abelian varieties through stability and Chern class inequalities.

Contribution

It introduces a new criterion for projective flatness over klt spaces and applies it to classify quotients of projective spaces and Abelian varieties via stability and Chern class conditions.

Findings

Criterion for projectivisation induced by fundamental group representation

Characterisation of quotients of projective spaces and Abelian varieties

Comparison of stability condition with K-semistability

Abstract

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by $\mathbb{Q}$-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.