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

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.

v2: minor corrections as suggested by referee; accepted for publication by Journal of Algebraic Geometry

References in corpus (5)

• K-stability of Fano varieties: an algebro-geometric approach
• The local fundamental group of a Kawamata log terminal singularity is finite
• Projectively flat KLT varieties
• Semistability and restrictions of tangent bundle to curves
• A decomposition theorem for $\mathbb{Q}$-Fano Kähler-Einstein varieties

Cited by in corpus (4)

• Projectively flat KLT varieties
• A decomposition theorem for $\mathbb{Q}$-Fano Kähler-Einstein varieties
• Miyaoka-Yau inequalities and the topological characterization of certain klt varieties
• On the structure of a log smooth pair in the equality case of the Bogomolov-Gieseker inequality