Projectively flat klt varieties

Daniel Greb,Stefan Kebekus,Thomas Peternell

doi:10.5802/jep.164

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.

Highlights

There exists an étale cover γX : X → X such that the fibration a : A → Y obtained as the Stein factorisation of (f ◦ γX ) : X → Y is birational to an Abelian group scheme over a proper base
We argue by contradiction and assume that there exists a contraction of a KX -negative extremal ray
Step 3: Kodaira dimension. — We aim to show that the minimal variety X is quasiAbelian, which implies in particular that κ(X) = 0

Summary

Introduction

– If X is Fano and Kähler-Einstein, again the equality (1.0.2) cannot occur, owing to the Chen–Oguie inequality In their remarkable paper [JR13], Jahnke and Radloff proved the following complete characterisation of manifolds with semistable cotangent bundle for which Equality (1.0.2) holds. Theorem 1.1 (Characterisation of torus quotients, [JR13, Ths. 0.1 & 1.1]) Let X be a projective manifold of dimension n and assume that Ω1X is H-semistable for some ample line bundle H. — Work of Narasimhan, Seshadri and others, summarised for example in [JR13, Th. 1.1] and explained in detail by Nakayama in [Nak[98], Th. A], can be used to reformulate Jahnke-Radloff’s result in terms of positivity properties of natural tensor sheaves: a projective manifold X of dimension n is quasi-Abelian if and only if the normalised cotangent bundle, Symn Ω1X ⊗OX (−KX ), is nef.

Assume

Projective flatness

Varieties with splitting cotangent sheaves