On varieties with trivial tangent bundle

TL;DR

This paper characterizes abelian varieties with trivial tangent bundle in characteristic p>0 using crystalline cohomology, proving a conjecture for surfaces and proposing modifications based on new insights.

Contribution

It provides a crystalline characterization of abelian varieties with trivial tangent bundle in positive characteristic, including proofs of conjectures for surfaces and new refinements.

Findings

A smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety iff its second crystalline cohomology is torsion free.

Confirmed Li's conjecture for surfaces with trivial tangent bundle.

Proposed modifications to Li's conjecture based on the new characterization.

Abstract

I prove a crystalline characterization of abelian varieties in characteristic $p>0$ amongst the class of varieties with trivial tangent bundle. I show using my characterization that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion free. I also show that a conjecture of Ke-Zheng Li about characteristic $p>0$ varieties with trivial tangent bundles is true for surfaces. I give a new proof of a result of Li and prove a refinement of it. Based on my characterization of abelian varieties I propose modifications of Li's conjecture which I expect to be true.