Projective varieties with nef tangent bundle in positive characteristic

TL;DR

This paper studies smooth projective varieties over fields of positive characteristic with nef tangent bundles, showing they admit special fibrations with Fano fibers and relating their structure to abelian varieties under Frobenius liftability.

Contribution

It proves the existence of smooth fibrations with Fano fibers and flat tangent bundles for such varieties, and links Frobenius liftability to their structure as fibered products over abelian varieties.

Findings

Existence of smooth morphisms with Fano fibers and flat tangent bundles.

Frobenius liftability implies the variety is fibered over an abelian variety.

Varieties are, up to finite cover, fibered over ordinary abelian varieties.

Abstract

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite \'etale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.