Positivity of the exterior power of the tangent bundles

TL;DR

This paper proves that complex smooth projective varieties with a nef exterior power of the tangent bundle are, up to an étale cover, Fano fiber spaces over Abelian varieties, extending previous structure theorems.

Contribution

It generalizes the structure theorem for varieties with nef tangent bundles to those with nef exterior powers of the tangent bundle.

Findings

Varieties with nef exterior power of tangent bundle are Fano fiber spaces over Abelian varieties.

Extends the structure theorem of Demailly, Peternell, and Schneider.

Provides an answer to a question by Li, Ou, and Yang for strictly nef exterior powers.

Abstract

Let $X$ be a complex smooth projective variety such that the exterior power of the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. We prove that, up to an \'etale cover, $X$ is a Fano fiber space over an Abelian variety. This gives generalizations of the structure theorem of varieties with nef tangent bundle by Demailly, Peternell and Schneider and that of varieties with nef $\bigwedge^{2} T_X$ by the author. Our result also gives an answer to a question raised by Li, Ou and Yang for varieties with strictly nef $\bigwedge^{r} T_X$ when $r < \dim X$.