On projective varieties with strictly nef tangent bundles

TL;DR

This paper investigates smooth complex projective varieties with strictly nef tangent bundles or their exterior powers, establishing their rational connectedness and classifying cases where the tangent bundle or its second exterior power is strictly nef.

Contribution

It proves that such varieties are rationally connected and classifies those with strictly nef tangent bundle or its second exterior power.

Findings

Varieties with strictly nef exterior powers are rationally connected.

If the tangent bundle is strictly nef, the variety is projective space.

If the second exterior power is strictly nef and dimension ≥ 3, the variety is projective space or a quadric.

Abstract

In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.