Positivity of higher exterior powers of the tangent bundle

Abstract

We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under the assumption that $\rho (X)\ne 1$.

[v4]: final version, updated after referee report