Positivity of higher exterior powers of the tangent bundle

TL;DR

This paper proves that certain positivity conditions on higher exterior powers of the tangent bundle imply the variety is Fano, and classifies some cases when the Picard number is not one.

Contribution

It establishes a link between the strict nefness of higher exterior powers of the tangent bundle and the Fano property, providing classification results under specific conditions.

Findings

Varieties with strictly nef third, fourth, or (n-1)-th exterior powers of tangent bundle are Fano.

Classification of such varieties when the Picard number is not one.

New insights into the positivity properties of tangent bundles and their geometric implications.

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 $\rho(X)\ne 1$.