Positivity of the second exterior power of the tangent bundles

Abstract

Let $X$ be a smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite \'etale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety $F$ with nef $\bigwedge^2 T_F$. As a bi-product, we see that either $T_X$ is nef or $X$ is a Fano variety. Moreover we study a contraction of a $K_X$-negative extremal ray $\varphi: X \to Y$. In particular, we prove that $X$ is isomorphic to the blow-up of a projective space at a point if $\varphi$ is of birational type. We also prove that $\varphi$ is a smooth morphism if $\varphi$ is of fiber type. As a consequence, we give a structure theorem of varieties with nef $\bigwedge^2 T_X$.