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 $\dim X\ge 3$. We prove that, up to a finite étale cover $\tilde{X}\to X$, the Albanese map $\tilde{X}\to \mathrm{Alb}(\tilde{X})$ 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 $\phi :X\to Y$. In particular, we prove that $X$ is isomorphic to the blow-up of a projective space at a point if $\phi$ is of birational type. We also prove that $\phi$ is a smooth morphism if $\phi$ is of fiber type. As a consequence, we give a structure theorem of varieties with nef ${\bigwedge }^2{T}_X$.

23 pages, some typos are fixed, to appear in Adv. Math

References in corpus (3)

• Log homogeneous varieties
• Fano manifolds having (n-1,0)-type extremal rays with large Picard number
• On the second and third exterior power of tangent bundles of Fano manifolds with birational contractions