Positivity of the second exterior power of the tangent bundles

Abstract

Let X be a smooth complex projective variety with nef ⋀2TX and dim⁡X≥3. We prove that, up to a finite étale cover X˜→X, the Albanese map X˜→Alb(X˜) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef ⋀2TF. As a bi-product, we see that TX is nef or X is a Fano variety. Moreover we study a contraction of a KX-negative extremal ray φ:X→Y. In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if φ is of birational type. We also prove that φ is a smooth morphism if φ is of fiber type. As a consequence, we give a structure theorem of varieties with nef ⋀2TX.