Positivity of the second exterior power of the tangent bundles

Abstract

Let X be a smooth complex projective variety with nef 2 T X 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 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 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 2 T X .