Positivity of the second exterior power of the tangent bundles

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