On projective varieties with strictly nef tangent bundles

Abstract

In this paper, we study smooth complex projective varieties X such that some exterior power ⋀rTX of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If TX is strictly nef, then X isomorphic to the projective space Pn. If ⋀2TX is strictly nef and if X has dimension at least 3, then X is either isomorphic to Pn or a quadric Qn.