Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Abstract

Suppose that X X is a projective manifold whose tangent bundle T X T_X contains a locally free strictly nef subsheaf. We prove that X X is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group π 1 ( X ) \pi _1(X) is virtually solvable, then X X is isomorphic to a projective space.