Big vector bundles and compact complex manifolds with semi-positive tangent bundles

Abstract

We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^{\ast }X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such that $P(T^{\ast }X)$ has nef tangent bundle, then $X\cong P^n$.