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

Abstract

We classify compact Kahler 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 \({\mathbb P}(T^*X)\) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such that \({\mathbb P}(T^*X)\) has nef tangent bundle, then \(X\cong {\mathbb P}^n\).