ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, give a structure theorem projective manifold $X$ with pseudo-effective tangent bundle: admits smooth fibration $X \to Y$ to flat $Y$ such that its general fiber is rationally connected. Moreover, by applying theorem, classify all minimal surfaces bundle and study non-minimal surfaces, which provide examples (possibly singular) positively curved