Hermitian manifolds with semi-positive holomorphic sectional curvature

Xiaokui Yang

Published 2014-12-17Version 1

We prove that a compact Hermitian manifold with semi-positive but not identitically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although they have nef tangent bundles. We also give examples of projective manifolds in all dimensions such that they have smooth K\"ahler metrics with strictly positive holomorphic sectional curvature, but their anti-canonical line bundles are not even pseudo-effective.