Hermitian manifolds with semi-positive holomorphic sectional curvature

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\mathrm{\infty }$. 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.