On real bisectional curvature for Hermitian manifolds

Abstract: Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.