On real bisectional curvature for Hermitian manifolds

By Xiaokui Yang, Fang Zheng. Transactions of the American Mathematical Society, 2016

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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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e87a8541664d4f5a5ff26d4c1bc8c313eae8b58f