On real bisectional curvature \newline for Hermitian manifolds

杨晓奎 MCM, CAS Fangyang Zheng

Differential Geometry mathscidoc:1703.10002

Trans. Am. Math. Soc., 2019
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 {\em 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.
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  title={On real bisectional curvature \newline  for Hermitian manifolds},
  author={杨晓奎, and Fangyang Zheng},
  booktitle={Trans. Am. Math. Soc.},
杨晓奎, and Fangyang Zheng. On real bisectional curvature \newline for Hermitian manifolds. 2019. In Trans. Am. Math. Soc.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170307201640167280621.
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