Ricci Curvatures on Hermitian manifolds

杨晓奎 MCM of Chinese Academy of Science Kefeng Liu UCLA

Differential Geometry mathscidoc:1608.10049

Tran. AMS
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.
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  title={Ricci Curvatures on Hermitian manifolds},
  author={杨晓奎, and Kefeng Liu},
  booktitle={Tran. AMS},
杨晓奎, and Kefeng Liu. Ricci Curvatures on Hermitian manifolds. In Tran. AMS. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821142122652163360.
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