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Ricci curvatures on Hermitian manifolds


Authors: Kefeng Liu and Xiaokui Yang
Journal: Trans. Amer. Math. Soc. 369 (2017), 5157-5196
MSC (2010): Primary 53C55, 32Q25; Secondary 32Q20
DOI: https://doi.org/10.1090/tran/7000
Published electronically: March 17, 2017
MathSciNet review: 3632564
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Abstract: 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-Kähler 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 $ \mathbb{S}^{2n-1}\times \mathbb{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 non-negative, 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 non-negative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.


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Additional Information

Kefeng Liu
Affiliation: Department of Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China — and — Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095
Email: liu@math.ucla.edu

Xiaokui Yang
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China — and — Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,100190, People’s Republic of China
Email: xkyang@amss.ac.cn

DOI: https://doi.org/10.1090/tran/7000
Received by editor(s): April 23, 2015
Received by editor(s) in revised form: April 16, 2016, and May 20, 2016
Published electronically: March 17, 2017
Additional Notes: The first author was supported in part by an NSF Grant.
The second author was partially supported by China’s Recruitment Program of Global Experts and National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences
Article copyright: © Copyright 2017 American Mathematical Society