Scalar curvature on compact complex manifolds

Xiaokui Yang Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China; HCMS, CEMS, NCNIS, HLM, UCAS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Differential Geometry mathscidoc:2207.10001

Transactions of the American Mathematical Society, 371, (3), 2073-2087, 2019.2
In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if $K_X$ (resp., $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti, and B. Weinkove.
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@inproceedings{xiaokui2019scalar,
  title={Scalar curvature on compact complex manifolds},
  author={Xiaokui Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703152040545845524},
  booktitle={Transactions of the American Mathematical Society},
  volume={371},
  number={3},
  pages={2073-2087},
  year={2019},
}
Xiaokui Yang. Scalar curvature on compact complex manifolds. 2019. Vol. 371. In Transactions of the American Mathematical Society. pp.2073-2087. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703152040545845524.
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