Scalar curvature on compact complex manifolds

Yang, Xiaokui

doi:10.1090/tran/7409

Scalar curvature on compact complex manifolds
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Trans. Amer. Math. Soc. 371 (2019), 2073-2087 Request permission

Abstract:

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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