Scalar curvature on compact complex manifolds

In this paper, we prove that, a compact complex manifold XX admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if KXK_X (resp., KX−1K_X^{-1}) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold XX with comple...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-02, Vol.371 (3), p.2073-2087
Main Author: Yang, Xiaokui
Format: Article
Language:English
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Research article
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Summary:In this paper, we prove that, a compact complex manifold XX admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if KXK_X (resp., KX−1K_X^{-1}) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold XX with complex dimension ≥2\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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7409