The J-equation and the supercritical deformed Hermitian–Yang–Mills equation

Abstract

In this paper, we prove that for any Kähler metrics \(\omega _0\) and \(\chi \) on M, there exists a Kähler metric \(\omega _\varphi =\omega _0+\sqrt{-1}\partial {\bar{\partial }}\varphi >0\) satisfying the J-equation \({\mathrm {tr}}_{\omega _\varphi }\chi =c\) if and only if \((M,[\omega _0],[\chi ])\) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with \(c_1<0\). Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

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Correspondence to Gao Chen.

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The author wishes to thank Xiuxiong Chen for suggesting this problem and providing valuable comments. The author is also grateful to Jingrui Cheng for pointing out a gap in the first version of this paper; to Helmut Hofer for a discussion about symplectic geometry; to anonymous referees for useful comments that made this article more readable; and to Simone Calamai, Jiyuan Han, Long Li, Yaxiong Liu, Vamsi Pingali, and Ryosuke Takahashi for minor suggestions. Sections 1–4 were based upon work supported by the National Science Foundation under Grant No. 1638352 and by a fund from the S. S. Chern Foundation for Mathematics Research when the author was a member of the Institute for Advanced Study. Section 5 was supported by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation when the author was an assistant professor of the University of Wisconsin-Madison.

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Chen, G. The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. math. (2021). https://doi.org/10.1007/s00222-021-01035-3

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