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

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