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.

在本文中，我们证明了对于M上的任何Kähler度量\（\ omega _0 \）和\（\ chi \），存在一个Kähler度量\（\ omega _ \ varphi = \ omega _0 + \ sqrt {-1} \ partial {\ bar {\ partial}} \ varphi> 0 \）满足J方程\（{\ mathrm {tr}} _ {\ omega _ \ varphi} \ chi = c \）当且仅当\（ （M，[\ omega _0]，[\ chi]）\）始终是J稳定的。作为推论，我们找到存在\（c_1 <0 \）的恒定标量曲率Kähler度量的充分条件。使用相同的方法，我们还证明了超临界变形Hermitian-Yang-Mills方程的相似结果。