On J-equation

In this paper, we prove that for any K"ahler metrics ${\omega }_0$ and $\chi$ on $M$, there exists ${\omega }_{\phi }={\omega }_0+\sqrt{-1}\mathrm{\partial }\bar{\mathrm{\partial }}\phi >0$ satisfying the J-equation ${\mathrm{t}\mathrm{r}}_{{\omega }_{\phi }}\chi =c$ if and only if $(M,[{\omega }_0],[\chi ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature K"ahler metrics with $c_1<0$. Using the same method, we also prove a similar result for the deformed Hermitian-Yang-Mills equation when the angle is in $(\frac{n\pi }{2}-\frac{\pi }{4},\frac{n\pi }{2})$.