# On J-equation

## TL;DR

This paper establishes a link between the existence of solutions to the J-equation and uniform J-stability on Kähler manifolds, and extends the approach to the deformed Hermitian-Yang-Mills equation.

## Contribution

It proves the equivalence between solving the J-equation and uniform J-stability, and applies the method to the deformed Hermitian-Yang-Mills equation.

## Key findings

- Existence of solutions to the J-equation is equivalent to uniform J-stability.
- Many constant scalar curvature Kähler metrics with negative first Chern class are constructed.
- A similar stability result is proved for the deformed Hermitian-Yang-Mills equation.

## Abstract

In this paper, we prove that for any K\"ahler metrics $\omega_0$ and $\chi$ on $M$, there exists $\omega_\varphi=\omega_0+\sqrt{-1}\partial\bar\partial\varphi>0$ 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 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 $(\frac{n\pi}{2}-\frac{\pi}{4},\frac{n\pi}{2})$.