A nakai-moishezon type criterion for supercritical deformed hermitian-yang-mills equation

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact Kahler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry [SYZ96]. Recently, Chen [Che21] proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the J-equation by Song [Son20], which makes further progress toward Collins-Jacob-Yau's original conjecture [CJY15] in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case.