A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation

Abstract

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen 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, which makes further progress toward Collins-Jacob-Yau's original conjecture in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case.

43 pages, added new references and Corollary 1.5 on Collins-Jacob-Yau's conjecture for projective manifolds