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 $\href{https://doi.org/10.1016/0550-3213(96)00434-8 }{[\textrm{SYZ96}]}$. Recently, Chen $\href{ https://doi.org/10.1007/s00222-021-01035-3 }{[\textrm{Che21}]}$ 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 $\href{https://doi.org/10.48550/arXiv.2012.07956}{[\textrm{Son}20]}$, which makes further progress toward Collins–Jacob–Yau’s original conjecture $\href{https://dx.doi.org/10.4310/CJM.2020.v8.n2.a4 }{[\textrm{CJY15}]}$ in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case.