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

TL;DR

This paper establishes a Nakai-Moishezon type criterion for the supercritical deformed Hermitian-Yang-Mills equation, linking stability conditions to the existence of solutions on compact K"ahler and projective manifolds, advancing the understanding of mirror symmetry.

Contribution

It introduces a new stability criterion for the supercritical phase case, confirming a conjecture for projective manifolds and extending previous results without requiring a uniform constant.

Findings

Proves a Nakai-Moishezon type criterion for the supercritical deformed Hermitian-Yang-Mills equation.

Confirms the conjecture for projective manifolds in the supercritical phase.

Establishes a stability condition analogous to recent work on the J-equation.

Abstract

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact K\"ahler 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.