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 [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. Contents 1. Introduction 1 2. Preliminaries 6 3. The higher dimensional case 14 4. The twisted dHYM equation on the product space Y 20 5. The mass concentration and local smoothing 22 6. Gluing construction 33 7. Completion of the proof of Theorem 1.3 39 References 41

The Deformed Hermitian--Yang--Mills Equation, the Positivstellensatz, and the Solvability

“…Jacob-Sheu [21] showed that the numerical conjecture holds on the blowup of complex projective space. Chu-Lee-Takahashi [7] improved the result without assuming a uniform lower bound for these intersection numbers. The method by Chu-Lee-Takahashi was inspired by a remarkable work of Song [30] in the study of the J-equation.…”

“…The author wants to point out that this work shows that when the complex dimension equals three or four, the existence of C-subsolution is equivalent to the solvability of the dHYM equation. In addition, it is also equivalent to the existence of supercritical C-subsolution by Collins-Jacob-Yau [8] and the numerical criterion by Chen [6] (see also Chu-Lee-Takahashi [7]).…”

Hypercritical deformed Hermitian-Yang-Mills equation

“…In [10], Collins-Jacob-Yau characterized the existence of solutions to dHYM in the supercritical phase case in the analytic viewpoint, see also the work of Székelyhidi [41] for the study of a very general class of Hessian type equations on compact Hermitian manifolds under similar analytic assumption. It is also conjectured by Collins-Jacob-Yau [10] that the solvability to dHYM equation is equivalent to a Nakai-Moishezon type criterion in the supercritical phase case, see [4,18,29,9] for the recent progress to this conjecture. For more related works, we refer readers to [27, 43,35,36,26,42] and the references therein.…”

The supercritical deformed Hermitian Yang--Mills equation on compact projective manifolds

“…However, one can do better, and in fact, it was also shown in [Che21] that these point-wise positivity conditions are also equivalent to certain seemingly weaker "numerical" conditions similar to uniform versions of the ones in [DP04, Theorem 4.2]. Recently, these numerical conditions were further weakened to their non-uniform versions in [CLT21] using the methods in [Son20] and [Che21] and consequently [CJY20, Conjecture 1.4] was proved in the projective case. In this paper, we will provide a proof of an analogue of this conjecture in the projective case for the twisted dHYM equation using the methods of [DP20] and [Che21].…”