Nakai-Moishezon criterions for complex Hessian equations

Abstract

The J-equation proposed by Donaldson is a complex Hessian quotient equation on Kähler manifolds. The solvability of the J-equation is proved by Song-Weinkove to be equivalent to the existence of a subsolution. It is also conjectured by Lejmi-Szekelyhidi to be equivalent to a stability condition in terms of holomorphic intersection numbers as an analogue of the Nakai-Moishezon criterion in algebraic geometry. The conjecture is recently proved by Chen under a stronger uniform stability condition. In this paper, we establish a Nakai-Moishezon type criterion for pairs of Kähler classes on analytic Kähler varieties. As a consequence, we prove Lejmi-Szekelyhidi's original conjecture for the J-equation. We also apply such a criterion to obtain a family of constant scalar curvature Kähler metrics on smooth minimal models.

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

“…Very recently, a remarkable progress for the study of the J-equation was made by Song [Son20]. He further extended the method of [Che21], and showed a Nakai-Moishezon type criterion for the J-equation without assuming a uniform lower bound for the intersection numbers, which confirms Lejmi-Székelyhidi's conjecture [LS15].…”

“…In this paper, we strengthen Chen's result [Che21] by using the same method as [Son20]. For the reader's convenience, let us recall some terminologies and notations first.…”

“…Motivated by [Son20], it is very natural to ask if the uniform stability condition in Theorem 1.2 can be replaced by the stability condition alone. We give an affirmative answer to this question.…”

“…Indeed, the assumptions of Corollary 1. We should mention that there are some significant differences from the arguments in [Son20]. First, we consider the stability condition not only for a single class α but also for a family of cohomology classes α t as in [Che21].…”

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].…”