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.

References in corpus (1)

• On the constant scalar curvature Kähler metrics, existence results

Cited by in corpus (3)

• A decomposition formula for J-stability and its applications
• Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds
• Convergence of cscK metrics on smooth minimal models of general type