Nakai-Moishezon criterions for complex Hessian equations
Abstract
The -equation proposed by Donaldson is a complex Hessian quotient equation on K\"ahler manifolds. The solvability of the -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 K\"ahler classes on analytic K\"ahler varieties. As a consequence, we prove Lejmi-Szekelyhidi's original conjecture for the -equation. We also apply such a criterion to obtain a family of constant scalar curvature K\"ahler metrics on smooth minimal models.
Cite
@article{arxiv.2012.07956,
title = {Nakai-Moishezon criterions for complex Hessian equations},
author = {Jian Song},
journal= {arXiv preprint arXiv:2012.07956},
year = {2020}
}