Nakai-Moishezon criterions for complex Hessian equations
Nakai-Moishezon criterions for complex Hessian equations
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复杂 Hessian 方程的 Nakai-Moishezon 准则
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Jian Song
中科院分区：
文献类型：
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作者：
Jian Song
唐纳森提出的$J$-方程是Kahler流形上的一个复Hessian商方程。Song-Weinkove证明了$J$-方程的可解性等价于其子解的存在性. Lejmi-Szekelyhidi也证明了它等价于代数几何中Nakai-Moishezon准则的一个类似的全纯相交数的稳定性条件。最近，Chen在一个更强的一致稳定性条件下证明了这个猜想。本文建立了解析Kahler簇上Kahler类对的Nakai-Moishezon型判别准则。因此，我们证明了Lejmi-Szekelyhidi的原始猜想的$J$-方程。我们也应用这样的标准，以获得家庭的常数标量曲率Kahler度量光滑极小模型。
The $J$-equation proposed by Donaldson is a complex Hessian quotient equation on Kahler 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 Kahler classes on analytic Kahler 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 Kahler metrics on smooth minimal models.