Nakai-Moishezon criterions for complex Hessian equations

By Jian Song. 2020

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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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/177752438e2a87bf92744b389bdeb78861a6fbe2

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In Applied Mathematics

1. V. Datar
2. Vamsi Pingali
3. Chi Li

In Geometry and Topology

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3. Chi Li

In Mathematical Physics

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Citations in the corpus

• A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, background): “In [36], Jian Song proved a version of Theorem 1.”