NakaiMoishezon 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 SongWeinkove to be equivalent to the existence of a subsolution. It is also conjectured by LejmiSzekelyhidi to be equivalent to a stability condition in terms of holomorphic intersection numbers as an analogue of the NakaiMoishezon criterion in algebraic geometry. The conjecture is recently proved by Chen under a stronger uniform stability condition. In this paper, we establish a NakaiMoishezon type criterion for pairs of Kähler classes on analytic Kähler varieties. As a consequence, we prove LejmiSzekelyhidi'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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.07956
 arXiv:
 arXiv:2012.07956
 Bibcode:
 2020arXiv201207956S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Analysis of PDEs