A remark on constant scalar curvature Kähler metrics on minimal models

Abstract

In this short note, we prove the existence of constant scalar curvature Kähler metrics on compact Kähler manifolds with semi-ample canonical bundles.

Nakai-Moishezon criterions for complex Hessian equations

“…Finally, we extend the result of [23,19,14] for constant scalar curvature Kähler metrics on smooth minimal models. Theorem 1.2.…”

“…Theorem 1.2 immediately implies the result of [14,18] on the existence of cscK metrics near the canonical class on a smooth minimal model because any sufficiently small perturbation of K X by a Kähler class will satisfy the assumption in Theorem 1.2.…”

A decomposition formula for J-stability and its applications

“…There is another application. We extend [23,Theorem 1.2] to the case for deminormal varieties in [21]. On the other hand, we prove in [21] the following another extension of [23]:…”

“…Remark 8.17. Jian-Shi-Song [23] prove Theorem 8.15 when ∆ = 0 and K X is semiample. Theorem 8.15 is also proved without the assumption that K X is semiample by Z. Sjöström Dyrefelt [43] and by J.…”

“…

For algebro-geometric study of J-stability, a variant of K-stability, we prove a decomposition formula of non-archimedean J -energy of n-dimensional varieties into n-dimensional intersection numbers rather than (n + 1)-dimensional ones, and show the equivalence of slope J H -(semi)stability and J H -(semi)stability for surfaces when H is pseudoeffective. Among other applications, we also give a purely algebro-geometric proof of a uniform K-stability of minimal surfaces due to [23], and provides examples which are J-stable (resp., K-stable) but not uniformly J-stable (resp., uniformly K-stable).

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