具有超越上同调类的常数量曲率Kähler流形的K-半稳定性

K-Semistability of cscK Manifolds with Transcendental Cohomology Class.

作者信息

Sjöström Dyrefelt Zakarias

机构信息

1Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France.

2Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France.

出版信息

J Geom Anal. 2018;28(4):2927-2960. doi: 10.1007/s12220-017-9942-9. Epub 2017 Oct 16.

DOI:10.1007/s12220-017-9942-9

PMID:30595639

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC6294182/

Abstract

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.

摘要

我们证明了具有超越上同调类的常数量曲率凯勒（cscK）流形是K-半稳定的，这自然地推广了极化流形的情形。基于R. 伯曼、T. 达尔瓦斯和C. 卢最近关于K能量恰当性的一个结果，进而可得具有离散自同构群的cscK流形是一致K-稳定的。作为证明的主要步骤，在一般的凯勒情形下，我们建立了一个将（广义的）唐纳森 - 富塔基不变量与K能量沿弱测地线射线的渐近斜率联系起来的公式。