Equivariant calculus on $μ$-character and $μ$K-stability of polarized schemes
Eiji Inoue
备注：
v2. Expressions are drastically revised. Sign convention is clarified. A section on a proposal on application to moduli problem, originally section 3.3, is removed, but the same content can be found in my thesis. I intend to write a separate article for these applications to moduli
We introduce and study $μ$K-stability of polarized schemes with respect to general test configurations as an algebro-geometric aspect of the existence of $μ$-cscK metrics, which is introduced in the paper arXiv:1902.00664 as a framework unifying the frameworks of Kähler-Ricci solitons and cscK metrics. This article consists of two ingredients.
On one hand, we develop a foundational framework concerning `derivative of relative equivariant intersection', which we call equivariant calculus. A core claim in equivariant calculus is a convergence result for some infinite series in equivariant cohomology, which is obtained by relative equivariant intersections. Our proof is based on some observations on deRham-Cartan model of equivariant locally finite homology. This framework furnishes a language to describe $μ$K-stability. We in particular conclude the $μ$K-semistability of $μ$-cscK manifolds with respect to general test configurations.
On the other hand, we introduce an equivariant character $\mathbf{\checkμ}^λ$ called $μ$-character for equivariant family of polarized schemes, motivated by the $μ$-volume functional introduced in the paper arXiv1902.00664 as a generalization of Tian-Zhu's functional in the theory of Kähler-Ricci soliton. The equivariant derivative of the $μ$-character derive $μ$-Futaki invariant for general test configuration, and furthermore, it also produces an analogue of the equivariant first Chern class of CM line bundle for family of polarized schemes, which is irrational and hence cannot be realized as a $\mathbb{Q}$-line bundle in our general $μ$K-stability setup.
本文讨论[A. Buch '02]和[A.克努森-T Tao '03]。也就是说，我们证明了一个组合规则的环面等变K-理论的格拉斯曼关于基础的舒伯特结构层的结构系数。我们由此推出[H]的代数规则。A-A Yong '13]的相同系数。这两个规则在[D]的意义上都是积极的。安德森Griffeth-E.米勒'11]（而且在更强的形式）。我们的工作是基于组合的基因组tableaux和推广[M.- P. Schutzenberger '77]'s jeu de taquin.
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Qiuzhen College, Tsinghua University（清华大学丘成桐学院）；University of Science and Technology of China（中国科学技术大学）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）；Center for combinatorics, Nankai University（南开大学组合数学中心）
