Study of group actions on complex manifolds

复流形上的群作用研究

• 批准号: 09640145
• 负责人: SHIMIZU Satoru
• 金额: $ 2.24万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640145/
• 关键词: Complex bounded domain; Holomorphic equivalence problem; Holomorphic automorphism group; Reinhardt domain; Torus action; CR-structure; Futaki character; Tangential Cauchy-Riemann equation; 同変調和写像; アーベル多様体の退化; 接コ-シ-・リーマン方程式

In this research, we have obtained the following results related to group actions on complex manifolds.1. Concerning group actions on complex manifolds, we have obtained the fundamental result on the normalization of torus actions on unbounded Reinhardt domains. More precisely speaking, we have given an answer to the holomorphic equivalence problem for a basic class of unbounded Reinhardt domains and, as an application, we have shown the conjugacy of torus actions on such a class of Reinhardt domains.2. Related to the study of group actions on complex domains, we have given a characterization of generalized complex ellipsoids from the viewpoint of biholomorphic automorphism groups by making use of well-known extension theorems on holomorphic mappings and CR-mappings and applying Webster's CR-invariant metrics.3. Related to the study of group actions on the boundaries of complex domains, we have investigated CR structures. In particular, we have shown that it is possible to formulate the concepts of dual spaces and tensor products in the category of CR vector spaces under some additional conditions.4. As a study of torus actions, we have investigated the Bando-Calabi-Futaki character that is a generalization of the Futaki character. Also, we have calculated signature defects of degenerations-of abelian varieties5. Related to the analysis of the boundaries of complex domains, we have proved new Morrey-Holder estimates for the Cauchy-Riemann equations on strongly pseudoconvex CR manifolds. Also, to study group actions in a wider mathematical framework, we have developed the investigation of equivariant harmonic mappings.

在本研究中，我们获得了以下与复流形上的群作用相关的结果： 1．关于复流形上的群作用，我们得到了无界 Reinhardt 域上环面作用归一化的基本结果。更准确地说，我们已经给出了基本类无界莱因哈特域的全纯等价问题的答案，并且作为一个应用，我们展示了此类莱因哈特域上环面作用的共轭性。2．与复杂域上群作用的研究相关，我们利用著名的全纯映射和CR映射的可拓定理，并应用韦氏CR不变度量，从双全纯自同构群的角度给出了广义复椭球的刻画。 3．与复杂域边界上的群体行为研究相关，我们研究了 CR 结构。特别地，我们证明了在一些附加条件下，可以在CR向量空间范畴中表述对偶空间和张量积的概念。 4．作为对环面作用的研究，我们研究了 Bando-Calabi-Futaki 字符，它是 Futaki 字符的概括。此外，我们还计算了阿贝尔变种的退化特征缺陷5。与复杂域边界的分析相关，我们证明了强赝凸 CR 流形上柯西-黎曼方程的新 Morrey-Holder 估计。此外，为了在更广泛的数学框架中研究群体行为，我们开展了等变调和映射的研究。

项目成果

S.Ogata: "Signature defects and eta functions of degenerations of abelian varieties" Japan.J.Math.23. 319-364 (1997)

S.Ogata：“阿贝尔变种的特征缺陷和 eta 函数”Japan.J.Math.23。

S.Takeuchi: "Canonical D/CR structures of tensor spaces for abstract CR manifolds" Proceeding of the Sixth International Colloquium on Complex Analysis, Andong. 128-133 (1998)

S.Takeuchi：“抽象 CR 流形张量空间的规范 D/CR 结构”第六届国际复分析学术研讨会论文集，安东。

Y.Nakagawa: "Bando-Calabi-Futaki characters of Kahler orbifolds (発表予定)" Math.Ann.

Y.Nakakawa：“Kahler orbifolds 的 Bando-Calabi-Futaki 字符（待呈现）”Math.Ann。

H.Arai: "Generalized Dirichlet growth theorem and applications to hypoelliptic and ab equations" Comm.in Partial Differential Equations. 22. 2061-2088 (1997)

H.Arai：“广义狄利克雷增长定理及其在亚椭圆和 ab 方程中的应用”Comm.in 偏微分方程。

H.Arai: "Generalized Dixichlet growth theorem and applications to Qb and hypoelliptic equations" Communications in Partial Differential Equations. 22. 2061-2088 (1997)

H.Arai：“广义 Dixichlet 增长定理及其在 Qb 和亚椭圆方程中的应用”偏微分方程中的通信。