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Lie group-theoretic approach to the holomorphic equivalence problem and related various problems in several complex variables

全纯等价问题的李群理论方法以及多个复变量中的相关各种问题

基本信息

批准号：
14540149
负责人：
SHIMIZU Satoru
金额：
$ 2.3万
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：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

In this research, centering our study in the holomorphic equivalence problem, holomorphic automorphism groups, Reinhardt domains, and torus actions, we have obtained the following results.1.Recently, an answer was given to the holomorphic equivalence problem for certain unbounded Reinhardt domains called elementary Reinhardt domains, whose method uses pluricomplex Green functions. We have given another approach to this problem that uses the theory of holomorphic automorphism groups. Also, related to the study of Reinhardt domains, we have developed the study of a group-theoretic characterization of the space X obtained by omitting the coordinate hyperplanes from the complex Euclidean space, and succeeded in characterizing X in the category of manifolds that are not necessarily Stein. As a by-product of this study, we have obtained a result on the standardization of rank n compact Lie group actions on n-dimensional complex manifolds, and moreover, as an application, we have succeeded in … More characterizing the space given as the direct product of the ball and the complex Euclidean space by its holomorphic automorphism group as well.2.As part of the study of the holomorphic equivalence problem and holomorphic automorphism groups, we have made one formulation to generalize the Riemann mapping theorem to the higher-dimensional case by utilizing holomorphic automorphism groups. According to this formulation, we have characterized the direct of balls by its holomorphic automorphism group.3.As part of the study of torus actions, we have studied the Bando-Calabi-Futaki character. In particular, we have made a detailed ovservation of the connection between the stability of polarized algebraic varieties and the existence of Kaehler metrics of constant scalar curvature (what is called Hitchin-Kobayashi correspondence for manifolds). Also, related to the study of one-dimensional torus actions on two-dimensional complex manifolds, we have obtained a new knowledge of the holomorphic automorphism groups of two-dimensional quasi-circular domains. Less

本文主要研究了全纯等价问题、全纯自同构群、Reinhardt域和环面作用，得到了如下结果：1.最近，利用复绿色函数，给出了一类无界Reinhardt域（称为初等Reinhardt域）的全纯等价问题的一个解答.我们已经给出了另一种方法来解决这个问题，使用全纯自同构群的理论。此外，与莱因哈特域的研究有关，我们已经开展了对空间X的群论特征的研究，该特征是通过从复欧氏空间中省略坐标超平面而获得的，并且成功地在流形范畴中表征了X，而流形不一定是Stein。作为这一研究的副产品，我们得到了n维复流形上的秩n紧李群作用的标准化结果，而且，作为应用，我们成功地 ...更多信息作为全纯等价问题和全纯自同构群研究的一部分，我们利用全纯自同构群给出了一个公式，将Riemann映射定理推广到高维情形.在此基础上，我们利用球的全纯自同构群刻画了球的直。3.作为研究环面作用的一部分，我们研究了Bando-Calabi-Futaki特征标。特别地，我们已经详细地讨论了极化代数簇的稳定性与常数量曲率的Kaehler度量的存在性之间的联系（流形上的Hitchin-Kobayashi对应）。同时，与二维复流形上的一维环面作用的研究相关，我们得到了二维拟圆域的全纯自同构群的一个新的认识。少