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

在这项研究中，将我们的研究集中在全体形态当量问题，霍尔态自动构成群体，莱因哈特域和托鲁斯的行动中，我们获得了以下结果。1。creply.1. creply，对于某些无绑定的Reinhardt域而言，给出了一个被称为Reinhardt reinhardtary Reinhardt domains domains的固定型的答案。我们已经为这个问题提供了另一种方法，该方法使用了尸体形态自动形态群体的理论。同样，与Reinhardt领域的研究相关，我们通过省略来自复杂欧几里得空间的坐标超平面而获得了对空间X的群体理论表征的研究，并成功地在不需必要的Stein的歧管类别中表征X。作为这项研究的副产品，我们已经获得了等级n紧凑型谎言组对n维复杂流形的标准化的结果自动形态群体，我们制作了一个公式，通过利用霍明型汽车组，将riemann映射到更高尺寸的情况下。根据这个公式，我们通过其圆形自动形态组的直接表征了球的直接。3。在《圆环动作研究》的一部分中，我们研究了Bando-Calabi-Futaki特征。特别是，我们对极化代数变化的稳定性与恒定标量曲率的Kaehler指标的存在（所谓的Hitchin-Kobayashi对应关系）之间的连接进行了详细的卵形化。同样，与研究一维圆环作用对二维复杂歧管的研究有关，我们已经获得了有关二维准圆形域的全态自动构成群体的新知识。较少的