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

本文围绕着全纯等价问题、全纯自同构群、Reinhardt域和环面作用量的研究，得到了如下结果：1.最近，我们给出了一类称为初等Reinhardt域的无界Reinhardt域的全纯等价问题的一个解答，其方法是利用多复格林函数。我们给出了利用全纯自同构群理论解决这个问题的另一种方法。此外，与Reinhardt域的研究相关，我们还发展了通过省略复欧氏空间中的坐标超平面而得到的空间X的群论特征的研究，并成功地刻画了X在不一定是Stein的流形范畴中的特征。作为研究的一个副产品，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，并且作为应用，我们成功地实现了…2.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将黎曼映射定理推广到高维情形。3.作为环面作用研究的一部分，我们研究了Bando-Calabi-Futaki特征标。特别地，我们对极化代数簇的稳定性与常数量曲率的Kaehler度量(流形上的Hitchin-Kobayashi对应)的存在性之间的关系作了详细的观察。同时，结合二维复流形上一维环面作用的研究，我们得到了二维拟圆域的全纯自同构群的新知识。较少

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