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

在本研究中，我们围绕全纯等价问题、全纯自同构群、Reinhardt域和环面作用进行了研究，得到了以下结果：摘要利用复数格林函数，给出了一类无界Reinhardt域的全纯等价问题的解。我们用全纯自同构群的理论给出了解决这个问题的另一种方法。此外，与Reinhardt域的研究相关，我们发展了通过从复欧几里德空间中省略坐标超平面而得到的空间X的群论表征的研究，并成功地在不一定是Stein的流形范畴中表征了X。作为本研究的一个副产品，我们得到了n维复流形上n阶紧李群作用的标准化结果，并且作为一个应用，我们成功地用球与复欧几里德空间的全纯自同构群进一步刻画了其直接积的空间。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将黎曼映射定理推广到高维情况的公式。根据这个公式，我们用球的全纯自同构群刻画了球的直性。作为环面动作研究的一部分，我们研究了Bando-Calabi-Futaki角色。特别地，我们详细地观察了极化代数变体的稳定性与常标量曲率的Kaehler度量的存在性（流形的Hitchin-Kobayashi对应关系）之间的联系。此外，在二维复流形上一维环面作用的研究中，我们对二维拟圆域的全纯自同构群有了新的认识。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