Study of group actions on complex manifolds

在本研究中，我们得到了关于复流形上的群作用的如下结果。关于复流形上的群作用，我们得到了无界Reinhardt域上环面作用的正规化的基本结果。更准确地说，我们已经给出了一类基本的无界Reinhardt域的全纯等价问题的答案，并且作为应用，我们证明了这类Reinhardt域上的环面作用的共轭性。结合复数整环上的群作用的研究，利用关于全纯映射和CR-映射的著名延拓定理以及Webster的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一个刻画。与研究复杂结构域边界上的基团作用有关，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向量空间范畴中建立对偶空间和张量积的概念。作为环面作用的研究，我们研究了Bando-Calabi-Futaki特征标，它是Futaki特征标的推广。此外，我们还计算了阿贝尔变种的简并的特征缺陷。通过对复域边界的分析，证明了强伪凸CR流形上Cauchy-Riemann方程的Morrey-Holder估计。此外，为了在更广泛的数学框架内研究群作用，我们发展了等变调和映射的研究。

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.