Study of special domains

在本研究中，我们获得了以下与特殊领域相关的结果.关于管域，我们建立了管域上完备多项式向量场延拓的基本结果，称为延拓定理。此外，利用延拓定理，我们得到了n维复数空间中管域仿射自同构群的李代数中n维交换理想的一个刻画结果.与Reinhardt域或弱伪凸域的研究相联系，我们研究了以球面为边界点的广义复椭球的特征问题，得到了大多数广义复椭球与球重合的Riemann映射定理型结果.此外，为了说明管域的研究是对Reinhardt域研究的补充，我们尝试利用Dadok和Yang对球管流形的分类给出了这一结果的另一种证明，并在一类广义复椭球的情况下获得了成功.与特殊区域边界的研究相关，我们研究了CR结构。特别是，我们已经澄清了DR李代数的集合论表示的一个特征，对应于CR李代数的对偶范畴。作为对环面作用的研究，我们得到了非奇异音调簇上的大量线丛是射影正规的结果。此外，我们还详细研究了Fano流形的两个稳定性概念--K稳定性和CM稳定性--这两个概念是由Tian引入的，与流形的Hitchin-Kobayashi对应有关。与特殊区域的边界几何有关，我们已经给出了常平均曲率曲面的各种推广。

In this research, we have obtained the following results related to special domains.1. Concerning tube domains, we have established the fundamental result on the prolongation of complete polynomial vector fields on a tube domain, which is called the Prolongation Theorem. Besides, using the Prolongation Theorem, we have obtained a result on the characterization of n-dimensional abelian ideals in the Lie algebra of the affine automorphism group of a tube domain in the n-dimensional complex number space.2. Related to the study of Reinhardt domains or weakly pseudoconvex domains, we have made a study of the problem of characterizing generalized complex ellipsoids with spherical boundary points, and obtained the Riemann mapping theorem type result that most of such generalized complex ellipsoids coincide with the balls. Moreover, to clarify the aspect that the study of tube domains complements the study of Reinhardt domains, we have tried to give an another proof of this result by making use of the classification of spherical tube manifolds due to Dadok and Yang, and succeeded in the case of a class of generalized complex ellipsoids.3. Related to the study of the boundaries of special domains, we have investigated the CR structures. In particular, we have clarified a characteristic of set-theoretic representations of DR Lie algebras that correspond to the category dual to CR Lie algebras.4. As a study of torus actions, we have obtained the result that ample line bundles on nonsingular tone varieties are projectively normal. Also, we have made a detailed study of the two notions of stability for Fano manifolds - the K stability and the CM stability - introduced by Tian related to the Hitchin-Kobayashi correspondence for manifolds.5. Related to the geometry of the boundaries of special domains, we have given various generalizations of constant mean curvature surfaces.