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Moduli theory of strongly pseudo-convex CR structure and its application to higher dimensional isolated singularities

强赝凸CR结构的模理论及其在高维孤立奇点中的应用

基本信息

批准号：
17540087
负责人：
MIYAJIMA Kimio
金额：
$ 2.3万
依托单位：
Kagoshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540087/
关键词：
CR structure isolated singularity moduli deformation コジュライ

项目摘要

The research purpose is to analytically and differential geometrically investigate the moduli of isolated singularities, especially various phenomena e.g. smoothing of singularities, so-called Brieskorn resolution, etc., relying on interrelation between "complex structure on its resolution", "complex structure on its regular part" and "CR structure on its boundary". And, we also intend to develop approach to moduli theory of metric structure or symplectic structure in connection with deformation of singularities. By this research, we constructed versal deformation of resolution of normal isolated singularities in terms of deformation of complex structures as well as deformations of singularities itself by means of deformation of complex structures on its regular part. These results provide us interrelation between deformation of boundary CR structure of and Brieskorn resolution of rational double point. In addition, we established a method constructing versal family using only subellipticity and optimal estimate for d-bar Neumann problem over a bounded domain with strongly pseudo convex boundaries which is not enough for standard method constructing the versal family. On the other hand, T. Akahori (a co-investigator)proved that the versal deformation space of a rational double point is smooth relying on the Hamiltonian flow over the boundary, which is an argument different form our previous research using the stable deformation theory of CR structures. And, Mitsuhiro Itoh (a co-investigator)proved a Serre duality theorem for holomorphic vector bundles over strongly pseudo-convex compact CR manifolds, which is generalized to an open strongly pseudo-convex compact CR manifolds. Next subject will be investigate interrelation between stable deformation of CR structure (or complex structure)and moduli of various geometric structure on the boundary of normal isolated singularities.

研究目的是从分析和微分几何角度研究孤立奇点的模量，特别是各种现象，例如奇异点的平滑，即所谓的Brieskorn分辨率等，依赖于“其分辨率上的复杂结构”、“其规则部分上的复杂结构”和“其边界上的CR结构”之间的相互关系。而且，我们还打算开发与奇点变形相关的度量结构或辛结构的模理论方法。通过这项研究，我们构建了从复杂结构变形角度解析法向孤立奇点的通用变形，以及通过复杂结构规则部分的变形来构造奇点本身的变形。这些结果为我们提供了边界 CR 结构变形与有理双点 Brieskorn 分辨率之间的相互关系。此外，我们建立了一种仅使用次椭圆性和对具有强伪凸边界的有界域上的 d-bar Neumann 问题的最优估计来构造Versal族的方法，这对于构造Versal族的标准方法是不够的。另一方面，T. Akahori（合作研究员）依靠边界上的哈密顿流证明了有理双点的通用变形空间是光滑的，这与我们之前使用CR结构稳定变形理论的研究不同。并且，Mitsuhiro Itoh（共同研究员）证明了强赝凸紧CR流形上的全纯向量丛的Serre对偶定理，该定理被推广到开强赝凸紧CR流形。下一课题将研究CR结构（或复杂结构）的稳定变形与法向孤立奇点边界上各种几何结构的模量之间的相互关系。

项目成果

期刊论文数量（0）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

The asymptotic behavior of the Takhtajan-Zograf metric

Takhtajan-Zograf 度量的渐近行为

DOI：
发表时间：
2008
期刊：
Communications in Mathematical Physics 284
影响因子：
0
作者：
K. Obitsu;W. -K. To;L. Weng
通讯作者：
L. Weng

A survey of characteristic classes of singular spaces

奇异空间特征类的调查

DOI：
发表时间：
2007
期刊：
影响因子：
0
作者：
K. Obitsu;W. -K. To;L. Weng;K. Obitsu and S. A. Wolpert;K. Miyajima;J.Schurmann and S.Yokura
通讯作者：
J.Schurmann and S.Yokura

On Grothendieck transformations in Fulton-MacPherson's bivariant theory

论富尔顿-麦克弗森二变理论中的格洛腾迪克变换