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Research on moduli of the boundary structure of isolated singularities

孤立奇点边界结构模的研究

基本信息

批准号：
12640080
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：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

Under the guiding principle that the stably embeddable deformation of CR structures is the boundary analogue of the deformation of normal isolated singularities, we investigate description of stably embeddable deformation of CR structures on a link of normal isolated singularities. In particular, we concentrated on establishing the method describing it in the following three cases; (i) complete intersection singularities, (ii) quasi-homogeneous singularities and (iii) quotient singularities. The main results are as follows, (i) Complete intersection singularities: Deformation of singularities is controlled by means of CR functions on its link, and we can deduce so-called tha Kas-Schlessinger theorem, (ii) Quasi-homogeneous singularities: A grading induced by the S^1-action associ ating with the quasi-homogenety is introduced in the deformation space. We realized that the grading controlls the grading of deformations of defining equations of the singularities. Furthermore, if the singularities are cone singularities, the grading controlls deformation of resolution of singularities. These graded arguments provide the CR-version of the the orems of H. Pinkham and J. Wahl on deformation of quasi-homogenous singularities, (iii) Higher dimensional quotient singularities: Based on the sphere analysis, our construction of semi-universal family of stably embeddable deformation of CR structures provides the Sch lessingers rigidity theorem, (iv) Cyclic quotient surface singularities: (these singularities are the origin of several interesting geometries, e.g. twistor space, hyper Kaler manifold and quiver): In the case of the degree <__- 4, we obtained the CR description of the semi-universal deformation of the singularity and also description of the simultaneous resolution; in the case of degree >__- 5, we obtained an algorithm constructing the semi-universal deformation.

在CR结构的稳定嵌入变形是法向孤立奇点变形的边界模拟的指导原则下，我们研究了CR结构在法向孤立奇点链接上的稳定嵌入变形的描述。特别是，我们集中精力建立了在以下三种情况下描述它的方法； (i) 完全交集奇点，(ii) 准齐次奇点和 (iii) 商奇点。主要结果如下：（i）完全相交奇点：奇点的变形由其链接上的CR函数控制，并可以推导出所谓的Kas-Schlessinger定理；（ii）拟齐次奇点：在变形空间中引入与准齐次性相关的S^1作用引起的分级。我们意识到分级控制着奇点定义方程的变形分级。此外，如果奇点是圆锥奇点，则分级控制奇点分辨率的变形。这些分级论证提供了 H. Pinkham 和 J. Wahl 关于准齐次奇点变形的定理的 CR 版本，(iii) 高维商奇点：基于球面分析，我们对 CR 结构稳定嵌入变形的半通用族的构造提供了 Sch lessingers 刚性定理，(iv) 循环商表面奇点：（这些奇点是几个有趣的几何形状的起源，例如扭曲空间、超卡勒流形和箭袋）：在度<__- 4的情况下，我们得到了奇点半万能变形的CR描述以及同时解析的描述；在度>__- 5的情况下，我们得到了构造半通用变形的算法。