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.

在指导原理中，稳定的Cr结构可嵌入变形是正常孤立奇点的变形的边界类似物，我们研究了在正常孤立奇异性链路上稳定嵌入Cr结构的稳定嵌入变形的描述。特别是，我们专注于在以下三种情况下建立描述该方法的方法。（i）完整的交点奇异性，（ii）准同质奇异性和（iii）商奇异性。 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.我们意识到，等级控制了定义奇点方程的变形的分级。此外，如果奇异性是圆锥形奇异性，则分级控制奇异性分辨率的变形。这些分级论点提供了H. Pinkham和J. Wahl关于准遗传奇异性变形的OREM的cr-（iii）较高维度的奇异性：基于领域的分析，我们的半正常的构建稳定的cr层结构的稳定性变形提供了sch schs sche schers cristers corpirection cyore theore thy cyoreme cyerem cencectectices（这些cy）（这些cy）（IV）（IV）（iv）（IV）（IV）（IV）（IV）（IV）（IV）（IV）（IV）（IV）（IV）（IV）。是几个有趣的几何形状的起源，例如扭曲器空间，超级卡勒歧管和颤抖）：在程度<__- 4的情况下，我们获得了对奇异性的半宇宙变形的CR描述以及同时分辨率的描述；在程度> __-5的情况下，我们获得了构建半宇宙变形的算法。