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Research on moduli of strongly pseudo-convex CR manifolds embedded in algebraic varieties

嵌入代数簇的强赝凸CR流形模研究

基本信息

批准号：
09640123
负责人：
MIYAJIMA Kimio
金额：
$ 1.92万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640123/
关键词：
CR manifold moduli deformation stability isolated singularty Hodge structure 正規孤立特異点モ-ジュライ

项目摘要

Strongly pseudo-convex CR manifolds of real dimension greater than or equals to five are realized as boundaries of normal Stein spaces and moreover as real hypersurfaces in algebraic varieties with only normal isolated singularities. Based on this connection between strongly pseudo-convex CR manifolds and normal isolated singularities, in the late '70, M. Kuranishi proposed a problem to approach to moduli of normal isolated singularities by means of CR structures on its boundaries. The main part of this research consists of the final accomplishment of the Kuranishi's problem together with establishment three dimensional geometric ∂ィイD2bィエD2-analysis which is crucial for that accomplishment. And this approach is applied to some typical singularities. The main result obtained in this research is as follows. (1) Let V be an analytic subvariety with dimィイD2CィエD2V≧2 in CィイD1NィエD1 with only singularity at the origin and M = V∩SィイD32n-1(/)εィエD3 where we denote SィイD32n-1(/)εィエD3 a sphere centered at the origin and with a small radius ε. There exists the Kuranishi semi-universal family of stably embeddable deformations of CR structures on M and it is realized as a family of boundaries of the semi-universal family of deformations of the germ ( V, o). (2) Let M be a strongly pseudo-convex boundary of a bounded subdomain of a complex surface and E a holomorphic vector bundle over M. If E is extendable to a holomorphic vector bundle on that bounded domain then the tangential Cauchy Riemann operator ∂ィイD2bィエD2: LィイD12ィエD1(E)→LィイD32(/)(0,1)ィエD3(E) has a closed range. (This analytical result is crucial for the construction of the Kuranishi semi-universal family in (1) in the case of dimィイD2RィエD2 M = 3.)

实维数大于或等于5的强伪凸CR流形被实现为正规Stein空间的边界，并且被实现为只有正规孤立奇点的代数变体中的实超曲面。基于强伪凸CR流形与正规孤立奇点之间的这种联系，70年代末，M. Kuranishi提出了一个利用CR结构在其边界上逼近正规孤立奇点模的问题。本研究的主要部分包括最终完成Kuranishi问题，以及建立三维几何∂D2b d2分析，这对完成该问题至关重要。这种方法应用于一些典型的奇点。本研究的主要结果如下：(1)设V是一个解析子变种，在C γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ。存在着Kuranishi semi-universal家庭稳定的可嵌入CR结构的变形在M和意识到作为一个家庭的界限semi-universal变形的家庭生殖(V, o)。(2)让M是有界的强烈pseudo-convex边界子域的复杂表面和E全纯矢量包/ M .如果E可扩展到全纯矢量束有限域上的切向柯西黎曼运营商∂ィイD2bィエD2:LィイD12ィエD1 (E)→LィイD32(/)(0, 1)ィエD3 (E)有一个封闭的范围内。（这一分析结果对于(1)中在dim φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ M = 3的情况下Kuranishi半泛族的构造是至关重要的。）