The application of the Rumin complex to isolated sirgularities

瘤胃复合体在孤立性规则中的应用

• 批准号: 10640211
• 负责人: AKAHORI Takao
• 金额: $ 0.32万
• 依托单位: Himeji Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640211/
• 关键词: CR structure; pseudo-convex; isolated singularity; 接触幾何; 変形理論

Let (v, x) be an isolated singularity in a complex enclidean space CィイD1NィエD1. We consider the intersection of v and the real hypersphere SィイD2εィエD2ィイD12N-1ィエD1(x) (we write M for this intersection). Then M is a non singular real odd dimensional CィイD2NィエD2 manifold. Furthermore, on M, aCR structure is induced from M. This CR structure is of interest. In fact, this CR structure determines the normal isolated singularity. Noting this point, Kuranishi initiated his research on isolated singularities from the point of vie of Cr structures (especially, deformation theory). I followed his line, and in 1981, I succeeded if dimィイD21R ィエD2M=zn-1≧7. But the case dimィイD21R ィエD2M=zn-1≧5 has been left open. Inspired by Rumin's work on contact manifolds, we (with J.M.Lee and Gurfiled constructed a CR version of Rumin's work , and by using the Rumin complex, we settle the case dimィイD21R ィエD2M=zn-1=5. Our complex is efficient in studying the complex structures of isolated singularities from the point of view of Hodge structure. But this is still in preparation.

设（v，x）是复恩克莱德空间CイD1 N D1中的孤立奇点。我们考虑v与真实的超球面S的交集D 2 ε ε R D 2 ε R D 12 N-1 ε R D 1（x）（我们将这个交集记为M）。则M是一个非奇异的真实的奇维C流形D2 N流形D2。此外，在M上，由M诱导出aCR结构.这种CR结构是有意义的。事实上，这种CR结构决定了正常的孤立奇点。注意到这一点，仓西从Cr结构的观点（特别是形变理论）开始了他对孤立奇点的研究。我遵循他的路线，在1981年，我成功了，如果dim d21 R d2 M =zn-1 7。但是dim dim d21 R dim d2 M =zn-1 dim d2 M的情况一直没有解决。受Rumin在接触流形上的工作的启发，我们（与J.M.Lee和Gurfiled一起）构建了Rumin工作的CR版本，并通过使用Rumin复形，我们解决了dim dim D21 R dim D2 M =zn-1=5。从Hodge结构的观点出发，我们的复形在研究孤立奇点的复形结构方面是有效的。但这仍在准备中。

项目成果

赤堀隆夫: "Real analyticity of the canonical versal deformation of CR-strnetures" Pacific J.of Math.Vol.185-1. 33-45 (1998)

Takao Akahori：“CR 结构的规范通用变形的真实分析”Pacific J.of Math.Vol.185-1（1998）。

Takao Akahori: "Real analyticity of the canonical deformations of CR-structures"Pacific Journal of Mathematics. Vol.185, No.1. 33-45 (1998)

Takao Akahori：“CR 结构规范变形的真实分析”太平洋数学杂志。

赤堀隆夫: "Real analyticity of the canonical versal deformations of CR-structures"Pacific Journal of Mathematics. 185. 33-45 (1998)

Takao Akahori：“CR 结构的规范通用变形的真实分析性”太平洋数学杂志 185. 33-45 (1998)。

赤堀隆夫: "A smoothness of deformations of normal strongly pseudo-convex CR structures"Kyushu J. of Mathematics. 52. 413-438 (1998)

Takao Akahori：“正常强伪凸 CR 结构变形的平滑性”九州数学杂志 52. 413-438 (1998)。

赤堀隆夫: "A Smoothness of deformations of normal strongly Psendo-cornex CR structures"Kyushu Journal of Mathematics. 52・2. 413-418 (1998)

Takao Akahori：“通常强Psendo-cornex CR结构的变形的平滑性”九州数学杂志52・2（1998）。