Research on Complex Analytic Geometry and Singularity Theory

复解析几何与奇异性理论研究

• 批准号: 02452001
• 负责人: SUWA Tatsuo
• 金额: $ 4.29万
• 依托单位: Hokkaido University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1990
• 资助国家: 日本
• 起止时间: 1990 至 1991
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-02452001/
• 关键词: Complex analytic geometry; Singularity theory; Singular foliation; Unfolding theory; Singular set; 特異点の留数; 時異葉層構造; 開析理論; 解複体; 特性多様体

The head investigators and the investigators did research in Complex Analytic Geometry and Singularity Theory. Especially, on the research of the singularities of complex analytic foliations which was proposed in the research plans for this project, a survey article was written summarizing the results on the unfolding theory, the determinacy problem, the structure of the singular set and the invariants associated to it which have been obtained mainly by the head investigator. This was presented at the College on Singularity Theory held in Trieste, Italy during the summer of 1991. Also, as to the invariants associated to the singular set, the Baum-Bott residues and the newly found Lehmann residues are studied and the fundamental principle underlying the appearance of these invariants is clarified and some generalizations of them are considered. As to the application of the D-module theory, which was proposed in the previous year, the investigations of the characteristic variety and the … More local structure of the-solution complex of the D-module associated to a complex analytic singular foliation and their relation with the aforementioned invariants are continued.Besides the collaboration of the above research, each of the investigators did research on his own subject as well and obtained many results on the-following subject :Holonomy groupoids for generalized foliations and the Chern-Simons-Maslov classes for symplectic boundles (Suzuki), applications of the singularity theory to the theory of differential equations, especially study of systems of completely integrable first order differential equations (Izumiya), singularity which appear in the projective geometry of curves and the maslov classes of Lagrangian varieties (Ishikawa), the microlocal analysis and its applications, especially the Morse inequality for R-constructible sheaves (Tose), analytic group actions on the complex plane, especially the existence of the separatrix and the proof of the rigidity theorem (Nakai). Less

课题组组长和研究人员进行了复解析几何和奇点理论的研究。特别是针对本课题研究计划中提出的复杂解析叶的奇异性研究，撰写了一篇综述文章，总结了主要研究者在展开理论、确定性问题、奇异集结构及其相关不变量方面的研究成果。这是1991年夏天在意大利的里雅斯特举行的奇点理论学院上提出的。此外，对于与奇点集相关的不变量，研究了鲍姆-博特留数和新发现的莱曼留数，阐明了这些不变量出现的基本原理，并考虑了它们的一些概括。对于去年提出的D-模理论的应用，继续对与复解析奇叶理相关的D-模解复合体的特征多样性和局部结构及其与上述不变量的关系进行研究。除了上述研究的合作外，每个研究者也对自己的课题进行了研究，并在以下课题上取得了许多成果：Holonomy群形 广义叶状结构和辛边界的 Chern-Simons-Maslov 类（Suzuki），奇点理论在微分方程理论中的应用，特别是完全可积一阶微分方程组的研究（Izumiya），曲线射影几何中出现的奇点和拉格朗日簇的 maslov 类（Ishikawa），微局域分析及其应用， 特别是 R 构造滑轮的莫尔斯不等式（Tose）、复平面上的解析群作用，特别是分界线的存在性和刚性定理的证明（Nakai）。较少的

项目成果

Nobuyuki Tose: "Morse inequalities for Rーconstructible sheaves" Advances in Math.

Nobuyuki Tose：“R—可构造滑轮的莫尔斯不等式”数学进展。

Isao Nakai: "Separatrices for conformal transformation groups of C,O" Ann.I'Intst.Fourier.

Isao Nakai：“C、O 的共形变换群的分离”Ann.IIntst.Fourier。

Goo Ishikawa: "Parametrization of a singular Lagrangian variety" Trans.Amer.Math.Soc.

Goo Ishikawa：“奇异拉格朗日簇的参数化” Trans.Amer.Math.Soc。

Haruo Suzuki: "ChernーSimonsーMaslov classes in some symplectic vector bundles" Proc.Amer.Math.Soc.

Haruo Suzuki：“某些辛向量丛中的 Chern-Simons-Maslov 类”Proc.Amer.Math.Soc。

Tatsuo Suwa: "Finite determinacy of analytic foliation germs without formal integrating factors" Proc.Amer.Math.Soc.112. 989-997 (1991)

Tatsuo Suwa：“没有正式积分因子的分析叶芽细菌的有限确定性”Proc.Amer.Math.Soc.112。