Research on Complex Analytic Geometry and Singularity Theory

首席研究人员和研究人员做了复杂解析几何和奇点理论的研究。特别是在本课题研究计划中提出的复解析叶的奇异性问题的研究上，作者撰写了一篇综述文章，总结了主要由首席研究员在展开理论、确定性问题、奇异集的结构及其相关不变量等方面所取得的成果。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发表的。此外，对于奇异集的不变量，研究了Baum-Bott残数和新发现的Lehmann残数，阐明了这些不变量出现的基本原理，并考虑了它们的一些推广。对于上一年提出的d模理论的应用，继续研究复解析奇异叶理的d模解复的特征变化和更局部结构及其与上述不变量的关系。除了以上的合作研究外，两位研究者也在自己的课题上进行了研究，并在以下课题上取得了许多成果：广义叶形的完整群和辛束的chen - simons - maslov类（Suzuki），奇异理论在微分方程理论中的应用，特别是对完全可积一阶微分方程系统的研究（Izumiya），曲线投影几何中的奇异性和拉格朗日变分的马斯洛夫类（Ishikawa）；微局部分析及其应用，特别是r -可构轴的Morse不等式（Tose）；复平面上的解析群作用，特别是分离矩阵的存在性和刚性定理的证明（Nakai）。少

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