Research on Complex Analytic Geometry and Singularity Theory

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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