Moduli theory of strongly pseudo-convex CR structure and its application to higher dimensional isolated singularities

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The research purpose is to analytically and differential geometrically investigate the moduli of isolated singularities, especially various phenomena e.g. smoothing of singularities, so-called Brieskorn resolution, etc., relying on interrelation between "complex structure on its resolution", "complex structure on its regular part" and "CR structure on its boundary". And, we also intend to develop approach to moduli theory of metric structure or symplectic structure in connection with deformation of singularities. By this research, we constructed versal deformation of resolution of normal isolated singularities in terms of deformation of complex structures as well as deformations of singularities itself by means of deformation of complex structures on its regular part. These results provide us interrelation between deformation of boundary CR structure of and Brieskorn resolution of rational double point. In addition, we established a method constructing versal family using only subellipticity and optimal estimate for d-bar Neumann problem over a bounded domain with strongly pseudo convex boundaries which is not enough for standard method constructing the versal family. On the other hand, T. Akahori (a co-investigator)proved that the versal deformation space of a rational double point is smooth relying on the Hamiltonian flow over the boundary, which is an argument different form our previous research using the stable deformation theory of CR structures. And, Mitsuhiro Itoh (a co-investigator)proved a Serre duality theorem for holomorphic vector bundles over strongly pseudo-convex compact CR manifolds, which is generalized to an open strongly pseudo-convex compact CR manifolds. Next subject will be investigate interrelation between stable deformation of CR structure (or complex structure)and moduli of various geometric structure on the boundary of normal isolated singularities.