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

研究目的是分析和微分几何研究孤立奇点的模量，特别是各种现象，例如奇点的平滑，所谓的Brieskorn分辨率等，依靠“解析复结构”、“规则复结构”和“边界复结构”之间的相互关系。并且，我们也打算发展与奇异变形有关的度量结构或辛结构的模理论的方法。通过这一研究，我们构造了由复杂结构变形引起的孤立奇点分解的变形，以及由复杂结构在其规则部分的变形引起的奇点自身的变形。这些结果提供了边界CR结构的变形与有理二重点的Brieskorn分解之间的相互关系。此外，对于具有强伪凸边界的有界区域上的d-bar Neumann问题，我们建立了一种仅利用次椭圆性和最优估计构造函数族的方法，这种方法对于构造函数族的标准方法是不够的.另一方面，T. Akahori（合作研究者）证明了有理双点的双变形空间是光滑的，依赖于边界上的Hamilton流，这与我们以前使用CR结构的稳定变形理论的研究不同。Mitsuhiro Itoh证明了强伪凸紧CR流形上全纯向量丛的Serre对偶定理，并将其推广到开的强伪凸紧CR流形上.接下来的课题是研究CR结构（或复杂结构）的稳定变形与各种几何结构在法向孤立奇点边界上的模量之间的相互关系。

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.