Research on moduli of the boundary structure of isolated singularities

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Under the guiding principle that the stably embeddable deformation of CR structures is the boundary analogue of the deformation of normal isolated singularities, we investigate description of stably embeddable deformation of CR structures on a link of normal isolated singularities. In particular, we concentrated on establishing the method describing it in the following three cases; (i) complete intersection singularities, (ii) quasi-homogeneous singularities and (iii) quotient singularities. The main results are as follows, (i) Complete intersection singularities: Deformation of singularities is controlled by means of CR functions on its link, and we can deduce so-called tha Kas-Schlessinger theorem, (ii) Quasi-homogeneous singularities: A grading induced by the S^1-action associ ating with the quasi-homogenety is introduced in the deformation space. We realized that the grading controlls the grading of deformations of defining equations of the singularities. Furthermore, if the singularities are cone singularities, the grading controlls deformation of resolution of singularities. These graded arguments provide the CR-version of the the orems of H. Pinkham and J. Wahl on deformation of quasi-homogenous singularities, (iii) Higher dimensional quotient singularities: Based on the sphere analysis, our construction of semi-universal family of stably embeddable deformation of CR structures provides the Sch lessingers rigidity theorem, (iv) Cyclic quotient surface singularities: (these singularities are the origin of several interesting geometries, e.g. twistor space, hyper Kaler manifold and quiver): In the case of the degree <__- 4, we obtained the CR description of the semi-universal deformation of the singularity and also description of the simultaneous resolution; in the case of degree >__- 5, we obtained an algorithm constructing the semi-universal deformation.