Research on the application of the boundary analysis and geometry to the moduli of isolated singularities

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There are two approaches to singularities ; an algebraic approach and analytic one. The aim of this research is to approach the moduli of singularities of complex analytic spaces by means of the boundary CR structure, the complex structure on the regular part and the complex structure on its resolution. Our leading principle was that the stably embeddability is the key property in order for those deformations to be acconpanied with deformation of singularities and we researched maily on the following subjects ; (i)deformation of the regular part and its application to the moduli space of singularities, (ii)deformation of the resolution of singularities and its application to the construction of the versal family for the Res-functor and (iii)their application to the moduli of singularities. The main results are as follows ;(i)By considering the stable deformation of complex structures and applying the sharp estimate of the Neumann operator, we found a way to clear the serious analytical difficulty in the construction of the moduli space of normal isolated singularities by means of deformation of the complex structure on the regular part.(ii)We established the analysis for construction of the versal family of deformation of the resolution of normal isolated singularities.(iii)We constructed the versal family of cone singularities in terms of the boundary CR structure and give a CR-theoretic explanation of the simultaneous resolution of cone singularities.