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

有两种方法来研究奇点;代数方法和解析方法。利用边界CR结构、正则部分上的复结构和复分解上的复结构来逼近复解析空间的奇点模。我们的主要原则是稳定可嵌入性是使这些变形与奇异变形相一致的关键性质，主要研究了以下几个问题：（i）正则部分的变形及其在奇点模空间中的应用，（二）变形的解决方案的奇异性和它的应用，以建设的res-函子和（iii）的res家庭。它们在奇点模上的应用。主要结果如下：（i）通过考虑复杂结构的稳定变形，并应用Neumann算子的精确估计，我们找到了一种方法，通过复杂结构在正则部分的变形来解决构造正常孤立奇点模空间的严重分析困难。（ii）建立了构造法向孤立奇点分解的变形族的分析。（iii）利用边界CR结构构造了锥奇点的CR族，并给出了锥奇点同时分解的CR理论解释。

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.