Research on moduli of strongly pseudo-convex CR manifolds embedded in algebraic varieties

强有力的伪卷积CR manifolds of real dimension greater than or equals to five are realized as boundaries of normal Stein spaces and moreover as real hypersurfaces in algebraic varieties with only normal solated singularities。基于这种强烈的伪卷积CR流形和正常孤立的单体之间的连接，在70年代后期， M.。Kuranishi proposed a problem to approach to moduli of normal isolated singularities by means of CR structures on its boundaries。The main part of the final ac？ishment of the Kuranishi's proManagement together with establishment three dimManagement al geometric·D2-analysis which is crucial for that ac？ishment。这一方法被应用到了一些典型的单一性。本研究中获得的主要结果如下。（1） Let V be an analytic subvariety with dim ― D2C ― D2V ≥ 2 in C ― D1N ― D1 with only singularity at the origin and M = V ― S ― D32n-1（±）― D ― 3 where we denote S ― D32n-1（±）― D ― 3 a sphere centered at the origin and with a small radius ε。M上CR结构稳定嵌入式变形的Kuranishi半通用家族存在，它被现实化为一个半通用家族变形的半通用家族（ V， o）。（2）让M成为一个复杂表面的边界子域和E在M上形成一个更强大的伪卷积边界。If E is extendable to a holomorphic vector bundle on that bounded domain then the tangential Cauchy Riemann operator·D2： LイイD12イエD1（E）￣ LイD32（\）（0，1）イD3（E） has a closed range。（这一分析结果是Kuranishi半通用家族在（1）的情况下，在Dim D2 Rye D2 M = 3的情况下，是关键的。）

Strongly pseudo-convex CR manifolds of real dimension greater than or equals to five are realized as boundaries of normal Stein spaces and moreover as real hypersurfaces in algebraic varieties with only normal isolated singularities. Based on this connection between strongly pseudo-convex CR manifolds and normal isolated singularities, in the late '70, M. Kuranishi proposed a problem to approach to moduli of normal isolated singularities by means of CR structures on its boundaries. The main part of this research consists of the final accomplishment of the Kuranishi's problem together with establishment three dimensional geometric ∂ィイD2bィエD2-analysis which is crucial for that accomplishment. And this approach is applied to some typical singularities. The main result obtained in this research is as follows. (1) Let V be an analytic subvariety with dimィイD2CィエD2V≧2 in CィイD1NィエD1 with only singularity at the origin and M = V∩SィイD32n-1(/)εィエD3 where we denote SィイD32n-1(/)εィエD3 a sphere centered at the origin and with a small radius ε. There exists the Kuranishi semi-universal family of stably embeddable deformations of CR structures on M and it is realized as a family of boundaries of the semi-universal family of deformations of the germ ( V, o). (2) Let M be a strongly pseudo-convex boundary of a bounded subdomain of a complex surface and E a holomorphic vector bundle over M. If E is extendable to a holomorphic vector bundle on that bounded domain then the tangential Cauchy Riemann operator ∂ィイD2bィエD2: LィイD12ィエD1(E)→LィイD32(/)(0,1)ィエD3(E) has a closed range. (This analytical result is crucial for the construction of the Kuranishi semi-universal family in (1) in the case of dimィイD2RィエD2 M = 3.)