A SURVEY OF DEFORMATION THEORY OF CR-STRUCTURES

This paper is a survey of deformation theory of CR-structures, which is studied in (Al),(A2),(A3),(Ku),(Mi). Let (V,o) be an $n $ dimensional normal isolated singularity in $(C^{N}, 0) $. We set $M=V\cap S_{\epsilon}^{2N-1}(0)$ where $S_{\epsilon}^{2N-1}(0) $ is the $\epsilon $-sphere in $C^{N} $. Then we have a real odd dimen-sional, compact manifold, which is obviously real analytic. Furthermore, over this $M $ , a CR-structure is naturally induced from V. By Rossi(see $(R) $), this CR-structure $(M^{0}T’) $ determines the normal isolated singularoty (V, $0$) , uniquely. Kuranishi noted this point, and in order to study deformation theory of isolated singularities, he initiated deformation theory of CR-structures. This method is improved by (A3),(Mi). Namely, in (A3), it is shown that there is a versal fam-ily $(M^{\phi(t)}T’) $ which satisfies that $\phi(t) $ is a $C^{k} $ element of $T’\otimes(0T’)^{*} $ valued form, which depends on $t $ , complex analytically, and $\phi(0)=0 $. Later, Miyajima proved that $\phi(t) $ is actually $0\infty $ in (Mi). Now our $\phi(t) $ satisfies the following non-linear partial differential equation