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Parametrization of local biholomorphisms of real analytic hypersurfaces. (English) Zbl 0943.32021
Let $$M$$ be a real analytic hypersurface in $${\mathbb C}^{N}$$, $$p$$ a point in $$M$$, $$\rho(Z,\overline{Z})$$ a local defining function of $$M$$ at $$p$$, $$L_1, \dots, L_{N-1}$$ a basis of CR vector fields near $$p$$, and $$L^\alpha=L_1^{\alpha_1}\cdots L_{N-1}^{\alpha_{N-1}}$$ for a multiindex $$\alpha$$. Then the hypersurface $$M$$ is said to be finitely nondegenerate at $$p$$ if there exists a positive integer $$k$$ such that span$$\{L^\alpha \rho_Z(p,\overline{p}): |\alpha|\leq k\} = {\mathbb C}^N$$ and it is said to be holomorphically nondegenrate if there is no nontrivial germ of holomorphic vector field tangent to $$M$$.
This paper shows that, for germs of finitely nondegenerate real analytic hypersurface $$(M,p)$$ and $$(M',p')$$, formal equivalence of germs are equivalent to holomorphic equivalence. In particular, the stability group at $$(M,p)$$ equipped with natural topology is a real Lie group if $$M$$ is finitely nondegenerate at $$p$$.
Reviewer: T.Sasaki (Kobe)

MSC:
 32V40 Real submanifolds in complex manifolds 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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