Parametrization of local biholomorphisms of real analytic hypersurfaces. (English) Zbl 0943.32021

Baouendi, M. S.; Ebenfelt, P.; Rothschild, Linda Preiss

Asian J. Math. 1, No. 1, 1-16 (1997).

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

Keywords:

local biholomorphism; holomorphic nondegeneracy; analytic hypersurface; finite nondegeneracy