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Linearization of local automorphisms of pseudoconvex surfaces. (English. Russian original) Zbl 0582.32040
Sov. Math., Dokl. 28, 70-72 (1983); translation from Dokl. Akad. Nauk SSSR 271, 280-282 (1983).
Let M be a real analytic hypersurface in $${\mathbb{C}}^{n+1}$$ with coordinate system $$(z_ 1,...,z_ n,w)$$, $$w=u+iv$$, given by the equation $$v=F(z,\bar z,u)$$. Here it is assumed that F is real analytic with $$F(0)=0$$, $$dF|_ 0=0$$. Assume further that the Levi form of M is positive definite at the origin 0. Let G(M) be the group of local CR- automorphisms of M leaving the origin 0 fixed.
The author proves the following theorem: Suppose further that the hypersurface M is not locally equivalent to a sphere. Then, for some choice of coordinate system in $${\mathbb{C}}^{n+1}$$, the group G(M) is realized as a (closed) subgroup of U(n). The proof is done by making use of the Chern-Moser normal form of such a hypersurface.

##### MSC:
 32M05 Complex Lie groups, group actions on complex spaces 32T99 Pseudoconvex domains 32V40 Real submanifolds in complex manifolds 32C05 Real-analytic manifolds, real-analytic spaces