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Finite-dimensionality of the group of automorphisms of a real analytic surface. (Russian) Zbl 0647.32010
Let M be a real analytic surface of codimension k in $${\mathbb{C}}^ N$$(k$$\geq 1)$$. We denote by $$G_{\xi}(M)$$ the set of mappings with the following properties: h is biholomorphic in some neighborhood $$U_ h$$ of the point $$\xi\in M$$; $$h(\xi)=\xi$$, and $$h(M\cap U_ h)\subset M$$. N. Tanaka [J. Math. Soc. Japan 14, 397-429 (1962; Zbl 0113.063)] and S. S. Chern and J. K. Moser [Acta Math. 133(1974), 219-271 (1975; Zbl 0302.32015)] independently proved that in the case $$k=1$$ the dimension of $$G_{\xi}(M)$$ is finite.
Let $$<z,z>$$ be the Hermitian form of M in the point $$\xi$$ and $$Q=\{(z,w)\in {\mathbb{C}}^ N:$$ $$v=<z,z>\}$$. There are two main results in the paper:
(A) dim $$G_{\xi}(M)\leq \dim G_ 0(Q);$$
(B) dim $$G_ 0(Q)<\infty$$ if and only if the form $$<z,z>$$ is nondegenerate.
Reviewer: A.D.Mednych

##### MSC:
 32C05 Real-analytic manifolds, real-analytic spaces 32H35 Proper holomorphic mappings, finiteness theorems
##### Keywords:
real analytic surface