Finite-dimensionality of the group of automorphisms of a real analytic surface. (Russian) Zbl 0647.32010

Beloshapka, V. K.

Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 2, 437-442 (1988).

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