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Representation of the group of holomorphic symmetries of a real germ in the symmetry group of its model surface. (English. Russian original) Zbl 1146.32018

Math. Notes 82, No. 4, 461-463 (2007); translation from Mat. Zametki 82, No. 4, 515-518 (2007).
Summary: Local polynomial models of real submanifolds of complex spaces were constructed and studied in a series of papers. Among the main features of model surfaces, there is the property that the dimension of the local group of holomorphic symmetries of a germ does not exceed that of the same group of the tangent model surface of this germ.
In the paper, this assertion is rendered much stronger; namely, it is proved that the connected component of the identity element in the symmetry group of a nondegenerate germ is isomorphic as a Lie group to a subgroup of the symmetry group of its tangent model surface.

MSC:

32V40 Real submanifolds in complex manifolds
32B10 Germs of analytic sets, local parametrization
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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[1] V. K. Beloshapka, ”Real submanifolds of a complex space: Their polynomial models, automorphisms, and classification problems.” Usp. Mat. Nauk 57(1), 3–44 (2002) [Russian Math. Surveys 57 (1), 1–41 (2002)]. · Zbl 1053.32022
[2] V. K. Beloshapka, ”A universal model for a real submanifold,” Mat. Zametki 75(4), 507–522 (2004) [Math. Notes 75 (3–4), 475–488 (2004)].
[3] M. S. Baouendi, L. P. Rothschild, J. Winkelmann, and D. Zaitsev, ”Lie group structures of diffeomorphisms and applications to CR manifolds,” Ann. Inst. Fourier (Grenoble) 54, 1279–1303 (2004). · Zbl 1062.22046
[4] A. G. Vitushkin, ”Holomorphic mappings and the geometry of surfaces,” Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat. Fundam. Napravleniya 7, 167–226 (1985) · Zbl 0781.32013
[5] R. V. Gammel’ and I. G. Kossovskii, ”The envelope of holomorphy of a model third-degree surface and the ’rigidity’ phenomenon,” in Trudy Mat. Inst. Steklov (Nauka, Moscow, 2006), Vol. 253, pp. 30–45 [Proc. Steklov Inst. Math., Vol. 253, pp. 22–36 (2006)]. · Zbl 1351.32059
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