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.

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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\textit{V. K. Beloshapka}, Math. Notes 82, No. 4, 461--463 (2007; Zbl 1146.32018); translation from Mat. Zametki 82, No. 4, 515--518 (2007)

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##### References:

[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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