Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface

Vladimir V. Ežov

Annales Polonici Mathematici (1998)

  • Volume: 70, Issue: 1, page 85-97
  • ISSN: 0066-2216

Abstract

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The Chern-Moser (CM) normal form of a real hypersurface in ℂ N can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in ℂ 4 is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.

How to cite

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Vladimir V. Ežov. "Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface." Annales Polonici Mathematici 70.1 (1998): 85-97. <http://eudml.org/doc/262799>.

@article{VladimirV1998,
abstract = {The Chern-Moser (CM) normal form of a real hypersurface in $ℂ^N$ can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in $ℂ^4$ is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.},
author = {Vladimir V. Ežov},
journal = {Annales Polonici Mathematici},
keywords = {CR automorphisms; canonical form; scalar linear type isotropy; nonquadratic hypersurface},
language = {eng},
number = {1},
pages = {85-97},
title = {Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface},
url = {http://eudml.org/doc/262799},
volume = {70},
year = {1998},
}

TY - JOUR
AU - Vladimir V. Ežov
TI - Triviality of scalar linear type isotropy subgroup by passing to an alternative canonical form of a hypersurface
JO - Annales Polonici Mathematici
PY - 1998
VL - 70
IS - 1
SP - 85
EP - 97
AB - The Chern-Moser (CM) normal form of a real hypersurface in $ℂ^N$ can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in $ℂ^4$ is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.
LA - eng
KW - CR automorphisms; canonical form; scalar linear type isotropy; nonquadratic hypersurface
UR - http://eudml.org/doc/262799
ER -

References

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  1. [Be] V. K. Beloshapka, On the dimension of the automorphism group of an analytic hypersurface, Izv. Akad. Nauk SSSR Ser. Mat. 93 (1979), 243-266 (in Russian). 
  2. [CM] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. Zbl0302.32015
  3. [ES] V. V. Ežov and G. Schmalz, Holomorphic automorphisms of quadrics, Math. Z. 216 (1994), 453-470. Zbl0806.32007
  4. [Lo1] A. V. Loboda, On local automorphisms of real-analytic hypersurfaces, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 620-645 (in Russian). Zbl0473.32016
  5. [Lo2] A. V. Loboda, Generic real-analytic manifolds of codimension 2 in ℂ 4 and their biholomorphic mappings, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 970-990 (in Russian). Zbl0662.32018
  6. [Sch] G. Schmalz, Über die Automorphismen einer streng pseudokonvexen CR-Mannigfaltigkeit der Kodimension 2 im ℂ 4 , Math. Nachr., to appear. 

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