On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions

Xiaojun Huang

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 2, page 433-463
  • ISSN: 0373-0956

Abstract

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In this paper, we show that if M 1 and M 2 are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if f is a holomorphic mapping defined near a neighborhood of M 1 so that f ( M 1 ) ⊂ M 2 , then f is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings that are holomorphic on one side and C k + 1 smooth up to M 1 where k is the codimension.

How to cite

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Huang, Xiaojun. "On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions." Annales de l'institut Fourier 44.2 (1994): 433-463. <http://eudml.org/doc/75069>.

@article{Huang1994,
abstract = {In this paper, we show that if $M_1$ and $M_2$ are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if $f$ is a holomorphic mapping defined near a neighborhood of $M_1$ so that $f(M_1)\subset M_2$, then $f$ is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings that are holomorphic on one side and $C^\{k+1\}$ smooth up to $M_1$ where $k$ is the codimension.},
author = {Huang, Xiaojun},
journal = {Annales de l'institut Fourier},
keywords = {algebraic real hypersurfaces; holomorphic mapping; reflection principle},
language = {eng},
number = {2},
pages = {433-463},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions},
url = {http://eudml.org/doc/75069},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Huang, Xiaojun
TI - On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 2
SP - 433
EP - 463
AB - In this paper, we show that if $M_1$ and $M_2$ are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if $f$ is a holomorphic mapping defined near a neighborhood of $M_1$ so that $f(M_1)\subset M_2$, then $f$ is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings that are holomorphic on one side and $C^{k+1}$ smooth up to $M_1$ where $k$ is the codimension.
LA - eng
KW - algebraic real hypersurfaces; holomorphic mapping; reflection principle
UR - http://eudml.org/doc/75069
ER -

References

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Citations in EuDML Documents

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  1. Joël Merker, Note on double reflection and algebraicity of holomorphic mappings
  2. M. S. Baouendi, L. P. Rothschild, Holomorphic mappings between algebraic hypersurfaces in complex space
  3. Joël Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets
  4. Xiaojun Huang, Shanyu Ji, Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains
  5. Francine Meylan, Nordine Mir, Dimitri Zaitsev, On some rigidity properties of mappings between CR-submanifolds in complex space
  6. Nordine Mir, Germs of holomorphic mappings between real algebraic hypersurfaces
  7. Rasul Shafikov, Kausha Verma, Extension of holomorphic maps between real hypersurfaces of different dimension

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