Germs of holomorphic mappings between real algebraic hypersurfaces

Nordine Mir

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 4, page 1025-1043
  • ISSN: 0373-0956

Abstract

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We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If ( M , p 0 ) and ( M ' , p 0 ' ) are two germs of real algebraic hypersurfaces in ℂ N + 1 , N ≥ 1 , M is not Levi-flat and H is a germ at p 0 of a holomorphic mapping such that H ( M ) ⊆ M ' and Jac ( H ) ≢ 0 then the so-called reflection function associated to H is always holomorphic algebraic. As a consequence, we obtain that if M ' is given in the so-called normal form, the transversal component of H is always algebraic. Another corollary of our main result is that any biholomorphism between holomorphically nondegenerate algebraic hypersurfaces is always algebraic, a result which was previously proved by Baouendi and Rothschild.

How to cite

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Mir, Nordine. "Germs of holomorphic mappings between real algebraic hypersurfaces." Annales de l'institut Fourier 48.4 (1998): 1025-1043. <http://eudml.org/doc/75307>.

@article{Mir1998,
abstract = {We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If $(M,p_0)$ and $(M^\{\prime \},p^\{\prime \}_0)$ are two germs of real algebraic hypersurfaces in $\{\Bbb C\}^\{N+1\}$, $N\ge 1$, $M$ is not Levi-flat and $H$ is a germ at $p_0$ of a holomorphic mapping such that $H(M) \subseteq M^\{\prime \}$ and $\{\rm Jac\}(H)\nequiv0$ then the so-called reflection function associated to $H$ is always holomorphic algebraic. As a consequence, we obtain that if $M^\{\prime \}$ is given in the so-called normal form, the transversal component of $H$ is always algebraic. Another corollary of our main result is that any biholomorphism between holomorphically nondegenerate algebraic hypersurfaces is always algebraic, a result which was previously proved by Baouendi and Rothschild.},
author = {Mir, Nordine},
journal = {Annales de l'institut Fourier},
keywords = {algebraic real hypersurfaces; holomorphic mappings; Segre varieties; holomorphic nondegeneracy},
language = {eng},
number = {4},
pages = {1025-1043},
publisher = {Association des Annales de l'Institut Fourier},
title = {Germs of holomorphic mappings between real algebraic hypersurfaces},
url = {http://eudml.org/doc/75307},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Mir, Nordine
TI - Germs of holomorphic mappings between real algebraic hypersurfaces
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 4
SP - 1025
EP - 1043
AB - We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If $(M,p_0)$ and $(M^{\prime },p^{\prime }_0)$ are two germs of real algebraic hypersurfaces in ${\Bbb C}^{N+1}$, $N\ge 1$, $M$ is not Levi-flat and $H$ is a germ at $p_0$ of a holomorphic mapping such that $H(M) \subseteq M^{\prime }$ and ${\rm Jac}(H)\nequiv0$ then the so-called reflection function associated to $H$ is always holomorphic algebraic. As a consequence, we obtain that if $M^{\prime }$ is given in the so-called normal form, the transversal component of $H$ is always algebraic. Another corollary of our main result is that any biholomorphism between holomorphically nondegenerate algebraic hypersurfaces is always algebraic, a result which was previously proved by Baouendi and Rothschild.
LA - eng
KW - algebraic real hypersurfaces; holomorphic mappings; Segre varieties; holomorphic nondegeneracy
UR - http://eudml.org/doc/75307
ER -

References

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