Nous étudions les germes d’applications holomorphes entre hypersurfaces algébriques réelles de . Plus précisément, nous considérons deux germes d’hypersurfaces algébriques et dans , , et : une application holomorphe de rang générique maximal telle que et . Nous montrons que si n’est pas Lévi-plate, alors la fonction dite de réflexion associée à est toujours algébrique. Par conséquent, si l’hypersurface cible est donnée sous une forme normale, la composante transverse de est algébrique (sans aucune autre hypothèse de non-dégénérescence sur les hypersurfaces). Une autre conséquence de notre résultat est le théorème bien connu de Baouendi et Rothschild qui affirme que tout biholomorphisme entre hypersurfaces algébriques réelles holomorphiquement non dégénérées de est algébrique.
We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If and are two germs of real algebraic hypersurfaces in , , is not Levi-flat and is a germ at of a holomorphic mapping such that and then the so-called reflection function associated to is always holomorphic algebraic. As a consequence, we obtain that if is given in the so-called normal form, the transversal component of 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.
@article{AIF_1998__48_4_1025_0, author = {Mir, Nordine}, title = {Germs of holomorphic mappings between real algebraic hypersurfaces}, journal = {Annales de l'Institut Fourier}, pages = {1025--1043}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {48}, number = {4}, year = {1998}, doi = {10.5802/aif.1647}, mrnumber = {2000c:32059}, zbl = {0914.32009}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1647/} }
TY - JOUR AU - Mir, Nordine TI - Germs of holomorphic mappings between real algebraic hypersurfaces JO - Annales de l'Institut Fourier PY - 1998 SP - 1025 EP - 1043 VL - 48 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1647/ DO - 10.5802/aif.1647 LA - en ID - AIF_1998__48_4_1025_0 ER -
%0 Journal Article %A Mir, Nordine %T Germs of holomorphic mappings between real algebraic hypersurfaces %J Annales de l'Institut Fourier %D 1998 %P 1025-1043 %V 48 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1647/ %R 10.5802/aif.1647 %G en %F AIF_1998__48_4_1025_0
Mir, Nordine. Germs of holomorphic mappings between real algebraic hypersurfaces. Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 1025-1043. doi : 10.5802/aif.1647. http://www.numdam.org/articles/10.5802/aif.1647/
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