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.
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.
@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}, zbl = {0914.32009}, mrnumber = {2000c:32059}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 48 (1998) no. 4, pp. 1025-1043. doi : 10.5802/aif.1647. https://aif.centre-mersenne.org/articles/10.5802/aif.1647/
[1] On the solutions of analytic equations, Invent. Math., 5 (1968), 277-291. | MR | Zbl
,[2] Algebraic approximations of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math., 36 (1969), 23-58. | Numdam | MR | Zbl
,[3] Algebraicity of holomorphic mappings between real algebraic sets in Cn, Acta Math., 177 (1996), 225-273. | MR | Zbl
, and ,[4] On the analyticity of CR mappings, Annals of Math., 122 (1985), 365-400. | MR | Zbl
, and ,[5] Holomorphic mappings between algebraic hypersurfaces in complex space, Séminaire Equations aux dérivées partielles, Ecole Polytechnique, Palaiseau, 1994-1995. | Numdam | Zbl
and ,[6] Germs of CR maps between real analytic hypersurfaces, Invent. Math., 93 (1988), 481-500. | MR | Zbl
and ,[7] Mappings of real algebraic hypersurfaces, J. Amer. Math. Soc., 8 (1995), 997-1015. | MR | Zbl
and ,[8] A geometric characterization of points of type m on real submanifolds of Cn, J. Diff. Geom., 12 (1977), 171-182. | MR | Zbl
and ,[9] Applications holomorphes propres entre domaines à bord analytique réel, C.R. Acad. Sci. Paris, 307 (1988), 321-324. | MR | Zbl
and ,[10] Proper holomorphic mappings between real analytic pseudoconvex domains in Cn, Math. Annalen, 282 (1988), 681-700. | MR | Zbl
and ,[11] Proper holomorphic maps in dimension two extend, Indiana Univ. Math. J., 44 (4) (1995), 1089-1126. | MR | Zbl
and ,[12] A reflection principle for degenerate real hyper-surfaces, Duke Math. J., 47 (1980), 835-843. | MR | Zbl
and ,[13] Local complex foliation of real submanifolds, Math. Annalen, 209 (1974), 1-30. | MR | Zbl
,[14] Analytic functions of several complex variables, Prentice-Hall, Englewoods Cliffs, N.J., 1965. | MR | Zbl
and ,[15] Methods of algebraic geometry, Cambridge University Press, Cambridge, 1953.
and ,[16] On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier, Grenoble, 44-2 (1994), 433-463. | Numdam | MR | Zbl
,[17] Schwarz reflection principle in complex spaces of dimension two, Comm. P.D.E., 21 (11-12) (1996), 1781-1828. | MR | Zbl
,[18] Boundary behaviour of ∂ on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542. | MR | Zbl
,[19] An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings, Math. Z., to appear, 1997. | Zbl
,[20] Field and Galois theory, Springer Verlag, 1996. | MR | Zbl
,[21] A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zam., 15 (1974), 205-212. | MR | Zbl
,[22] On CR mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions, Trans. Amer. Math. Soc., 348 (2) (1996), 767-780. | MR | Zbl
and ,[23] Intorno al problema di Poincaré della rappresentazione pseudo-conforme, Atti R. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (6), 13 (1931), 676-683. | JFM | Zbl
,[24] Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. Math. J., 117 (1995), 141-167. | MR | Zbl
,[25] Extending CR functions on a manifold of finite type over a wedge, Math. USSR Sbornik, 64 (1989), 129-140. | MR | Zbl
,[26] On the mapping problem for algebraic real hypersurfaces, Invent. Math., 43 (1977), 53-68. | MR | Zbl
,[27] Commutative algebra, volume 1, Van Nostrand, 1958. | Zbl
and ,Cited by Sources: