# Germs of holomorphic mappings between real algebraic hypersurfaces

Annales de l'institut Fourier (1998)

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

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topMir, 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

top- [1] M. ARTIN, On the solutions of analytic equations, Invent. Math., 5 (1968), 277-291. Zbl0172.05301MR38 #344
- [2] M. ARTIN, Algebraic approximations of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math., 36 (1969), 23-58. Zbl0181.48802MR42 #3087
- [3] M.S. BAOUENDI, P. EBENFELT and L.P. ROTHSCHILD, Algebraicity of holomorphic mappings between real algebraic sets in Cn, Acta Math., 177 (1996), 225-273. Zbl0890.32005MR99b:32030
- [4] M.S. BAOUENDI, H. JACOBOWITZ and F. TREVES, On the analyticity of CR mappings, Annals of Math., 122 (1985), 365-400. Zbl0583.32021MR87f:32044
- [5] M.S. BAOUENDI and L.P. ROTHSCHILD, Holomorphic mappings between algebraic hypersurfaces in complex space, Séminaire Equations aux dérivées partielles, Ecole Polytechnique, Palaiseau, 1994-1995. Zbl0886.32007
- [6] M.S. BAOUENDI and L.P. ROTHSCHILD, Germs of CR maps between real analytic hypersurfaces, Invent. Math., 93 (1988), 481-500. Zbl0653.32020MR90a:32036
- [7] M.S. BAOUENDI and L.P. ROTHSCHILD, Mappings of real algebraic hypersurfaces, J. Amer. Math. Soc., 8 (1995), 997-1015. Zbl0869.14025MR96f:32039
- [8] T. BLOOM and I. GRAHAM, A geometric characterization of points of type m on real submanifolds of Cn, J. Diff. Geom., 12 (1977), 171-182. Zbl0436.32013MR58 #11495
- [9] K. DIEDERICH and J.E. FORNAESS, Applications holomorphes propres entre domaines à bord analytique réel, C.R. Acad. Sci. Paris, 307 (1988), 321-324. Zbl0656.32013MR89i:32052
- [10] K. DIEDERICH and J.E. FORNAESS, Proper holomorphic mappings between real analytic pseudoconvex domains in Cn, Math. Annalen, 282 (1988), 681-700. Zbl0661.32025MR89m:32045
- [11] K. DIEDERICH and S. PINCHUK, Proper holomorphic maps in dimension two extend, Indiana Univ. Math. J., 44 (4) (1995), 1089-1126. Zbl0857.32015MR97g:32031
- [12] K. DIEDERICH and S. WEBSTER, A reflection principle for degenerate real hyper-surfaces, Duke Math. J., 47 (1980), 835-843. Zbl0451.32008MR82j:32046
- [13] M. FREEMAN, Local complex foliation of real submanifolds, Math. Annalen, 209 (1974), 1-30. Zbl0267.32006MR49 #10911
- [14] R.C. GUNNING and H. ROSSI, Analytic functions of several complex variables, Prentice-Hall, Englewoods Cliffs, N.J., 1965. Zbl0141.08601MR31 #4927
- [15] W.H.D. HODGE and D. PEDOE, Methods of algebraic geometry, Cambridge University Press, Cambridge, 1953. Zbl0055.38705
- [16] X. HUANG, On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier, Grenoble, 44-2 (1994), 433-463. Zbl0803.32011MR95i:32030
- [17] X. HUANG, Schwarz reflection principle in complex spaces of dimension two, Comm. P.D.E., 21 (11-12) (1996), 1781-1828. Zbl0886.32010MR97m:32043
- [18] J.J. KOHN, Boundary behaviour of ∂ on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542. Zbl0256.35060MR48 #727
- [19] N. MIR, An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings, Math. Z., to appear, 1997. Zbl0930.32020
- [20] P. MORANDI, Field and Galois theory, Springer Verlag, 1996. Zbl0865.12001MR97i:12001
- [21] S. PINCHUK, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zam., 15 (1974), 205-212. Zbl0292.32002MR50 #2558
- [22] R. SHARIPOV and A. SUKHOV, On CR mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions, Trans. Amer. Math. Soc., 348 (2) (1996), 767-780. Zbl0851.32017MR96g:32019
- [23] B. SEGRE, 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. Zbl0003.21302JFM57.0404.05
- [24] N.K. STANTON, Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. Math. J., 117 (1995), 141-167. Zbl0826.32013MR96a:32036
- [25] A. TUMANOV, Extending CR functions on a manifold of finite type over a wedge, Math. USSR Sbornik, 64 (1989), 129-140. Zbl0692.58005MR89m:32027
- [26] S.M. WEBSTER, On the mapping problem for algebraic real hypersurfaces, Invent. Math., 43 (1977), 53-68. Zbl0348.32005MR57 #3431
- [27] O. ZARISKI and P. SAMUEL, Commutative algebra, volume 1, Van Nostrand, 1958. Zbl0081.26501

## Citations in EuDML Documents

top- Joël Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets
- Francine Meylan, Nordine Mir, Dimitri Zaitsev, On some rigidity properties of mappings between CR-submanifolds in complex space
- Joël Merker, On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle
- Joël Merker, Étude de la régularité analytique de l'application de réflexion CR formelle

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