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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  1. [Al] H. ALEXANDER, Holomorphic mappings from ball and polydisc, Math. Ann., 209 (1974), 245-256. Zbl0272.32006MR50 #5018
  2. [BBR] S. BAOUENDI, S. BELL, and L. ROTHSCHILD, Mappings of three-dimensional CR manifolds and their holomorphic extension, Duke Math. J., 56 (1988), 503-530. Zbl0655.32015MR90a:32035
  3. [BR] S. BAOUENDI and L. ROTHSCHILD, Germs of CR maps between real analytic hypersurfaces, Invent. Math., 93 (1988), 481-500. Zbl0653.32020MR90a:32036
  4. [Be] E. BEDFORD, Proper holomorphic mappings, Bull. Amer. Math. Soc., 10 (1984), 157-175. Zbl0534.32009MR85b:32041
  5. [BN] S. BELL and R. NARASIMHAN, Proper holomorphic mappings of complex spaces, EMS 69, Several Complex Variables VI (edited by W. Barth and R. Narasimhan), Springer-Verlag, 1990. Zbl0733.32021MR92m:32046
  6. [BM] S. BOCHNER and W. T. MARTIN, Several Complex Variables, Princeton University Press, 1948. Zbl0041.05205MR10,366a
  7. [CS1] J. CIMA and T. J. SUFFRIGE, A reflection principle with applications to proper holomorphic mappings, Math Ann., 265 (1983), 489-500. Zbl0525.32021MR84m:32033
  8. [CKS] J. CIMA, S. KRANTZ, and T. J. SUFFRIGE, A reflection principle for proper holomorphic mappings of strictly pseudoconvex domains and applications, Math. Z., 186 (1984), 1-8. Zbl0518.32009
  9. [DF1] K. DIEDERICH and E. FORNAESS, Proper holomorphic mappings between real-analytic domains in Cn, Math. Ann., 282 (1988), 681-700. Zbl0661.32025
  10. [DF2] K. DIEDERICH and E. FORNAESS, Applications holomorphes propres entre domaines à bord analytique réel, C.R.A.S., Ser.I-Math., 307, No7 (1988), 321-324. Zbl0656.32013MR89i:32052
  11. [Fa1] J. FARAN, A reflection principle for proper holomorphic mappings and geometric invariants, Math. Z., 203 (1990), 363-377. Zbl0664.32021MR90k:32082
  12. [Fa2] J. FARAN, Maps from the two ball to the three ball, Invent Math., 68 (1982), 441-475. Zbl0519.32016MR83k:32038
  13. [Fe] C. FEFFERMAN, The Bergman kernel and biholomorphic mappings pseudo-convex domains, Invent. Math., 26 (1974), 1-65. Zbl0289.32012MR50 #2562
  14. [Fr1] F. FORSTNERIC, Extending proper holomorphic mappings of positive codimension, Invent. Math., 95 (1989), 31-62. Zbl0633.32017MR89j:32033
  15. [Fr2] F. FORSTNERIC, A survey on proper holomorphic mappings, Proceeding of Year in SCVs at Mittag-Leffler Institute, Math. Notes 38, Princeton, NJ : Princeton University Press, 1992. 
  16. [Le] H. LEWY, On the boundary behavior of holomorphic mappings, Acad. Naz., Lincei, 3 (1977), 1-8. 
  17. [Kr] S. KRANTZ, Function Theory of Several Complex Variables, 2nd Ed., Wadsworth Publishing, Belmont, 1992. Zbl0776.32001MR93c:32001
  18. [Pi] PINCHUK, On analytic continuation of biholomorphic mappings, Mat. USSR Sb., 105 (1978), 574-593. 
  19. [Po] H. POINCARÉ, Les fonctions analytiques de deux variables et la représentation conforme, Ren. Cire. Mat. Palermo, II. Ser. 23 (1907), 185-220. Zbl38.0459.02JFM38.0459.02
  20. [Ta] N. TANAKA, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan, 14 (1962), 397-429. Zbl0113.06303MR26 #3086
  21. [We1] S. H. WEBSTER, On the mapping problem for algebraic real hypersurfaces, Invent. Math., 43 (1977), 53-68. Zbl0348.32005MR57 #3431
  22. [We2] S. H. WEBSTER, On mappings an (n + 1)-ball in the complex space, Pac. J. Math., 81 (1979), 267-272. Zbl0379.32018MR81h:32022

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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