On the partial algebraicity of holomorphic mappings between two real algebraic sets

Joël Merker

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 4, page 547-591
  • ISSN: 0037-9484

Abstract

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The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for holomorphic mappings between two arbitrary real algebraic sets.

How to cite

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Merker, Joël. "On the partial algebraicity of holomorphic mappings between two real algebraic sets." Bulletin de la Société Mathématique de France 129.4 (2001): 547-591. <http://eudml.org/doc/272361>.

@article{Merker2001,
abstract = {The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for holomorphic mappings between two arbitrary real algebraic sets.},
author = {Merker, Joël},
journal = {Bulletin de la Société Mathématique de France},
keywords = {local holomorphic mappings; real algebraic sets; transcendence degree; local algebraic foliations; minimality in the sense of Tumanov; Segre chains; Tumanov's minimality},
language = {eng},
number = {4},
pages = {547-591},
publisher = {Société mathématique de France},
title = {On the partial algebraicity of holomorphic mappings between two real algebraic sets},
url = {http://eudml.org/doc/272361},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Merker, Joël
TI - On the partial algebraicity of holomorphic mappings between two real algebraic sets
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 4
SP - 547
EP - 591
AB - The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for holomorphic mappings between two arbitrary real algebraic sets.
LA - eng
KW - local holomorphic mappings; real algebraic sets; transcendence degree; local algebraic foliations; minimality in the sense of Tumanov; Segre chains; Tumanov's minimality
UR - http://eudml.org/doc/272361
ER -

References

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  1. [1] M. Artin – « Algebraic approximation of structures over complete local rings », Inst. Hautes Études Sci. Publ. Math.36 (1969), p. 23–58. Zbl0181.48802MR268188
  2. [2] M. S. Baouendi, P. Ebenfelt & L. P. Rothschild – « Algebraicity of holomorphic mappings between real algebraic sets in n », Acta Math. 177 (1996), no. 2, p. 225–273. Zbl0890.32005MR1440933
  3. [3] S. Bochner & W. Martin – « Several complex variables », Princeton Math. Ser., vol. 10, Princeton Univ. Press, Princeton, N.J., 1949. Zbl0041.05205MR27863
  4. [4] E. Chirka – « An introduction to the geometry of CR manifolds », Russian Math. Surveys 46 (1991), no. 1, p. 95–197. Zbl0742.32006MR1109037
  5. [5] B. Coupet, F. Meylan & A. Sukhov – « Holomorphic maps of algebraic CR manifolds », Int. Math. Research Notices1 (1999), p. 1–29. Zbl0926.32044MR1666972
  6. [6] B. Coupet, S. Pinchuk & A. Sukhov – « On the partial analyticity of CR mappings », Math. Z.235 (2000), p. 541–557. Zbl0972.32008MR1800211
  7. [7] S. Damour – « Sur l’algébricité des applications holomorphes », C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 6, p. 491–496. Zbl1066.14509MR1834056
  8. [8] X. Huang – « On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimension », Ann. Inst. Fourier Grenoble44 (1994), p. 433–463. Zbl0803.32011MR1296739
  9. [9] J. Merker – « Vector field construction of Segre sets », e-print : http://arxiv.org/abs/math.cv/9901010. 
  10. [10] —, « Note on double reflection and algebraicity of holomorphic mappings », Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 4, p. 689–721. Zbl0998.32020MR1838145
  11. [11] J. Merker & F. Meylan – « On the Schwarz symmetry principle in a model case », Proc. Amer. Math. Soc.127 (1999), p. 1197–1102. Zbl0919.32011MR1476379
  12. [12] N. Mir – « Germs of holomorphic mappings between real algebraic hypersurfaces », Ann. Inst. Fourier Grenoble48 (1998), p. 1025–1043. Zbl0914.32009MR1656006
  13. [13] S. Pinchuk – « CR transformations of real manifolds in n », Indiana University Math. J.41 (1992), p. 1–16. Zbl0766.32021MR1160899
  14. [14] R. Sharipov & A. Sukhov – « On CR mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions », Trans. Amer. Math. Soc.348 (1996), p. 767–780. Zbl0851.32017MR1325920
  15. [15] N. Stanton – « Infinitesimal CR automorphisms of real hypersurfaces », Amer. J. Math.118 (1996), p. 209–233. Zbl0849.32012MR1375306
  16. [16] A. Sukhov – « On the mapping problem for quadric Cauchy-Riemann manifolds », Indiana Univ. Math. J.42 (1993), p. 27–32. Zbl0848.32016MR1218705
  17. [17] H. J. Sussmann – « Orbits of families of vector fields and integrability of distributions », Trans. Amer. Math. Soc.180 (1973), p. 171–188. Zbl0274.58002MR321133
  18. [18] S. M. Webster – « On the mapping problem for algebraic real hypersurfaces », Invent. Math. 43-1 (1977), p. 53–68. Zbl0348.32005MR463482
  19. [19] D. Zaitsev – « Algebraicity of local holomorphisms between real algebraic submanifolds in complex spaces », Acta Math.183 (1999), p. 273–305. Zbl1005.32014MR1738046

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