[Algébricité partielle des applications holomorphes entre deux ensembles algébriques réels]
La rigidité des invariants locaux des structures de Cauchy-Riemann réelles algébriques impose aux applications holomorphes des propriétés globales de rationalité (Poincaré 1907), ou plus généralement d'algébricité (Webster 1977). Notre objectif principal sera d'unifier les résultats classiques ou récents, grâce à une étude du degré de transcendance, de discuter aussi l'hypothèse habituelle de minimalité au sens de Tumanov, et ce en dimension quelconque, sans hypothèse de rang et pour des applications holomorphes quelconques entre deux ensembles algébriques réels arbitraires.
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.
Mots-clés : local holomorphic mappings, real algebraic sets, transcendence degree, local algebraic foliations, minimality in the sense of Tumanov, Segre chains
@article{BSMF_2001__129_4_547_0,
author = {Merker, Jo\"el},
title = {On the partial algebraicity of holomorphic mappings between two real algebraic sets},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {547--591},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {129},
number = {4},
year = {2001},
doi = {10.24033/bsmf.2408},
zbl = {0998.32019},
language = {en},
url = {http://archive.numdam.org/articles/10.24033/bsmf.2408/}
}
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 SP - 547 EP - 591 VL - 129 IS - 4 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/bsmf.2408/ DO - 10.24033/bsmf.2408 LA - en ID - BSMF_2001__129_4_547_0 ER -
%0 Journal Article %A Merker, Joël %T On the partial algebraicity of holomorphic mappings between two real algebraic sets %J Bulletin de la Société Mathématique de France %D 2001 %P 547-591 %V 129 %N 4 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/bsmf.2408/ %R 10.24033/bsmf.2408 %G en %F BSMF_2001__129_4_547_0
Merker, Joël. On the partial algebraicity of holomorphic mappings between two real algebraic sets. Bulletin de la Société Mathématique de France, Tome 129 (2001) no. 4, pp. 547-591. doi : 10.24033/bsmf.2408. http://archive.numdam.org/articles/10.24033/bsmf.2408/
[1] - « Algebraic approximation of structures over complete local rings », Inst. Hautes Études Sci. Publ. Math. 36 (1969), p. 23-58. | Numdam | MR | Zbl
[2] , & - « Algebraicity of holomorphic mappings between real algebraic sets in », Acta Math. 177 (1996), no. 2, p. 225-273. | MR | Zbl
[3] & - « Several complex variables », Princeton Math. Ser., vol. 10, Princeton Univ. Press, Princeton, N.J., 1949. | MR | Zbl
[4] - « An introduction to the geometry of CR manifolds », Russian Math. Surveys 46 (1991), no. 1, p. 95-197. | MR | Zbl
[5] , & - « Holomorphic maps of algebraic CR manifolds », Int. Math. Research Notices 1 (1999), p. 1-29. | MR | Zbl
[6] , & - « On the partial analyticity of CR mappings », Math. Z. 235 (2000), p. 541-557. | MR | Zbl
[7] - « Sur l'algébricité des applications holomorphes », C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 6, p. 491-496. | MR | Zbl
[8] - « On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimension », Ann. Inst. Fourier Grenoble 44 (1994), p. 433-463. | Numdam | MR | Zbl
[9] - « Vector field construction of Segre sets », e-print : http://arxiv.org/abs/math.cv/9901010.
[10] -, « Note on double reflection and algebraicity of holomorphic mappings », Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 4, p. 689-721. | Numdam | MR | Zbl
[11] & - « On the Schwarz symmetry principle in a model case », Proc. Amer. Math. Soc. 127 (1999), p. 1197-1102. | MR | Zbl
[12] - « Germs of holomorphic mappings between real algebraic hypersurfaces », Ann. Inst. Fourier Grenoble 48 (1998), p. 1025-1043. | Numdam | MR | Zbl
[13] - « CR transformations of real manifolds in », Indiana University Math. J. 41 (1992), p. 1-16. | MR | Zbl
[14] & - « On CR mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions », Trans. Amer. Math. Soc. 348 (1996), p. 767-780. | MR | Zbl
[15] - « Infinitesimal CR automorphisms of real hypersurfaces », Amer. J. Math. 118 (1996), p. 209-233. | MR | Zbl
[16] - « On the mapping problem for quadric Cauchy-Riemann manifolds », Indiana Univ. Math. J. 42 (1993), p. 27-32. | MR | Zbl
[17] - « Orbits of families of vector fields and integrability of distributions », Trans. Amer. Math. Soc. 180 (1973), p. 171-188. | MR | Zbl
[18] - « On the mapping problem for algebraic real hypersurfaces », Invent. Math. 43-1 (1977), p. 53-68. | MR | Zbl
[19] - « Algebraicity of local holomorphisms between real algebraic submanifolds in complex spaces », Acta Math. 183 (1999), p. 273-305. | MR | Zbl
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