On étudie les remplissages d’une variété CR de dimension trois par une surface complexe, sous une hypothèse d’équivariance : on suppose que beaucoup d’automorphismes CR du bord se prolongent en des biholomorphismes de tout le remplissage. On démontre dans le cas strictement pseudoconvexe un résultat d’unicité (à éclatement près).
We study the fillings of a three-dimensional CR manifold by a complex surface, under an equivariance hypothesis. Namely, we assume that many automorphisms of the CR manifold admit a biholomorphic extension to the whole filling. When the CR manifold is strictly pseudoconvex, we prove a uniqueness result (up to a blow-up).
@article{AIF_2007__57_6_2041_0,
author = {Kloeckner, Beno\^\i t},
title = {Sur les remplissages holomorphes \'equivariants},
journal = {Annales de l'Institut Fourier},
volume = {57},
year = {2007},
pages = {2041-2061},
doi = {10.5802/aif.2323},
zbl = {1132.32014},
mrnumber = {2377896},
language = {fr},
url = {http://dml.mathdoc.fr/item/AIF_2007__57_6_2041_0}
}
Kloeckner, Benoît. Sur les remplissages holomorphes équivariants. Annales de l'Institut Fourier, Tome 57 (2007) pp. 2041-2061. doi : 10.5802/aif.2323. http://gdmltest.u-ga.fr/item/AIF_2007__57_6_2041_0/
[1] Compact complex surfaces, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Tome 4 (2004) | MR 2030225 | Zbl 1036.14016
[2] Several complex variables and the geometry of real hypersurfaces, CRC Press, Boca Raton, FL, Studies in Advanced Mathematics (1993) | Zbl 0854.32001
[3] Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2 (Durham, 1989), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 151 (1990), pp. 45-67 | MR 1171908 | Zbl 0731.53036
[4] Sur certaines fonctions automorphes de deux variables, Ann. Sci. École Norm. Sup. (3), Tome 38 (1921), pp. 43-164 | Numdam | MR 1509233
[5] Complex hyperbolic geometry, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1999) (, Oxford Science Publications) | MR 1695450 | Zbl 0939.32024
[6] Principles of algebraic geometry, John Wiley & Sons Inc., New York, Wiley Classics Library (1994) (Reprint of the 1978 original) | MR 1288523 | Zbl 0836.14001
[7] Several complex variables. III, Springer-Verlag, Berlin, Encyclopaedia of Mathematical Sciences, Tome 9 (1989) (Geometric function theory, A translation of Sovremennye problemy matematiki. Fundamentalnye napravleniya, Tom 9, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986 [ MR0860607 (87m :32011)], Translation by J. Peetre, Translation edited by G. M. Khenkin) | MR 995071 | Zbl 0658.00007
[8] Conjugacy classes in , Conformal geometry (Bonn, 1985/1986), Vieweg, Braunschweig (Aspects Math., E12) (1988), pp. 41-64 | MR 979788 | Zbl 0659.53014
[9] Dynamics on surfaces : Salem numbers and Siegel disks, J. Reine Angew. Math., Tome 545 (2002), pp. 201-233 | Article | MR 1896103 | Zbl 1054.37026
[10] Dynamics on blowups of the projective plane (2006) (prépublication)
[11] Existence of subgroups isomorphic to the real numbers, Ann. of Math. (2), Tome 53 (1951), pp. 298-326 | Article | MR 47668 | Zbl 0045.31202
[12] On the conformal and CR automorphism groups, Geom. Funct. Anal., Tome 5 (1995) no. 2, pp. 464-481 | Article | MR 1334876 | Zbl 0835.53015
[13] Purely elliptic Möbius groups, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), Springer, New York (Math. Sci. Res. Inst. Publ.) Tome 11 (1988), pp. 173-178 | MR 955839 | Zbl 0661.57015