We will classify -dimensional real submanifolds in which have a set of parabolic complex tangents of real dimension . All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an -dimensional submanifold in such that its images under biholomorphisms , , are not equivalent to via any local volume-preserving holomorphic map.
Nous classifions les sous-variétés réelles analytiques de dimension dans , qui ont un ensemble de points de tangence complexe paraboliques de dimension réelle . Ces sous variétés sont toutes équivalentes via biholomorphisme formel. Nous montrons que les classes d’équivalence sous changement de variables par biholomorphisme local (convergent) forment un ’espace de modules’ de dimension infinie. Nous montrons aussi qu’il existe une sous-variété de dimension dans , dont les images par les biholomorphismes , , ne sont pas équivalentes à via biholomorphisme local préservant le volume.
@article{AFST_2009_6_18_1_1_0, author = {Ahern, Patrick and Gong, Xianghong}, title = {Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {1--64}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 18}, number = {1}, year = {2009}, doi = {10.5802/afst.1204}, zbl = {1182.32013}, mrnumber = {2518102}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1204/} }
TY - JOUR AU - Ahern, Patrick AU - Gong, Xianghong TI - Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2009 SP - 1 EP - 64 VL - 18 IS - 1 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1204/ DO - 10.5802/afst.1204 LA - en ID - AFST_2009_6_18_1_1_0 ER -
%0 Journal Article %A Ahern, Patrick %A Gong, Xianghong %T Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2009 %P 1-64 %V 18 %N 1 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1204/ %R 10.5802/afst.1204 %G en %F AFST_2009_6_18_1_1_0
Ahern, Patrick; Gong, Xianghong. Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 1, pp. 1-64. doi : 10.5802/afst.1204. http://archive.numdam.org/articles/10.5802/afst.1204/
[1] Bishop (E.).— Differentiable manifolds in complex Euclidean space, Duke Math. J., 32, p. 1-22 (1965). | MR | Zbl
[2] Écalle (J.).— Les fonctions résurgentes, I, II. Publications Mathématiques d’Orsay 81, 5, 6, 1-247, p. 248-531, Université de Paris-Sud, Département de Mathématique, Orsay (1981). | Zbl
[3] Gong (X.).— On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents, Comment. Math. Helv. 69, no.4, p. 549-574 (1994). | MR | Zbl
[4] Gong (X.).— Real analytic submanifolds under unimodular transformations, Proc. Amer. Math. Soc. 123, no.1, p. 191-200 (1995). | MR | Zbl
[5] Gong (X.).— Divergence of the normalization for real Lagrangian surfaces near complex tangents, Pacific J. Math. 176, no. 2, p. 311–324 (1996). | MR | Zbl
[6] Huang (X.) and Yin (W.).— A Bishop surface with a vanishing Bishop invariant, preprint.
[7] Malgrange (B.).— Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, pp. 59-73, Astérisque, p. 92-93, Soc. Math. France, Paris (1982). | Numdam | MR | Zbl
[8] Moser (J.K.) and Webster (S.M.).— Normal forms for real surfaces in near complex tangents and hyperbolic surface transformations, Acta Math., 150, p. 255-296 (1983). | MR | Zbl
[9] Voronin (S.M.).— Analytic classification of germs of conformal mappings , Functional Anal. Appl. 15, no. 1, p. 1-13 (1981). | MR | Zbl
[10] Voronin (S.M.).— The Darboux-Whitney theorem and related questions, in Nonlinear Stokes phenomena, p. 139–233, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, (1993). | MR | Zbl
[11] Webster (S.M.).— Holomorphic symplectic normalization of a real function, Ann. Scuola Norm. Sup. di Pisa, 19, p. 69-86 (1992). | Numdam | MR | Zbl
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