[Quantification du cône canonique d'une courbe algébrique]
Nous construisons des quantifications de l’algèbre de Poisson des fonctions sur le cône canonique d’une courbe algébrique , qui s’appuie sur la théorie des opérateurs pseudodifférentiels formels. Quand est une courbe complexe munie d’une uniformisation de Poincaré, nous proposons une construction équivalente, basée sur le travail de Cohen- Manin-Zagier sur les crochets de Rankin-Cohen. Quand est une courbe rationnelle, nous donnons une présentation de l’algèbre quantique, et nous discutons le problème de la construction algébrique de “relèvements différentiels”.
We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve , based on the theory of formal pseudodifferential operators. When is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.
Keywords: algebraic curves, canonical cones, formal pseudodifferential operators, Rankin-Cohen brackets, Poincaré uniformization
Mot clés : courbes algébriques, cônes canoniques, opérateurs pseudodifférentiels formels, de Rankin-Cohen, uniformisation de Poincaré
@article{AIF_2002__52_6_1629_0,
author = {Enriquez, Benjamin and Odesskii, Alexander},
title = {Quantization of canonical cones of algebraic curves},
journal = {Annales de l'Institut Fourier},
pages = {1629--1663},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {52},
number = {6},
year = {2002},
doi = {10.5802/aif.1929},
mrnumber = {1952526},
zbl = {1052.14035},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.1929/}
}
TY - JOUR AU - Enriquez, Benjamin AU - Odesskii, Alexander TI - Quantization of canonical cones of algebraic curves JO - Annales de l'Institut Fourier PY - 2002 SP - 1629 EP - 1663 VL - 52 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1929/ DO - 10.5802/aif.1929 LA - en ID - AIF_2002__52_6_1629_0 ER -
%0 Journal Article %A Enriquez, Benjamin %A Odesskii, Alexander %T Quantization of canonical cones of algebraic curves %J Annales de l'Institut Fourier %D 2002 %P 1629-1663 %V 52 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1929/ %R 10.5802/aif.1929 %G en %F AIF_2002__52_6_1629_0
Enriquez, Benjamin; Odesskii, Alexander. Quantization of canonical cones of algebraic curves. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1629-1663. doi : 10.5802/aif.1929. http://archive.numdam.org/articles/10.5802/aif.1929/
[1] On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV equation, Invent. Math, Volume 50 (1979), pp. 219-248 | DOI | MR | Zbl
[2] Systèmes hamiltoniens complètement intégrables associés aux surfaces K3, Problems in the theory of surfaces and their classification (Cortona, 1988) (Sympos. Math.), Volume XXXII (1991), pp. 25-31 | MR | Zbl
[3] Automorphic pseudodifferential operators, paper in memory of Irene Dorfman, Algebraic aspects of integrable systems (Progr. Nonlinear Diff. Eqs. Appl), Volume 26 (1997), pp. 17-47 | MR | Zbl
[4] Complex star algebras (Math. Physics, Analysis and Geometry) (1999), pp. 1-27 | MR | Zbl
[5] Sklyanin's elliptic algebras, Functional Anal. Appl, Volume 23 (1990) no. 3, pp. 207-214 | DOI | MR | Zbl
[6] Principles of algebraic geometry, Wiley Classics Library, J. Wiley and Sons, Inc., New York, 1994 | MR | Zbl
[7] Algebraic geometry. A first course, Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1985 | MR | Zbl
[8] Deformation quantization of Poisson manifolds, I (e-print, math.QA/9709040) | Zbl
[9] Algebraic aspects of differential equations, J. Sov. Math, Volume 11 (1979), pp. 1-128 | DOI | Zbl
[10] Polynomial Poisson algebras with regular structure of symplectic leaves (2001) (Preprint) | MR | Zbl
[11] Exotic deformation quantization, J. Differential Geom, Volume 45 (1997) no. 2, pp. 390-406 | MR | Zbl
Cité par Sources :
