Quantization of canonical cones of algebraic curves  [ Quantification du cône canonique d’une courbe algébrique ]
Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1629-1663.

Nous construisons des quantifications de l’algèbre de Poisson des fonctions sur le cône canonique d’une courbe algébrique C, qui s’appuie sur la théorie des opérateurs pseudodifférentiels formels. Quand C 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 C 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 C, based on the theory of formal pseudodifferential operators. When C 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 C is a rational curve, and discuss the problem of constructing algebraically "differential liftings".

DOI : https://doi.org/10.5802/aif.1929
Classification : 14Hxx
Mots 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'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     doi = {10.5802/aif.1929},
     zbl = {1052.14035},
     mrnumber = {1952526},
     language = {en},
     url = {aif.centre-mersenne.org/item/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. https://aif.centre-mersenne.org/item/AIF_2002__52_6_1629_0/

[1] M. Adler On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV equation, Invent. Math, Tome 50 (1979), pp. 219-248 | Article | MR 520927 | Zbl 0393.35058

[2] A. Beauville Systèmes hamiltoniens complètement intégrables associés aux surfaces K3, Problems in the theory of surfaces and their classification (Cortona, 1988) (Sympos. Math.) Tome XXXII (1991), pp. 25-31 | MR 1273370 | Zbl 0827.58022

[3] P. Beazley Cohen; Yu. Manin; D. Zagier Automorphic pseudodifferential operators, paper in memory of Irene Dorfman, Algebraic aspects of integrable systems (Progr. Nonlinear Diff. Eqs. Appl) Tome 26 (1997), pp. 17-47 | MR 1418868 | Zbl 1055.11514

[4] L. Boutet de Monvel Complex star algebras (Math. Physics, Analysis and Geometry) (1999), pp. 1-27 | MR 1733883 | Zbl 0980.53106

[5] B. Feigin; A. Odesskii Sklyanin's elliptic algebras, Functional Anal. Appl, Tome 23 (1990) no. 3, pp. 207-214 | Article | MR 1026987 | Zbl 0713.17009

[6] P. Griffiths; J. Harris Principles of algebraic geometry, Wiley Classics Library, J. Wiley and Sons, Inc., New York, 1994 | MR 1288523 | Zbl 0836.14001

[7] J. Harris Algebraic geometry. A first course, Graduate Texts in Mathematics, Tome 133, Springer-Verlag, New York, 1985 | MR 1182558 | Zbl 0779.14001

[8] M. Kontsevich Deformation quantization of Poisson manifolds, I (e-print, math.QA/9709040) | Zbl 1058.53065

[9] Y. Manin Algebraic aspects of differential equations, J. Sov. Math, Tome 11 (1979), pp. 1-128 | Article | Zbl 0419.35001

[10] A. Odesskii; V. Rubtsov Polynomial Poisson algebras with regular structure of symplectic leaves (2001) (Preprint) | MR 1992166 | Zbl 1138.53314

[11] V. Ovsienko Exotic deformation quantization, J. Differential Geom, Tome 45 (1997) no. 2, pp. 390-406 | MR 1449978 | Zbl 0879.58028