Quantization of canonical cones of algebraic curves
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, p. 1629-1663

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”.

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”.

DOI : https://doi.org/10.5802/aif.1929
Classification:  14Hxx
Keywords: algebraic curves, canonical cones, formal pseudodifferential operators, Rankin-Cohen brackets, Poincaré uniformization
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
     pages = {1629-1663},
     doi = {10.5802/aif.1929},
     zbl = {1052.14035},
     mrnumber = {1952526},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_6_1629_0}
}
Enriquez, Benjamin; Odesskii, Alexander. Quantization of canonical cones of algebraic curves. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1629-1663. doi : 10.5802/aif.1929. http://www.numdam.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), Academic Press (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, Birkhäuser Boston, Boston, MA (Progr. Nonlinear Diff. Eqs. Appl) Tome 26 (1997), pp. 17-47 | MR 1418868 | Zbl 1055.11514

[4] L. Boutet De Monvel Complex star algebras, Kluwer Acad. Publishers, the Netherlands (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, J. Wiley and Sons, Inc., New York, Wiley Classics Library (1994) | MR 1288523 | Zbl 0836.14001

[7] J. Harris Algebraic geometry. A first course, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 133 (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