no. 6
Quantization of canonical cones of algebraic curves
Enriquez, Benjamin1; Odesskii, Alexander2
1 Université Louis Pasteur, IRMA, 7 rue René Descartes, 67084 Strasbourg Cedex (France)
2 Landau Institute of Theoretical Physics, 2 Kosygina str., 117334 Moscow (Russie)
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 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: 10.5802/aif.1929
Classification: 14Hxx
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

Enriquez, Benjamin 1; Odesskii, Alexander 2

1 Université Louis Pasteur, IRMA, 7 rue René Descartes, 67084 Strasbourg Cedex (France)
2 Landau Institute of Theoretical Physics, 2 Kosygina str., 117334 Moscow (Russie) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1929/

[1] M. Adler 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] A. Beauville 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] 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), Volume 26 (1997), pp. 17-47 | MR | Zbl

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

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

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

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

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

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

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

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

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