no. 6
Quantization of canonical cones of algebraic curves
[Quantification du cône canonique d'une courbe algébrique]
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, 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 : 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, Tome 52 (2002) no. 6, pp. 1629-1663. doi : 10.5802/aif.1929. https://aif.centre-mersenne.org/articles/10.5802/aif.1929/

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