Letters in Mathematical Physics

, Volume 56, Issue 3, pp 271–294 | Cite as

Deformation Quantization of Algebraic Varieties

  • Maxim Kontsevich


The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.

noncommutative algebraic varieties deformation quantization stacks quadratic algebras filtrations 


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  1. [AV]
    Artin, M. and Van den Bergh, M.: Twisted homogeneous coordinate rings, J. Algebra 133 (1990), 249-271.Google Scholar
  2. [AZh]
    Artin, M. and Zhang, J.: Noncommutative projective schemes, Adv. Math. 109 (1994), 228-287.Google Scholar
  3. [BW]
    Berest, Yu. and Wilson, J.: Automorphisms and ideals of the Weyl algebra, Math. Ann. 318 (2000), 127-147.Google Scholar
  4. [Bou]
    Bourbaki, N.: Commutative Algebra, Hermann, Paris, 1972.Google Scholar
  5. [CFT]
    Cattaneo, A., Felder, G. and Tomassini, L.: From local to global deformation quantization of Poisson manifolds, Preprint math/0012228.Google Scholar
  6. [Dr1]
    Drinfeld, V.: On quasi-triangular quasi-Hopf algebras and a group closely related with Gal (\(\overline {\mathbb{Q}} \)/ℚ), Leningrad Math. J. 2 (1991), 826-860.Google Scholar
  7. [Dr2]
    Drinfeld, V.: On quadratic commutation relations in the quasiclassical case, Selecta Math. Soviet 11 (1992), 317-326.Google Scholar
  8. [Gi]
    Giraud, J.: Cohomologie non abélienne, Springer-Verlag, Berlin, 1971.Google Scholar
  9. [HS]
    Hinich, V. and Schechtman, V.: Deformation theory and Lie algebra cohomology, Algebra Colloq. 4 (1997), No. 2, 213-240 and No. 3, 291–316.Google Scholar
  10. [KS]
    Kogorodski, L. and Soibelman, Y.: Algebras of Functions on Quantum Groups, Math. Surveys Monogr. 56, Amer. Math. Soc., Providence, 1998.Google Scholar
  11. [Ko1]
    Kontsevich, M.: Deformation quantization of Poisson manifolds, I, Preprint math/9709180.Google Scholar
  12. [Ko2]
    Kontsevich, M.: Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35-72.Google Scholar
  13. [KR]
    Kontsevich, M. and Rosenberg, A.: Smooth noncommutative spaces, In: The Gelfand Mathematical Seminars 1996–1999, Birkhäuser, Boston, 2000, 85-108.Google Scholar
  14. [NT]
    Nest, R. and Tsygan, B.: Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Preprint math/9906020.Google Scholar
  15. [RS]
    Roche, Ph. and Szenes, A.: Trace functionals on non-commutative deformations of moduli spaces of flat connections, Preprint math/0008149.Google Scholar
  16. [Sh]
    Sharpe, E.: Discrete torsion and gerbes II, Preprint hep-th/9909120.Google Scholar
  17. [Ta]
    Tamarkin, D.: Another proof of M. Kontsevich formality theorem, Preprint math/9809164.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Maxim Kontsevich
    • 1
  1. 1.Bures-sur-YvetteFrance

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