Skip to main content
SpringerLink
Log in

Modules Over Deformation Quantization Algebroids: An Overview

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper is essentially an overview of a forthcoming paper in which we study coherent modules over deformation quantization algebroids on complex Poisson manifolds. First, we construct the convolution of coherent kernels over such algebroids, and prove that this convolution preserves coherency and commutes with duality. Next, we define the Hochschild class of coherent modules and prove that the Hochschild class of the convolution of two coherent kernels is the convolution of their Hochschild classes. Finally, we study with some details the case of symplectic deformations and apply these results to the Euler class of coherent \({\fancyscript{D}}\) -modules.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berezin F.A.: Quantization (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 38, 1116–1175 (1974)

    MathSciNet  Google Scholar 

  2. Berezin F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  3. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization I, II. Ann. Phys. 111, 61–110, 111–151 (1978)

    Google Scholar 

  4. Bressler, P., Nest, R., Tsygan, B.: Riemann–Roch theorems via deformation quantization. I, II. Adv. Math. 167, 1–25, 26–73 (2002)

    Google Scholar 

  5. Brylinski J.-L., Getzler E.: The homology of algebras of pseudodifferential symbols and the noncommutative residue. K-Theory 1, 385–403 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caldararu, A.: The Mukai Pairing I: the Hochschild Structure. arXiv:math.AG/ 0308079

  7. Caldararu A.: The Mukai pairing II: the Hochschild–Kostant–Rosenberg isomorphism. Adv. Math. 194, 34–66 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caldararu, A., Willerton, S.: The Mukai pairing I: a categorical approach. arXiv:0707. 2052

  9. Calaque, D., Halbout, G.: Weak quantization of Poisson structures. arXiv:math.QA/ 0707.1978

  10. Dolgushev, V.: The Van den Bergh duality and the modular symmetry of a Poisson variety. arXiv:math.QA/0612288

  11. Dolgushev, V., Rubtsov, V.N.: An algebraic index theorem for Poisson manifolds. arXiv:math.QA/0711.0184

  12. Feigin, B., Tsygan, B.: Riemann–Roch theorem and Lie algebra cohomology. I. In: Proceedings of the Winter School on Geometry and Physics (Srní, 1988). Rend. Circ. Mat. Palermo (2) Suppl. 21, pp. 15–52 (1989)

  13. Huybrechts D.: Fourier–Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs, Oxford (2006)

    Book  MATH  Google Scholar 

  14. Kashiwara M.: Quantization of contact manifolds. Publ. RIMS Kyoto Univ. 32, 1–5 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kashiwara, M.: D-modules and microlocal calculus. Translations of Mathematical Monographs, vol. 217. American Mathematical Society (2003)

  16. Kashiwara, M., Schapira, P.: Categories and Sheaves. Grundlehren der Math. Wiss., vol. 332. Springer, Heidelberg (2005)

  17. Kashiwara, M., Schapira, P.: Deformation quantization modules I: Finiteness and duality. arXiv:math.QA/0802.1245

  18. Kashiwara, M., Schapira, P.: Deformation quantization modules II: Hochschild class. arXiv:0809.4309

  19. Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Kontsevich M. Deformation quantization of algebraic varieties. In: EuroConférence Moshé Flato, Part III (Dijon, 2000). Lett. Math. Phys. 56(3), 271–294 (2001)

  21. Laumon, G.: Sur la catégorie dérivée des \({\fancyscript{D}}\) -modules filtrés. In: Raynaud, M., Shioda T. (eds.) Algebraic geometry. Lecture Notes in Math, vol. 1016, pp. 151–237. Springer, Heidelberg (1983)

  22. Markarian, N.: The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. arXiv:math.AG/0610553

  23. May J.-P.: The additivity of traces in triangulated categories. Adv. Math. 163, 34–73 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Polesello P., Schapira P.: Stacks of quantization–deformation modules over complex symplectic manifolds. Int. Math. Res. Notices 49, 2637–2664 (2004)

    Article  MathSciNet  Google Scholar 

  25. Ramadoss, A.C.: The relative Riemann–Roch theorem from Hochschild homology. arXiv:math/0603127

  26. Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. In: Komatsu (ed.) Hyperfunctions and pseudo-differential equations. Proceedings Katata 1971, Lecture Notes in Math, vol. 287, pp. 265–529. Springer, Heidelberg (1973)

  27. Schapira, P., Schneiders, J.-P.: Index theorem for elliptic pairs II. Euler class and relative index theorem. Astérisque 224 Soc. Math. France (1994)

  28. Shklyarov, D.: Hirzebruch–Riemann–Roch theorem for DG algebras. arXiv:math/ 0710.1937

  29. Van den Bergh M.: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126, 1345–1348 (1998) [erratum 130, 2809–2810 (2002)]

    Article  MATH  Google Scholar 

  30. Yekutieli, A.: Twisted deformation quantization of algebraic varieties. arXiv:math.AG/ 0801.3233v1

Download references

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Institut de Mathématiques, Université Pierre et Marie Curie, 175, rue du Chevaleret, 75013, Paris, France

    Pierre Schapira

Authors
  1. Masaki Kashiwara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Schapira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masaki Kashiwara.

Additional information

Copyright to this article is held by Masaki Kashiwara and Pierre Schapira, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kashiwara, M., Schapira, P. Modules Over Deformation Quantization Algebroids: An Overview. Lett Math Phys 88, 79–99 (2009). https://doi.org/10.1007/s11005-009-0297-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-009-0297-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation