Letters in Mathematical Physics

, Volume 100, Issue 3, pp 309–325 | Cite as

A Variant of the Mukai Pairing via Deformation Quantization



Let X be a smooth projective complex variety. The Hochschild homology HH(X) of X is an important invariant of X, which is isomorphic to the Hodge cohomology of X via the Hochschild–Kostant–Rosenberg isomorphism. On HH(X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH(X). The latter pairing on HH(X) is intimately linked to the study of Fourier–Mukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu’s Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the “Noncommutative Riemann–Roch” implies the classical Riemann–Roch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties.

Mathematics Subject Classification (2000)

19L10 14C40 19D55 53D55 


Mukai pairing Hochschild homology periodic cyclic homology algebraic index theorem Euler class deformation quantization 


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  1. 1.
    Bressler P., Nest R., Tsygan B.: A Riemann–Roch formula for the microlocal Euler class. IMRN 20, 1033–1044 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bressler P., Nest R., Tsygan B.: Riemann–Roch theorems via deformation quantization I. Adv. Math. 167(1), 1–25 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bressler P., Nest R., Tsygan B.: Riemann–Roch theorems via deformation quantization II. Adv. Math. 167(1), 26–73 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brylinski J.-L.: A differential complex for Poisson manifolds. J. Diff. Geom. 28(1), 93–114 (1988)MathSciNetMATHGoogle Scholar
  5. 5.
    Caldararu A.: The Mukai pairing I: a categorical approach. N. Y. J. Math. 16, 61–98 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Caldadaru A.: The Mukai pairing II: the Hochschild–Kostant–Rosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Engeli M., Felder G.: A Riemann–Roch–Hirzebruch formula for traces of differential operators. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 621–653 (2008)MathSciNetGoogle Scholar
  8. 8.
    Feigin B., Felder G., Shoikhet B.: Hochschild cohomology of the Weyl algebra and traces in deformation quantization. Duke Math. J. 127(3), 487–517 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Grivaux, J.: On a conjecture of Kashiwara relating Chern and Euler classes of O-modules. preprint, arxiv:0910.5384Google Scholar
  10. 10.
    Grothendieck A.: On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 29, 95–103 (1966)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Huybrechts D., Macri E., Stellari P.: Derived equivalences of K3 surfaces and orientation. Duke Math. J. 149(3), 461–507 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Keller B.: On the cyclic homology of ringed spaces and schemes. Doc. Math. 3, 231–259 (1998)MathSciNetMATHGoogle Scholar
  13. 13.
    Kashiwara, M.: Letter to Pierre Schapira dated 18 Nov 1991Google Scholar
  14. 14.
    Kashiwara M., Schapira P.: Modules over deformation quantization algebroids: an overview. Lett. Math. Phys. 88(1–3), 79–99 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Kashiwara, M., Schapira, P.: Deformation quantization modules. preprint, arxiv: 1003.3304Google Scholar
  16. 16.
    Lunts, V.: Lefschetz fixed point theorems for algebraic varieties and DG algebras. preprint, arxiv:1102.2884Google Scholar
  17. 17.
    Macri E., Stellari P.: Infinitesinal derived Torelli theorem for K3 surfaces. With an appendix by Sukhendu Mehrotra. IMRN 2009(17), 3190–3220 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    Markarian, N.: Poincare–Birkhoff–Witt isomorphism, Hochschild homology and Riemann–Roch theorem. Max Planck Institute MPI 2001-52 (2001)Google Scholar
  19. 19.
    Markarian N.: The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. J. Lond. Math. Soc. 79(1), 129–143 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pflaum M., Posthuma H., Tang X.: Cyclic cocycles in deformation quantization and higher index theorems. Adv. Math. 223(6), 1958–2021 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ramadoss A.: The relative Riemann–Roch theorem from Hochschild homology. N. Y. J. Math. 14, 643–717 (2008)MathSciNetMATHGoogle Scholar
  22. 22.
    Ramadoss A.: Some notes on the Feigin–Losev–Shoikhet integral conjecture. J. Noncommut. Geom. 2, 405–448 (2008)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Ramadoss A.: The Mukai pairing and integral transforms in Hochschild homology. Mosc. Math. J. 10(3), 629–645 (2010)MathSciNetMATHGoogle Scholar
  24. 24.
    Ramadoss A.: A generalized Hirzebruch Riemann–Roch theorem. C. R. Math. Acad. Sci. Paris 347(5–6), 289–292 (2009)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Ramadoss A.: The big Chern classes and the Chern character. Int. J. Math. 19(6), 699–746 (2008)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Schapira P., Schneiders J.-P.: Elliptic pairs I. Relative finiteness and duality. Index theorem for elliptic pairs. Astérisque 224, 5–60 (1994)MathSciNetGoogle Scholar
  27. 27.
    Shklyarov, D.: Hirzebruch Riemann–Roch theorem for DG-algebras. preprint, arxiv: 0710.1937Google Scholar
  28. 28.
    Willwacher, T.: Cyclic Cohomology of the Weyl algebra. preprint, Arxiv:0804.2812Google Scholar
  29. 29.
    Töen B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, vol. III, 247–435, Progr. Math., vol. 88. Birkhauser, Boston (1990)Google Scholar
  31. 31.
    Tsygan, B.: Cyclic homology. Cyclic Homology in Non-Commutative Geometry, 73–113. In: Encyclopaedia Math. Sci. vol. 121. Springer, Berlin (2004)Google Scholar
  32. 32.
    Yao D.: Higher algebraic K-theory of admissible abelian categories and localization theorems. J. Pure Appl. Algebra 77(3), 263–339 (1992)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Yekutieli A.: The continuous Hochschild cochain complex of a scheme. Can. J. Math. 54(6), 1319–1337 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Departement MathematikETH ZürichZurichSwitzerland

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