Abstract
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.
Similar content being viewed by others
References
Bressler P., Nest R., Tsygan B.: A Riemann–Roch formula for the microlocal Euler class. IMRN 20, 1033–1044 (1997)
Bressler P., Nest R., Tsygan B.: Riemann–Roch theorems via deformation quantization I. Adv. Math. 167(1), 1–25 (2002)
Bressler P., Nest R., Tsygan B.: Riemann–Roch theorems via deformation quantization II. Adv. Math. 167(1), 26–73 (2002)
Brylinski J.-L.: A differential complex for Poisson manifolds. J. Diff. Geom. 28(1), 93–114 (1988)
Caldararu A.: The Mukai pairing I: a categorical approach. N. Y. J. Math. 16, 61–98 (2010)
Caldadaru A.: The Mukai pairing II: the Hochschild–Kostant–Rosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005)
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)
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)
Grivaux, J.: On a conjecture of Kashiwara relating Chern and Euler classes of O-modules. preprint, arxiv:0910.5384
Grothendieck A.: On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 29, 95–103 (1966)
Huybrechts D., Macri E., Stellari P.: Derived equivalences of K3 surfaces and orientation. Duke Math. J. 149(3), 461–507 (2009)
Keller B.: On the cyclic homology of ringed spaces and schemes. Doc. Math. 3, 231–259 (1998)
Kashiwara, M.: Letter to Pierre Schapira dated 18 Nov 1991
Kashiwara M., Schapira P.: Modules over deformation quantization algebroids: an overview. Lett. Math. Phys. 88(1–3), 79–99 (2009)
Kashiwara, M., Schapira, P.: Deformation quantization modules. preprint, arxiv: 1003.3304
Lunts, V.: Lefschetz fixed point theorems for algebraic varieties and DG algebras. preprint, arxiv:1102.2884
Macri E., Stellari P.: Infinitesinal derived Torelli theorem for K3 surfaces. With an appendix by Sukhendu Mehrotra. IMRN 2009(17), 3190–3220 (2009)
Markarian, N.: Poincare–Birkhoff–Witt isomorphism, Hochschild homology and Riemann–Roch theorem. Max Planck Institute MPI 2001-52 (2001)
Markarian N.: The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. J. Lond. Math. Soc. 79(1), 129–143 (2009)
Pflaum M., Posthuma H., Tang X.: Cyclic cocycles in deformation quantization and higher index theorems. Adv. Math. 223(6), 1958–2021 (2010)
Ramadoss A.: The relative Riemann–Roch theorem from Hochschild homology. N. Y. J. Math. 14, 643–717 (2008)
Ramadoss A.: Some notes on the Feigin–Losev–Shoikhet integral conjecture. J. Noncommut. Geom. 2, 405–448 (2008)
Ramadoss A.: The Mukai pairing and integral transforms in Hochschild homology. Mosc. Math. J. 10(3), 629–645 (2010)
Ramadoss A.: A generalized Hirzebruch Riemann–Roch theorem. C. R. Math. Acad. Sci. Paris 347(5–6), 289–292 (2009)
Ramadoss A.: The big Chern classes and the Chern character. Int. J. Math. 19(6), 699–746 (2008)
Schapira P., Schneiders J.-P.: Elliptic pairs I. Relative finiteness and duality. Index theorem for elliptic pairs. Astérisque 224, 5–60 (1994)
Shklyarov, D.: Hirzebruch Riemann–Roch theorem for DG-algebras. preprint, arxiv: 0710.1937
Willwacher, T.: Cyclic Cohomology of the Weyl algebra. preprint, Arxiv:0804.2812
Töen B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)
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)
Tsygan, B.: Cyclic homology. Cyclic Homology in Non-Commutative Geometry, 73–113. In: Encyclopaedia Math. Sci. vol. 121. Springer, Berlin (2004)
Yao D.: Higher algebraic K-theory of admissible abelian categories and localization theorems. J. Pure Appl. Algebra 77(3), 263–339 (1992)
Yekutieli A.: The continuous Hochschild cochain complex of a scheme. Can. J. Math. 54(6), 1319–1337 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ramadoss, A.C. A Variant of the Mukai Pairing via Deformation Quantization. Lett Math Phys 100, 309–325 (2012). https://doi.org/10.1007/s11005-011-0541-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-011-0541-6