A Variant of the Mukai Pairing via Deformation Quantization
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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
KeywordsMukai pairing Hochschild homology periodic cyclic homology algebraic index theorem Euler class deformation quantization
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