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?CKostant?CRosenberg 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?CMukai 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?CRoch?? implies the classical Riemann?CRoch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties.