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The Mukai pairing, I: the Hochschild structure

Andrei Caldararu
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Abstract

We study the Hochschild structure of a smooth space or orbifold, emphasizing the importance of a pairing defined on Hochschild homology which generalizes a similar pairing introduced by Mukai on the cohomology of a K3 surface. We discuss those properties of the structure which can be derived without appealing to the Hochschild-Kostant-Rosenberg isomorphism and Kontsevich formality, namely: -- functoriality of homology, commutation of push-forward with the Chern character, and adjointness with respect to the generalized pairing; -- formal Hirzebruch-Riemann-Roch and the Cardy condition from physics; -- invariance of the full Hochschild structure under Fourier-Mukai transforms. Connections with homotopy theory and TQFT's are discussed in an appendix. A separate paper treats consequences of the HKR isomorphism. Applications of these results to the study of a mirror symmetric analogue of Chen-Ruan's orbifold product will be presented in a future paper.

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Andrei Caldararu. 2003-10-10. The Mukai pairing, I: the Hochschild structure. https://arxiv.org/abs/math/0308079

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