The Mukai pairing. I: A categorical approach
The purpose of the article under review is to develop an abstract framework to define and study Hochschild homology and cohomology in various geometric contexts (e.g. smooth complex manifolds, smooth projective varieties or smooth proper Deligne-Mumford stacks over a field of characteristic zero). This is done using the formalism of 2-categories. In the classical situation of projective varieties, the 2-category which is used is the category of kernels: objects are smooth projective varieties, 1-morphisms are kernels (i.e., \(\text{1-Hom}(X, Y)=\text{D}(X \times Y))\) and 2-morphisms are morphisms of kernels. For any smooth projective variety \(X\), the Serre functor \(S_X\) can be interpreted as an \(1\)-isomorphism of \(X\). Then the Hochschild homology and cohomology groups of \(X\) can be defined by \(\text{HH}^i(X)=\text{2-Hom}(\text{id}_X, \text{id}_X[i])\) and \(\text{HH}_i(X)=\text{2-Hom}(S^{-1}_X, \text{id}_X[-i])\). The authors also define the action of kernels by pullback and push-forward on Hochschild homology. Then they construct a generalized Mukai pairing on \(\text{HH}_{\bullet}(X)\) which extends the classical Mukai pairing on the cohomology of \(K3\)-surfaces, they define a Chern character in Hochschild homology which factorizes through \(K\)-theory and they prove a Riemann-Roch type index theorem. The paper is written with great care and does not assume any preliminary knowledge of 2-category theory. It is a foundational article in the field, together (for instance) with the papers of \textit{D. Shklyarov} [Hirzebruch-Riemann-Roch theorems for DG-algebras, \url{arXiv:0710.1937}], \textit{B. Keller} [J. Pure Appl. Algebra 136, No.1, 1--56 (1999; Zbl 0923.19004)] and chapters 4 and 5 of \textit{M. Kashiwara-P. Schapira}'s book ``Deformation quantization modules'' [cf. \url{arXiv:1003.3304}].
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