The Mukai Pairing and Integral Transforms in Hochschild Homology

Abstract

Let X be a smooth proper scheme over a field of characteristic 0. Following Shklyarov [10] , we construct a (non-degenerate) pairing on the Hochschild homology of perf (X), and hence, on the Hochschild homology of X. On the other hand the Hochschild homology of X also has the Mukai pairing (see [1]). If X is Calabi-Yau, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in H0(M0(2,0)) on the algebra of closed states of a version of the B-Model on X. We show that these pairings ”almost” coincide. This is done via a different view of the construction of integral transforms in Hochschild homology that originally appeared in Caldararu’s work [1]. This is used to prove that the more ”natural” construction of integral transforms in Hochschild homology by Shklyarov [10] coincides with that of Caldararu [1]. These results give rise to a Hirzebruch Riemann-Roch theorem for the sheafification of the Dennis trace map.