The Mukai pairing and integral transforms in Hochschild homology

TL;DR

This paper compares two pairings on Hochschild homology of smooth proper schemes, showing they nearly coincide, and proves the equivalence of two integral transform constructions, leading to a Hirzebruch Riemann-Roch theorem.

Contribution

It demonstrates the near equivalence of the Mukai pairing and Shklyarov's pairing on Hochschild homology, and shows the consistency of different integral transform constructions.

Findings

The pairings on Hochschild homology nearly coincide for Calabi-Yau schemes.

Shklyarov's integral transforms coincide with Caldararu's constructions.

Results imply a Hirzebruch Riemann-Roch theorem for the sheafified Dennis trace map.

Abstract

Let $X$ be a smooth proper scheme over a field of characteristic 0. Following D. Shklyarov [10], we construct a (non-degenerate) pairing on the Hochschild homology of $\per{X}$, 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 $\text{H}_{0}({\mathcal M}_0(2,0))$ 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.