The relative Riemann-Roch theorem from Hochschild homology

TL;DR

This paper clarifies and elaborates on Markarian's proof of the relative Riemann-Roch theorem, focusing on Hochschild homology and the HKR map's properties, and confirms a conjecture of Caldararu.

Contribution

It provides detailed computations that show the HKR map's near-preservation of the Mukai pairing, advancing understanding of the relative Riemann-Roch theorem.

Findings

HKR map twisted by the square root of the Todd genus nearly preserves the Mukai pairing

Clarification of Markarian's proof of the relative Riemann-Roch theorem

Partial resolution of Caldararu's conjecture

Abstract

This write up attempts to clarify a preprint by Markarian [2] which proves This paper attempts to clarify a preprint of Markarian [2]. The preprint by Markarian [2] proves the relative Riemann-Roch theorem using a result describing how the HKR map fails to respect comultiplication. This paper elaborates on the core computations in [2]. These computations show that the HKR map twisted by the square root of the Todd genus "almost preserves" the Mukai pairing. This settles a part of a conjecture of Caldararu[3]. The relative Riemann-Roch theorem follows from this and a result of Caldararu[4].