Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

TL;DR

This paper generalizes the Grothendieck-Riemann-Roch theorem and Atiyah-Bott fixed point formula using higher category traces and formal deformation theory, providing new insights into Todd classes and equivariant cases.

Contribution

It introduces a novel proof leveraging higher category traces and formal deformation theory, unifying fixed point formulas and Riemann-Roch in a new framework.

Findings

Unified proof of Atiyah-Bott and Grothendieck-Riemann-Roch theorems

Description of Todd class via formal group structures on derived loop schemes

Reduction of equivariant case to non-equivariant via localization

Abstract

We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.