Cambridge Journal of Mathematics

Volume 9 (2021)

Number 4

Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory

Pages: 809 – 899



Grigory Kondyrev (Northwestern University, Evanston, Illinois, U.S.A.; and National Research University Higher School of Economics, Russian Federation)

Artem Prikhodko (National Research University Higher School of Economics, Center for Advanced Studies, Skoltech, Russian Federation)


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 $\mathcal{L}X$. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah–Bott localization theorem.

Grigory Kondyrev is supported by the HSE University Basic Research Program.

Artem Prikhodko has been funded within the framework of the HSE University Basic Research Program, and was also supported by the Simons Foundation stipend for PhD students.

Received 17 December 2020

Published 22 March 2022