Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

Grigory Kondyrev; Artem Prikhodko

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arXiv · 1906.00172

Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

Grigory Kondyrev·Artem Prikhodko

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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.

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BibTeX RIS

Grigory Kondyrev, Artem Prikhodko. 2019-06-01. Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory. https://arxiv.org/abs/1906.00172

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