Equivariant GrothendieckRiemannRoch theorem via formal deformation theory
Abstract
We use the formalism of traces in higher categories to prove a common generalization of the holomorphic AtiyahBott fixed point formula and the GrothendieckRiemannRoch 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 GaitsgoryRozenblyum. 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 nonequivariant one by a variant of the AtiyahBott localization theorem.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.00172
 Bibcode:
 2019arXiv190600172K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory
 EPrint:
 54 pages