2019/06/01 by Grigory Kondyrev, Kondyrev, Grigory, Artem Prikhodko +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.CT
paper · pdf · doi:10.48550/arxiv.1906.00172
54 pages; Minor update: added references and corrected typos
openalex publication_date 2019/06/01 · arxiv created 2019/06/25 · arxiv updated 2019/06/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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.