{"title":"Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory","authors":"G. Kondyrev, A. Prikhodko","doi":"10.4310/cjm.2021.v9.n4.a1","DOIUrl":null,"url":null,"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 quasiand indcoherent 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 LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2021.v9.n4.a1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
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 quasiand indcoherent 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 LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.