The equivariant Hirzebruch-Riemann-Roch theorem and the geometry of derived loop spaces. (English. Russian original) Zbl 1440.14044
Math. Notes 107, No. 6, 1029-1033 (2020); translation from Mat. Zametki 107, No. 6, 940-944 (2020).
From the introduction: In [“Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory”, Preprint, arXiv:1906.00172], G. Kondyrev and the author proved a theorem, which generalizes both the Hirzebruch-Riemann-Roch theorem and the holomorphic Atiyah-Bott formula.
A similar result for a projective scheme and a finite-order endomorphism was obtained in [P. Donovan, Bull. Soc. Math. Fr. 97, 257–273 (1969; Zbl 0185.49401)]. Of most interest is not the result but the new proof, which offers potential possibilities of generalization in other contexts. In particular, we do not apply Grothendieck’s trick of factoring a projective morphism into a composition of a closed embedding and a projection, do not use Chow’s lemma, and do not assume that the scheme is projective.
Since the proof given in [Kondyrev and the author, loc. cit.] is technically involved, in this paper, we shall try to clarify the main idea underlying it, omitting technical details. The proof consists of two parts. In the nonequivariant part, we reprove the usual Hirzebruch-Riemann-Roch formula by studying the geometry of derivative loop spaces. Our approach is close to that of N. Markarian [J. Lond. Math. Soc., II. Ser. 79, No. 1, 129–143 (2009; Zbl 1167.14005)] but more geometric. The equivariant part is reduced to the non-equivariant one by using functoriality considerations and a formality criterion (Proposition 9), which is of independent interest.
A similar result for a projective scheme and a finite-order endomorphism was obtained in [P. Donovan, Bull. Soc. Math. Fr. 97, 257–273 (1969; Zbl 0185.49401)]. Of most interest is not the result but the new proof, which offers potential possibilities of generalization in other contexts. In particular, we do not apply Grothendieck’s trick of factoring a projective morphism into a composition of a closed embedding and a projection, do not use Chow’s lemma, and do not assume that the scheme is projective.
Since the proof given in [Kondyrev and the author, loc. cit.] is technically involved, in this paper, we shall try to clarify the main idea underlying it, omitting technical details. The proof consists of two parts. In the nonequivariant part, we reprove the usual Hirzebruch-Riemann-Roch formula by studying the geometry of derivative loop spaces. Our approach is close to that of N. Markarian [J. Lond. Math. Soc., II. Ser. 79, No. 1, 129–143 (2009; Zbl 1167.14005)] but more geometric. The equivariant part is reduced to the non-equivariant one by using functoriality considerations and a formality criterion (Proposition 9), which is of independent interest.
MSC:
| 14C40 | Riemann-Roch theorems |
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |
| 18N10 | 2-categories, bicategories, double categories |
References:
| [1] | G. Kondyrev and A. Prikhodko, Equivariant Grothendieck-Riemann-Roch Theorem Via Formal Deformation Theory, arXiv: 1906.00172(2019). |
| [2] | Donovan, P., Bull. Soc. Math. France, 97, 257 (1969) · Zbl 0185.49401 · doi:10.24033/bsmf.1680 |
| [3] | Markarian, N., J. London Math. Soc. (2), 79, 1, 129 (2008) · Zbl 1167.14005 · doi:10.1112/jlms/jdn064 |
| [4] | G. Kondyrev and A. Prikhodko, J. Inst. Math. Jussieu (2018). |
| [5] | Gaitsgory, D.; Rozenblyum, N., A Study in Derived Algebraic Geometry Vol. I. Correspondences and Duality (2017), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 1408.14001 |
| [6] | Gaitsgory, D.; Rozenblyum, N., A Study in Derived Algebraic Geometry Vol. I. Deformations, Lie Theory and Formal Geometry (2017), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 1409.14003 |
| [7] | Hennion, B., J. Reine Angew. Math., 741, 1 (2018) · Zbl 1423.14011 · doi:10.1515/crelle-2015-0065 |
| [8] | D. Arinkin, A. Caldararu, and M. Hablicsek, Formality of Derived Intersections and the Orbifold HKR Isomorphism, arXiv: 1412.5233(2014). · Zbl 1445.14009 |
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