Deligne–Riemann–Roch and intersection bundles

Eriksson, Dennis; Freixas i Montplet, Gerard

doi:10.5802/jep.254

Deligne–Riemann–Roch and intersection bundles
[Deligne-Riemann-Roch et fibrés d’intersection]

Eriksson, Dennis ¹ ; Freixas i Montplet, Gerard ²

¹Department of Mathematics, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
²C.N.R.S., Centre de Mathématiques Laurent Schwartz, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 247-361

Abstract (VO)
Résumé

This article is part of a series of works by the authors with the goal of completing a far-reaching program propounded by Deligne, aiming to extend the codimension one part of the Grothendieck–Riemann–Roch theorem from isomorphism classes of line bundles to canonical isomorphisms thereof. The paper develops a relative functorial intersection theory with values in line bundles, together with a formalism that generalizes previous constructions by Deligne and Elkik, related to the right-hand side of the theorem.

Cet article fait partie d’une série de travaux des auteurs ayant pour objectif de compléter un vaste programme énoncé par Deligne, visant à relever la partie de codimension $1$ du théorème de Grothendieck-Riemann-Roch des classes d’isomorphisme de fibrés en droites à des isomorphismes canoniques. L’article développe une théorie d’intersection fonctorielle relative à valeurs dans les fibrés en droites, avec un formalisme qui généralise les constructions précédentes de Deligne et Elkik, liées au côté droit du théorème.

Reçu le : 2023-06-08
Accepté le : 2024-01-12
Publié le : 2024-01-26

Zbl

DOI : 10.5802/jep.254

Classification : 14C17, 19D99, 14C40, 19D23
Keywords: Deligne program, virtual categories, intersection bundles, Grothendieck–Riemann–Roch, categorification
Mots-clés : Programme de Deligne, catégories virtuelles, fibrés d’intersection, Grothendieck-Riemann-Roch, catégorification

Affiliations des auteurs :

Eriksson, Dennis ¹ ; Freixas i Montplet, Gerard ²

¹ Department of Mathematics, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
² C.N.R.S., Centre de Mathématiques Laurent Schwartz, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Deligne{\textendash}Riemann{\textendash}Roch and intersection~bundles},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {247--361},
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Eriksson, Dennis; Freixas i Montplet, Gerard. Deligne–Riemann–Roch and intersection bundles. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 247-361. doi: 10.5802/jep.254

Bibliographie
Cité par

[1] Baas, N. A.; Dundas, B. I.; Richter, B.; Rognes, J. Ring completion of rig categories, J. reine angew. Math., Volume 674 (2013), pp. 43-80 | DOI | MR | Zbl

[2] Bismut, J.-M.; Freed, D. The analysis of elliptic families. I. Metrics and connections on determinant bundles, Comm. Math. Phys., Volume 106 (1986) no. 1, pp. 159-176 | DOI | MR | Zbl

[3] Bismut, J.-M.; Freed, D. The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys., Volume 107 (1986) no. 1, pp. 103-163 | MR | Zbl

[4] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys., Volume 115 (1988) no. 1, pp. 49-78 | DOI | MR | Zbl

[5] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms, Comm. Math. Phys., Volume 115 (1988) no. 1, pp. 79-126 | DOI | MR | Zbl

[6] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351 | DOI | MR

[7] Borel, A.; Serre, J.-P. Le théorème de Riemann-Roch, Bull. Soc. math. France, Volume 86 (1958), pp. 97-136 | DOI | Numdam | Zbl | MR

[8] Boucksom, S.; Eriksson, D. Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, Adv. Math., Volume 378 (2021), 107501, 124 pages | Zbl | DOI | MR

[9] Boucksom, S.; Gubler, W.; Martin, F. Differentiability of relative volumes over an arbitrary non-Archimedean field, Internat. Math. Res. Notices (2022) no. 8, pp. 6214-6242 | Zbl | DOI | MR

[10] Deligne, P. Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) (Contemp. Math.), Volume 67, American Mathematical Society, Providence, RI, 1987, pp. 93-177 | DOI | Zbl | MR

[11] Ducrot, F. Cube structures and intersection bundles, J. Pure Appl. Algebra, Volume 195 (2005) no. 1, pp. 33-73 | Zbl | DOI | MR

[12] Elkik, R. Fibrés d’intersections et intégrales de classes de Chern, Ann. Sci. École Norm. Sup. (4), Volume 22 (1989) no. 2, pp. 195-226 | Numdam | Zbl | DOI | MR

[13] Elkik, R. Métriques sur les fibrés d’intersection, Duke Math. J., Volume 61 (1990) no. 1, pp. 303-328 | MR | Zbl | DOI

[14] Eriksson, D. Un isomorphisme de type Deligne-Riemann-Roch, Comptes Rendus Mathématique, Volume 347 (2009) no. 19-20, pp. 1115-1118 | Zbl | Numdam | DOI | MR

[15] Eriksson, D.; Freixas i Montplet, G.; Wentworth, R. A. Complex Chern–Simons bundles in the relative setting, 2021 | arXiv

[16] Franke, J. Chow categories, Compositio Math., Volume 76 (1990) no. 1-2, pp. 101-162 Algebraic geometry (Berlin, 1988) | MR | Numdam | Zbl

[17] Franke, J. Chern functors, Arithmetic algebraic geometry (Texel, 1989) (Progress in Math.), Volume 89, Birkhäuser Boston, Boston, MA, 1991, pp. 75-152 | DOI | MR | Zbl

[18] Franke, J. Riemann–Roch in functorial form (1992) (Unpublished)

[19] Fulton, W. Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1998 | DOI

[20] Fulton, W.; Lang, S. Riemann-Roch algebra, Grundlehren Math. Wissen., 277, Springer-Verlag, New York, 1985, x+203 pages | DOI | MR

[21] Gabriel, P.; Zisman, M. Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb. (2), 35, Springer-Verlag New York, Inc., New York, 1967 | DOI | MR

[22] Muñoz García, E. Fibrés d’intersection, Compositio Math., Volume 124 (2000) no. 3, pp. 219-252 | MR | Zbl

[23] Gillet, H.; Soulé, S. An arithmetic Riemann-Roch theorem, Invent. Math., Volume 110 (1992) no. 3, pp. 473-543 | DOI | MR | Zbl

[24] Goerss, P.; Jardine, J. Simplicial homotopy theory, Progress in Math., 174, Birkhäuser Verlag, Basel, 1999, xvi+510 pages | DOI | MR

[25] Grayson, D. Higher algebraic $K$ -theory. II (after Daniel Quillen), Algebraic $K$ -theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) (Lect. Notes in Math.), Volume 551, Springer, 1976, pp. 217-240 | Zbl | MR

[26] Grothendieck, A. Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci., Volume 8 (1961), pp. 5-222 | Numdam | MR

[27] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci., Volume 20 (1964), pp. 5-251 | DOI | MR

[28] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci., Volume 24 (1965), pp. 5-231 | Zbl | MR

[29] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci., Volume 28 (1966), pp. 5-255 | Zbl | MR

[30] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci., Volume 32 (1967), pp. 5-361 | Numdam | Zbl | MR

[31] Théorie des intersections et théorème de Riemann-Roch (SGA 6), Lect. Notes in Math., 225, Springer-Verlag, Berlin-New York, 1971 Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre | DOI

[32] Johnson, N.; Yau, D. Bimonoidal categories, $E_{n}$ -monoidal categories, and algebraic $K$ -theory, 2021 | arXiv

[33] Knudsen, F. Determinant functors on exact categories and their extensions to categories of bounded complexes, Michigan Math. J., Volume 50 (2002) no. 2, pp. 407-444 Erratum: Ibid. 50 (2002), no. 2, p. 407–444 | MR | Zbl

[34] Knudsen, F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., Volume 39 (1976) no. 1, pp. 19-55 | MR | Zbl | DOI

[35] Laplaza, M. L. Coherence for distributivity, Coherence in categories (Lect. Notes in Math.), Volume 281, Springer-Verlag, Berlin-Heidelberg-New York, 1972, pp. 29-65 | DOI | MR | Zbl

[36] Lipman, J.; Neeman, A. Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor, Illinois J. Math., Volume 51 (2007) no. 1, pp. 209-236 | Zbl | MR

[37] Mac Lane, S. Natural associativity and commutativity, Rice Univ. Stud., Volume 49 (1963) no. 4, pp. 28-46 | MR | Zbl

[38] Mac Lane, S. Categories for the working mathematician, Graduate Texts in Math., 5, Springer-Verlag, New York-Berlin, 1971 | MR

[39] Mumford, D. The red book of varieties and schemes, Lect. Notes in Math., 1358, Springer-Verlag, Berlin, 1999 | DOI | MR

[40] Muro, F.; Tonks, A.; Witte, M. On determinant functors and $K$ -theory, Publ. Mat., Volume 59 (2015) no. 1, pp. 137-233 | MR | Zbl | DOI

[41] Nakai, Y. Some fundamental lemmas on projective schemes, Trans. Amer. Math. Soc., Volume 109 (1963), pp. 296-302 | DOI | MR | Zbl

[42] Nakayama, N. Intersection sheaves over normal schemes, J. Math. Soc. Japan, Volume 62 (2010) no. 2, pp. 487-595 | MR | Zbl

[43] Patel, D. de Rham $ℰ$ -factors, Invent. Math., Volume 190 (2012) no. 2, pp. 299-355 | Zbl | DOI | MR

[44] Quillen, D. Higher algebraic $K$ -theory. I, Algebraic $K$ -theory, I: Higher $K$ -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (Lect. Notes in Math.), Volume 341, Sprinegr (1973), pp. 85-147 | MR | Zbl

[45] Richter, B. From categories to homotopy theory, Cambridge Studies in Advanced Math., 188, Cambridge University Press, Cambridge, 2020 | DOI | MR

[46] Rössler, D. A local refinement of the Adams-Riemann-Roch theorem in degree one, Arithmetic L-functions and differential geometric methods (Progress in Math.), Volume 338, Birkhäuser/Springer, Cham, 2021, pp. 213-246 | DOI | MR | Zbl

[47] Saavedra Rivano, N. Catégories tannakiennes, Lect. Notes in Math., 265, Springer-Verlag, Berlin-New York, 1972 | MR

[48] Sính, H. X. $Gr$ -catégories, Ph. D. Thesis, Institut pédagogique no. 2 de Hanoi (1975) (https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/SinhThesis.pdf)

[49] Suslin, A.; Voevodsky, Vl. Relative cycles and Chow sheaves, Cycles, transfers, and motivic homology theories (Annals of Math. Studies), Volume 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 10-86 | MR | Zbl

[50] The Stacks Project Authors Stacks Project, https://stacks.math.columbia.edu, 2023

[51] Thomason, R. Beware the phony multiplication on Quillen’s $𝒜^{- 1} 𝒜$ , Proc. Amer. Math. Soc., Volume 80 (1980) no. 4, pp. 569-573 | MR | Zbl

[52] Thomason, R. First quadrant spectral sequences in algebraic $K$ -theory via homotopy colimits, Comm. Algebra, Volume 10 (1982) no. 15, pp. 1589-1668 | DOI | MR | Zbl

[53] Thomason, R. W. Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc., Volume 85 (1979) no. 1, pp. 91-109 | Zbl | MR | DOI

[54] Thomason, R. W.; Trobaugh, T. Higher algebraic $K$ -theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III (Progress in Math.), Volume 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435 | MR | Zbl | DOI

[55] Waldhausen, F. Algebraic $K$ -theory of spaces, Algebraic and geometric topology (New Brunswick, NJ, 1983) (Lect. Notes in Math.), Volume 1126, Springer, 1985, pp. 318-419 | DOI | MR | Zbl

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