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The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 3, pp. 513-559.

In this paper we extend the arithmetic Grothendieck-Riemann-Roch Theorem to projective morphisms between arithmetic varieties that are not necessarily smooth over the complex numbers. The main ingredient of this extension is the theory of generalized holomorphic analytic torsion classes previously developed by the authors.

Dans cet article on étend le théorème de Grothendieck-Riemann-Roch arithmétique aux morphismes projectifs entre variétés arithmétiques qui ne sont pas nécessairement lisses sur les nombres complexes. L’outil principal pour établir cette extension est la théorie des classes généralisées de torsion analytique holomorphe, développée dans les travaux précédents des auteurs.

Published online:
DOI: 10.5802/afst.1415
@article{AFST_2014_6_23_3_513_0,
     author = {Jos\'e Ignacio Burgos Gil and Gerard Freixas i Montplet and R\u{a}zvan Li\c{t}canu},
     title = {The arithmetic {Grothendieck-Riemann-Roch} theorem for general projective morphisms},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {513--559},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {3},
     year = {2014},
     doi = {10.5802/afst.1415},
     zbl = {06374879},
     mrnumber = {3266704},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1415/}
}
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José Ignacio Burgos Gil; Gerard Freixas i Montplet; Răzvan Liţcanu. The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 3, pp. 513-559. doi : 10.5802/afst.1415. https://afst.centre-mersenne.org/articles/10.5802/afst.1415/

[1] Bismut (J.-M.).— Holomorphic families of immersions and higher analytic torsion forms, Astérisque, vol. 244, SMF, 1997. | MR: 1623496 | Zbl: 0899.32013

[2] Bismut (J.-M.), Gillet (H.), and Soulé (C.).— Analytic torsion and holomorphic determinant bundles I, Comm. Math. Phys. 115 (1988), 49-78. | MR: 929146 | Zbl: 0651.32017

[3] Bismut (J.-M.), Gillet (H.), and Soulé (C.).— Analytic torsion and holomorphic determinant bundles II, Comm. Math. Phys. 115 (1988), 79-126. | MR: 929147 | Zbl: 0651.32017

[4] Bismut (J.-M.), Gillet (H.), and Soulé (C.).— Analytic torsion and holomorphic determinant bundles III, Comm. Math. Phys. 115 (1988), 301-351. | MR: 931666 | Zbl: 0651.32017

[5] Bismut (J.-M.) and Köhler (K.).— Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom. 1 (1992), 647-684. | MR: 1174905 | Zbl: 0784.32023

[6] Bismut (J.-M.) and Lebeau (G.).— Complex immersions and Quillen metrics, Publ. Math. IHES 74 (1991), 1-298. | Numdam | MR: 1188532 | Zbl: 0784.32010

[7] Burgos Gil (J. I.).— Green forms and their product, Duke Math. J. 75 (1994), 529-574. | MR: 1291696 | Zbl: 1044.14505

[8] Burgos Gil (J. I.).— Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Alg. Geom. 6 (1997), 335-377. | MR: 1489119 | Zbl: 0922.14002

[9] Burgos Gil (J. I.), Freixas i Montplet (G.), and Liţcanu (R.).— Generalized holomorphic analytic torsion, J. Eur. Math. Soc. 16 (2014), p. 463-535. | MR: 3165730 | Zbl: pre06273108

[10] Burgos Gil (J. I.), Freixas i Montplet (G.), and Liţcanu (R.).— Hermitian structures on the derived category of coherent sheaves, J. Math. Pures Appl. (9) 97 (2012), no. 5, 424-459. | MR: 2914943 | Zbl: 1248.18011

[11] Burgos Gil (J. I.), Kramer (J.), and Kühn (U.).— Arithmetic characteristic classes of automorphic vector bundles, Documenta Math. 10 (2005), 619-716. | MR: 2218402 | Zbl: 1080.14028

[12] Burgos Gil (J. I.), Kramer (J.), and Kühn (U.).— Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), no. 1, 1-172. | MR: 2285241 | Zbl: 1115.14013

[13] Burgos Gil (J. I.) and Liţcanu (R.).— Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions, Doc. Math. 15 (2010), 73-176. | MR: 2628847 | Zbl: 1192.14019

[14] Faltings (G.).— Calculus on arithmetic surfaces, Annals of Math. 119 (1984), 387-424. | MR: 740897 | Zbl: 0559.14005

[15] Faltings (G.).— Lectures on the arithmetic Riemann-Roch theorem, Annals of Math. Studies, vol. 127, Princeton University Press, 1992. | MR: 1158661 | Zbl: 0744.14016

[16] Gillet (H.), Rössler (D.), and Soulé (C.).— An arithmetic Riemann-Roch theorem in higher degrees, Ann. Inst. Fourier 58 (2008), 2169-2189. | Numdam | MR: 2473633 | Zbl: 1152.14023

[17] Gillet (H.) and Soulé (C.).— Arithmetic intersection theory, Publ. Math. I.H.E.S. 72 (1990), 94-174. | Numdam | MR: 1087394 | Zbl: 0741.14012

[18] Gillet (H.) and Soulé (C.).— Characteristic classes for algebraic vector bundles with hermitian metric I, II, Annals of Math. 131 (1990), 163-203, 205-238. | MR: 1038362 | Zbl: 0715.14006

[19] Gillet (H.) and Soulé (C.).— Analytic torsion and the arithmetic Todd genus, Topology 30 (1991), no. 1, 21-54, With an appendix by D. Zagier. | MR: 1081932 | Zbl: 0787.14005

[20] Gillet (H.) and Soulé (C.).— An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473-543. | MR: 1189489 | Zbl: 0777.14008

[21] Griffiths (P.) and Harris (J.).— Principles of algebraic geometry, John Wiley & Sons, Inc., 1994. | MR: 1288523 | Zbl: 0836.14001

[22] Hörmander (L.).— The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis. | MR: 1065993 | Zbl: 0712.35001

[23] Kawaguchi (S.) and Moriwaki (A.).— Inequalities for semistable families of arithmetic varieties, J. Math. Kyoto Univ. 41 (2001), no. 1, 97-182. | MR: 1844863 | Zbl: 1041.14007

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