An arithmetic Riemann-Roch theorem in higher degrees
[Un théorème de Riemann-Roch arithmétique en degrés supérieurs]
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2169-2189.

Nous démontrons un analogue du théorème de Grothendieck-Riemann-Roch en géométrie d’Arakelov.

We prove an analog in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.

DOI : 10.5802/aif.2410
Classification : 14G40, 14C40, 58J52
Keywords: Arakelov Geometry, Grothendieck-Riemann-Roch theorem, analytic torsion form, arithmetic intersection theory
Mot clés : géométrie d’Arakelov, théorème de Grothendieck-Riemann-Roch, forme de torsion analytique, théorie de l’intersection arithmétique
Gillet, Henri 1 ; Rössler, Damian 2 ; Soulé, Christophe 3

1 University of Illinois at Chicago Department of Mathematics Box 4348 Chicago IL 60680 (USA)
2 Institut de Mathématiques de Jussieu 2 place Jussieu Case Postale 7012 75251 Paris cedex 05 (France)
3 IHÉS 35 route de Chartres 91440 Bures-Sur-Yvette (France)
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Gillet, Henri; Rössler, Damian; Soulé, Christophe. An arithmetic Riemann-Roch theorem in higher degrees. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2169-2189. doi : 10.5802/aif.2410. http://archive.numdam.org/articles/10.5802/aif.2410/

[1] Arakelov, S. Ju. An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat., Volume 38 (1974), pp. 1179-1192 | MR | Zbl

[2] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004 (Corrected reprint of the 1992 original) | MR | Zbl

[3] Berthelot, P.; Grothendieck, A.; Illusie, L. Théorie des intersections et théorème de Riemann-Roch, Springer-Verlag, Berlin, 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, Lecture Notes in Mathematics, Vol. 225 | MR | Zbl

[4] Bismut, J.-M.; Gillet, H.; Soulé, C. Bott-Chern currents and complex immersions, Duke Math. J., Volume 60 (1990) no. 1, pp. 255-284 | DOI | MR | Zbl

[5] Bismut, Jean-Michel Holomorphic families of immersions and higher analytic torsion forms, Astérisque (1997) no. 244, pp. viii+275 | Numdam | MR | Zbl

[6] Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe 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 | Zbl

[7] Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe 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

[8] Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe 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 | Zbl

[9] Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe Complex immersions and Arakelov geometry, The Grothendieck Festschrift, Vol. I (Progr. Math.), Volume 86, Birkhäuser Boston, Boston, MA, 1990, pp. 249-331 | MR | Zbl

[10] Bismut, Jean-Michel; Köhler, Kai Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom., Volume 1 (1992) no. 4, pp. 647-684 | MR | Zbl

[11] Bost, Jean-Benoît Analytic torsion of projective spaces and compatibility with immersions of Quillen metrics, Internat. Math. Res. Notices (1998) no. 8, pp. 427-435 | DOI | MR | Zbl

[12] Faltings, Gerd Calculus on arithmetic surfaces, Ann. of Math. (2), Volume 119 (1984) no. 2, pp. 387-424 | DOI | MR | Zbl

[13] Faltings, Gerd Lectures on the arithmetic Riemann-Roch theorem, Annals of Mathematics Studies, 127, Princeton University Press, Princeton, NJ, 1992 (Notes taken by Shouwu Zhang) | MR | Zbl

[14] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984 | MR | Zbl

[15] Gillet, H.; Soulé, C. Analytic torsion and the arithmetic Todd genus, Topology, Volume 30 (1991) no. 1, pp. 21-54 (With an appendix by D. Zagier) | DOI | MR | Zbl

[16] Gillet, Henri; Soulé, Christophe Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. (1990) no. 72, p. 93-174 (1991) | Numdam | MR | Zbl

[17] Gillet, Henri; Soulé, Christophe Characteristic classes for algebraic vector bundles with Hermitian metric. I, Ann. of Math. (2), Volume 131 (1990) no. 1, pp. 163-203 | DOI | MR | Zbl

[18] Gillet, Henri; Soulé, Christophe Characteristic classes for algebraic vector bundles with Hermitian metric. II, Ann. of Math. (2), Volume 131 (1990) no. 2, pp. 205-238 | DOI | MR | Zbl

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

[20] Gubler, Walter Moving lemma for K 1 -chains, J. Reine Angew. Math., Volume 548 (2002), pp. 1-19 | DOI | MR | Zbl

[21] Lang, Serge Introduction to Arakelov theory, Springer-Verlag, New York, 1988 | MR | Zbl

[22] Lelong, Pierre Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France, Volume 85 (1957), pp. 239-262 | Numdam | MR | Zbl

[23] Roessler, Damian An Adams-Riemann-Roch theorem in Arakelov geometry, Duke Math. J., Volume 96 (1999) no. 1, pp. 61-126 | DOI | MR | Zbl

[24] Soulé, C. Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, 33, Cambridge University Press, Cambridge, 1992 (With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer) | MR | Zbl

[25] Zha, Y. A General Arithmetic Riemann-Roch Theorem, Chicago University (1998) (Ph. D. Thesis)

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