A fixed point formula of Lefschetz type in Arakelov geometry II: A residue formula
Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 81-103.

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "à la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

Cet article est le second d'une série, dont l'objet est un analogue en géométrie d'Arakelov de la formule du point fixe de Lefschetz holomorphe. Nous utilisons le résultat principal du premier article pour prouver une formule des résidus "à la Bott" pour des classes caractéristiques existant sur des variétés arithmétiques munies d'une action de tore; de récents résultats de Bismut-Goette sur la torsion analytique équivariante (de Ray-Singer) jouent un rôle clé dans la preuve.

DOI: 10.5802/aif.1877
Classification: 14G40, 58J52, 14C40, 14L30, 58J20, 14K15
Keywords: Arakelov, analytic torsion, Bott, fixed point formula, height, Hermitian bundle
Mot clés : Arakelov, torsion analytique, Bott, formule du point fixe, hauteur, fibré hermitien
Köhler, Kai 1; Roessler, Damien 2

1 Mathematisches Institut, Einsteinstr. 62, 48149 Münster (Allemagne)
2 ETH-Zentrum, Mathematik Department, 8092 Zurich (Suisse)
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Köhler, Kai; Roessler, Damien. A fixed point formula of Lefschetz type in Arakelov geometry II: A residue formula. Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 81-103. doi : 10.5802/aif.1877. http://archive.numdam.org/articles/10.5802/aif.1877/

[AS] M.F. Atiyah; I.M. Singer The index of elliptic operators I, II, III, Ann. of Math., Volume 87 (1967), pp. 484-604 | DOI | MR | Zbl

[B1] J.-M. Bismut Equivariant Bott-Chern currents and the Ray-Singer analytic torsion, Math. Ann., Volume 287 (1990), pp. 495-507 | DOI | MR | Zbl

[B2] J.-M. Bismut Equivariant short exact sequences of vector bundles and their analytic torsion forms, Comp. Math., Volume 93 (1994), pp. 291-354 | Numdam | MR | Zbl

[B3] J.-M. Bismut Equivariant immersions and Quillen metrics, J. Diff. Geom., Volume 41 (1995), pp. 53-157 | MR | Zbl

[BeGeV] N. Berline; E. Getzler; M. Vergne Heat kernels and Dirac operators, Springer, 1992 | MR | Zbl

[BGo] J.-M. Bismut; S. Goette Holomorphic equivariant analytic torsions, Geom. Funct. Anal., Volume 10 (2001), pp. 1289-1422 | DOI | MR | Zbl

[BGS3] J.-M. Bismut Analytic torsion and holomorphic determinant bundles III, Comm. Math. Phys., Volume 115 (1988), pp. 301-351 | DOI | MR | Zbl

[DG] J. Dieudonné; A. Grothendieck Éléments de Géométrie Algébrique I, Grundlehren, 166, Springer, 1971 | Numdam | Zbl

[Do] H. Donnelly Spectrum and the fixed point set of isometries I, Math. Ann., Volume 224 (1976), pp. 161-176 | DOI | MR | Zbl

[DoP] H. Donnelly; V.K. Patodi Spectrum and the fixed point sets of isometries II, Topology, Volume 16 (1977), pp. 1-11 | DOI | MR | Zbl

[EdGr] D. Edidin; W. Graham Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math., Volume 120 (1998) no. 3, pp. 619-636 | DOI | MR | Zbl

[GS3] H. Gillet; C. Soulé Characteristic classes for algebraic vector bundles with Hermitian metrics I, II, Annals of Math., Volume 131 (1990), p. 163-203 ; 205--238 | DOI | MR | Zbl

[GS8] H. Gillet; C. Soulé An arithmetic Riemann-Roch theorem, Inv. Math., Volume 110 (1992), pp. 473-543 | DOI | MR | Zbl

[Ha] R. Hartshorne Algebraic geometry, Springer, 1977 | MR | Zbl

[K1] K. Köhler Equivariant analytic torsion on n (), Math. Ann., Volume 297 (1993), pp. 553-565 | DOI | MR | Zbl

[K2] K. Köhler Holomorphic torsion on Hermitian symmetric spaces, J. reine angew. Math., Volume 460 (1995), pp. 93-116 | DOI | MR | Zbl

[K4] K. Köhler; N. Schappacher, A. Reznikov (ed.) Complex analytic torsion forms for torus fibrations and moduli spaces, Regulators in Analysis, Geometry and Number Theory, pp. 167-205 | Zbl

[K5] K. Köhler A Hirzebruch proportionality principle in Arakelov geometry (April 2001) (Preprint Jussieu, 284) | MR

[KK] Ch. Kaiser; K. Köhler A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties (to appear) | Zbl

[KR1] K. Köhler; D. Roessler A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof, Inv. Math., Volume 145 (2001), pp. 333-396 | DOI | MR | Zbl

[Ma] X. Ma Submersions and equivariant Quillen metrics, Ann. Inst. Fourier (Grenoble) (2000), pp. 1539-1588 | DOI | Numdam | MR | Zbl

[MR] V. Maillot; D. Roessler Conjectures sur les dérivées logarithmiques des fonctions L d'Artin aux entiers négatifs (to appear) | Zbl

[Thom] R.W. Thomasson Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory (Princeton, 1983) (1987) | Zbl

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