Dans cet article, on calcule le comportement de la métrique de Quillen équivariante par submersions. On étend ainsi une formule de Berthomieu-Bismut au cas équivariant.
In this paper, we calculate the behaviour of the equivariant Quillen metric by submersions. We thus extend a formula of Berthomieu-Bismut to the equivariant case.
@article{AIF_2000__50_5_1539_0, author = {Ma, Xiaonan}, title = {Submersions and equivariant {Quillen} metrics}, journal = {Annales de l'Institut Fourier}, pages = {1539--1588}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {50}, number = {5}, year = {2000}, doi = {10.5802/aif.1800}, mrnumber = {2002c:58053}, zbl = {0964.58025}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1800/} }
TY - JOUR AU - Ma, Xiaonan TI - Submersions and equivariant Quillen metrics JO - Annales de l'Institut Fourier PY - 2000 SP - 1539 EP - 1588 VL - 50 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1800/ DO - 10.5802/aif.1800 LA - en ID - AIF_2000__50_5_1539_0 ER -
Ma, Xiaonan. Submersions and equivariant Quillen metrics. Annales de l'Institut Fourier, Tome 50 (2000) no. 5, pp. 1539-1588. doi : 10.5802/aif.1800. http://www.numdam.org/articles/10.5802/aif.1800/
[ABo] A Lefchetz fixed point formula for elliptic complexes, I, Ann. of Math., 86 (1968), 374-407. | MR | Zbl
, ,[ABoP] On the heat equation and the Index Theorem, Invent. Math., 19 (1973), 279-330. | MR | Zbl
, , ,[BeGeV] Heat kernels and the Dirac operator, Grundl. Math. Wiss., 298 (1992). | MR | Zbl
, , ,[BerB] Quillen metric and higher analytic torsion forms, J. reine angew. Math, 457 (1994), 85-184. | MR | Zbl
, ,[B1] The Index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 83 (1986), 91-151. | MR | Zbl
,[B2] Superconnection currents and complex immersions, Invent. Math., 99 (1990), 59-113. | MR | Zbl
,[B3] Koszul complexes, harmonic oscillators and the Todd class, J. Amer. Math. Soc., 3 (1990), 159-256. | MR | Zbl
,[B4] Equivariant short exact sequences of vector bundles and their analytic torsion forms, Comp. Math., 93 (1994), 291-354. | Numdam | MR | Zbl
,[B5] Equivariant immersions and Quillen metrics, J. Diff. Geom., 41 (1995), 53-159. | MR | Zbl
,[B6] Families of immersions, and higher analytic torsion, Astérisque, 244 (1997). | Numdam | MR | Zbl
,[B7] The Atiyah-Singer Index Theorems: a probabilistic approach, II. The Lefschetz fixed point formulas, J. Funct. Anal., 57 (1984), 329-348. | MR | Zbl
,[BCh] n-invariants and their adiabatic limits, J. Amer. Math. Soc., 2 (1989), 33-70. | MR | Zbl
, ,[BGS1] Analytic torsion and holomorphic determinant bundles, I, Comm. Math. Phys., 115 (1988), 49-78. | MR | Zbl
, , ,[BGS2] Analytic torsion and holomorphic determinant bundles, II, Comm. Math. Phys., 115 (1988), 79-126. | MR | Zbl
, , ,[BGS3] Analytic torsion and holomorphic determinant bundles, III, Comm. Math. Phys., 115 (1988), 301-351. | MR | Zbl
, , ,[BKö] Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom., 1 (1992), 647-684. | MR | Zbl
, ,[BL] Complex immersions and Quillen metrics, Publ. Math. IHES, 74 (1991), 1-297. | Numdam | MR | Zbl
, ,[CP] Introduction à la théorie des équations aux dérivées partielles, Paris: Gauthier-Villars, 1981. | Zbl
, ,[D] Adiabatic limits, non multiplicativity of signature and Leray spectral sequence, J. Amer. Math. Soc., 4 (1991), 265-321. | MR | Zbl
,[DM] Adiabatic limit of the analytic torsion, Preprint.
, ,[Ge] A short proof of the Atiyah-Singer Index Theorem, Topology, 25 (1986), 111-117. | MR | Zbl
,[GrH] Principles of Algebraic Geometry, New York, Wiley, 1978. | MR | Zbl
, ,[GS1] Analytic torsion and the arithmetic Todd genus, Topology, 30 (1991), 21-54. | MR | Zbl
, ,[GS2] An arithmetic Riemann-Roch Theorem, Invent. Math., 110 (1992), 473-543. | MR | Zbl
, ,[KM] The projectivity of the moduli space of stable curves, I, Preliminaries on “det” and “div”, Math. Scand., 39 (1976), 19-55. | Zbl
, ,[K] Complex analytic torsion forms for torus fibrations and moduli spaces, in Regulars in analysis, Geometry and Number Theory, N. Schappacher, A. Reznikov (eds.), Progress in Math., Birkhäuser, 171 (1999). | Zbl
,[KRo] Un théorème du point fixe de Lefschetz en géométrie d'Arakerov, C. R. Acad. Sci. Paris, 326, série I (1998), 719-722. | Zbl
, ,[KRo1] A fixed point formula of Lefschetz type in Arakelov Geometry I: statement and proof, Invent. Math., to appear. | Zbl
, ,[Ma] Formes de torsion analytique et familles de submersions, C. R. Acad. Sci. Paris, 324, série I (1997), 205-210. | MR | Zbl
,[Ma1] Formes de torsion analytique et familles de submersions, I, Bull. Soc. Math. France, 127 (1999), 541-62 ; II, Asian J. Math., 4 (2000), 633-668. | Numdam | Zbl
,[MKS] Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43-69. | MR | Zbl
, ,[Q1] Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., 14 (1985), 31-34. | MR | Zbl
,[Q2] Superconnections and the Chern character, Topology, 24 (1986), 89-95. | MR | Zbl
,[RS] Analytic torsion for complex manifolds, Ann. of Math., 98 (1973), 154-177. | MR | Zbl
, Singer I.M.,[T] Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. | MR | Zbl
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