Submersions and equivariant Quillen metrics
Ma, Xiaonan
Annales de l'Institut Fourier, Tome 50 (2000), p. 1539-1588 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {50},
     year = {2000},
     pages = {1539-1588},
     doi = {10.5802/aif.1800},
     mrnumber = {2002c:58053},
     zbl = {0964.58025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_5_1539_0}
}

Ma, Xiaonan. Submersions and equivariant Quillen metrics. Annales de l'Institut Fourier, Tome 50 (2000) pp. 1539-1588. doi : 10.5802/aif.1800. http://gdmltest.u-ga.fr/item/AIF_2000__50_5_1539_0/

[ABo] M.F. Atiyah, R. Bott, A Lefchetz fixed point formula for elliptic complexes, I, Ann. of Math., 86 (1968), 374-407. | MR 35 #3701 | Zbl 0161.43201

[ABoP] M.F. Atiyah, R. Bott, V.K. Patodi, On the heat equation and the Index Theorem, Invent. Math., 19 (1973), 279-330. | MR 58 #31287 | Zbl 0257.58008

[BeGeV] N. Berline, E. Getzler, M. Vergne, Heat kernels and the Dirac operator, Grundl. Math. Wiss., 298 (1992). | MR 94e:58130 | Zbl 0744.58001

[BerB] A. Berthomieu, J.-M. Bismut, Quillen metric and higher analytic torsion forms, J. reine angew. Math, 457 (1994), 85-184. | MR 96d:32036 | Zbl 0804.32017

[B1] J.-M. Bismut, The Index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 83 (1986), 91-151. | MR 87g:58117 | Zbl 0592.58047

[B2] J.-M. Bismut, Superconnection currents and complex immersions, Invent. Math., 99 (1990), 59-113. | MR 91b:58240 | Zbl 0696.58006

[B3] J.-M. Bismut, Koszul complexes, harmonic oscillators and the Todd class, J. Amer. Math. Soc., 3 (1990), 159-256. | MR 91b:58245 | Zbl 0702.58071

[B4] J.-M. Bismut, Equivariant short exact sequences of vector bundles and their analytic torsion forms, Comp. Math., 93 (1994), 291-354. | Numdam | MR 96g:58201 | Zbl 0817.32014

[B5] J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Diff. Geom., 41 (1995), 53-159. | MR 96m:58261 | Zbl 0826.32024

[B6] J.-M. Bismut, Families of immersions, and higher analytic torsion, Astérisque, 244 (1997). | MR 2000b:58057 | Zbl 0899.32013

[B7] J.-M. Bismut, The Atiyah-Singer Index Theorems: a probabilistic approach, II. The Lefschetz fixed point formulas, J. Funct. Anal., 57 (1984), 329-348. | MR 86g:58128b | Zbl 0556.58027

[BCh] J.-M. Bismut, J. Cheeger, n-invariants and their adiabatic limits, J. Amer. Math. Soc., 2 (1989), 33-70. | MR 89k:58269 | Zbl 0671.58037

[BGS1] J.-M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles, I, Comm. Math. Phys., 115 (1988), 49-78. | MR 89g:58192a | Zbl 0651.32017

[BGS2] J.-M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles, II, Comm. Math. Phys., 115 (1988), 79-126. | MR 89g:58192b | Zbl 0651.32017

[BGS3] J.-M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles, III, Comm. Math. Phys., 115 (1988), 301-351. | MR 89g:58192c | Zbl 0651.32017

[BKö] J.-M. Bismut, K. Köhler, Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom., 1 (1992), 647-684. | MR 94a:58209 | Zbl 0784.32023

[BL] J.-M. Bismut, G. Lebeau, Complex immersions and Quillen metrics, Publ. Math. IHES, 74 (1991), 1-297. | Numdam | MR 94a:58205 | Zbl 0784.32010

[CP] J. Charazain, A. Piriou, Introduction à la théorie des équations aux dérivées partielles, Paris: Gauthier-Villars, 1981. | Zbl 0446.35001

[D] X. Dai, Adiabatic limits, non multiplicativity of signature and Leray spectral sequence, J. Amer. Math. Soc., 4 (1991), 265-321. | MR 92f:58169 | Zbl 0736.58039

[DM] X. Dai, R.B. Melrose, Adiabatic limit of the analytic torsion, Preprint.

[Ge] E. Getzler, A short proof of the Atiyah-Singer Index Theorem, Topology, 25 (1986), 111-117. | MR 87h:58207 | Zbl 0607.58040

[GrH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, New York, Wiley, 1978. | MR 80b:14001 | Zbl 0408.14001

[GS1] H. Gillet, C. Soulé, Analytic torsion and the arithmetic Todd genus, Topology, 30 (1991), 21-54. | MR 92d:14015 | Zbl 0787.14005

[GS2] H. Gillet, C. Soulé, An arithmetic Riemann-Roch Theorem, Invent. Math., 110 (1992), 473-543. | MR 94f:14019 | Zbl 0777.14008

[KM] P.F. Knudsen, D. Mumford, The projectivity of the moduli space of stable curves, I, Preliminaries on “det” and “div”, Math. Scand., 39 (1976), 19-55. | Zbl 0343.14008

[K] K. Köhler, 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 1071.58026

[KRo] K. Köhler, D. Roessler, Un théorème du point fixe de Lefschetz en géométrie d'Arakerov, C. R. Acad. Sci. Paris, 326, série I (1998), 719-722. | Zbl 0934.14017

[KRo1] K. Köhler, D. Roessler, A fixed point formula of Lefschetz type in Arakelov Geometry I: statement and proof, Invent. Math., to appear. | Zbl 01655727

[Ma] Xiaonan Ma, Formes de torsion analytique et familles de submersions, C. R. Acad. Sci. Paris, 324, série I (1997), 205-210. | MR 98b:32026 | Zbl 0880.32015

[Ma1] Xiaonan Ma, 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 0956.58017

[MKS] H. Mckean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43-69. | MR 36 #828 | Zbl 0198.44301

[Q1] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., 14 (1985), 31-34. | MR 86g:32035 | Zbl 0603.32016

[Q2] D. Quillen, Superconnections and the Chern character, Topology, 24 (1986), 89-95. | MR 86m:58010 | Zbl 0569.58030

[RS] D.B. Ray, Singer I.M., Analytic torsion for complex manifolds, Ann. of Math., 98 (1973), 154-177. | MR 52 #4344 | Zbl 0267.32014

[T] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. | MR 82i:35172 | Zbl 0453.47026