Submersions and equivariant Quillen metrics
Annales de l'Institut Fourier, Volume 50 (2000) no. 5, pp. 1539-1588.

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.

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.

@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},
     zbl = {0964.58025},
     mrnumber = {2002c:58053},
     language = {en},
     url = {http://archive.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
DA  - 2000///
SP  - 1539
EP  - 1588
VL  - 50
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1800/
UR  - https://zbmath.org/?q=an%3A0964.58025
UR  - https://www.ams.org/mathscinet-getitem?mr=2002c:58053
UR  - https://doi.org/10.5802/aif.1800
DO  - 10.5802/aif.1800
LA  - en
ID  - AIF_2000__50_5_1539_0
ER  - 
%0 Journal Article
%A Ma, Xiaonan
%T Submersions and equivariant Quillen metrics
%J Annales de l'Institut Fourier
%D 2000
%P 1539-1588
%V 50
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1800
%R 10.5802/aif.1800
%G en
%F AIF_2000__50_5_1539_0
Ma, Xiaonan. Submersions and equivariant Quillen metrics. Annales de l'Institut Fourier, Volume 50 (2000) no. 5, pp. 1539-1588. doi : 10.5802/aif.1800. http://archive.numdam.org/articles/10.5802/aif.1800/

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

[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

[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

[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

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

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

[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

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

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

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

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

[T] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. | MR | Zbl

Cited by Sources: