Submersions and equivariant Quillen metrics

Xiaonan Ma

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 5, page 1539-1588
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Ma, Xiaonan. "Submersions and equivariant Quillen metrics." Annales de l'institut Fourier 50.5 (2000): 1539-1588. <http://eudml.org/doc/75464>.

@article{Ma2000,
abstract = {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.},
author = {Ma, Xiaonan},
journal = {Annales de l'institut Fourier},
keywords = {Quillen metric; determinant line bundle; cohomology; equivariant analytic torsion forms; complex manifold; submersion},
language = {eng},
number = {5},
pages = {1539-1588},
publisher = {Association des Annales de l'Institut Fourier},
title = {Submersions and equivariant Quillen metrics},
url = {http://eudml.org/doc/75464},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Ma, Xiaonan
TI - Submersions and equivariant Quillen metrics
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 5
SP - 1539
EP - 1588
AB - 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.
LA - eng
KW - Quillen metric; determinant line bundle; cohomology; equivariant analytic torsion forms; complex manifold; submersion
UR - http://eudml.org/doc/75464
ER -

References

top
  1. [ABo] M.F. ATIYAH, R. BOTT, A Lefchetz fixed point formula for elliptic complexes, I, Ann. of Math., 86 (1968), 374-407. Zbl0161.43201MR35 #3701
  2. [ABoP] M.F. ATIYAH, R. BOTT, V.K. PATODI, On the heat equation and the Index Theorem, Invent. Math., 19 (1973), 279-330. Zbl0257.58008MR58 #31287
  3. [BeGeV] N. BERLINE, E. GETZLER, M. VERGNE, Heat kernels and the Dirac operator, Grundl. Math. Wiss., 298 (1992). Zbl0744.58001MR94e:58130
  4. [BerB] A. BERTHOMIEU, J.-M. BISMUT, Quillen metric and higher analytic torsion forms, J. reine angew. Math, 457 (1994), 85-184. Zbl0804.32017MR96d:32036
  5. [B1] J.-M. BISMUT, The Index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 83 (1986), 91-151. Zbl0592.58047MR87g:58117
  6. [B2] J.-M. BISMUT, Superconnection currents and complex immersions, Invent. Math., 99 (1990), 59-113. Zbl0696.58006MR91b:58240
  7. [B3] J.-M. BISMUT, Koszul complexes, harmonic oscillators and the Todd class, J. Amer. Math. Soc., 3 (1990), 159-256. Zbl0702.58071MR91b:58245
  8. [B4] J.-M. BISMUT, Equivariant short exact sequences of vector bundles and their analytic torsion forms, Comp. Math., 93 (1994), 291-354. Zbl0817.32014MR96g:58201
  9. [B5] J.-M. BISMUT, Equivariant immersions and Quillen metrics, J. Diff. Geom., 41 (1995), 53-159. Zbl0826.32024MR96m:58261
  10. [B6] J.-M. BISMUT, Families of immersions, and higher analytic torsion, Astérisque, 244 (1997). Zbl0899.32013MR2000b:58057
  11. [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. Zbl0556.58027MR86g:58128b
  12. [BCh] J.-M. BISMUT, J. CHEEGER, n-invariants and their adiabatic limits, J. Amer. Math. Soc., 2 (1989), 33-70. Zbl0671.58037MR89k:58269
  13. [BGS1] J.-M. BISMUT, H. GILLET, C. SOULÉ, Analytic torsion and holomorphic determinant bundles, I, Comm. Math. Phys., 115 (1988), 49-78. Zbl0651.32017MR89g:58192a
  14. [BGS2] J.-M. BISMUT, H. GILLET, C. SOULÉ, Analytic torsion and holomorphic determinant bundles, II, Comm. Math. Phys., 115 (1988), 79-126. Zbl0651.32017MR89g:58192b
  15. [BGS3] J.-M. BISMUT, H. GILLET, C. SOULÉ, Analytic torsion and holomorphic determinant bundles, III, Comm. Math. Phys., 115 (1988), 301-351. Zbl0651.32017MR89g:58192c
  16. [BKö] J.-M. BISMUT, K. KÖHLER, Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom., 1 (1992), 647-684. Zbl0784.32023MR94a:58209
  17. [BL] J.-M. BISMUT, G. LEBEAU, Complex immersions and Quillen metrics, Publ. Math. IHES, 74 (1991), 1-297. Zbl0784.32010MR94a:58205
  18. [CP] J. CHARAZAIN, A. PIRIOU, Introduction à la théorie des équations aux dérivées partielles, Paris: Gauthier-Villars, 1981. Zbl0446.35001
  19. [D] X. DAI, Adiabatic limits, non multiplicativity of signature and Leray spectral sequence, J. Amer. Math. Soc., 4 (1991), 265-321. Zbl0736.58039MR92f:58169
  20. [DM] X. DAI, R.B. MELROSE, Adiabatic limit of the analytic torsion, Preprint. 
  21. [Ge] E. GETZLER, A short proof of the Atiyah-Singer Index Theorem, Topology, 25 (1986), 111-117. Zbl0607.58040MR87h:58207
  22. [GrH] P. GRIFFITHS, J. HARRIS, Principles of Algebraic Geometry, New York, Wiley, 1978. Zbl0408.14001MR80b:14001
  23. [GS1] H. GILLET, C. SOULÉ, Analytic torsion and the arithmetic Todd genus, Topology, 30 (1991), 21-54. Zbl0787.14005MR92d:14015
  24. [GS2] H. GILLET, C. SOULÉ, An arithmetic Riemann-Roch Theorem, Invent. Math., 110 (1992), 473-543. Zbl0777.14008MR94f:14019
  25. [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. Zbl0343.14008
  26. [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). Zbl1071.58026
  27. [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. Zbl0934.14017
  28. [KRo1] K. KÖHLER, D. ROESSLER, A fixed point formula of Lefschetz type in Arakelov Geometry I: statement and proof, Invent. Math., to appear. Zbl0999.14002
  29. [Ma] XIAONAN MA, Formes de torsion analytique et familles de submersions, C. R. Acad. Sci. Paris, 324, série I (1997), 205-210. Zbl0880.32015MR98b:32026
  30. [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. Zbl0956.58017
  31. [MKS] H. MCKEAN, I.M. SINGER, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43-69. Zbl0198.44301MR36 #828
  32. [Q1] D. QUILLEN, Determinants of Cauchy-Riemann operators over a Riemann surface, Funct. Anal. Appl., 14 (1985), 31-34. Zbl0603.32016MR86g:32035
  33. [Q2] D. QUILLEN, Superconnections and the Chern character, Topology, 24 (1986), 89-95. Zbl0569.58030MR86m:58010
  34. [RS] D.B. RAY, Singer I.M., Analytic torsion for complex manifolds, Ann. of Math., 98 (1973), 154-177. Zbl0267.32014MR52 #4344
  35. [T] M. TAYLOR, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981. Zbl0453.47026MR82i:35172

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.