Equivariant de Rham torsions

Annals of Mathematics

Volumn 159, Issue 1, 2004, Pages 53-216

  Bismut, Jean Michel ^a   Goette, Sebastian ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b UNIVERSITY OF REGENSBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442526796     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2004.159.53     Document Type: Article

References (52)

1. [ABoP] M. F. ATIYAH, R. BOTT, and V. K. PATODI, On the heat equation and the index theorem, Invent. Math. 19 (1973), 247-330.
2. [BeGeV] N. BERLINE, E. GETZLER, and M. VERGNE, Heat Kernels and Dirac Operators, Grundl. Math. Wiss. 298, Springer-Verlag, New York, 1992.
3. [BeV] N. BERLINE and M. VERONE, Zéros d'un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983), 539-549.
4. [B1] J.-M. BISMUT, The Atiyah-Singer theorems: a probabilistic approach. I, II J. Func. Anal. 57 (1984), 56-99, 329-348.
5. [B2] _, The infinitesimal Lefschetz formulas: a heat equation proof, J. Funct. Anal. 62 (1985), 435-457.
6. [B3] _, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math, 83 (1986), 91-151.
7. [B4] _, Localization formulas, superconnections and the index theorem for families, Comm. Math. Phys. 103 (1986), 127-166.
8. [B5] _, Formules de Lichnerowicz et théorème de l'indice, Proc. of the Conference in honour of A. Lichnerowicz (Paris, 1986), (D. Bernard, Y. Choquet-Bruhat, eds.), in Géometrie Différentielle, 11-31, Travaux en Cours 33, Hermann, Paris, 1988.
9. [B6] _, Superconnection currents and complex immersions, Invent. Math. 99 (1990), 59-113.
10. [B7] _, Equivariant Bott-Chern currents and the Ray-Singer analytic torsion, Math. Ann. 287 (1990), 495-507.
11. [B8] _, Complex equivariant intersection, excess normal bundles and Bott-Chern currents, Comm. Math. Phys. 148 (1992), 1-55.
12. [B9] _, J.-M. Bismut, Bott-Chern currents, excess normal bundles and the Chern character, Geom. Funct. Anal. 2 (1992), 285-340.
13. [B10] _, Equivariant short exact sequences of vector bundles and their analytic torsion forms, Compositio Math. 93 (1994), 291-354.
14. [B11] _, J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), 53-157.
15. [B12] _, J.-M. Bismut, Holomorphic families of immersions and higher analytic torsion forms, Astérisque 244, Soc. Math. France, Paris, 1997.
16. [BGS] J.-M. BISMUT, H. GILLET, and H. SOULÉ, Analytic torsion and holomorphic determinant bundles. I, Comm. Math. Phys. 115 (1988), 49-78.
17. [BGo1] J.- M. BISMUT and S. GOETTE, Holomorphic equivariant analytic torsions, Geom. Funct. Anal. 10 (2000), 1289-1422.
18. [BGo2] _, Families Torsion and Morse Functions, Astérisque 275, Soc. Math. France, Paris, 2001.
19. [BGo3] _, Torsions analytiques équivariantes en théorie de de Rham, C. R. Acad Sci Paris Ser. I - bf 332 (2001), 33-39.
20. [BL] J.- M. BISMUT and G. LEBEAU, Complex immersions and Quillen metrics, Publ. Math. IHES 74 (1991), 1-297.
21. [BLo] J.-M. BISMUT and J. LOTT, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), 291-363.
22. [BZ1] J.-M. BISMUT and W. ZHANG, An Extension of a Theorem by Cheeger and Müller, Astérisque 205 (1992).
23. [BZ2] _, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Fund. Anal. 4 (1994), 136-212.
24. [Bu1] U. BUNKE, Equivariant torsion and G-CW-complexes, Geom. Funct. Anal. 9 (1999), 67-89.
25. [Bu2] U. BUNKE, Equivariant higher analytic torsion and equivariant Euler characteristic, Amer. J. Math. 122 (2000), 377-401.
26. [CaE] H. CARTAN and S. EILSENBERG, Homological Algebra, Princeton Univ. Press, Princeton, NJ 1956.
27. [ChP] J. CHAZARAIN and A. PIRIOU, Introduction à la Théorie des 'Equations aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.
28. [C] J. CHEEGER, Analytic torsion and the heat equation, Ann. of Math. 109 (1979), 259-322.
29. [DuH] J. J. DUISTERMAAT and G. HECKMAN, On the variation of the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268; Addendum 72 (1983), 153-158.
30. [Ge] E. GETZLER, A short proof of the Atiyah-Singer Index Theorem, Topology 25 (1986), 111-117.
31. [Gi] P. B. GILKEY, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Publish or Perish, Inc., Wilmington, 1984.
32. [Go] S. GOETTE, Equivariant η-invariants and η-forms, J. reine angew. Math. 526 (2000), 181-236.
33. [Ho] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators I, Grundl. Math. Wiss. 256, Springer-Verlag, New York, 1983.
34. [I] S. ILLMAN, Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Ann. 233 (1978), 199-220.
35. [KnMu] F. F. KNUDSON and D. MUMFORD, The projectivity of the moduli space of stable curves. I. Preliminaries on 'det' and 'div', Math. Scand. 39 (1976), 19-55.
36. s, Acta Math. 11 (1887-1888), 19-24.
37. [Lo] J. LOTT, Equivariant analytic torsion for compact Lie group actions, J. Funct. Anal. 125 (1994), 438-451.
38. [LoRo] J. LOTT and M. ROTHENBERG, Analytic torsion for group actions, J. Differential Geom. 34 (1991), 431-481.
39. [Lü] W. LÜCK, Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer-Verlag, New York, 1989.
40. [Ma] X. MA, Functoriality of real analytic torsion forms, Israel J. Math. 131 (2002), 1-50.
41. [MQ] V. MATHAI and D. QUILLEN, Superconnections, Thorn classes, and equivariant differential forms, Topology 25 (1986), 85-110.
42. [MKeS] H. MCKEAN and I. M. SINGER, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43-69.
43. [Mü1] W. MÜLLER, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233-305.
44. [Mü2] _, Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721-753.
45. [Mi] J. MILNOR, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, Princeton, NJ, 1965.
46. [Q1] D. QUILLEN, Superconnections and the Chern character, Topology 24 (1985), 89-95.
47. [Q2] _, Determinants of Cauchy-Riemann operators on Riemannian surfaces, Funct. Anal. Appl. 19 (1985), 31-34.
48. [RS] D. B. RAY and I. M. SINGER, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210.
49. [Sm1] S. SMALE, On gradient dynamical systems, Ann. of Math. 74 (1961), 199-206.
50. [Sm2] _, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.
51. [T] M. TAYLOR, Pseudodifferential Operators, Princeton Math. Series 34, Princeton Univ. Press, Princeton, NJ, 1981.
52. [W] H. WEISS, Die Variationsformeln der äquivarianten analytischen Torsion nach Bismut und Zhang, Diplomarbeit, Universität Göttingen, 1999.