Communications in Mathematical Physics

, Volume 115, Issue 1, pp 49–78 | Cite as

Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion

  • J. -M. Bismut
  • H. Gillet
  • C. Soulé


We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillen's superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the determinant of the cohomology of a holomorphic vector bundle.


Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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  1. [A1]
    Alvarez, O.: Conformal anomalies and the Index Theorem, to appear in Nucl. Phys.Google Scholar
  2. [AHS]
    Atiyah, M. F., Hitchin, N. J., Singer, I. M.: Self-duality in four dimensional Riemannian geometry. Proc. Royal Soc. Lond. A362, 425–461 (1978)Google Scholar
  3. [AS]
    Atiyah, M. F., Singer, I. M.: The index of elliptic operators IV. Ann. Math.93, 119–138 (1971)Google Scholar
  4. [BeK]
    Belavin, A. A., Knizhnik, V. G.: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B168, 201–206 (1986)Google Scholar
  5. [BF1]
    Bismut, J.-M., Freed, D. S.: The analysis of elliptic families I, Metrics and connections on determinant bundles. Commun. Math. Phys.106, 159–176 (1986)Google Scholar
  6. [BF2]
    Bismut, J.-M., Freed, D. S.: The analysis of elliptic families II, Dirac operators, êta invariants and the holonomy Theorem. Commun. Math. Phys.107, 103–163 (1986)Google Scholar
  7. [BGS1]
    Bismut, J.-M., Gillet, H., Soulé, C.: Torsion analytique et fibrés déterminants holomorphes. C. R. Acad. Sci. Paris305, Série I, 81–84 (1987)Google Scholar
  8. [BGS2]
    Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles, II: Direct images and Bott-Chern forms. Commun. Math. Phys.115, 79–126 (1988)Google Scholar
  9. [BGS3]
    Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles, III: Quillen metrics on holomorphic determinants. To appear in Commun. Math. Phys.Google Scholar
  10. [Bo]
    Bost, J. B.: Conformal and holomorphic anomalies on Riemann surfaces and determinant line bundles. Preprint (1986)Google Scholar
  11. [BotC]
    Bott, R., Chern, S. S.: Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections. Acta Math.114, 71–112 (1968)Google Scholar
  12. [D1]
    Donaldson, S.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc.50, 1–26 (1985)Google Scholar
  13. [D2]
    Donaldson, S.: Infinite determinants, stable bundles and curvature. Preprint 1986Google Scholar
  14. [F]
    Freed, D.: Determinants, torsion and strings. Commun. Math. Phys.107, 483–513 (1986)Google Scholar
  15. [GS1]
    Gillet, H., Soulé, C.: Classes caractéristiques en théorie d'Arakelov. C. R. Acad. Sci. Paris, Série I,301, 439–442 (1985)Google Scholar
  16. [GS2]
    Gillet, H., Soulé, C.: Direct images of Hermitian holomorphic bundles. Bull. AMS15, 209–212 (1986)Google Scholar
  17. [KM]
    Knudsen, F. F., Mumford, D.: The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “div”. Math. Scand.39, 19–55 (1976)Google Scholar
  18. [P]
    Polyakov, A. M.: Quantum geometry of bosonic strings. Phys. Lett. B103, 207–210 (1981)Google Scholar
  19. [Q1]
    Quillen, D.: Superconnections and the Chern character. Topology24, 89–95 (1985)Google Scholar
  20. [Q2]
    Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–34 (1985)Google Scholar
  21. [RS]
    Ray, D. B., Singer, I. M.: Analytic torsion for complex manifolds. Ann. Math.98, 154–177 (1973)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • J. -M. Bismut
    • 1
  • H. Gillet
    • 2
  • C. Soulé
    • 3
  1. 1.Département de MathématiqueUniversité Paris-SudOrsay CedexFrance
  2. 2.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  3. 3.CNRS LA212 and I.H.E.S.Bures-Sur-YvetteFrance

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