Communications in Mathematical Physics

, Volume 115, Issue 2, pp 301–351 | Cite as

Analytic torsion and holomorphic determinant bundles

III. Quillen metrics on holomorphic determinants
  • Jean-Michel Bismut
  • Henri Gillet
  • Christophe Soulé


In this paper, we prove that in the case of holomorphic locally Kähler fibrations, the analytic and algebraic geometry constructions of determinant bundles for direct images coincide. We calculate the curvature of the holomorphic Hermitian connection for the Quillen metric on the determinant bundle. We study the behavior of the Quillen metric under change of metrics in the fibers, and also on the twisting vector bundles. We thus generalize the conformal anomaly formula to Kähler manifolds of arbitrary dimension. We also study the Quillen metrics on determinants associated with exact sequences of vector bundles. We prove that the Quillen metric is smooth on the Grothendieck-Knudsen-Mumford determinant for arbitrary holomorphic fibrations.


Neural Network Manifold Nonlinear Dynamics Exact Sequence Vector Bundle 
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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Jean-Michel Bismut
    • 1
  • Henri Gillet
    • 2
  • Christophe Soulé
    • 3
  1. 1.Département de MathématiqueUniversité Paris-SudOrsay CedexFrance
  2. 2.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  3. 3.CNRS LA 212 and IHESBures/YvetteFrance

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