Quillen metrics and higher analytic torsion forms. (English) Zbl 0804.32017

Berthomieu, Alain; Bismut, Jean-Michel

J. Reine Angew. Math. 457, 85-184 (1994).

Let \(\pi : M \to B\) be a holomorphic submersion of compact complex manifolds. Let \(\xi\) be a holomorphic vector bundle. Assume that the \(R^ k \pi_ *\xi'\)’s are locally free. Let \(\lambda (\xi)\) and \(\lambda (R^ \bullet \pi_ * \xi)\) be the inverses of the determinants of the cohomology of \(\xi\) and \(R^ \bullet \pi_ * \xi\), and let \(\sigma \in \lambda^{-1} (R^ \bullet \pi_ * \xi) \otimes \lambda (\xi)\) be the canonical nonzero section identifying \(\lambda (R^ \bullet \pi_ * \xi)\) and \(\lambda (\xi)\).
The purpose of this paper is to give a formula for the Quillen norm of \(\sigma\) with respect to Quillen metrics on \(\lambda (\xi)\) and \(\lambda (R^ \bullet \pi_ * \xi)\) in term of Bott-Chern classes, and of the higher analytic torsion forms of Bismut-Köhler.

Reviewer: A.Berthomieu

MSC:

32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R20	Characteristic classes and numbers in differential topology
58J20	Index theory and related fixed-point theorems on manifolds

Keywords:

Quillen norm; Quillen metrics; Bott-Chern classes; analytic torsion