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Submersions and equivariant Quillen metrics. (English) Zbl 0964.58025

This paper fits in the program, developed by J.-M. Bismut, of calculating Quillen metrics of the line bundle sections arising canonically from geometrically defined isomorphisms of inverse determinant cohomology bundles on compact complex manifolds. Previous examples of such work include: J.-M. Bismut and G. Lebeau [Publ. Math., Inst. Hautes Etud. Sci. 74, 1-297 (1991; Zbl 0784.32010)] on immersions; A. Berthomieu and J.-M. Bismut [J. Reine Angew. Math. 457, 85-184 (1994; Zbl 0804.32017)] on submersions; and J.-M. Bismut [J. Differ. Geom. 41, 53-157 (1995; Zbl 0826.32024)] on equivariant immersions. The paper under review handles the case of a submersion of compact complex manifolds, equivariant with respect to the holomorphic action of a compact Lie group. The formula for the Quillen metric involves equivariant analytic torsion forms.

MSC:

58J52 Determinants and determinant bundles, analytic torsion
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
58J35 Heat and other parabolic equation methods for PDEs on manifolds
32L05 Holomorphic bundles and generalizations
57S15 Compact Lie groups of differentiable transformations
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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