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Type: Journal article
Title: Correspondences, von Neumann algebras and holomorphic L2 torsion
Author: Carey, A.
Farber, M.
Varghese, M.
Citation: Canadian Journal of Mathematics-Journal Canadien de Mathematiques, 2000; 52(4):695-736
Publisher: Canadian Mathematical Soc
Issue Date: 2000
ISSN: 0008-414X
Statement of
A. Carey, M. Farber and V. Mathai
Abstract: Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L2 torsion, which lies in the determinant line of the twisted L2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in [CFM]. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute ametric variation formula for the holomorphic L2 torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic L2 torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.
Keywords: holomorphic L2 torsion, correspondences, local index theorem, almost Kahler manifolds, von Neumann algebras, determinant lines
Description: Copyright © Canadian Mathematical Society 2000
RMID: 0001000670
DOI: 10.4153/CJM-2000-030-7
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Appears in Collections:Pure Mathematics publications

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