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Correspondences, von Neumann Algebras and Holomorphic L2 Torsion

Published online by Cambridge University Press:  20 November 2018

A. Carey ,
M. Farber  and
V. Mathai
A. Carey
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide 500, Australia email: acarey@maths.adelaide.edu.au
M. Farber
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel email: farber@math.tau.ac.il
V. Mathai
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia email: vmathai@maths.adelaide.edu.au
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Abstract

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Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic ${{L}^{2}}$ torsion, which lies in the determinant line of the twisted ${{L}^{2}}$ 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 $[\text{CFM}]$. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic ${{L}^{2}}$ 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 ${{L}^{2}}$ 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

58J5258J3558J20Holomorphic L2 torsioncorrespondenceslocal index theoremalmost Kähler manifoldsvon Neumann algebrasdeterminant lines.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 4 , 01 August 2000 , pp. 695 - 736
Copyright
Copyright © Canadian Mathematical Society 2000

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