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# Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion

Published:2000-08-01
Printed: Aug 2000
• A. Carey
• M. Farber
• V. Mathai
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## Abstract

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 \cite{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 {\it 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: holomorphic $L^2$ torsion, correspondences, local index theorem, almost Kähler manifolds, von~Neumann algebras, determinant lines
 MSC Classifications: 58J52 - Determinants and determinant bundles, analytic torsion 58J35 - Heat and other parabolic equation methods 58J20 - Index theory and related fixed point theorems [See also 19K56, 46L80]

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