Correspondences, von Neumann algebras and holomorphic L^2 torsion

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 an earlier paper by the authors. 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.