Correspondences, Von Neumann Algebras and Holomorphic L 2 Torsion

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 [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. §0. Introduction Ray and Singer (cf. [RS]) introduced the notion of holomorphic torsion of a holomorphic bundle over a compact complex manifold. In [Q], Quillen viewed the holomorphic torsion as an element in the real determinant line of the twisted Dolbeault cohomology, or equivalently, as a metric in the dual of the determinant line of the twisted Dolbeault cohomology. Since then there have been many generalisations in the finite dimensional case, particularly by Bismut, Freed, Gillet and Soule, [BF, BGS]. In this paper, we investigate generalisations of aspects of this previous work to the case of infinite dimensional representations of the fundamental group. Our approach is to introduce the concepts of holomorphic Hilbertian bundles and of connections compatible with the holomorphic structure. These bundles have fibres which are von Neumann algebra modules. We are able to define the determinant line bundle of a holomorphic Hilbertian bundle over a compact complex manifold, generalising the construction of the determinant line of a finitely generated Hilbertian module that was developed in our earlier paper [CFM]. A nonzero element of the determinant line bundle can be naturally viewed as a volume form on the Hilbertian bundle. This enables us to make sense of the notions of volume form and determinant line bundle in this infinite dimensional and non-commutative situation. Given an isomorphism of the determinant line bundles of holomorphic Hilbertian bundles, we introduce the concept of a correspondence between the determinant lines …