  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.
