We characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. By using such frames we also give a simpler and shorter proof of a theorem of Goldberg. Furthermore, we study almost-Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi-Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

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.