Berndtsson-Lempert-Szőke fields associated to proper holomorphic families of vector bundles

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but nevertheless have a reasonable definition of curvature that agrees with the curvature of the Chern connection when the family of Hilbert spaces is locally trivial. As one consequence, we obtain a new proof of a theorem of Berndtsson on the curvature of direct images of semi-positively twisted relative canonical bundles (i.e., adjoint bundles), and of its higher-rank generalization by Liu-Yang.