Berndtsson-Lempert-Sz\H{o}ke Fields Associated to Proper Holomorphic Families of Vector Bundles

TL;DR

This paper introduces a new complex analytic framework for families of Hilbert spaces, enabling the analysis of curvature properties and providing alternative proofs for key theorems in the curvature of direct image bundles.

Contribution

It defines a novel structure called Berndtsson-Lempert-Sz"H{o}ke fields for Hilbert space families, extending curvature analysis beyond traditional holomorphic vector bundles.

Findings

Provides a new proof of Berndtsson's theorem on curvature of direct images.

Generalizes the curvature results to higher-rank bundles.

Establishes a framework applicable to non-trivially fibered Hilbert spaces.

Abstract

Drawing on work of Berndtsson and of Lempert and Sz\H{o}ke, we define a kind of 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. We thus obtain a new proof of a celebrated 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 due to Liu and Yang.