Twisted Nakano-Positivity of Fields of Hilbert Spaces

TL;DR

This paper extends the concept of twisted Nakano-positivity to fields of Hilbert spaces on unbounded Stein manifolds, leading to new log-plurisubharmonic variation results for complex manifolds.

Contribution

It generalizes the twisted Nakano-positivity theorem to broader settings involving unbounded Stein manifolds using exhaustion techniques.

Findings

Established twisted Nakano-positivity for Hilbert space fields on unbounded Stein manifolds

Proved log-plurisubharmonic variation results for families of Stein manifolds

Abstract

In a previous article (arXiv:2111.03143), we generalized Berndtsson's Nakano-positivity by retaining the same consequences under weaker hypotheses. In this article, we propose to further generalize our "twisted" Nakano-positivity theorem to fields of Hilbert spaces associated to possibly unbounded Stein manifolds. We achieve this result using exhaustion arguments. As direct applications, we prove $\log$-plurisubharmonic variation results for a certain class of non-trivial families of Stein manifolds.