Twisted Nakano-Positivity of Fields of Hilbert Spaces

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.