A Twisted Complex Brunn-Minkowski Theorem

TL;DR

This paper generalizes Berndtsson's Nakano positivity results for holomorphic vector bundles by employing a variant of H"ormander's theorem, enabling curvature negativity in a twisted setting with broader curvature assumptions.

Contribution

It introduces a new approach using a variant of H"ormander's theorem to extend positivity results under more general curvature conditions, including negative curvature scenarios.

Findings

Positivity results hold under broader curvature assumptions.

The method accommodates manifolds with negative non-constant plurisubharmonic functions.

Extension of Berndtsson's theorem to a twisted setting with curvature negativity.

Abstract

In his Annals of Mathematics paper (2009), Berndtsson proves an important result on the Nakano positivity of holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting of holomorphic $L^2$-functions with respect to a family of weight functions $\left\{e^{-\varphi(\cdot,t)}\right\}_{t \in U}$, varying in $t \in U \in \mathbb{C}^m$, over a pseudoconvex domain. Using a variant of H\"ormander's theorem due to Donnelly and Fefferman, we show that Berndtsson's Nakano positivity result holds under different (in fact, more general) curvature assumptions. This is of particular interest when the manifold admits a negative non-constant plurisubharmonic function, as these curvature assumptions then allow for some curvature negativity. We describe this setting as a "twisted" setting