  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
