A twisted complex Brunn-Minkowski theorem

In his Annals of Mathematics paper [2], Berndtsson proves an important result on the Nakano positivity holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting L2-functions with respect to a family weight functions {e−φ(t,⋅)}t∈U, varying in t∈U⊂Cm, over pseudoconvex domain. Using variant Hörmander's theorem due Donnelly and Fefferman, we show that Berndtsson's holds under different (in fact, more general) curvature assumptions. This is particular interest when manifold admits negative non-constant plurisubharmonic function, as these assumptions then allow for some negativity. We describe this setting “twisted” setting.