In his Annals of Mathematics paper (2009), Berndtsson proves an important result on the curvature 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 $e^{-{\varphi}^(t,\cdot)}$, varying in $t$, over a pseudoconvex domain. Using Hörmander's classical theorem on $L^2$-estimates for the $\bar{\partial}$-operator, he shows that such bundles are positively curved in the sense of Nakano provided that the function $\varphi$ is plurisubharmonic with respect to both variables. This result is at the center of a long-standing project of Berndtsson aiming at formulating a complex analogue of Brunn-Minkowski theory, which first started with his result on the log-plurisubharmonicity of Bergman kernels over pseudoconvex domains (2006). In addition to extending the Brunn-Minkowski theorem and its generalization -the Prékopa-Leindler theorem- to the complex setting, Berndtsson's result has deep applications in complex analysis and geometry. For example, his result leads to alternative proofs of existence and uniqueness theorems for Kähler-Einstein metrics, as well as optimal $L^2$-extension (or Ohsawa-Takegoshi type) theorems.

Berndtsson’s result can easily be extended to the geometric setting by taking the pseudoconvex domain to be a Stein manifold and the family of weights $e^{-{\varphi}^(t,\cdot)}$ to be a family of positively curved hermitian metrics for a line bundle over the manifold. Using a variant of Hörmander’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 plurisubharmonic function, as these curvature assumptions then allow for some curvature negativity. We describe this setting as a "twisted" setting. In particular, we extend Berndtsson’s Nakano-positivity result to general trivial families of Stein manifolds. As immediate applications of this result, we prove $log$-plurisubharmonic variation theorems for Bergman kernels, as well as families of compactly supported measures and currents, over general trivial families of Stein manifolds. We then generalize these $log$-plurisubharmonic variation results to a certain class of non-trivial families of Stein manifolds. Finally, we also discuss Prékopa-Leindler type theorems showing, for instance, that the consequence of the Prékopa-Leindler theorem holds under weaker convexity assumptions.