On the converse of Prékopa's theorem and Berndtsson's theorem

Abstract

Extending the framework of the converse $L^2$ theory, we establish converse results for Berndtsson's theorem on the plurisubharmonic variation of Bergman kernels, showing that both the plurisubharmonicity of functions and the pseudoconvexity of domains are necessary conditions in twisted senses. We also prove analogous results for Prékopa's theorem from convex analysis.