The log-plurisubharmonicity of fiberwise $\xi -$Bergman kernels for variant functionals

Abstract

In this paper, we establish the log-plurisubharmonicity of fiberwise $\xi$-Bergman kernels for a family of variant functionals, thereby addressing a question posed by Bo Berndtsson to the authors. As an application, we prove that for a plurisubharmonic function $\varphi$ and a locally finitely generated ideal sheaf $\mathcal{I}$ on the polydisc ${\Delta }^{n+m}={\Delta }^n\times {\Delta }^m$, the set of points $w\in {\Delta }^m$ for which $\mathcal{I}(\varphi {|}_{H_w}{)}_{(o,w)}\subseteq (\mathcal{I}{|}_{H_w}{)}_{(o,w)}$ is a pluripolar set, where $H_w$ represents the fiber over $w$, and $\mathcal{I}(\varphi {|}_{H_w})$ denotes the multiplier ideal sheaf of $\varphi {|}_{H_w}$ on $H_w$.

15 pages. We add an application of the main theorem in the latest edition