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

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 $\phi$ and a locally finitely generated ideal sheaf $\mathcal{I}$ on the polydisc ${\mathrm{\Delta }}^{n+m}={\mathrm{\Delta }}^n\times {\mathrm{\Delta }}^m$, the set of points $w\in {\mathrm{\Delta }}^m$ for which $\mathcal{I}(\phi {\mathrm{\mid }}_{H_w}{)}_{(o,w)}\subseteq (\mathcal{I}{\mathrm{\mid }}_{H_w}{)}_{(o,w)}$ is a pluripolar set, where $H_w$ represents the fiber over $w$, and $\mathcal{I}(\phi {\mathrm{\mid }}_{H_w})$ denotes the multiplier ideal sheaf of $\phi {\mathrm{\mid }}_{H_w}$ on $H_w$.