Parameter dependence of the Bergman kernels

Let $\{Ω_t:-1<t<1\}$ be a family of bounded pseudoconvex domains and $φ_t\in PSH(Ω_t)$. Let $K_t(z,w)$ denote the Bergman kernel with weight $φ_t$ on $Ω_t$. We study the continuity and Hölder continuity of $K_t(z,w)$ in $t$. Several applications to singularity theory of psh functions are given, including a new proof of the openness theorem.

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