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Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between K\"ahlermanifolds. Let $D$ be a bounded smooth domain in $X$ such that every genericfiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in$X_y:=p^{-1}(y)$, which admits the complete K\"ahler-Einstein metric. Thisfamily of K\"ahler-Einstein metrics induces a smooth $(1,1)$-form $\rho$ on$D$. In this paper, we prove that $\rho$ is positive-definite on $D$ if $D$ isstrongly pseudoconvex. We also discuss the extensioin of $\rho$ as a positivecurrent across singular fibers.

Holomorphic family of strongly pseudoconvex domains in a K\"ahler manifold