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

## TL;DR

This paper proves that a family of strongly pseudoconvex domains in a K"ahler manifold admits a positive-definite K"ahler form induced by K"ahler-Einstein metrics, and explores its extension across singular fibers.

## Contribution

It establishes the positivity of the induced form on the family of domains and discusses its extension as a positive current across singular fibers.

## Key findings

- The form $
ho$ is positive-definite on strongly pseudoconvex domains.
- $
ho$ extends as a positive current across singular fibers.
- The results connect K"ahler-Einstein metrics with the geometry of fiber families.

## Abstract

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between K\"ahler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete K\"ahler-Einstein metric. This family 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 $D$ is strongly pseudoconvex. We also discuss the extensioin of $\rho$ as a positive current across singular fibers.