Let $\pi:\mathcal X\to B$ be a proper holomorphic submersion of complex manifolds, and let $0\in B$. Set $X:=X_0:=\pi^{-1}(0)$. If $X$ is a compact Kähler manifold, then there exists an open neighbourhood $U\subset B$ of $0$ such that, for every $b\in U$, the compact complex manifold $\mathcal X_b:=\pi^{-1}(b)$ is Kähler.