Let $S$ be a complex manifold, let $T_S$ denote its holomorphic tangent bundle, and let $\Omega_S^1$ denote its holomorphic cotangent bundle. Let $\pi:\mathcal X\to S$ be a smooth proper holomorphic submersion of complex manifolds whose fibres $X_s:=\pi^{-1}(s)$ are compact Kähler manifolds, and assume that $\pi$ is a Kähler family in the sense that, locally on $S$, there is a smooth relative real $(1,1)$-form whose restriction to each fibre is a Kähler form. Fix an integer $k\ge 0$. Let $\mathbb V:=R^k\pi_*\mathbb C$ be the degree-$k$ complex local system, let $\mathcal H^k:=\mathbb V\otimes_{\mathbb C}\mathcal O_S$ be the associated holomorphic vector bundle, and let $\nabla:\mathcal H^k\to \mathcal H^k\otimes_{\mathcal O_S}\Omega_S^1$ be the Gauss-Manin connection. For each $s\in S$, let $H^{a,b}(X_s)$ denote the Hodge summand in the Kähler [Hodge decomposition](/theorems/2745) of $H^k(X_s,\mathbb C)$, with $H^{a,b}(X_s)=0$ unless $a+b=k$. For each integer $p$, let $F^p\mathcal H^k\subseteq \mathcal H^k$ denote the holomorphic Hodge filtration subbundle whose fibre at $s\in S$ is $F^pH^k(X_s,\mathbb C):=\bigoplus_{a\ge p}H^{a,k-a}(X_s)$. Then, for every integer $p$, one has $\nabla(F^p\mathcal H^k)\subseteq F^{p-1}\mathcal H^k\otimes_{\mathcal O_S}\Omega_S^1$.