[proofplan] The assertion is local on the base, so we work over a small coordinate neighbourhood and test the connection against one holomorphic tangent vector. A local section of $F^p\mathcal H^k$ is represented by a smooth family of fibrewise closed forms whose fibrewise Hodge components have holomorphic degree at least $p$. Differentiating such a representative along the base gives the Gauss-Manin derivative; the only part of this derivative that changes Hodge type is contraction with the Kodaira-Spencer tensor, and this lowers holomorphic degree by exactly one. Hence no cohomology class of type with holomorphic degree below $p-1$ appears. [/proofplan]

[step:Reduce the containment to a local derivative along one holomorphic vector field]The statement is local on $S$ as an inclusion of sheaves. Let $U\subset S$ be a coordinate neighbourhood, let \begin{align*} \sigma\in \Gamma(U,F^p\mathcal H^k) \end{align*} be a holomorphic local section. Let $T_S$ denote the holomorphic tangent bundle of $S$, and let \begin{align*} v\in \Gamma(U,T_S) \end{align*} be a holomorphic vector field. It is enough to prove that \begin{align*} \nabla_v\sigma\in \Gamma(U,F^{p-1}\mathcal H^k), \end{align*} because the $\mathcal O_U$-linear contraction of $\nabla\sigma$ with arbitrary local holomorphic vector fields determines whether $\nabla\sigma$ lies in $F^{p-1}\mathcal H^k\otimes_{\mathcal O_U}\Omega_U^1$. Since $\pi$ is proper and a smooth submersion, Ehresmann local triviality applies. After replacing $U$ by a smaller connected coordinate neighbourhood, choose a smooth trivialisation \begin{align*} \Phi:\pi^{-1}(U)\to X_0\times U \end{align*} over $U$, where $X_0$ is the fibre over a fixed base point of $U$. Under this trivialisation the local system $\mathbb V|_U$ is identified with the constant [vector space](/page/Vector%20Space) $H^k(X_0,\mathbb C)$, and the Gauss-Manin connection becomes the flat connection obtained by differentiating the corresponding family of de Rham cohomology classes.[/step]

[guided]The inclusion \begin{align*} \nabla(F^p\mathcal H^k)\subseteq F^{p-1}\mathcal H^k\otimes_{\mathcal O_S}\Omega_S^1 \end{align*} is a local statement on the base. Therefore we may restrict to a coordinate neighbourhood $U\subset S$. We choose a section \begin{align*} \sigma\in \Gamma(U,F^p\mathcal H^k) \end{align*} and a holomorphic vector field \begin{align*} v\in \Gamma(U,T_S). \end{align*} To prove the sheaf inclusion, it suffices to prove \begin{align*} \nabla_v\sigma\in \Gamma(U,F^{p-1}\mathcal H^k) \end{align*} for every such $v$, because a one-form with values in $\mathcal H^k$ lies in the subbundle $F^{p-1}\mathcal H^k\otimes\Omega_U^1$ exactly when all of its contractions with local holomorphic tangent fields lie in $F^{p-1}\mathcal H^k$. Now we fix the differentiable framework in which the Gauss-Manin derivative is computed. Since $\pi:\mathcal X\to S$ is a smooth proper submersion, Ehresmann local triviality gives, after shrinking $U$ if necessary, a smooth map \begin{align*} \Phi:\pi^{-1}(U)\to X_0\times U \end{align*} which is a diffeomorphism over $U$. This does not identify the complex structures on the fibres holomorphically; it only identifies their underlying smooth manifolds. That is exactly what the Gauss-Manin connection uses: under this smooth trivialisation, the local system $R^k\pi_*\mathbb C$ becomes constant, and $\nabla_v\sigma$ is computed by differentiating the family of de Rham cohomology classes represented by $\sigma$ in the direction $v$.[/guided]

[step:Choose a smooth relative representative in the Hodge filtration]For each $s\in U$, write $X_s:=\pi^{-1}(s)$. Let $A^m(X_s)$ denote the complex vector space of smooth complex-valued $m$-forms on $X_s$, let $A^{a,b}(X_s)$ denote the smooth forms of Hodge type $(a,b)$ on $X_s$, and let \begin{align*} d_{X_s}:A^m(X_s)\to A^{m+1}(X_s) \end{align*} denote the de Rham differential on the fibre $X_s$. Let $A^k_{\mathcal X/U}$ denote the smooth vector bundle over $\pi^{-1}(U)$ whose restriction to $X_s$ is $\Lambda^kT^*X_s\otimes_{\mathbb R}\mathbb C$; its smooth sections are smooth relative complex-valued $k$-forms along the fibres. The fibrewise de Rham differential is the map \begin{align*} d_{\mathcal X/U}:\Gamma(\pi^{-1}(U),A^k_{\mathcal X/U})\to\Gamma(\pi^{-1}(U),A^{k+1}_{\mathcal X/U}) \end{align*} whose restriction to $X_s$ is $d_{X_s}$. By the local relative Kähler hypothesis in the statement, after shrinking $U$ if necessary there is a smooth relative Kähler form on $\pi^{-1}(U)$, meaning a smooth relative real $(1,1)$-form whose restriction to each fibre $X_s$ is a Kähler form. The associated fibrewise Hodge Laplacians form a smooth elliptic family on compact fibres. Because the local system $\mathbb V|_U$ is locally constant, the dimensions of the degree-$k$ harmonic spaces are locally constant and equal to $\dim_{\mathbb C}H^k(X_s,\mathbb C)$. Hence the zero eigenspaces form a smooth finite-rank bundle, and the harmonic projection operators vary smoothly with $s$. Since $\sigma$ is a holomorphic local section of the Hodge filtration subbundle $F^p\mathcal H^k$, applying these smoothly varying harmonic projections gives a smooth relative $k$-form \begin{align*} \alpha\in \Gamma(\pi^{-1}(U),A^k_{\mathcal X/U}) \end{align*} such that, for every $s\in U$, the restriction $\alpha_s\in A^k(X_s)$ is $d_{X_s}$-closed, represents the class $\sigma(s)$, and has type decomposition \begin{align*} \alpha_s=\sum_{r=p}^{k}\alpha_s^{r,k-r} \end{align*} with $\alpha_s^{r,k-r}\in A^{r,k-r}(X_s)$. This uses the fibrewise [Hodge decomposition for compact Kähler manifolds](/theorems/8066), equivalently the scalar case of the Hodge identification for holomorphic vector bundles [citetheorem:9104].[/step]

[guided]We need a representative of $\sigma$ that varies smoothly in $s$ and already lies in the Hodge filtration on each fibre. First fix the notation. For a fibre $X_s$, the space $A^m(X_s)$ consists of smooth complex-valued $m$-forms, the subspace $A^{a,b}(X_s)$ consists of forms of Hodge type $(a,b)$, and \begin{align*} d_{X_s}:A^m(X_s)\to A^{m+1}(X_s) \end{align*} is the fibrewise de Rham differential. The bundle $A^k_{\mathcal X/U}$ is the smooth bundle of relative complex-valued $k$-forms: a section $\alpha\in\Gamma(\pi^{-1}(U),A^k_{\mathcal X/U})$ restricts on each fibre to a form $\alpha_s\in A^k(X_s)$. Its relative differential $d_{\mathcal X/U}\alpha$ is defined fibrewise by $(d_{\mathcal X/U}\alpha)_s=d_{X_s}\alpha_s$. Because the fibres are compact Kähler manifolds, Hodge theory identifies $H^k(X_s,\mathbb C)$ with harmonic $k$-forms and decomposes harmonic forms into types $(r,k-r)$. The scalar case of the Hodge identification [citetheorem:9104] gives precisely this fibrewise description. By the local relative Kähler hypothesis, after shrinking $U$ if necessary there is a smooth relative Kähler form on $\pi^{-1}(U)$, meaning a smooth relative real $(1,1)$-form whose restriction to each fibre $X_s$ is a Kähler form. This form gives a smooth family of elliptic Hodge Laplacians on the compact fibres. The ranks of their harmonic spaces are locally constant because they compute the fibres of the locally constant system $\mathbb V|_U$. Therefore the zero eigenspaces form a smooth finite-rank bundle, standard smooth dependence of spectral projections for smooth elliptic families applies, and the harmonic projections vary smoothly with $s$. Hence the cohomology section $\sigma$ has a smooth harmonic representative $\alpha$ along the fibres. Since $\sigma(s)$ lies in $F^pH^k(X_s,\mathbb C)$ for every $s$, its harmonic representative has no Hodge components with first index below $p$. Thus, after writing the type decomposition of $\alpha_s$ on $X_s$, we have \begin{align*} \alpha_s=\sum_{r=p}^{k}\alpha_s^{r,k-r} \end{align*} with $\alpha_s^{r,k-r}\in A^{r,k-r}(X_s)$. The form $\alpha_s$ is $d_{X_s}$-closed because harmonic forms are closed, and it represents exactly the de Rham class $\sigma(s)$ by the [Hodge theorem](/theorems/3942) on the compact Kähler fibre.[/guided]

[step:Compute the Hodge type of the Gauss-Manin derivative]Let $T_{\mathcal X}^{\infty}$ denote the smooth real tangent bundle of the underlying smooth manifold of $\mathcal X$, complexified when acting on complex-valued forms. Choose \begin{align*} \widetilde v\in \Gamma(\pi^{-1}(U),T_{\mathcal X}^{\infty}\otimes_{\mathbb R}\mathbb C) \end{align*} to be a smooth vector field which is $\pi$-related to $v$ under the chosen smooth trivialisation. Choose an ordinary smooth complex-valued $k$-form $\widehat\alpha$ on $\pi^{-1}(U)$ whose restriction to vertical tangent vectors is the relative form $\alpha$. The Gauss-Manin derivative $\nabla_v\sigma$ is represented on each fibre $X_s$ by the cohomology class of \begin{align*} (\mathcal L_{\widetilde v}\widehat\alpha)|_{X_s} \end{align*} where $\mathcal L_{\widetilde v}$ denotes the Lie derivative of ordinary smooth differential forms along $\widetilde v$; changing $\widehat\alpha$ changes this representative by a fibrewise exact form. Let $T_{X_s}$ denote the holomorphic tangent bundle of $X_s$. The [Kodaira-Spencer correspondence](/theorems/9117) [citetheorem:9117] assigns to $v(s)\in T_sS$ a Dolbeault class represented by a tensor \begin{align*} \kappa(v)_s\in A^{0,1}(X_s,T_{X_s}). \end{align*} In local holomorphic coordinates $z_1,\dots,z_n$ on $X_s$, write \begin{align*} \kappa(v)_s=\sum_{i,j}\kappa_{ij}\,d\overline z_i\otimes \frac{\partial}{\partial z_j}. \end{align*} For $\eta\in A^{r,k-r}(X_s)$, contraction is defined by \begin{align*} \kappa(v)_s\lrcorner\eta:=\sum_{i,j}\kappa_{ij}\,d\overline z_i\wedge \iota_{\partial/\partial z_j}\eta. \end{align*} The operator $\iota_{\partial/\partial z_j}$ removes one holomorphic covector and $d\overline z_i\wedge(\cdot)$ adds one antiholomorphic covector, so \begin{align*} \kappa(v)_s\lrcorner(\cdot):A^{r,k-r}(X_s)\to A^{r-1,k-r+1}(X_s). \end{align*} The local Gauss-Manin derivative formula, in the explicit first-order form supplied by the [formula for the infinitesimal period map](/theorems/9131) [citetheorem:9131], applies to the present smooth proper Kähler family, the point $s\in U$, and the tangent vector $v(s)\in T_sS$. We use only its local type computation: the component of the derivative on a class of type $(r,k-r)$ is represented, modulo the fibrewise exact image $d_{X_s}A^{k-1}(X_s)$, by the coefficient-derivative term of type $(r,k-r)$ together with the Kodaira-Spencer contraction term of type $(r-1,k-r+1)$. Hence \begin{align*} [(\mathcal L_{\widetilde v}\widehat\alpha)|_{X_s}]\in \bigoplus_{r\ge p}H^{r,k-r}(X_s)\oplus \bigoplus_{r\ge p}H^{r-1,k-r+1}(X_s). \end{align*} Since every index occurring on the right has first coordinate at least $p-1$, this direct sum is contained in \begin{align*} F^{p-1}H^k(X_s,\mathbb C)=\bigoplus_{a\ge p-1}H^{a,k-a}(X_s). \end{align*}[/step]

[guided]We now identify the one place where Hodge type can change. Let $T_{\mathcal X}^{\infty}$ denote the smooth real tangent bundle of the underlying smooth manifold of $\mathcal X$, complexified when acting on complex-valued forms. Choose \begin{align*} \widetilde v\in \Gamma(\pi^{-1}(U),T_{\mathcal X}^{\infty}\otimes_{\mathbb R}\mathbb C) \end{align*} to be a smooth lift of $v$, meaning that $d\pi(\widetilde v)=v$. Because $\alpha$ is a relative form, we choose an ordinary smooth complex-valued $k$-form $\widehat\alpha$ on $\pi^{-1}(U)$ whose restriction to vertical tangent vectors is $\alpha$. The Gauss-Manin connection is the flat connection on de Rham cohomology induced by the smooth trivialisation. Therefore $\nabla_v\sigma$ is represented on $X_s$ by the derivative of this ordinary form along $\widetilde v$, namely by \begin{align*} (\mathcal L_{\widetilde v}\widehat\alpha)|_{X_s}. \end{align*} A different choice of $\widehat\alpha$ changes the displayed fibrewise form only by an element of $d_{X_s}A^{k-1}(X_s)$, so the de Rham cohomology class is independent of the extension. The subtle point is that the decomposition into types changes with $s$. If the complex structure were fixed, differentiating a family of $(r,k-r)$-forms would keep the same type. In a holomorphic family, the first-order variation of the fibre complex structure in the direction $v(s)$ is measured by the Kodaira-Spencer tensor. By the Kodaira-Spencer correspondence [citetheorem:9117], this tensor is represented by an element \begin{align*} \kappa(v)_s\in A^{0,1}(X_s,T_{X_s}), \end{align*} where $T_{X_s}$ is the holomorphic tangent bundle of $X_s$. We now check the type shift in coordinates. Choose local holomorphic coordinates $z_1,\dots,z_n$ on $X_s$ and write \begin{align*} \kappa(v)_s=\sum_{i,j}\kappa_{ij}\,d\overline z_i\otimes \frac{\partial}{\partial z_j}. \end{align*} For $\eta\in A^{r,k-r}(X_s)$, define \begin{align*} \kappa(v)_s\lrcorner\eta:=\sum_{i,j}\kappa_{ij}\,d\overline z_i\wedge \iota_{\partial/\partial z_j}\eta. \end{align*} The contraction $\iota_{\partial/\partial z_j}\eta$ removes one $dz$-factor, so it has type $(r-1,k-r)$. Wedge multiplication by $d\overline z_i$ adds one antiholomorphic covector, so the result has type $(r-1,k-r+1)$. Therefore contraction by the Kodaira-Spencer tensor defines a map \begin{align*} \kappa(v)_s\lrcorner(\cdot):A^{r,k-r}(X_s)\to A^{r-1,k-r+1}(X_s). \end{align*} We now invoke the local Gauss-Manin derivative formula in the explicit first-order form supplied by the formula for the infinitesimal period map [citetheorem:9131]. Its hypotheses are satisfied here: $\pi$ is a smooth proper Kähler family, $s\in U$ is a base point, and $v(s)\in T_sS$ is the tangent vector being tested. We use the theorem only for its local type computation: the derivative of a class of type $(r,k-r)$ has graded component obtained by cup product with the Kodaira-Spencer class and contraction of forms. In the coordinate notation above, this is exactly the contraction operator $\kappa(v)_s\lrcorner(\cdot)$, so the possible new component has type $(r-1,k-r+1)$; the coefficient derivative remains in type $(r,k-r)$, and terms depending on the choice of representative are fibrewise exact and vanish in de Rham cohomology. Thus, modulo $d_{X_s}A^{k-1}(X_s)$, a component of $\alpha_s$ of type $(r,k-r)$ contributes only to Hodge types $(r,k-r)$ and $(r-1,k-r+1)$. Our chosen representative satisfies \begin{align*} \alpha_s=\sum_{r=p}^{k}\alpha_s^{r,k-r}. \end{align*} Every contributing $r$ therefore satisfies $r\ge p$, so the possible first Hodge indices after differentiation are $r$ and $r-1$, both at least $p-1$. Consequently \begin{align*} [(\mathcal L_{\widetilde v}\widehat\alpha)|_{X_s}]\in F^{p-1}H^k(X_s,\mathbb C). \end{align*}[/guided]

[step:Pass from fibrewise classes to the bundle inclusion]The preceding step proves that, for every $s\in U$, \begin{align*} (\nabla_v\sigma)(s)\in F^{p-1}H^k(X_s,\mathbb C). \end{align*} Since $s$ was arbitrary and $F^{p-1}\mathcal H^k$ is the holomorphic subbundle with these fibres, this gives \begin{align*} \nabla_v\sigma\in \Gamma(U,F^{p-1}\mathcal H^k). \end{align*} Because $U$, $\sigma$, and $v$ were arbitrary, the contraction of $\nabla\sigma$ with every local holomorphic vector field lies in $F^{p-1}\mathcal H^k$. Therefore \begin{align*} \nabla\sigma\in \Gamma(U,F^{p-1}\mathcal H^k\otimes_{\mathcal O_U}\Omega_U^1). \end{align*} This proves \begin{align*} \nabla(F^p\mathcal H^k)\subseteq F^{p-1}\mathcal H^k\otimes_{\mathcal O_S}\Omega_S^1 \end{align*} which is Griffiths transversality.[/step]

[guided]The previous step proved a fibrewise statement: for every point $s\in U$, \begin{align*} (\nabla_v\sigma)(s)\in F^{p-1}H^k(X_s,\mathbb C). \end{align*} The bundle $F^{p-1}\mathcal H^k$ is defined as the holomorphic subbundle of $\mathcal H^k$ with precisely these fibres, so this fibrewise containment says exactly that \begin{align*} \nabla_v\sigma\in \Gamma(U,F^{p-1}\mathcal H^k). \end{align*} Since $v$ was an arbitrary local holomorphic vector field, every contraction of the $\mathcal H^k$-valued one-form $\nabla\sigma$ with a local holomorphic tangent field lands in $F^{p-1}\mathcal H^k$. Therefore the one-form itself lies in the [tensor product](/page/Tensor%20Product) subbundle \begin{align*} F^{p-1}\mathcal H^k\otimes_{\mathcal O_U}\Omega_U^1. \end{align*} Because the neighbourhood $U$ and the section $\sigma\in\Gamma(U,F^p\mathcal H^k)$ were arbitrary, the desired sheaf inclusion holds on $S$: \begin{align*} \nabla(F^p\mathcal H^k)\subseteq F^{p-1}\mathcal H^k\otimes_{\mathcal O_S}\Omega_S^1. \end{align*}[/guided]