[proofplan]
Fix a point $s\in S$ and a tangent vector $v\in T_sS$. The differential of the period map is computed by differentiating local sections of the Hodge filtration using the Gauss-Manin connection. [Griffiths transversality](/theorems/9129) says that the covariant derivative of a section of $F^p\mathcal H^k$ lies in $F^{p-1}\mathcal H^k$, so the resulting tangent map lowers the filtration by at most one step. This is exactly the defining condition for membership in the horizontal tangent subspace.
[/proofplan]
[step:Identify the period derivative through the Gauss-Manin connection]
Let $H$ denote the fixed complex [vector space](/page/Vector%20Space) obtained from the marked local system $R^k\pi_*\mathbb C$. For $s\in S$, write $F_s^p\subset H$ for the fibre at $s$ of the Hodge subbundle $F^p\mathcal H^k$, transported to $H$ by the marking. Thus
\begin{align*}
\Phi(s)=F_s^\bullet.
\end{align*}
Fix $v\in T_sS$. By the tangent-space description for filtrations in [citetheorem:9128], the tangent vector $d\Phi_s(v)\in T_{\Phi(s)}D$ is represented by the family of linear maps
\begin{align*}
A_p(v):F_s^p/F_s^{p+1}\to H/F_s^p
\end{align*}
defined as follows. Given $a\in F_s^p$, choose a local holomorphic section
\begin{align*}
\sigma:U\to F^p\mathcal H^k
\end{align*}
on an open neighbourhood $U\subset S$ of $s$ such that $\sigma(s)=a$. Then set
\begin{align*}
A_p(v)(a+F_s^{p+1})=(\nabla_v\sigma)(s)+F_s^p.
\end{align*}
This is the standard infinitesimal description of the variation of a subbundle inside the fixed bundle $H\times S$ determined by the marking.
[guided]
The marking lets us regard all cohomology fibres as one fixed complex vector space $H$. Under this identification, the period map is the map
\begin{align*}
\Phi:S\to D
\end{align*}
that sends a point $s\in S$ to the Hodge filtration $F_s^\bullet$ on $H$.
We now compute what the derivative means. Fix $v\in T_sS$. The tangent-space description for filtrations in [citetheorem:9128] identifies an infinitesimal variation of $F_s^\bullet$ with maps
\begin{align*}
F_s^p/F_s^{p+1}\to H/F_s^p
\end{align*}
for the relevant indices $p$. For $a\in F_s^p$, choose a local holomorphic section
\begin{align*}
\sigma:U\to F^p\mathcal H^k
\end{align*}
defined on an open neighbourhood $U\subset S$ of $s$ and satisfying $\sigma(s)=a$. The first-order movement of this section in the tangent direction $v$ is measured by the Gauss-Manin covariant derivative $(\nabla_v\sigma)(s)$. Since tangent vectors to the flag variety are taken modulo the original subspace $F_s^p$, the corresponding component of $d\Phi_s(v)$ is
\begin{align*}
A_p(v)(a+F_s^{p+1})=(\nabla_v\sigma)(s)+F_s^p.
\end{align*}
The quotient by $F_s^p$ is essential: changing the frame inside $F^p$ itself should not count as moving the point of the flag variety. Thus the derivative records only the normal component of the covariant derivative of $F^p$ inside the fixed ambient vector space $H$.
[/guided]
[/step]
[step:Check that the representative is independent of the chosen section]
Let $\sigma_1:U_1\to F^p\mathcal H^k$ and $\sigma_2:U_2\to F^p\mathcal H^k$ be local holomorphic sections with $\sigma_1(s)=\sigma_2(s)=a$. On $U=U_1\cap U_2$, define
\begin{align*}
\tau:U\to F^p\mathcal H^k
\end{align*}
by $\tau=\sigma_1-\sigma_2$. Then $\tau(s)=0$. Choose a local holomorphic frame $e_1,\dots,e_m$ for $F^p\mathcal H^k$ on a neighbourhood of $s$, and write
\begin{align*}
\tau=\sum_{j=1}^m f_j e_j
\end{align*}
with holomorphic functions $f_j:U\to\mathbb C$ satisfying $f_j(s)=0$. Therefore
\begin{align*}
(\nabla_v\tau)(s)=\sum_{j=1}^m v(f_j)e_j(s)+\sum_{j=1}^m f_j(s)(\nabla_v e_j)(s).
\end{align*}
The second sum is zero because $f_j(s)=0$, and the first sum lies in $F_s^p$. Hence
\begin{align*}
(\nabla_v\sigma_1)(s)-(\nabla_v\sigma_2)(s)\in F_s^p.
\end{align*}
Thus the class $(\nabla_v\sigma)(s)+F_s^p\in H/F_s^p$ is independent of the chosen local section $\sigma$.
If $a\in F_s^{p+1}$, the same construction using a local section of $F^{p+1}\mathcal H^k\subset F^p\mathcal H^k$ shows that the induced map factors through $F_s^p/F_s^{p+1}$. Therefore $A_p(v)$ is a well-defined component of $d\Phi_s(v)$.
[/step]
[step:Use Griffiths transversality to force the image into the horizontal quotient]
By [citetheorem:9129], the Gauss-Manin connection satisfies Griffiths transversality:
\begin{align*}
\nabla(F^p\mathcal H^k)\subset F^{p-1}\mathcal H^k\otimes\Omega_S^1.
\end{align*}
Evaluating this inclusion on the tangent vector $v\in T_sS$ gives
\begin{align*}
(\nabla_v\sigma)(s)\in F_s^{p-1}
\end{align*}
for every local holomorphic section $\sigma:U\to F^p\mathcal H^k$. Consequently the component $A_p(v)$ has image contained in the subquotient
\begin{align*}
F_s^{p-1}/F_s^p\subset H/F_s^p.
\end{align*}
Thus
\begin{align*}
A_p(v)\in \operatorname{Hom}(F_s^p/F_s^{p+1},F_s^{p-1}/F_s^p)
\end{align*}
for every $p$.
[guided]
The only input about a polarized Kähler family used here is Griffiths transversality. In the notation of the Hodge bundle $\mathcal H^k$ and its filtration $F^\bullet\mathcal H^k$, [citetheorem:9129] states that
\begin{align*}
\nabla(F^p\mathcal H^k)\subset F^{p-1}\mathcal H^k\otimes\Omega_S^1.
\end{align*}
This means the following concrete statement: if
\begin{align*}
\sigma:U\to F^p\mathcal H^k
\end{align*}
is a local holomorphic section and $v\in T_sS$, then differentiating $\sigma$ by the Gauss-Manin connection in the direction $v$ produces a vector satisfying
\begin{align*}
(\nabla_v\sigma)(s)\in F_s^{p-1}.
\end{align*}
Now compare this with the formula for the derivative of the period map. The $p$-th component of $d\Phi_s(v)$ sends the class $a+F_s^{p+1}$ to the class
\begin{align*}
(\nabla_v\sigma)(s)+F_s^p\in H/F_s^p.
\end{align*}
Griffiths transversality improves the target: the representative $(\nabla_v\sigma)(s)$ is not merely in $H$, but in $F_s^{p-1}$. Therefore its class modulo $F_s^p$ lies in
\begin{align*}
F_s^{p-1}/F_s^p.
\end{align*}
So the component of $d\Phi_s(v)$ is not an arbitrary map
\begin{align*}
F_s^p/F_s^{p+1}\to H/F_s^p.
\end{align*}
It is a filtration-lowering map
\begin{align*}
F_s^p/F_s^{p+1}\to F_s^{p-1}/F_s^p.
\end{align*}
This is exactly the infinitesimal content of horizontality: the first-order change of the Hodge filtration may lower the filtration index by one, but not by two or more.
[/guided]
[/step]
[step:Conclude that the period derivative is horizontal]
By definition, the horizontal tangent subspace $T_{\Phi(s)}^{\mathrm{hor}}D$ consists of those tangent vectors whose filtration components send $F_s^p/F_s^{p+1}$ into $F_s^{p-1}/F_s^p$ for every $p$. The previous step proves precisely this condition for the components $A_p(v)$ of $d\Phi_s(v)$. Hence
\begin{align*}
d\Phi_s(v)\in T_{\Phi(s)}^{\mathrm{hor}}D.
\end{align*}
Since $v\in T_sS$ was arbitrary, we obtain
\begin{align*}
d\Phi_s(T_sS)\subset T_{\Phi(s)}^{\mathrm{hor}}D.
\end{align*}
Since $s\in S$ was arbitrary, the inclusion holds at every point of $S$.
[/step]
