Let $\pi:\mathcal X\to S$ be a smooth proper holomorphic submersion of complex manifolds whose fibres are compact Kähler manifolds, and fix $s\in S$. Set $X:=\mathcal X_s$. Let $k\ge 0$, and let

\begin{align*} \Phi:S\to D_k \end{align*}

denote the local period map for the weight-$k$ Hodge filtration on $H^k(X,\mathbb C)$ after choosing a local marking of the flat local system $R^k\pi_*\mathbb C$, where $D_k$ is the classifying space of filtrations with the Hodge numbers of $H^k(X,\mathbb C)$. For integers $p,q\ge 0$ with $p+q=k$ and $p\ge 1$, identify the graded Hodge piece $\operatorname{Gr}_F^p H^k(X,\mathbb C)$ with

\begin{align*} H^{p,q}(X)\cong H^q(X,\Omega_X^p) \end{align*}

by the Kähler [Hodge decomposition](/theorems/2745).

Let

\begin{align*} \rho_s:T_sS\to H^1(X,T_X) \end{align*}

be the Kodaira-Spencer map, normalized as follows. If a one-parameter deformation tangent to $v\in T_sS$ is represented on the fixed smooth fibre by a first-order Beltrami differential $\mu_v\in A^{0,1}(X,T_X)$ such that a local $J_t$-holomorphic covector has fixed $J_0$-type expansion $\zeta-t\,\mu_v\lrcorner\zeta+O(|t|^2)$, then $[\mu_v]=\rho_s(v)$ in $H^1(X,T_X)$. Then, for every $v\in T_sS$, the infinitesimal period map component

\begin{align*} d\Phi_s(v)_{p,q}:H^{p,q}(X)\to H^{p-1,q+1}(X) \end{align*}

is given by

\begin{align*} d\Phi_s(v)_{p,q}([\alpha])=-[\rho_s(v)\lrcorner \alpha]. \end{align*}

Here $\alpha$ is any Dolbeault-closed smooth $(p,q)$-form representing $[\alpha]\in H^q(X,\Omega_X^p)$, and the contraction denotes the natural cup-product pairing

\begin{align*} H^1(X,T_X)\otimes H^q(X,\Omega_X^p)\to H^{q+1}(X,\Omega_X^{p-1}). \end{align*}