Let $X$ be a compact connected Kähler complex manifold of complex dimension $n\ge 1$, and suppose that the canonical bundle $K_X=\Omega_X^n$ is holomorphically trivial. Let $\Omega\in H^0(X,K_X)$ be a nowhere-vanishing holomorphic $n$-form. Let $(\mathcal M,[X])$ be a local deformation space of $X$, represented after shrinking by a smooth proper holomorphic family

\begin{align*} \pi:\mathcal X\to \mathcal M \end{align*}

with central fibre $\pi^{-1}([X])=X$, whose Kodaira-Spencer map identifies $T_{[X]}\mathcal M$ with $H^1(X,T_X)$. Shrink $\mathcal M$ further, if necessary, so that every fibre of $\pi$ over $\mathcal M$ is compact Kähler. Choose a local marking of the flat local system $R^n\pi_*\mathbb C$, and let $\mathcal P$ denote the corresponding local period map sending a nearby complex structure to the line $H^{n,0}$ in the weight-$n$ Hodge filtration on $H^n(X,\mathbb C)$. Use the sign convention for the Kodaira-Spencer class and contraction under which the infinitesimal period-map formula is $[\varphi]\mapsto(\alpha\mapsto[\varphi\lrcorner\alpha])$.

Under the Griffiths-transversality identification

\begin{align*} T_{H^{n,0}(X)}\operatorname{Flag}\bigl(H^n(X,\mathbb C)\bigr)\supseteq \operatorname{Hom}\bigl(H^{n,0}(X),H^{n-1,1}(X)\bigr), \end{align*}

the derivative of the period map at $[X]$ is the [linear map](/page/Linear%20Map)

\begin{align*} d\mathcal P_{[X]}:H^1(X,T_X)\to \operatorname{Hom}\bigl(H^{n,0}(X),H^{n-1,1}(X)\bigr) \end{align*}

characterized by

\begin{align*} d\mathcal P_{[X]}([\varphi])(\Omega)=[\varphi\lrcorner\Omega], \end{align*}

where $\varphi\in A^{0,1}(X,T_X)$ is any $\bar\partial$-closed representative of the Kodaira-Spencer class $[\varphi]\in H^1(X,T_X)$, and $\varphi\lrcorner\Omega\in A^{n-1,1}(X)$ denotes contraction of the tangent-vector component of $\varphi$ into $\Omega$.