Let $X$ be a compact connected Kähler complex manifold of complex dimension $n\ge 1$, and suppose that $K_X \cong \mathcal O_X$ as a holomorphic line bundle. Choose a nowhere-vanishing holomorphic section $\Omega\in H^0(X,K_X)$.

Let $0\in S$ be a germ of a complex space and let $\pi:\mathcal X\to S$ be a smooth proper holomorphic family with central fibre $\mathcal X_0=X$. Suppose the family is equipped with a local marking of the cohomology local system, so that the cohomology groups $H^n(\mathcal X_s,\mathbb C)$ are identified with $H^n(X,\mathbb C)$ near $0$. Suppose also that the Kodaira-Spencer map $\rho_0:T_0S\to H^1(X,T_X)$ is an isomorphism.

Let $\mathcal P:S\to \mathbb P(H^n(X,\mathbb C))$ be the local period map defined by this marking, sending $s$ to the transported Hodge line $H^{n,0}(\mathcal X_s)$. Under the Kodaira-Spencer identification $T_0S\cong H^1(X,T_X)$, the $H^{n-1,1}(X)$ component of the differential $d\mathcal P_0$ is the map $H^1(X,T_X)\to H^{n-1,1}(X)$ given by $[\varphi]\mapsto [\varphi\lrcorner\Omega]$. This map is injective. Consequently, $d\mathcal P_0$ is injective.