Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c_{1,\mathbb{R}}=0$) or canonically polarised ($c_1<0$). The differential of the moduli map $\mu : B^{\circ} \to \mathcal{M}$ is the Kodaira--Spencer map $\tau = d\mu$, measuring in the complex structure of the smooth fibres of the family. The period map $p : B^{\circ} \to D$ is a holomorphic map which measures the variation of the Hodge decomposition of the fibres.

Is there a relationship between the Kodaira--Spencer map and the period map?