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 $tau = dmu$, 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?