The period map and the Kodaira--Spencer map

Question

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?

Answer 1

Differential of period map $d P^{p+q,p}$ is composition of KS-map $T_{B,0} \to H^1(X_0,T_{X_0})$ with natural map $H^1(X_0,T_{X_0}) \to Hom(H^{p,q}(X_0),H^{p-1,q+1}(X_0))$ (given by the cup product and the interior product). See Voisin "Hodge theory and complex algebraic geometry" Theorem 10.4