# The period map and the Kodaira--Spencer map

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?

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

• The Voisin book is very nice, thank you for reminding me about it Apr 11 at 4:27