# ag.algebraic geometry – 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 $$tau = dmu$$, 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?