 Solution

A pure integral variation of Hodge structure of weight $w$ on a complex manifold $S$ consists of a local system ${\mathbb{H}}_{\mathbb{Z}}$ of finite-rank free abelian groups and a decreasing filtration by holomorphic subbundlesEach fibre, with this filtration and the conjugation from its integral lattice, must be a Hodge structure of weight $w$ with locally constant Hodge numbers. The local system gives a flat connection $\nabla$: in a locally constant frame it differentiates the coefficient functions. The filtration must satisfy Griffiths transversality,A polarized variation additionally has a locally constant integral bilinear form $Q$ which polarizes every fibre. Flatness means $\nabla Q=0$. A collection of fibrewise Hodge decompositions without a holomorphic filtration or the transversality condition is not yet a variation of Hodge structure.