= 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 subbundles
$$
F^p\mathcal H\subseteq\mathcal H
=\mathbb H_{\mathbb Z}\otimes_{\mathbb Z}\mathcal O_S.
$$
Each 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>,
$$
\boxed{\nabla(F^p\mathcal H)\subseteq
F^{p-1}\mathcal H\otimes\Omega_S^1.}
$$
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>.
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