Variation of Hodge structure ($\nabla F^p\subseteq F^{p-1}\otimes {\Omega }_S^1$)

A pure integral variation consists of a local system of free abelian groups, holomorphic filtration subbundles of its associated flat bundle, and fibrewise Hodge structures of fixed weight and Hodge numbers. Its flat connection satisfies Griffiths transversality. A polarized variation additionally has a flat integral bilinear form polarizing every fibre.

In a variation of Hodge structure, differentiation lowers the filtration index by at most one. In the relative de Rham construction, an absolute form of degree at least $p$ contributes, after extracting one base differential, a relative form of degree at least $p-1$. This filtration shift proves the condition for geometric variations.

The flat connection on the complexified fibre-cohomology local system differentiates cohomology classes using topological parallel transport. For a smooth proper holomorphic family, relative de Rham cohomology identifies it with the connecting morphism obtained by retaining one base differential in the absolute de Rham complex.