Smooth proper family of compact complex manifolds ($f:\mathcal{X}\to S$)

A proper holomorphic submersion has compact smooth fibres, and the Ehresmann fibration theorem gives differentiable local trivializations. Their integral cohomology, modulo torsion, forms a local system. If the fibres are Kähler manifolds, their Hodge filtrations form holomorphic subbundles and give a variation of Hodge structure. Properness is needed for this usual conclusion.