Relative de Rham complex (${\Omega }_{\mathcal{X}/S}^{\bullet },d_{\mathcal{X}/S}$)

For a holomorphic submersion, relative differential forms are forms modulo those containing a base differential. The relative exterior derivative gives a complex whose fibrewise hypercohomology is complex cohomology for a smooth proper family. Its filtration by degree defines the Hodge filtration; retaining exactly one base differential in the absolute complex constructs the Gauss–Manin connection.