Gauss–Manin connection ($\nabla :\mathcal{H}\to \mathcal{H}\otimes {\Omega }_S^1$)

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.