For families of curves of genus at least 1 the Gauss-Manin connections are isomorphic to the group cohomology of the geometric relative fundamental group, yielding a cohomological interpretation and a de Rham K(π,1) surface after shrinking.
Zhang, The homotopy sequence of the algebraic fundamental group.Int
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Tannakian duality and Gauss-Manin connections for a family of curves
For families of curves of genus at least 1 the Gauss-Manin connections are isomorphic to the group cohomology of the geometric relative fundamental group, yielding a cohomological interpretation and a de Rham K(π,1) surface after shrinking.