Tannakian duality and Gauss-Manin connections for a family of curves

Abstract

Let $X/S$ be a smooth family of smooth projective varieties, where $S$ is a smooth affine curve over a field $k$ of characteristic $0.$ We relate the differential fundamental groupoid scheme of $X/k$ with the differential fundamental groupoid scheme of $S/k$ and the relative differential fundamental group of $X/S$ in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least $1,$ we prove that these maps are isomorphisms. This gives an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we can shrink $X$ (as a family on $S$) to obtain a de Rham $K(\pi,1)$ surface.