Gauß-Manin connection and Tannaka duality

Abstract

In analogy with the étale fundamental groups, we express the Gauß-Manin connection for $H^1$ in Tannaka terms. One difficulty is that unlike for fundamental groups, the Tannaka group scheme of relative connections, and the groupoid schemes of absolute connections and of connections on the base do not form an exact sequence. We added a theorem due to Pierre Deligne which improves the main result of the article. It allows to compare the Tannaka group scheme of relative connections with the kernel of the map from the groupoid scheme of absolute connections to the one of the base.