The Gauss-manin Connection and Tannaka Duality

If f : X → S is a submersive fibration of complex analytic manifolds, given a point x ∈ X, there is an exact sequence of fundamental groups 0 → π 1 F q , ρ) is an isomorphism, this action defines the cohomology of the fibers as a local system over the base, resp. a Galois action on the cohomology of ρ over X × Fq ¯ F q. A good analog in algebraic geometry of the topological fundamental group on one side and thé etale fundamental group on the other side is provided by the Tannaka group associated to the category of flat connections. The action of the fundamental group or of the Galois group of the base corresponds to the Gauß-Manin connection. The purpose of this article is to show that the analogy isn't straightforward, that those actions are difficult to define, partly because the homomorphism analogous to π 1 (X s , x) → π 1 (X, x) is not injective.