GAUSS-MANIN CONNECTION AND t-ADIC GEOMETRY

Let k be a field of characteristic zero, denote by R the ring k[[t]] of formal power series over k, and by K the field k((t)) of Laurent series over k. The aim of this note is to show that the de Rham cohomology of any separated and smooth rigid K-variety X carries a natural formal meromorphic connection ∂X , which we call the Gauss-Manin connection. It relates to the classical Gauss-Manin connection in the following way. Let S be a smooth k-curve, 0 a point of S(k), and t a local parameter on S at 0. We put S = S − {0}. The choice of t defines a morphism of k-schemes η̂ : SpecK → S. Let f : Y → S be a proper and smooth morphism, and put X = Y ×So η̂. If we denote by ∇ the classical Gauss-Manin connection on Rf∗(Ω • Y/So), then the covariant derivative ∂t = ∇∂ ∂t induces a formal meromorphic connection on the de Rham cohomology space H dR(X/K) for each i ≥ 0. We showed that the natural GAGA isomorphism H dR(X/K) ∼= H dR(X /K) commutes with the connections ∂t and ∂Xan (Theorem 3.1). In the local case, we conjecture that a similar comparison result holds. Let f : C → C be a complex analytic map with an isolated singularity at x ∈ f(0) and denote by Fx the analytic Milnor fiber of f at x. Our conjecture compares the formal meromorphic connection (H dR(Fx), ∂Fx) to the Gauss-Manin connection on the relative de Rham cohomology of the Milnor fibration of f at x. See Conjecture 4.4 for a precise statement. This note merely serves as an announcement of the principal results. Detailed proofs will appear in a forthcoming paper.