Gauss-Manin connection and t-adic geometry

TL;DR

This paper establishes that the de Rham cohomology of smooth rigid varieties over Laurent series fields naturally admits a formal meromorphic Gauss-Manin connection, linking it to classical and singularity theory contexts.

Contribution

It introduces a natural Gauss-Manin connection on de Rham cohomology for rigid varieties over Laurent series fields, extending classical notions to a non-archimedean setting.

Findings

De Rham cohomology of smooth rigid varieties admits a natural formal meromorphic connection.

Comparison with classical Gauss-Manin connections in algebraic geometry.

Connections with Milnor fibrations at hypersurface singularities.

Abstract

We show that the de Rham cohomology of any separated and smooth rigid variety over a field of Laurent series of characteristic zero carries a natural formal meromorphic connection, which we call the Gauss-Manin connection. We compare it with the Gauss-Manin connection of a proper and smooth variety over a curve, and with the Gauss-Manin connection of the Milnor fibration at an isolated complex hypersurface singularity.