Regular-singular connections on relative complex schemes

Deligne's celebrated "Riemann-Hilbert correspondence" appearing in [9] relates representations of the fundamental group of a smooth complex alge-braic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar statement in the case of a smooth scheme X over the spectrum of a ring R D CJt1, ... , trK=I. On one side of the correspondence we have representations on R-modules of the fundamental group of the special fibre, and on the other we have certain integrable R-connections admitting loga-rithmic models. The correspondence is then applied to give explicit examples of differential Galois groups of CJtK-connections.