Connections on trivial vector bundles over projective schemes

Abstract

Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain ``closure'' of the aforementioned Lie algebra. This is then applied to construct connections on curves with prescribed differential Galois group.