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Over a smooth and proper complex scheme, the differential Galois group of anintegrable connection may be obtained as the closure of the transcendentalmonodromy representation. In this paper, we employ a completely algebraicvariation of this idea by restricting attention to connections on trivialvector bundles and replacing the fundamental group by a certain Lie algebraconstructed from the regular forms. In more detail, we show that thedifferential Galois group is a certain ``closure'' of the aforementioned Liealgebra. This is then applied to construct connections on curves withprescribed differential Galois group.

Connections on trivial vector bundles over projective schemes