Connections on trivial vector bundles over projective schemes

TL;DR

This paper develops an algebraic approach to understanding the differential Galois group of connections on trivial vector bundles over projective schemes, replacing monodromy with a Lie algebra construction.

Contribution

It introduces an algebraic method to determine the differential Galois group using a Lie algebra derived from regular forms, extending classical transcendental techniques.

Findings

The differential Galois group is a closure of a Lie algebra constructed from regular forms.

Constructs connections on curves with specified differential Galois groups.

Provides an algebraic framework for analyzing integrable connections on trivial bundles.

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.