The original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential equations with prescribed monodromy representations.
Riemann–Hilbert correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X.
Thus such connections give a purely algebraic way to access the finite dimensional representations of the topological fundamental group.
The condition of regular singularities means that locally constant sections of the bundle (with respect to the flat connection) have moderate growth at points of Y − X, where Y is an algebraic compactification of X.
On the other hand, if we work with holomorphic (rather than algebraic) vector bundles with flat connection on a noncompact complex manifold such as A1 = C, then the notion of regular singularities is not defined.
A much more elementary theorem than the Riemann–Hilbert correspondence states that flat connections on holomorphic vector bundles are determined up to isomorphism by their monodromy.
This can be regarded as the positive characteristic analogue of the classical theory, where one can find a similar interplay of constructive vs. perverse t-structures.