Riemann–Hilbert correspondence

Statement

Let $X$ be a smooth complex algebraic variety. Analytic horizontal sections give an equivalence between the category of regular-singular algebraic flat connections on $X$ and the category of finite-dimensional complex local systems on $X^{\mathrm{an}}$.

For a smooth projective $X$, every algebraic flat connection is regular singular. On an open variety, the regular-singular condition refers to the boundary of a smooth compactification. Irregular connections require an enhanced correspondence incorporating Stokes data.