Algebraic fundamental group

Schemes over a field of characteristic zero

For a scheme ${\displaystyle X}$ that is of finite type over ${\displaystyle \mathbb {C} }$, the complex numbers, there is a close relation between the étale fundamental group of ${\displaystyle X}$ and the usual, topological, fundamental group of ${\displaystyle X(\mathbb {C} )}$, the complex analytic space attached to ${\displaystyle X}$. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of ${\displaystyle \pi _{1}(X)}$. This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of ${\displaystyle X(\mathbb {C} )}$ stem from ones of ${\displaystyle X}$. In particular, as the fundamental group of smooth curves over ${\displaystyle \mathbb {C} }$ (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.