# Calculating étale fundamental groups from the usual fundamental group

$$\newcommand{Spec}{\operatorname{Spec}}$$Let $$X$$ be a connected affine smooth variety over $$\mathbb{Q}$$, with a point $$x\in X(\Spec(\mathbb{Q})$$. For any algebraically closed field $$K$$ of characteristic zero, we may consider the base-change $$X_K=S\times_{\Spec(\mathbb{Q})} \Spec(K)$$ and the geometric point $$\overline{x}_K \in X_K(\Spec(K))$$ lying over $$x$$. Then we get the étale fundamental group $$\pi_1(X_K,\overline{x}_K)$$, which is the pro-finite group associated to the Galois category of finite étale schemes over $$X_K$$.

If $$K=\mathbb{C}$$, the group $$\pi_1(X_{\mathbb{C}}, \overline{x}_{\mathbb{C}})$$ is canonically isomorphic to the profinite completion of the fundamental group (in the topological sense) of the complex analytic variety associated to $$X_{\mathbb{C}}$$. That is, we have an isomorphism:

$$\pi_1(X_{\mathbb{C}},\overline{x}_{\mathbb{C}})\rightarrow \widehat{\pi_1(X_{\mathbb{C}}^{an},\overline{x}_{\mathbb{C}})}$$ where $$X_{\mathbb{C}}^{an}$$ is the complex analytification of $$X_{\mathbb{C}}$$. This is a nice isomorphism, as calculating the usual fundamental group seems easier than calculating its étale counterpart. Even if the profinite completion is not very explicit, a system of generators of $$\pi_1(X_{\mathbb{C}}^{an},\overline{x}_{\mathbb{C}})$$ yields a system of topological generators o its pro-finite completion, so we can somewhat understand the étale fundamental group from the usual one.

My question is whether there is a relation between the étale fundamental groups $$\pi_1(X_K,\overline{x}_K)$$ for an arbitrary algebraically closed field $$K$$ of characteristic zero, and the group $$\pi_1(X_{\mathbb{C}},\overline{x}_{\mathbb{C}})$$. I have seen that these seem to agree in some cases such as $$\mathbb{G}_{m,K}$$, $$\mathbb{A}^n_K$$ and $$\mathbb{P}_K^n$$, and in some other posts (see for example MathOverflow: étale fundamental group of the projective space) I have seen some people argue using the Lefschetz principle, but I have not been able to find anything that useful.

• One can indeed use the Lefschetz principle here: $\pi_1(X_K, \overline{x}_K)$ is independent of the choice of $K$ for $K$ algebraically closed of characteristic zero, so one may assume $K=\mathbb C$. This is contained in SGA 1 somewhere. Commented Mar 14, 2023 at 20:42
• The precise reference is SGA 1, Exp. X, Cor. 1.8. Commented Mar 14, 2023 at 20:59