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$\newcommand{Spec}{\operatorname{Spec}}$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.

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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented Mar 14, 2023 at 20:42
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    $\begingroup$ The precise reference is SGA 1, Exp. X, Cor. 1.8. $\endgroup$ Commented Mar 14, 2023 at 20:59

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