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It is proven in SGA1 that if $k$ is an algebraically closed field, if $X$ is a proper $k$-scheme and if $Y$ is a locally noetherian $k$-scheme (say, $X$ and $Y$ are non-empty and connected) then $\pi_1(X\times_k Y)=\pi_1(X)\times \pi_1(Y)$. This implies that if $L$ is any algebraically closed extension of $k$ then $\pi_1(X\times_k L)=\pi_1(X)$. All of these results are known to fail without the properness assumption, but all counter-examples are in positive characteristic, and I am almost sure that this assumption can be removed in char. $0$.

Question: is it written somewhere? I did not find any proof of it in SGA, nor in Stacks Project.

For the commutation with the product over $\mathbb C$ one can use the corresponding result in topology and the comparison theorem between étale and topological fundamental groups, but reducing the case of an arbitrary algebraically closed field of char. 0 to that of $\mathbb C$ would first require a proof of the invariance of the $\pi_1$ under algebraically closed extension.

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    $\begingroup$ The invariance of fundamental groups under extensions of alg. closed fields in char. 0 is discussed here. Since then Aaron Landesman has also written up a proof of the invariance here: arxiv.org/pdf/2005.09690.pdf $\endgroup$
    – Will Chen
    Dec 16, 2020 at 23:36
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    $\begingroup$ Thank you very much! The quasi-projectivity assumption in the ArXiv paper you refer to looks strange. I think it is probably not necessary? $\endgroup$
    – Antoine Ducros
    Dec 18, 2020 at 10:04
  • $\begingroup$ @AntoineDucros it seems that by van Kampen it suffices to treat the affine case? $\endgroup$
    – Piotr Achinger
    Dec 18, 2020 at 19:58

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