TITLE: Is the map on etale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?
QUESTION [13 upvotes]: Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the base change of $U$ to $Spec (L)$. Does anyone have a reference for why the map on etale fundamental groups $\pi_1(U_L) \rightarrow \pi_1(U)$ is an isomorphism?
Ultimately, I'm interested in the case that $U$ is normal, but I have a fairly easy argument to deduce that from the case that $U$ is smooth. Brian Conrad has also suggested a promising, though involved, avenue of attack using that topological $\pi_1$ is finitely generated. However since this seems like something that should be known, I was curious whether anyone knows of a reference (or perhaps a very short proof).
Here are some further remarks: It's fairly clear that the map is a surjection, and it is known to be an isomorphism if $U$ is projective (even in characteristic $p$). However, if $U$ is quasi-projective, the map need not be an isomorphism in characteristic $p$ (for example, it fails to be an isomorphism for $\mathbb A^1$, due to Artin Schreier covers). I suspect the isomorphism still holds true without the smoothness hypothesis on U, and true for prime to p parts of the etale fundamental group in characteristic p, but I really only want to apply this when U is smooth (or normal) and $k$ is a subfield of $\mathbb C$.

REPLY [2 votes]: I ended up writing a proof of the isomorphism $\pi_1(U_L)\rightarrow \pi_1(U)$ in my question above, including a characteristic p version. It is essentially a re-organization of Jason Starr's proof above, together with some suggestions by Brian Conrad.
You can find the write up at
https://arxiv.org/abs/2005.09690