Let $K$ be an algebraically closed field of characteristic zero. Let $X/K$ be a smooth variety. Is it true that the \'etale fundamental group $\pi_1(X)$ is topologically finitely generated.
I know that the answer is ``yes'' in the following two cases:
1) $X/K$ is proper.
2) $\dim(X)=1$. (In this case we can write down a presentation of $\pi_1(X)$ very explicitly.)
Can there anything go wrong in the general case? (It would be wonderful to have a reference which one can simply quote.)