# Finiteness property of fundamental groups

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.)

• Yes. Use the finite generation for the topological fund. grp (by using finite triangulability of complex varieties) + the comparison theorem SGA1 exp XII cor 5.2. – Donu Arapura Jan 31 '11 at 16:21
• @Donu: can it be proved 'purely algebraicly'? – Mariano Suárez-Álvarez Jan 31 '11 at 16:35
• Mariano, I don't know of a purely algebraic argument, but it would certainly be interesting. As an aside, I might point out that the "wild" part of $\pi_1(\matbb{A}^1)$ in positive characteristic need not be finitely generated. – Donu Arapura Jan 31 '11 at 18:09
• @Donu: This only covers the case $K = \mathbb{C}$, right? Or can we generalize this using model theory? – Martin Brandenburg Jan 31 '11 at 20:25
• Donu's argument works by the invariance of the etale $pi_1$ under base-change of algebraically closed fields in characteristic zero. I believe the required specialization argument can be found in Serre's lectures on Galois theory. The key point is that $\pi_1(X\times Y)=\pi_1(X)\times \pi_1(Y)$, which fails in positive characteristic. An amusing (or vexing) fact is that an 'algebraic proof', in any reasonable sense, of finite-generation is unknown even for curves. – Minhyong Kim Jan 31 '11 at 23:41