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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.)

flag	asked Jan 31 2011 at 16:11 Sebastian Petersen 1,261●2●9

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 2011 at 16:21

@Donu: can it be proved 'purely algebraicly'? – Mariano Suárez-Alvarez Jan 31 2011 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 2011 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 2011 at 20:25

3

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 2011 at 23:41

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Sebastian Petersen · Answer 1 · 2011-03-14 12:55:30Z UTC

Thanks, Dono Arapura, for explaining how one proves this statement, using the Riemann existance theorem in SGA1.

I mention for the sake of completeness: Today in the morning, I found that there is a SGA-reference for my question as it stands above: SGA 7.1, II.2.3.1. (I was surprised, that it is SGA 7.1, not SGA 1.)

Finiteness property of fundamental groups

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