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I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.

It is well known that the étale fundamental group in positive characteristics is a very difficult object, especially so in the non-projective case due to possibly wild ramification at infinity. I'm not even sure if there are examples of the kind above. Is this known?

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Note that any projective variety is quasi-projective. A comparison theorem of Grothendieck gives many examples of projective varieties in positive characteristic with trivial etale fundamental group. So probably you wish to amend your question slightly? – Pete L. Clark Feb 22 '10 at 11:59
I am fairly sure that removing a codimension $2$ subvariety from a projective variety doesn't change the fundamental group. So I can take any projective example of dimension $2$ or greater and make it nonprojective by yanking out a point. – David Speyer Feb 22 '10 at 12:21
In fact David's comment is SGA1 Corollary X.3.3 and has now reminded me of an old question of mine at… – Frank Feb 22 '10 at 12:29
@DS: Grothendieck's specialization theorem applies to any projective variety which lifts to characteristic $0$. So for instance $\mathbb{P}^2$ is simply connected in all characteristics. So based on what you say (which I haven't seen before but sounds good to me), $\mathbb{P}^2$ minus a point is an example. – Pete L. Clark Feb 22 '10 at 12:31
Technical point: I should have asked for the bigger variety to be regular. – David Speyer Feb 22 '10 at 12:48
up vote 9 down vote accepted

This is an answer to Pete's question on simply connected affine varieties (I can not put it in a comment because of space limitation).

I think that in positive characteristic $p$, no affine irreducible variety $X$ of positive dimension is simply connected. We can assume $X=\operatorname{Spec}(A)$ integral because $\pi_1$ is insensible to nipotent elements (SGA IX.4.10). Let $k[t_1,\ldots, t_d] \subseteq A$ be a finite extension with minimal degree $[k(A):k(t_1,\ldots, t_d)]$. Consider the étale cover $Y\to \mathbb A^d_k= \operatorname{Spec}(k[t_1,\ldots, t_d])$ defined by $s^p-s=t_1$. Then $X\times_{\mathbb A^d_k} Y\to X$ is an étale cover of degree $p$. As $k(Y)$ and $k(X)$ are linearly disjoint over $k(t_1,\ldots, t_d)$ ($k(Y)$ is Galois over $k({\bf t}):=k(t_1,\ldots, t_d)$ and $k(Y)\cap k(X)=k({\bf t})$), the tensor product $k(Y)\otimes_{k({\bf t})} k(X)$ is a field. This implies that $X\times_{\mathbb A^d_k} Y$ is connected.

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Here's a remark. It's a generalisation of a theorem of Katz-Lang given by Szamuely and Spieß that for a quasi-projective variety over a perfect field $k$, the abelianised tame fundamental group sits in the following exact sequence

$0\rightarrow T \rightarrow \pi_1^{t,ab}(X) \rightarrow T(Alb_X)\rightarrow 0$

where $T$ is a group related to the torsion subgroup in the Neron-Severi group $NS(X)$ and $T(Alb_X)$ is the full Tate module of the generalised Albanese variety of $X$. The definition of tame covers tries to control exactly this wild ramification at infinity (i.e control the function field for the points in $\mathfrak{X}\setminus X$ where $\mathfrak{X}$ is the corresponding projective). You can find more in Szamuely's new book for example.

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It is a direct consequence of Abhyankar's Conjecture (which was proved by Raynaud and Harbater) that if $k$ is an algebraically closed field of characteristic $p > 0$, then no affine curve $X_{/k}$ has trivial etale fundamental group. (Note that for curves, affine = quasi-projective, non-projective, by Riemann-Roch.)

I have some lecture notes on this subject from years back:

(In contrast to what it says on the first page, they are from 2002.)

Addendum: The comments above give plenty of examples of non-projective quasi-projective varieties with trivial etale fundamental group in characteristic $p$ (or really in characteristic quelconque). An interesting question left open by these examples is whether there are any (nontrivial) simply connected affine varieties in characteristic $p$. As I have said, the answer is "no" in dimension one.

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Regarding the question of finding a simply connected affine variety (of positive dimension): I'm tempted to say that it is not possible. Embed $X$ into $\mathbb{A}^n$ and take a nontrivial cover of $\mathbb{A}^n$ to get a nontrivial cover of $X$. The gap in this argument, of course, is that the cover of $X$ might not be connected. Any idea how to get around this? – David Speyer Feb 22 '10 at 14:36
If it is an open subset if $\mathbb{A}^n$, then they are birational and you can use extensions of $\mathbb{A}^n$ that introduce a common nontrivial extension of their function fields. – Tyler Lawson Feb 22 '10 at 14:50

$\mathbb{P}_k^1$, over an algebraically closed field, is such an example. You can adapt the proof that $\mathbb{Q}$ has no unramified extensions to show that $k(t)$ has no unramified extensions.

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Hi David, I am looking for non-projective examples. I'll edit the question to make that clearer. – Lars Feb 22 '10 at 12:00

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