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?
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.
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.
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:
http://alpha.math.uga.edu/~pete/fundincharp.pdf
(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.
$\mathbb{P}_k^1$, 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.
