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The étale homotopy type is a construction due to Artin and Mazur that generalizes the étale fundamental group. If $X$ is a scheme over a separably closed field $k$, then the étale homotopy type of $X$ is a profinite homotopy type that knows everything about the étale cohomology of $X$ with finite coefficients. If the base field is $\mathbb{C}$, and $X$ is not too bad, the etale homotopy type of $X$ is just the profinite completion of the topological space that underlies the set of $\mathbb{C}$-points of $X$. I am wondering what is known if $k$ is of positive characteristic ?

Here is a precise question : Are there examples of smooth and proper schemes $X$ over the algebraic closure of $\mathbb{F}_{p}$ that are such that the étale homotopy type of $X$ is not that of a complex manifold if we complete away from $p$ ? I am also willing to drop the assumption that $X$ is proper if this makes the question more interesting.

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    $\begingroup$ It is an interesting question, but probably very difficult. I would guess the answer is no. If $X$ lifts to characteristic zero, the answer should be yes by some homotopy form of smooth proper base change. So we should consider an $X$ which does not lift, and most (if not all!) known examples of nonliftable varieties admit a purely inseparable $f:Y\to X$ from a liftable smooth projective $Y$, such that there exists a $g:X\to Y$ with $fg = F_X^a$, $gf=F_Y^b$. But the Frobenius induces the identity on the etale topos, so $X$ and $Y$ have equivalent etale topoi, and the same homotopy type. $\endgroup$
    – Piotr Achinger
    Dec 23, 2016 at 11:59
  • $\begingroup$ @R.vanDobbendeBruyn That's true, but the question is about the prime-to-$p$ completion of the homotopy type. A map of profinite homotopy types is a weak equivalence if and only if it induces isomorphisms on $\pi_1$ and on the cohomology of local systems. Thus there is a chance that the homotopy form of smooth proper base change would follow formally from (a) the specialization theorem on prime-to-$p$ quotients of $\pi_1$ (SGA1), and (b) smooth proper base change for prime-to-$p$ lcc sheaves. $\endgroup$
    – Piotr Achinger
    Dec 23, 2016 at 20:38
  • $\begingroup$ Yes I agree with Piotr that if X is smooth and proper and lifts to characteristic zero then the etale homotopy type of X will be weakly equivalent to the etale homotopy type of the complex points of the lift. Such a statement can be found in Artin-Mazur's book or Friedlander's book. $\endgroup$
    – Geoffroy Horel
    Dec 23, 2016 at 21:16
  • $\begingroup$ I think that it is easy to produce varieties that do not lift to char 0. For example, take the supersingular elliptic curve in char 2, so that it has extra automorphisms that do not lift to char 0. Then take a Godeaux variety with that fundamental group. It has a bundle of elliptic curves over it with that monodromy. Its fundamental group $(\hat Z/Z_2)^2\rtimes Q_8$, which wants to lift to $\hat Z^2\rtimes Q_8$, which is nonsense, so it doesn't lift. . . This is not a purely prime-to-$p$ phenomenon, but it is a finite 2-group acting on an infinite odd group, so it's pretty close. $\endgroup$
    – Ben Wieland
    Dec 27, 2016 at 23:21

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