I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a $K(\pi,1)$.


For me, $X/\mathbb{C}$ should be a $K(\pi^{\mathrm{\acute{e}t}},1)$ if for all LCC $\mathcal{F}$ the natural map


$$H^i(\pi\_1^{\mathrm{\acute{e}t}}(X,\overline{x}),\mathcal{F}\_{\overline{x}})\to H^i(X\_{\mathrm{\acute{e}t}},\mathcal{F})$$


is an isomorphism. 


I believe this is equivalent to $X$ having vanishing higher étale homotopy groups. So, this question naturally led me to the following questions:


1. Does there exist $X/\mathbb{C}$ smooth projective and some $i>1$ such that $\pi\_i^\mathrm{\acute{e}t}(X)=0$ but $\pi\_i(X^\mathrm{an})\ne 0$?
2. Does there even exist $X/\mathbb{C}$ smooth projective with $\pi\_1^{\mathrm{\acute{e}t}}(X)=0$ but $\pi\_1(X^\mathrm{an})\ne 0$?
3. Does there even exist a connected compact Kähler manifold $X$ such that $\pi\_1(X)\ne 0$ but $\widehat{\pi\_1(X)}=0$ (profinite completion)?


Any help with these questions would be much appreciated!



Relationship between étale and topological $K(\pi,1)$s
Let $G$ be a group, $\iota: G\to \hat G$ its pro-finite completion. We call a $G$ a *good group* (cf. J.-P. Serre *Cohomologie galoisienne*, 2.6) if for every finite $\hat G$-module $M$, the maps
$$ \iota^\* : H^q(\hat G, M) \to H^q(G, M) $$ 
are isomorphisms for all $q\geq 0$.


Some examples of good groups:
(1) finite groups,
(2) finitely generated free groups and finitely generated free abelian groups,
(3) iterated extensions of groups of type (1) and (2),
(4) braid groups (a special case of (3)),


Arithmetic groups are not good in general, e.g., as Donu Arapura commented, ${\rm Sp}(2n,\mathbb{Z})$ is not a good group for $n > 1$. It is not known whether the mapping class groups $\Gamma\_{g ,n}$ (the orbifold fundamental group of $\mathcal{M}\_{g, n}$) are good groups (cf. Lochak–Schneps 2006, par. 3.4).


Now, say $X$ is a connected scheme of finite type over $\mathbb{C}$. Consider the following three conditions:


1. The scheme $X$ is a $K(\pi, 1)$ scheme.
2. The topological space $X^{\rm an}$ is a $K(\pi, 1)$ space.
3. The fundamental group $\pi\_1(X^{\rm an})$ is a good group.


Then (1)+(2) implies (3) and (2)+(3) implies (1). To prove this, consider an lcc sheaf $\mathcal{F}$ on $X$ and look at the commutative squares
$$ \begin{array}{c} H^q(\pi\_1(X^{\rm an}, x), \mathcal{F}\_x) & \to & H^q(X^{\rm an}, \mathcal{F}) \\ \uparrow & & \uparrow & \\ H^q(\pi\_1(X, x), \mathcal{F}\_x) & \to & H^q(X, \mathcal{F}). \end{array} $$ 


Note that we do not get that (1)+(3) implies (2), as neither (1) nor (3) gives us any information
about the cohomology of $X^{\rm an}$ with non-torsion coefficients. Still, we see that (1)+(3) implies
that $X^{\rm an}$ is a "$K(\pi, 1)$ for local systems of finite groups”.


The same should hold for fundamental groups of Deligne–Mumford stacks. In particular, the open question
whether $\Gamma\_{g ,n}$ is a good group would be equivalent to the question whether the
stack $\mathcal{M}\_{g, n}$ is a $K(\pi, 1)$ in the algebraic sense. Similarly (as in Donu Arapura's answer), as the orbifold fundamental group
of $\mathcal{A}\_g$ is ${\rm Sp}(2g ,\mathbb{Z})$, while the orbifold universal cover of $\mathcal{A}\_g$ is the Siegel upper-half space
(which is contractible), the stack $\mathcal{A}\_g$
 ($g > 1$) gives a probable example of a smooth Deligne–Mumford stack which is a $K(\pi, 1)$ in the analytic sense but not in the algebraic sense.


