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 $\pi_1(X^\mathrm{an})=0$?
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!

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!