Let $k$ be a number field with a fixed embedding to $ \mathbb{C}$. If $V$ is an algebraic variety over $k$ and $V_\mathbb{C} = V \times_k \mathbb{C}$ has nice enough properties one can show that the profinite completion of the topological fundamental group $\pi_1(V(\mathbb{C}))$ is the etale fundamental group $\pi^{et}_1(V_\mathbb{C})$. I'm omitting basepoints for simplicity. My question is:
Is there an example of a smooth projective geometrically connected algebraic variety $V$ over $k$ such that $\pi_1(V(\mathbb{C})) \neq 0$ and $\pi^{et}_1(V_\mathbb{C}) = 0$?
I suspect the answer is yes based on the example of the unit circle, which has $\mathbb{Z}$ as topological fundamental group and has trivial profinite completion.