Q: Fundamental group & Albanese Let $X$ be a variety (smooth, projective -- if convenient) over a field $k$. To $X$, one can associate


*

*the étale fundamental group $\pi_1^{\text{et}}(X)$

*the Albanese variety $\text{Alb}(X)$


What can be said about the relation between these two objects?
In particular, if the étale fundamental group is trivial, must $\text{Alb}(X)$ be a point?

A: Here's a comment about smooth projective varieties over $\mathbf C$. You might find it useful.
In this case, Hodge theory says that $\text{Alb}(X)$ is nontrivial if and only if $H_1(X)$, which is the abelianisation of $\pi_1^{top}(X)$, is infinite. But in that case, it is easy to construct nontrivial finite quotients of $\pi_1^{top}(X)$. Since $\pi_1^{\text{et}}(X)$ is the profinite completion of $\pi_1^{top}(X)$, it must then be nontrivial too. 
