For each smooth algebraic variety $X$ over some field $F$, does there exist an abelian variety $Y$ defined over $F$ (or some extension of $F$) with an unramified (or étale) morphism $Y \rightarrow X$?
If not, under what conditions does an abelian cover exist? If such a cover by an algebraic variety $Y$ is given, is there some way to check whether it is abelian?