Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers.
If $\dim X =1$, this holds if and only if the genus of $X$ is positive.
Do we have a similar criterion in dimension two?
Clearly, if $\dim X=2$ this holds if $X$ is an abelian variety,but not if it is a K3 surface. Also, if $X$ is a Jacobian elliptic fibration, we have infinitely many finite degree topological covers.
What if $X$ is of general type? That is, does a surface of general type have infinitely many etale covers?
Note that the topological fundamental group of $X$ is finitely generated, thus the existence of infinitely many etale covers exists that the degree of such an etale cover is not bounded.