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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.

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A surface of general type could be simply connected, contrary to the situation for ''curves of general type''. For instance, consider the Fermat surface $x_0^5+x_1^5+x_2^5+x_3^5 =0$ in $\mathbf P^3$. It is a smooth complete intersection of degree $5$, thus it is simply connected. It is of general type because its Kodaira dimension is easily seen to equal two (once you write down the canonical divisor).

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