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Prym varieties are abelian varieties that are associated to a double cover of algebraic curves.

Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way?

I think this is a natural question, but some googling didn't get me very far.

Of course, there is the Albanese variety. In the case of curves, if $Y\to X$ is a double cover, we have that $Jac(Y) = Jac(X) \times P(Y\to X)$, where $P(Y\to X)$ is the Prym variety of $Y\to X$. We could use this as a "definition" of the Prym variety I think, i.e., $P(Y\to X) := Alb(Y)/Alb(X)$, but I don't know if this is useful. I have a feeling the Albanese variety is almost always zero.

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