Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero (see for instance <https://arxiv.org/pdf/math/0006107.pdf>).


Let $A(n)\to S^n(A)$ be a resolution of singularities. Then, up to finite etale cover, $A(n)$ is a product of hyperkaehler varieties, an abelian variety, and simply connected strict Calabi-Yau varieties. (This should follow from the Beauville-Bogomolov decomposition theorem. Or does this require an additional hypothesis on $A(n)$.)


I am wondering how the decomposition of $A(n)$ looks like as $n$ grows. Is it always a strict Calabi-Yau variety? Could it be that $A(n)$ is an abelian variety in fact? 


I am looking for examples and would appreciate any comments.



Is the symmetric product of an abelian variety a CY variety?
When $\dim A = 1$, $S^nA$ is a $\mathbb{P}^{n-1}$-bundle over $A$, so its Kodaira dimension is $-\infty$.


When $\dim A = 2$, the minimal resolution of $S^nA$ is given by the Hilbert scheme $A^{[n]}$, there is a natural map
$$
A^{[n]} \to A
$$
(summation of points), which is smooth with fiber $K\_{n-1}A$, so-called higher Kummer variety, which is hyperkahler. 


