# Is the symmetric product of an abelian variety a CY variety?

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.

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.

You need for the dimension of $$A$$ to be even in order for the canonical sheaf on the quotient to be trivial (so that the resolution has a chance to be $$K$$-trivial). For $$\dim A =2$$, the story is as Sasha described. For $$\dim A$$ even and at least 4, there will not exist a crepant resolution. You will encounter singularities which locally look like $$\mathbb{C}^{2d}/\{\pm 1\}$$ and for $$d>1$$ these do not admit crepant resolutions.