Let $A$ be an abelian surface. And consider the Hilbert scheme $A^{[n]}$ of $n$ points on $A$, which is a desingularisation of the $n$-th symmetric product $S^nA$. We have a composed map $$ \varphi:A^{[n]}\to S^nA\to A\,, $$ where the second map is the sum.

We now define $K^{[[n]]}:=\varphi^{-1}(0)$.

Question:Has anybody computed the Hodge numbers of $K^{[[n]]}$?

I know that Lothar Göttsche has worked on closely related things. But I was unable to find precisely what I am looking for.

EDIT: 1) any information about low-dimension examples is very welcome!

2) a formula exists (by Göttsche-Soergel, see comments below) but in order to get actual numbers one should probably use a computer.