The Topology of Equivariant Hilbert Schemes

For $G$ a finite group acting linearly on ${\mathbb{A}}^2$, the equivariant Hilbert scheme ${\mathrm{Hilb}}^r[{\mathbb{A}}^2\mathrm{/}G]$ is a natural resolution of singularities of ${\mathrm{Sym}}^r({\mathbb{A}}^2\mathrm{/}G)$. In this paper we study the topology of ${\mathrm{Hilb}}^r[{\mathbb{A}}^2\mathrm{/}G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of ${\mathrm{Hilb}}^r[{\mathbb{A}}^2\mathrm{/}G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $G{L}_2$. This is done by using the Bialynicki-Birula decomposition to compute topological invariants in terms of the combinatorics of a certain set of partitions.