Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes

Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $\Gamma \subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}} = eH_{\mathbf{k}} e$ the spherical subalgebra of $H_{\mathbf{k}}$. We show that for suitable ${\mathbf{k}}$ there is a filtered $\Z^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\mathrm{qgr} \simeq U_{\bf k}$-mod, (2) there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_\Gamma \mathbb{C}^2)$. Here $\ttt{Coh}(Hilb_\Gamma \mathbb{C}^2)$ is the category of coherent sheaves on the $\Gamma$-Hilbert scheme. and for a graded algebra $\mathcal{R},$ we write $\mathcal{R}-\mathrm{qgr}$ for the quotient category of finitely generated graded $\mathcal{R}$-modules modulo torsion.