Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes

Abstract

Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $Γ\subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}}=e{H}_{\mathbf{k}}e$ the spherical subalgebra of $H_{\mathbf{k}}$. We show that for suitable $\mathbf{k}$ there is a filtered ${\mathbb{Z}}^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\mathrm{qgr}\simeq U_{\mathbf{k}}$-mod, (2) there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$. Here $ \ttt{Coh}(Hilb_Γ\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.

Cited by in corpus (1)

• Quantization of minimal resolutions of Kleinian singularities