Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes

TL;DR

This paper establishes a deep connection between symplectic reflection algebras, their spherical subalgebras, and the geometry of Hilbert schemes for cyclic groups, revealing new algebraic and geometric equivalences.

Contribution

It constructs a filtered algebra linking symplectic reflection algebras to the geometry of Hilbert schemes, providing categorical equivalences that bridge algebraic and geometric perspectives.

Findings

Categorical equivalence between $R$-modules and $U_{f k}$-modules.

Equivalence of graded categories with coherent sheaves on Hilbert schemes.

Identification of algebraic structures with geometric moduli spaces.

Abstract

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.