Author: Eiko

Tags: quiver variety, Hilbert scheme, Nakajima quiver variety, McKay correspondence, preprojective algebra, tautological bundle

Hilbert Scheme of Points on ADE Singularity

ADE Singularities

An ADE singularity, or Kleinian, Du Val singularities, we mean a singular surface of the form ${\mathbb{C}}^2/\mathrm{\Gamma }$ for a finite subgroup $\mathrm{\Gamma }\subset {\mathrm{SL}}_2(\mathbb{C})$.

1. These surfaces are well known to be hypersurfaces in ${\mathbb{C}}^3$ with an isolated singularity at the origin, and they are classified by the finite subgroups of ${\mathrm{SL}}_2(\mathbb{C})$, which are the cyclic, dihedral, tetrahedral, octahedral, and icosahedral groups

2. Their unique minimal resolution $S\to {\mathbb{C}}^2/\mathrm{\Gamma }$ contracts a tree of rational curves in a configuration encoded by ADE Dynkin diagrams.

By the way, the theorems about Hilbert schemes of points on ADE singularities seems to have assumed or works with the reduced Hilbert scheme.

It seems at first that Hilbert scheme ${\mathrm{Hilb}}^n({\mathbb{C}}^2)$ have nothing to do with Quiver Varieties, since its quiver has three arrows and is not a doubled quiver. But it is a quiver variety with a special choice of stability parameter. (i.e. under certain $\theta$, one of the arrows on the framing node becomes zero).

This article describes how Hilbert schemes of points on ADE singularities can be described in terms of some Nakajima quiver varieties with special stability parameters.

The Preprojective Algebra for the McKay Graph

Let $\mathrm{\Gamma }\subset {\mathrm{SL}}_2(\mathbb{C})$ be a finite subgroup. The McKay correspondence is a dictionary between the representation theory of $\mathrm{\Gamma }$ and certain extended Dynkin diagrams, and the geometry of the minimal resolution of ${\mathbb{C}}^2/\mathrm{\Gamma }$.

1. The McKay graph of $\mathrm{\Gamma }$ is defined to be the graph whose vertex set is $\mathrm{Irr}(\mathrm{\Gamma })$, the set of irreducible representations (up to isomorphism). In this graph, the number of edges $a_{ji}$ from ${\rho }_i\to {\rho }_j$ is defined as $\dim {\mathrm{Hom}}_{\mathrm{\Gamma }}({\rho }_j,{\rho }_i\otimes V)$ where $V={\mathbb{C}}^2$ is the canonical representation induced from $\mathrm{SL}({\mathbb{C}}^2)$, i.e. it gets acted by $\mathrm{\Gamma }\subset \mathrm{SL}({\mathbb{C}}^2)$.

2. The famous McKay correspondence says that $$\{\text{McKay graphs of finite subgroups of}{\mathrm{SL}}_2(\mathbb{C})\}\iff \{\text{extended Dynkin diagrams}\}$$

3. In the affine Dynkin diagram,

   • the extended node corresponds to the framing node $\mathrm{\infty }$, or the trivial representation ${\rho }_0$.

   • The irreducible representations of $\mathrm{\Gamma }$ provide a system of simple roots,

   • the regular representation becomes the imaginary root $\delta$.

4. Let $Q^{\mathrm{\Gamma }}$ be the doubled quiver of the McKay quiver of $\mathrm{\Gamma }$ and let $k{Q}^{\mathrm{\Gamma }}$ be the path algebra. The preprojective algebra ${\mathrm{\Pi }}_{\mathrm{\Gamma }}$ is obtained as the quotient algebra $$k{Q}^{\mathrm{\Gamma }}/⟨\sum \epsilon (a)a\bar{a}⟩$$ where $\epsilon (a)$ marks the sign of the incoming arrow. If the incoming arrow is obtained by doubling, it gives $-1$, otherwise it is $1$.

5. Define $$R_k={\mathrm{Hom}}_{\mathrm{\Gamma }}({\rho }_k,k[x,y])={\mathrm{Hom}}_{\mathrm{\Gamma }}({\rho }_0,k[x,y]{\otimes }_k{\rho }_k^{\ast })\cong (k[x,y]{\otimes }_k{\rho }_k^{\ast }{)}^{\mathrm{\Gamma }}.$$ for example $R_0=k[x,y{]}^{\mathrm{\Gamma }}$. This ring determines the number of representations ${\rho }_k$ in $k[x,y]$, as a result we should have an isomorphism $$k[x,y]{\cong }_{{}_{R_0}\mathbf{Mod}}\underset{0\le k\le r}{⨁}{R}_k{\otimes }_k{\rho }_k.$$

6. There is a $k$-algebra isomorphism $${\mathrm{\Pi }}_{\mathrm{\Gamma }}\cong {\mathrm{End}}_{R_0}\left(\underset{0\le k\le r}{⨁}{R}_k\right)$$ which provides a geometric interpretation of ${\mathrm{\Pi }}_{\mathrm{\Gamma }}$.

Nakajima Quiver Varieties for McKay Graph

Consider the McKay graph of $\mathrm{\Gamma }$, and fix dimension vectors $v,w\in \mathrm{Rep}(\mathrm{\Gamma })$, given by $v=n\delta$ and $w={\rho }_0$. This will create a quiver by adding framing vertex $\mathrm{\infty }$ and arrow $\mathrm{\infty }\to {\rho }_0$. The Nakajima quiver variety ${\mathfrak{M}}_{\theta }(n\delta ,{\rho }_0)$ is the moduli space of $\theta$-stable representations of the doubled quiver, where $\theta \in \mathrm{\Theta }=\{\theta \in \mathrm{Hom}(\mathbb{Z}\oplus \mathrm{Rep}(\mathrm{\Gamma }),\mathbb{Q}):\theta v=0\}$ is a stability parameter.

The over aboundant Hilbert scheme $n\mathrm{\Gamma }\text{-}\mathrm{Hilb}({\mathbb{A}}^2)$ is isomorphic to a quiver variety ${\mathfrak{M}}_{\theta }(v,w)$ for some stability parameters with all positive coefficients $\theta ({\rho }_k)>0$ $$n\mathrm{\Gamma }\text{-}\mathrm{Hilb}({\mathbb{A}}^2)=\{I=\mathrm{\Gamma }\cdot I\subset k[x,y]:k[x,y]/I\cong k[\mathrm{\Gamma }{]}^{(n)}\}.$$

Tautological Bundles

The rational vector space $\mathrm{\Theta }$ of GIT stability parameters admits a wall-and-chamber decomposition (which general GIT theory also has, we should learn it at some point). The interiors of the top-dimensional cones are chambers and their codim $1$ faces are walls. $\theta$ is generic if it is not on any wall. Since $v=(1,n\delta )$ is indivisible, a result of King says that ${\mathfrak{M}}_{\theta }(v,w)$ is the fine moduli space of $\theta$-stable representations or $\mathrm{\Pi }$-modules of dimension $v$.