Author: Eiko

Tags: geometry, algebraic geometry, quiver variety, representation theory, lcoal quiver, etale local

Time: 2025-01-09 11:31:52 - 2025-01-09 11:31:52 (UTC)

References:

• Local Quivers And Stable Representations by Jan Adriaenssens and Lieven Le Bruyn

Question: Why does the moduli space of $\theta$-semistable $\alpha$-dimensional representations of $Q$ corresponds to isomorphism classes of direct sums of $\theta$-stable representations of $Q$.

• $M_{\theta }^{ss}(Q,\alpha )$ denotes the GIT-quotient ${\mathrm{Rep}}_{\theta -ss}(Q,\alpha )/{/}_{\theta }{G}_{\alpha }$ or $\theta$-semistable representations by the group action.

• So by ordinary GIT, there is a bijection between the set of semi-equivalence classes of $G_{\alpha }$-orbits in ${\mathrm{Rep}}_{\theta }(Q,\alpha )$ and (geometric) points in $M_{\theta }^{ss}(Q,\alpha )$.

  $$\{\text{S-equiv classes }{G}_{\alpha }\text{-orbits }{\mathrm{Rep}}_{\theta }(Q,\alpha )\}\iff \{\text{points }{M}_{\theta }^{ss}(Q,\alpha )\}$$

Example

Let $K_n$ be the Kronecker quiver with $n$ arrows. Consider the dimension vector $\alpha =(1,r)$ where $r\le n$, we can find that the set of $\theta$-semistable representations of $K_n$ corresponds to $n$ vectors in $k^r$ which generates the space $k^r$.

This means the corresponding moduli space, $\mathrm{Hom}(k,k^r{)}^n=\mathrm{Hom}(k^n,k^r)$ under the action of $\mathrm{GL}(1,r)$, is the Grassmannian $\mathrm{Gr}(r,n)$.

• $\alpha$ is minimal in $\mathrm{\Sigma }$ so the space $\mathrm{Rep}(K_n,\alpha )/{/}_{\theta }{G}_{\alpha }$ is smooth.

• For $k\alpha$ we can have direct sums of stable representations corresponding to $W^{\oplus k}$.

Local Quiver

For a point $\xi \in M_{\theta }^{ss}(Q,\alpha )$ with its representation

$$V_{\xi }=W_1^{\oplus k_1}\oplus \cdots \oplus W_s^{\oplus k_s}$$

where $W_i$ are $\theta$-stable representations of $Q$ of dimensions ${\beta }_i$.

Then the local quiver $Q_{\xi }$ and a dimension vector ${\alpha }_{\xi }^{\prime }$ is described as the quiver with

• vertices $w_1,\dots ,w_s$

• whose adjacency is given by $\mathrm{\#}\{w_i\to w_j\}={\delta }_{ij}-⟨{\beta }_i,{\beta }_j⟩=\dim {\mathrm{Ext}}_Q^1(W_i,W_j)$

• with dimension vector ${\alpha }_{\xi }^{\prime }=(k_1,\dots ,k_s)$.

Theorem. There is an etale isomorphism between the neighborhood of $\xi$ in the space ${\mathrm{Rep}}_{\theta }(Q,\alpha )/{/}_{\theta }{G}_{\alpha }$ and some neighborhood of the origin (trivial representation with zero morphisms) in the space $\mathrm{Rep}(Q_{\xi },{\alpha }_{\xi }^{\prime })/G_{{\alpha }_{\xi }^{\prime }}$ where $Q_{\xi }$ is the local quiver associated to $\xi$.

$$M_{\theta }^{ss}(Q,\alpha ){|}_{U_{\xi }}\cong \mathrm{Rep}(Q_{\xi },{\alpha }_{\xi }^{\prime })/G_{{\alpha }_{\xi }^{\prime }}{|}_{U_0}$$

Let’s See Some Other Examples

Consider our familiar example of $({\mathbb{C}}^2{)}^{[n]}$, which is the quiver variety of a doubled framed Jordan quiver with dimension vector $(1,n)$.