Author: Eiko

Tags: Stability, GIT, Quiver Variety, Slope Stability, semistable

We summarize the different stability conditions in the context of quiver varieties and GIT.

$\chi$-stability

• Let $\chi :G\to {\mathbb{C}}^{\ast }$ be a character, a point $x\in X$ is called $\chi$-semistable if it is seen by a semi-invariant $f\in k[X{]}^{{\chi }^n}$, i.e. $f(x)\ne 0$ or $x\in X_f$.

• A point $x\in X$ is called $\chi$-stable if it is semistable, and the stablizer $G_x$ is finite. This also means orbit of stable point will be of maximal dimension, and such orbit is closed in $X^{ss}$.

• Two $\chi$-semistable points $x,x^{\prime }$ are called semi-equivalent if their orbit closures meet in $X^{ss}$, i.e. $$\overline{Gx}\cap \overline{G{x}^{\prime }}\cap X^{ss}\ne \mathrm{\varnothing }.$$ Semi-equivalent points define the same point in the quotient $X/{/}_{\chi }G=X^{ss}/G$.

Semistable locus and GIT

This gives a morphism from an open subset $X^{ss}$ of $X$ to the GIT quotient

$$X^{ss}=\bigcup _{n\ge 1,f\in k[X{]}^{{\chi }^n}}{X}_f\to X/{/}_{\chi }G$$

sending each point into an semi-equivalent class. The image of a $G$-orbit $Gx$ is a point corresponding to the maximal homogeneous ideal $I\subset \underset{n\ge 0}{⨁}k[X{]}^{{\chi }^n}$ of functions vanishing on $Gx$.

If $X$ is affine and $G$ acts freely on $X^{ss}$, then $|G_x|=1$ is finite so any semistable point is stable and $X/{/}_{\chi }G$ is a smooth variety, $X^{ss}\to X/{/}_{\chi }G$ is a principal $G$-bundle.

Slope stability

There is a purely representation-theoretic definition of stability, which in the quiver context is equivalent to the above GIT-theoretic $\chi$-stability. Let $Q$ be a quiver $\theta \in {\mathbb{R}}^Q$ which is an analogue of the character $\chi$ in GIT.

Consider a finite dimensional $Q$-representation $V$ with dimension vector $v=\bar{\dim }V$, the slope is defined as

$$\mathrm{slope}:{\mathbb{R}}^Q\times \mathrm{Rep}(Q)\to \mathbb{R},{\mathrm{slope}}_{\theta }V=\frac{\theta \cdot v}{(1_i)\cdot v}=\frac{\sum {\theta }_i{v}_i}{\sum v_i}.$$

Let $0\ne V\in \mathrm{Rep}(Q)$ be a non-zero representation, then

• $V$ is called $\theta$-semistable if for any subrepresentation $V^{\prime }\subset V$, we have ${\mathrm{slope}}_{\theta }{V}^{\prime }\le {\mathrm{slope}}_{\theta }V$,

  $$V^{\prime }{\subset }_{\mathrm{Rep}(Q)}V\Rightarrow {\mathrm{slope}}_{\theta }{V}^{\prime }\le {\mathrm{slope}}_{\theta }V.$$

• $V$ is called $\theta$-stable if the strict inequality holds for any proper subrepresentation $V^{\prime }⊊V$,

  $$V^{\prime }{⊊}_{\mathrm{Rep}(Q)}V\Rightarrow {\mathrm{slope}}_{\theta }{V}^{\prime }<{\mathrm{slope}}_{\theta }V.$$

General Concept of Stability

Consider arbitrary $\mathbb{C}$-algebra $A$, let $K_{fin}(A)$ be the Grothendieck group of finite dimensional $A$-modules, which is free abelian on the basis of finite dimensional simple modules. Any morphism $\varphi :K_{fin}(A)\to \mathbb{R}$ gives a concept of slope, which only depends on the class of the module in $K_{fin}(A)$.

$${\mathrm{slope}}_{\varphi }V=\frac{\varphi ([V])}{\dim V}.$$

And the concept of $\varphi$-stability is defined similarly as before, for any subrepresentation $V^{\prime }\subset V$, ${\mathrm{slope}}_{\varphi }{V}^{\prime }\le {\mathrm{slope}}_{\varphi }V$.

This easily goes back to our definition as we have a canonical morphism ${\dim }_I:K_{fin}(A)\to {\mathbb{Z}}^I$, any morphism of ${\mathbb{Z}}^I$ into $\mathbb{R}$ is given by a vector $\theta \in {\mathbb{R}}^I$.