We summarize the different stability conditions in the context of quiver varieties and GIT.
\(\chi\)-stability
Let \(\chi: G\to \mathbb{C}^*\) 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. \(f(x)\neq 0\) 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 \(x,x'\) are called semi-equivalent if their orbit closures meet in \(X^{ss}\), i.e. \[\overline{Gx}\cap \overline{Gx'}\cap X^{ss}\neq \emptyset.\] 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 \(I\subset \bigoplus_{n\ge 0} k[X]^{\chi^n}\) 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 \(v = \overline{\dim} V\), the slope is defined as
\[\mathrm{slope}: \mathbb{R}^Q\times \mathrm{Rep}(Q)\to \mathbb{R}, \quad \mathrm{slope}_\theta V = \frac{\theta\cdot v}{(1_i)\cdot v} = \frac{\sum \theta_i v_i}{\sum v_i}.\]
Let \(0\neq V\in \mathrm{Rep}(Q)\) be a non-zero representation, then
\(V\) is called \(\theta\)-semistable if for any subrepresentation \(V'\subset V\), we have \(\mathrm{slope}_\theta V'\le \mathrm{slope}_\theta V\),
\[V'\subset_{\mathrm{Rep}(Q)} V \Rightarrow \mathrm{slope}_\theta V'\le \mathrm{slope}_\theta V.\]
\(V\) is called \(\theta\)-stable if the strict inequality holds for any proper subrepresentation \(V'\subsetneq V\),
\[V'\subsetneq_{\mathrm{Rep}(Q)} V \Rightarrow \mathrm{slope}_\theta V'<\mathrm{slope}_\theta V.\]
Example 1. Let \(\theta=0\), then any representation is \(\theta\)-semistable since all slopes are \(0\). And a representation is \(\theta\)-stable if and only if it has no proper subrepresentation, i.e. it is simple.
Remarks 1. The slope is only a function of the dimension vector of a representation, it does not care about the arrows. But different representations having the same dimension vector can have different subrepresentations, and the stability is determined by the dimensions of these subrepresentations. \[\mathrm{Stability}(M) = \mathrm{Stability}\{\mathrm{slope}(\dim M), \mathrm{slope}(\dim M'):M'\subset M\}\]
Remarks 2. Since adding a real constant to the slope is the same as adding \(c\cdot (1_i)\) to \(\theta\), which does not affect the concept of stability, often cases we will normalize our \(\theta\) so that it satisfy \(\theta\cdot \overline{\dim} V = 0\). Then the semi-stability condition can be simplified to saying
\[V'\subset_{\mathrm{Rep}(Q)} V \Rightarrow \theta\cdot \overline{\dim} V'\le 0\]
and a similar condition holds for stability with strict inequality.
Remarks 3. There are different conventions on the sign of the inequality, some authors define stability with \(\ge\) instead of \(\le\). In order to distingush these two definitions we will call our case as ’negative stability convention’ and the other one as ’positive stability convention’.
Corollary 1. Let \(S = \{ v \in \mathbb{N}^Q \mid \theta\cdot v \le 0\}\) and \(L_v = \{w\in \mathbb{N}^\mathbb{Q}\mid w \le v\}\), then \[\theta\cdot v = 0, L_v\subset S \Rightarrow \text{representations with } \dim = v \text{ are semistable.}\]
Example 2. Consider the example of the following quiver representation with dimension \(v=(1,1)\) and \(\theta=(1,-1)\), clearly \(\theta \cdot v = 0\),
the dimension vectors that satisfy \(\theta\cdot v\le 0\) are \((0, 0)\), \((0, 1)\), \((1, 1)\).
If we require \(ba=0\) (which can happen in real life for example in the \(0\) locus of moment map), then the only semistable representations are those with \(b=0\), \(a\in \mathbb{R}\). We see that the (negative) stability convention prefers arrows that flows to the lower points of \(\theta\).
Example 3. Consider the following example of \(v = (1, n)\) and \(\theta = (n, -1)\), where there are extra arrows coming in and out from the \(k^n\). Note that the dimensions of the extra points does not affect \(\theta\)-stability here. We see the representation \(V\) is semistable iff all its subrepresentations have dimension vector in the first row \((1, n)\) or \((0, *)\).
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 \(\phi: K_{fin}(A)\to \mathbb{R}\) gives a concept of slope, which only depends on the class of the module in \(K_{fin}(A)\).
\[\mathrm{slope}_\phi V = \frac{\phi([V])}{\dim V}.\]
And the concept of \(\phi\)-stability is defined similarly as before, for any subrepresentation \(V'\subset V\), \(\mathrm{slope}_\phi V'\le \mathrm{slope}_\phi V\).
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 \(\theta\in \mathbb{R}^I\).
Theorem 1. Finite dimensional \(\phi\)-semistable modules form an abelian category, and \(\phi\)-stable modules are exactly the simple objects in this category.
Proof.
The first part states that stability is closed under taking kernel \(K\) and cokernels \(C\) of map \(f:M\to N\) between semistable modules. This is because of semi-stability implies inequality for submodules, where we canidentify \(M/K\subset N\) so \(\mu(M/K)\le \mu(N)\) which is equivalent to
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