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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 \(Q\).
In this article we use \(M_\theta(Q,\alpha)\) to denote the GIT-quotient \(\mathrm{Rep}_\theta(Q,\alpha)/\!\!/_\theta G_\alpha\) 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 \(M_\theta(Q,\alpha)\).
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 \(\Sigma\) 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 \(W^{\oplus k}\).
For a point \(\xi\in M_\theta(Q,\alpha)\) 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 is described as the quiver with vertices \(w_1,\dots, w_s\) whose adjacency is given by \(\delta_{ij}-\langle \beta_i,\beta_j\rangle = \dim \mathrm{Hom}_Q(W_i,W_j)\) with dimension vector \(\alpha'=(k_1,\dots,k_s)\).
Theorem. There is an etale isomorphism between the neighborhood of \(\xi\) in the space \(\mathrm{Rep}(Q,\alpha)/\!\!/_\theta G_\alpha\) and some neighborhood of the origin (trivial representation with zero morphisms) in the space \(\mathrm{Rep}_\theta(Q',\alpha')\) where \(Q'=Q'(Q,\alpha)\) is the local quiver.
Consider our familiar example of \((\mathbb{C}^2)^{[n]}\), which is the quiver variety of a doubled framed Jordan quiver with dimension vector \((1,n)\).