# Daniel Kaplan, Quiver varieties and symplectic resolutions of singularities (Wednesday, July 20)

:::{.remark}
Three main sections:

- Spaces you (yes you!) care about can be realized as quiver varieties,
- These are well-understood with respect to symplectic resolutions,
- (joint work with Travis Schedler) Even more spaces are "locally" quiver varieties.

:::

:::{.example title="?"}
Let $C_2\actson \CC^2$ by negation, then $C^2\actson \CC[\CC^2] = \CC[x, y]$ by $f(x, y) \mapsto f(-x, -y)$.
The invariants are $R\da \CC[x, y]^{C_2} = \CC[x^2, y^2, xy] = \CC[y,v,w]/\gens{uv-w^2}$ which is a type $A_1$ singularity at the origin.
Take a blowup $\Bl_0(X) \to X \da \spec R$.
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:::{.example title="?"}
Let's do the same thing but differently: consider the nilpotent cone $\Nil(\liesl_2) = \ts{m = \matt a b c d \st \tr(m) = \det(m) = 0} = \ts{\matt a b c d \st a^2-bc=0}$, which is the same equation as before.
Take pairs $(m, \ell\in \ker m)$, then the projection map $(m, \ell) \mapsto m$ is the cotangent bundle $\T\dual \PP^1\to \Nil(\liesl_2)$.
Now perhaps it's clear how to generalize, e.g. by taking other lie algebras.
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## Quivers

:::{.remark}
A quiver is the data $Q = (Q_0, Q_1, s,t: Q_1\to Q_0)$ where $Q_0$ is a vertex set and $Q_1$ is an arrow set, and everything is over a field $k=\CC$ in our case.
Fix $d\in \NN^{Q_0}$, a tuple $d = (d_i)_{i\in Q_0}$, then
\[
\Rep(Q, d) = \bigoplus _{q\in Q_1} \Hom_{\mods{\CC}}(\CC^{d_{s(q)} }, \CC^{d_{t(q)} })
.\]
There is an action $G_d \da \prod_{i\in Q_0} \GL_{d_i}(\CC)$ by conjugation, which we sometimes quotient by.
:::

:::{.remark}
The $G_d$ orbits are too big!
Let $A,B,C$ be matrices and consider the quiver $\CC^2 \mapsvia{A} \CC^2\selfmap_{B} \mapsvia{C} \CC$.
It's hard to find $G_d\dash$invariant functions, and in fact $\CC[\Rep(Q, d)]^{G_d} = \CC[\ts{ \tr(\alpha), \alpha \text{ a cycle} } ]$.
Note that $B$ is a cycle in this case.
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:::{.remark}
There is a process of doubling a quiver: add inverse arrows, and duplicate every loop:

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![](figures/2022-07-20_11-34-06.png)

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:::{.remark}
Note
\[
\Rep(\bar Q, d) \to \Rep(Q, d) \oplus \Rep(Q^*, d) = \Rep(Q, d) \oplus \Rep(Q, d)\dual \to \T\dual \Rep(Q, d)
,\]
which yields a cotangent bundle and puts us in the realm of symplectic geometry.
The moment map is
\[
\mu_d: \Rep(\bar Q, d) \to \lieg_d\dual &\from \lieg_d \\
\tr(X(\wait)) &\mapsfrom X
.\]
Write $\nu = \sum_{q\in Q_1} [q, q^*]$ where the commutator is $aa^* - a^*a$, then $p\in \mu_d\inv(0)$ if $p$ is a module for the path algebra $\CC\bar{\QQ}/\gens{\nu} = \Pi(Q)$.
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:::{.remark}
Let $\theta \in \ZZ^{Q_0} = \Hom_{\Alg\Grp}(\prod \GL_{d_i} (\CC), \CCstar )$ where $\theta \cdot \dim(\rho) = 0$.
Say $\rho' \subseteq \rho$ is destabilizing if $\dim(\rho') \cdot \theta> 0$, and $\rho$ is $\theta\dash$semistable if it does not contain any destabilizing $\rho'$.
One can then define
\[
\mcm(Q, d, \theta) = \Proj \bigoplus \CC[ \mu_d\inv(0) ]^{k\theta ?}
.\]
This is an important object: the extra $\theta$ parameter can be sent to zero, and generically will yield a smooth space.
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## Local to global obstructions

:::{.theorem title="Bellamy-Schedler"}
There is a decomposition into symmetric spaces:
\[
\mcm(Q, d, \theta) = \prod_{i = 1}^k \Sym^{n_i} \mcm(Q, d_i, \theta)
\]
where $d = \sum n_i d_i$ and $d_i \cdot \theta = 0$ for all $i$.
Moreover, this has a symplectic resolution iff each factor does iff $\gcd(d_i) = 1$ or $(\gcd(d_i), p\qty{d\over \gcd(d_i)} ) = (2, 2)$.
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:::{.theorem title="K-Schedler"}
Let $\mcm(A, d, \theta)$ be the moduli space of $\theta\dash$semisimple $A\dash$modules of dimension $d$, where e.g. $A = k\bar{Q} / \gens{R}$.
There is an isomorphism of formal Poisson varieties after completing:
\[
\hat\mcm(\Lambda(Q), d, \theta)_m \iso \hat\mcm(Q', d', \theta)_0
,\]
where e.g. $Q = Q'$ in type $\tilde A_n$.
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:::{.remark}
Local to global obstructions: since $X$ is Poisson, it has a stratification into symplectic leaves, $X = \disjoint_i S_i$.
There is an action of $\pi_1(S_i, x_i)$ on the set of local resolutions $\ts{ \tilde U_{s_i} \mapsvia{\pi} U_{s_i} }/W$ where $W$ is a Nakajima-Weyl group.
This action can be nontrivial, and there is a compatibility with leaf closures: if $\bar{S_i} \contains S_j$ then $\ro{ \pi_{\bar S_i} }{S_j} = \pi_{S_j}$.
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:::{.example title="?"}
In the situation $C_2\actson (\CCstar)^2$ there are four fixed points $(\pm 1, \pm 1)$ yielding simple singularities with local resolutions.
The theorem implies there is a global symplectic resolution.
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