Symplectic resolutions of quiver varieties

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

, [48], have become ubiquitous throughout representation theory.For instance, they play a key role in the categorification of representations of Kac-Moody Lie algebras and the corresponding theory of canonical bases.They provide étale-local models of singularities appearing in important moduli spaces, together with, in most cases, a canonical symplectic resolution given by varying the stability parameter.They give global constructions of certain moduli spaces, such as resolutions of du Val singularities [46], Hilbert schemes of points on them [36], and Uhlenbeck and Gieseker instanton moduli spaces [35,47,49].
Surprisingly, there seems to be no explicit criterion in the literature for when a quiver variety admits a symplectic resolution; often, in applications, suitable sufficient conditions for their existence are provided, but they do not appear always to be necessary.The main motivation of this article is to give such an explicit criterion.Following arguments of Kaledin, Lehn and Sorger (who consider the related case of moduli spaces of semistable sheaves on a K3 or abelian surface), our classification applies Drezet's criteria to show that certain GIT quotients are locally factorial.To do so we undertake a careful study of the local and global algebraic symplectic geometry of quiver varieties.
Our classification begins by generalizing Crawley-Boevey's decomposition theorem [17] of affine quiver varieties into products of such varieties, which we will call indecomposable, to the non-affine case; i.e., to quiver varieties with nonzero stability condition (Theorem 1.4).Along the way, we also generalize Le Bruyn's theorem, [38,Theorem 3.2], which computes the smooth locus of these varieties, again from the affine to nonaffine setting (Theorem 1.15).Then, our main result, Theorem 1.5, states that those quiver varieties admitting resolutions are exactly those whose indecomposable factors, as above, are one of the following types of varieties: (c) Varieties whose dimension vector are twice a root whose Cartan pairing with itself is −2 (i.e., the variety has dimension ten).
Here, a dimension vector α ∈ N I is called indivisible if gcd(α i ) = 1 for i ∈ I.The last type (c) is perhaps surprising: it is closely related to O'Grady's examples [55,56,31,39].In this case, one cannot fully resolve or smoothly deform via a quiver variety, but after maximally smoothing in this way, the remaining singularities are étale-equivalent to the product of V = C 4 with the locus of square-zero matrices in sp(V ) (as considered in preceding articles).Via the partial Springer resolution [11], the latter is resolved by the cotangent bundle of the Lagrangian Grassmannian of V .As explained in [31,Remark 4.6], [39], this resolution can also be obtained by blowing up the reduced singular locus (once), which makes sense globally on the quiver variety.
In the case of type (a), one can resolve or deform by varying the quiver (GIT) parameters.In fact, (for λ = 0) it is shown in [5] that all symplectic resolutions can be realised in this way.On the other hand, for quiver varieties of type (b), one cannot resolve in this way, but the variety is well-known to be isomorphic to another quiver variety (whose quiver is obtained by adding an additional vertex, usually called a framing, and arrows from it to the other vertices), which does admit a resolution via varying the parameters.Moreover, in this case, if the stability parameter is chosen to lie in the appropriate chamber, then the resulting resolution is a punctual Hilbert scheme of the minimal resolution of the original du Val singularity; see [36].The other chambers give in general different resolutions: in fact, thanks to [4], they again produce all symplectic resolutions of symmetric powers of du Val singularities.
It does not seem to be known whether M λ (α, θ), equipped with its natural scheme structure, is reduced (though we expect it is the case).Therefore, following Crawley-Boevey [18], we will consider throughout the paper all quiver varieties as reduced schemes.
Remark 1.1.The construction in [46,48] is apparently more general, depending on an additional dimension vector, called the framing.However, as observed by Crawley-Boevey [16], every framed variety can be identified with an unframed one.In more detail, for the variety as in [46,48] with framing β ∈ N Q 0 , it is observed in [16, Section 1] that the resulting variety can alternatively be constructed by replacing Q by the new quiver (Q 0 ∪ {∞}, Q 1 ), where Q 1 consists of Q 1 together with, for every i ∈ Q 0 , β i new arrows from ∞ to i; then Nakajima's β-framed variety is the same as M (λ,0) ((α, 1), (θ, −α • θ)).Thus, for the purposes of the questions addressed in this article, it is sufficient to consider the unframed varieties.
Let R + λ,θ denote those positive roots of Q that pair to zero with both λ and θ.If α / ∈ NR + λ,θ then M λ (α, θ) = ∅, therefore we assume α ∈ NR + λ,θ .As defined by Beauville [2], a normal variety X is said to be a symplectic singularity if there exists an (algebraic) symplectic 2-form ω on the smooth locus of X such that π * ω extends to a regular 2-form on the whole of Y , for any resolution of singularities π : Y → X.We say that π is a symplectic resolution if π * ω extends to a nondegenerate 2-form on Y .Note that a symplectic resolution does not always exist, and when it does exist, it is not always unique.
This theorem is important because symplectic singularities have become important in representation theory: on the one hand they include many of the most important examples (aside from quiver varieties, they include linear quotient singularities, nilpotent cones, orbit closures, Slodowy slices, hypertoric varieties, and so on), and on the other hand they exhibit important properties, at least in the conical case, such as the existence of a nice universal family of deformations [33,52,51] and of quantizations [8,12,40].
From both the representation theoretic and the geometric point of view, it is important to know when the variety M λ (α, θ) admits a symplectic resolution.In this article, we address this question, giving a complete answer.The first step is to reduce to the case where α is a root for which there exists a θ-stable point in µ −1 (λ).This is done via the canonical decomposition of α, as described by Crawley-Boevey; it is analogous to Kac's canonical decomposition.In this article, the term canonical decomposition will only refer to the former, which we now recall.Associated to λ, θ is a combinatorially defined set Σ λ,θ ⊂ R + λ,θ ; see Section 2 below.Then α admits a canonical decomposition with σ (i) ∈ Σ λ,θ pairwise distinct, such that any other decomposition of α into a sum of roots belonging to Σ λ,θ is a refinement of the decomposition (2).Closed points in µ −1 (λ) correspond to representations of the so-called deformed preprojective algebra Π λ (Q); see section 2.1 for details.
Then points of M λ (α, θ) are in bijection with isomorphism classes of θ-polystable representations of Π λ (Q) (equivalently, representations of the doubled quiver of moment λ which decompose as direct sums of θ-stable representations) of dimension α.Generalizing [16, Theorem 1.2], Proposition 3.18 implies Theorem 1.3.There exists a θ-stable representation of the deformed preprojective algebra Crawley-Boevey's Decomposition Theorem [17], which we will show holds in somewhat greater generality, then implies that the canonical decomposition gives a decomposition of the quiver variety as a product of varieties for each of the summands (the first statement of the next theorem).We show that the question of existence of symplectic resolutions of M λ (α, θ) can be reduced to the analogous question for each factor.
Here S n X denotes the nth symmetric product of X.
To finish the classification, it suffices to describe the case α ∈ Σ λ,θ .Write gcd(α) for the greatest common divisor of the integers {α i } i∈Q 0 ; it is divisible if gcd(α) > 1, and otherwise indivisible.
Our main theorem is then: The latter case in the theorem will be referred to as "the (2, 2) case".
If α ∈ Σ λ,θ is indivisible and anisotropic, then a projective symplectic resolution of M λ (α, θ) is given by moving θ to a generic stability parameter.However, this fails in the (2, 2) case.It seems unlikely that M λ (α, θ) can be resolved by another quiver variety in this case.Instead, we show that the 10-dimensional symplectic singularity M λ (α, θ) can be resolved by blowing up the singular locus.We will need the partial ordering ≥ on stability conditions, where θ ′ ≥ θ if every θ ′ -semistable representation is θ-semistable; see Section 2.4 below.
Remark 1.8.Most of the literature deals with projective rather than proper resolutions.However, there are interesting examples of proper symplectic resolutions that are not projective.For example, in [1] such examples are constructed admitting Hamiltonian torus actions of maximal dimension (this condition is called hypertoric there, which generalizes the usual definition of hypertoric variety).It seems to be an interesting question if, whenever a proper symplectic resolution exists, also a projective symplectic resolution exists.More generally, it seems reasonable to ask whether, if a proper symplectic resolution exists, then every proper Q-factorial terminalization is symplectic; if we restrict to projective resolutions and terminalizations, then the proof of [52,Theorem 5.5] shows that this holds at least when the singularity is conical with homogeneous generic symplectic form.Since we have shown that quiver varieties have symplectic singularities, thanks to [30,Theorem 2.3], they must necessarily be a finite union of symplectic leaves and the latter are algebraic.It has long been assumed that the leaves are precisely the strata M λ (α, θ) τ given by the representationtype stratification.Here τ is a decomposition of α in Σ λ,θ .Since this explicit identification of the symplectic leaves is crucial later in the article, we provide a complete proof that this is indeed the case.
This result follows from Proposition 3.15 and Corollary 3.25.The classification of symplectic leaves already appears in [43], however there seems to be a gap in the proof given there; see remark 3.26.Theorem 1.9 allows us to give a combinatorial classification, in Corollary 1.17 below, of those quiver varieties that are smooth.
An important tool in both the proof of Theorem 1.9 and later results on the factoriality of quiver varieties is an étale local description of the varieties.In the case of trivial stability parameter θ = 0, this étale local picture was described by Crawley-Boevey in [18], where it was used to prove that those quiver varieties are normal.In section 3.2 we show that this étale local description holds for all stability parameters.See Theorem 3.3 for the precise statement.
A relative version of Theorem 3.3, proving an étale local description of the morphism M(α, θ) → M(α, θ ′ ) is given in [4].This result allows the authors to completely classify (in the case of a framed affine Dynkin quiver) those walls in the space of stability parameters that are flops; resp.are divisorial contractions.This is a key step in showing (as mentioned above) that all symplectic resolutions of symmetric powers of du Val singularities are given by variation of GIT.1.3.Factoriality of quiver varieties.The real difficulty in the proof of Theorem 1.5 is in showing then M λ (α, θ) does not admit a projective symplectic resolution.Based upon a result of Drezet [20], who considered instead the moduli space of semistable sheaves on a rational surface, we show in Corollary 6.9 the following result.Recall that a variety is locally factorial if all of its local rings are unique factorization domains.
Observe that we did not require α to be divisible, although if were indivisible then we already noted that M λ (α, θ) is smooth for generic θ.On the other hand, in the divisible case, we will see that, for θ generic, the variety M λ (α, θ) has terminal singularities, using that, by [50], this is equivalent to having singularities in codimension at least four.Therefore, by a well-known fact, the above theorem implies that it cannot admit a proper symplectic resolution.
In fact, we prove in Corollary 6.9 a more precise statement than Theorem 1.10 which does not require that θ be generic.By the argument given in the proof of Theorem 6.There is one quiver variety in particular that captures the "unresolvable" singularities of M λ (α, θ).
This variety, which we denote X(g, n) with g, n ∈ N, has been studied in the works of Lehn, Kaledin and Sorger.Concretely, Viewed as a special case of Corollary 6.9, we see that X(g, n) does not admit a proper symplectic resolution if g, n ≥ 2 and (g, n) = (2, 2).
When g = 1, the Hilbert scheme of n points in the plane provides a symplectic resolution of Remark 1.12.It is interesting to note that [16, Theorem 1.1] implies that the moment map is flat when g > 1, in contrast to the case g = 1, which is easily seen not to be flat.
Remark 1.13.Generalizing the Geiseker moduli spaces that arise from framings of the Jordan quiver, it seems likely that the framed versions of X(g, n), which are smooth for generic stability parameters, should have interesting combinatorial and representation theoretic properties.
Remark 1.14.One does not need the full strength of Theorem 1.10 to prove that M λ (α, θ) does not admit a symplectic resolution: it suffices to show that a formal neighborhood of some point does not admit a symplectic resolution.This reduces the problem to the one-vertex case, i.e., to X(g, n).However, the techniques (following [32]) do not actually simplify in this case.Moreover, this would not be enough to imply Corollary 1.11.
1.4.Smooth versus canonically polystable points.In order to decide when the variety M λ (α, θ) is smooth, we describe the smooth locus in terms of θ-stable representations.Write the canonical , where a given β ∈ Σ λ,θ may appear multiple times.Recall that a representation is said to be θ-polystable if it is a direct sum of θ-stable representations.We say that a representation x is canonically θ-polystable if is a real root, i.e., p(β (i) ) = 0. Observe that the notion of canonical θ-polystability reduces to θ-stability precisely in the case that α ∈ Σ λ,θ .In general, the set of points of M λ (α, θ) which are the image of canonically θ-polystable representations is a dense open subset.When θ = 0, the result below is due to Le Bruyn [38,Theorem 3.2] (whose arguments we generalize).
Theorem 1.15.A point x ∈ M λ (α, θ) belongs to the smooth locus if and only if it is canonically θ-polystable.
Since, as recalled in Corollary 2.3 below, n i is always one if σ (i) is aniostropic, we could equivalently drop the assumption "is isotropic" at the end of the corollary.Corollary 1.17 is a crucial ingredient in the proof of the main result of [4], where a key step is the classification of stability parameters for which the corresponding quiver variety (associate to a framed affine Dynkin quiver) is smooth.1.5.Namikawa's Weyl group.When both λ and θ are zero, M 0 (α, 0) is an affine conic symplectic singularity.Associated to M 0 (α, 0) is Namikawa's Weyl group W [51], a finite reflection group.
In order to compute W , one needs to describe the codimension two symplectic leaves of M 0 (α, 0).
More generally, we consider the codimension two leaves in a general quiver variety M λ (α, θ).It is enough by Crawley-Boevey's canonical decomposition to consider the case α ∈ Σ λ,θ .We show that the codimension two symplectic leaves are parameterized by isotropic decompositions of α.
(c) The γ (i) are pairwise distinct real roots.
However, the isotropic β (i) need not be distinct.As an example, when Q is the quiver with two vertices 1, 2 and two arrows, one loop at 1 and one arrow from 1 to 2, then we can take α = (4, 2), 4) , and γ (1) = (0, 1).Then p(α) = 5 and α ∈ Σ 0,0 , and the quiver Q ′′ is of affine D 4 type with central vertex corresponding to γ (1) and external vertices corresponding to the β (i) .This example is also interesting since α ∈ Σ 0,0 is divisible, but not Σ-divisible (as 1 Thanks to Jasper van de Kreeke for pointing out this exceptional case.
Given an isotropic decomposition with affine Dynkin quiver Q ′′ , let Q ′′ f be the finite part, which is a Dynkin diagram.
Theorem 1.20.Let α ∈ Σ λ,θ be imaginary.Then the codimension two strata of M λ (α, θ) are in bijection with the isotropic decompositions of α.The singularity along each such stratum is étale-equivalent to the du Val singularity of the type A n , D n , E n corresponding to Q ′′ f .
As a consequence, for λ = 0 = θ, by [51, Theorem 1.1] the Namikawa Weyl group is a product over all isotropic decompositions B of a group W B .This group W B is either the Weyl group of the corresponding Dynkin diagram Q ′′ f , or else the centralizer therein of an automorphism of this diagram, corresponding to the monodromy around the fiber over a point of the stratum under a crepant resolution of the complement of the codimension > 2 strata.1.6.Character varieties.The methods we use seem to be applicable to many other situations.
Indeed, as we have noted previously, they were first developed by Kaledin-Lehn-Sorger in the context of semistable sheaves on a K3 or abelian surface.Any situation where the symplectic singularity is constructed as a Hamiltonian reduction with respect to a reductive group of type A is amenable to this sort of analysis.One such situation, which is of crucial importance throughout geometry, topology, and group theory, is that of character varieties of a Riemannian surface.
Let Σ be a compact Riemannian surface of genus g > 0 and π its fundamental group.The SL-character variety of Σ is the affine quotient Similarly, the GL-character variety is In the article [7] we show that X(g, n) and Y(g, n) are irreducible symplectic singularities.Moreover, we show: Theorem 1.21.[7] Assume that g > 1 and (g, n) = (2, 2).Then the varieties X(g, n) and Y(g, n) are locally factorial with terminal singularities and hence do not admit proper symplectic resolutions.

The same holds for any singular open subset.
Another very similar situation is that of moduli spaces of Higgs bundles on a genus g curve.The symplectic singularities of these moduli spaces are considered by A. Tirelli in [60].1.7.Applications.In joint work with A. Craw, the first author studies the symplectic resolutions of the symplectic quotient singularities C 2n /(S n ≀ Γ), where Γ ⊂ SL(2, C) is a finite group and is the associated wreath product.It is well-known that C 2n /(S n ≀ Γ) is a quiver variety and symplectic resolutions of the quotient singularity can be realised using variation of GIT for quiver varieties.Using the results from this article, it is shown in [4] that in fact all projective symplectic resolutions of the quotient singularity can be realised using quiver varieties.Moreover, one can say when stability parameters lying in different chambers give rise to the same symplectic resolutions.To prove these statements, it is crucial to have (a) the characterization of smooth quiver varieties given by Corollary 1.17 (b) the classification of symplectic leaves given in Corollary 3.25; and (c) the local normal form given by Theorem 3.3.
More generally, in joint work [5] with A. Craw, we use results of this paper to give a complete classification of projective symplectic resolutions of quiver varieties.
In [14], the authors prove that symmetric powers of minimal resolutions of du Val singularities are also quiver varieties, for non-generic stability parameters on affine Dynkin quivers.By Theorem 1.2, this implies that they are symplectic singularities.Our results in Section 3.4 are also employed in the proof of their main result.
1.8.Other related work.In joint work [58] with Tirelli, the second author has used similar methods to give a classification of those multiplicative quiver varieties and character varieties of open Riemann surfaces that admit symplectic resolutions.Though the methods are similar, the situation considered in [58] is considerably more complex that the additive case considered here (owing, for example, to the fact that it is unknown there when the varieties in question are nonempty).
Our classification explains which quiver varieties fall under the general framework of Springer theory as recently developed by McGerty-Nevins [44].Additionally, similar questions to ours are analyzed there in greater detail for the Dynkin cases.

Quiver varieties
In this section we fix notation.
2.1.Notation.Let N := Z ≥0 .We work over C throughout.All quivers considered will have a finite number of vertices and arrows.We allow Q to have loops at vertices.Let Q = (Q 0 , Q 1 ) be a quiver, where Q 0 denotes the set of vertices and Q 1 denotes the set of arrows.Given a ∈ Q 1 , let a s , a t ∈ Q 0 be the source and target, so a : a s → a t .For a dimension vector α Let Q be the doubled quiver of Q, where for each arrow a : i → j of Q we add a reverse arrow acts symplectically on Rep(Q, α) and the corresponding moment map is µ : where we have identified g(α) with its dual using the trace form.An element λ ∈ C Q 0 is identified with the tuple of scalar matrices ( , where p i is the length-zero path at the vertex i. See [16] for details. If M is a finite dimensional Π λ (Q)-module, then dim M will always denote the dimension vector of M , and not just its total dimension.

2.2.
Root systems.The coordinate vector at vertex i is denoted e i .The set N Q 0 of dimension vectors is partially ordered by α ≥ β if α i ≥ β i for all i and we say that α > β if α ≥ β with α = β.
The support of the vector α is the subquiver of Q obtained by deleting all vertices i ∈ Q 0 where α i = 0. Following [18,Section 8], α is called sincere if α i > 0 for all i i.e. the support of α equals Let (α, β) = α, β + β, α denote the corresponding Euler form and set p(α) = 1 − α, α .The fundamental region F(Q) is the set of 0 = α ∈ N Q 0 with connected support and with (α, e i ) ≤ 0 for all i.
If i is a loopfree vertex, so p(e i ) = 0, there is a reflection s i : There is also the dual reflection, r i : The real roots (respectively imaginary roots) are the elements of Z Q 0 which can be obtained from the coordinate vector at a loopfree vertex (respectively ± an element of the fundamental region) by applying some sequence of reflections at loopfree vertices.Let R + denote the set of positive roots.
Example 2. Again suppose that λ = 0 = θ, and now assume that α ∈ Σ λ,θ is isotropic i.e., p(α) = 1.Then as observed in the proof of [17, Proposition 1.2.(2)], α is supported on an affine Dynkin subquiver and there is the minimal imaginary root.We repeat the argument for the reader's convenience.First, α is indivisible, since α = kβ would imply p(α) < kp(β), and as β is also a root, this contradicts the assumption α ∈ Σ 0,0 .Next, α is in the fundamental region, since otherwise α = s i α + ke i for some i ∈ Q 0 and k ≥ 1, which implies 1 = p(α) = p(s i α) + kp(e i ), again contradicting the assumption that α ∈ Σ 0,0 .Now the support of α is connected.Letting Q ′ be its supporting quiver (i.e., the result of discarding all vertices not in the support and all incident arrows), we obtain a connected quiver for which α is in the kernel of the Cartan pairing.By [28, Lemma 1.9.(d)],Q ′ is affine (ADE) Dynkin and α is an imaginary root.Since it is also indivisible, it is the minimal imaginary root δ of Q ′ .
In several places below, we choose a parameter ν ∈ C Q 0 such that R + λ,θ = R + ν so that we can apply results of [17], where the case θ = 0 is considered.This is only for convenience, since the arguments of [17] can also be generalized directly to the context of the pair (θ, λ).Then [17, as a sum of element σ (i) ∈ Σ λ,θ such that any other decomposition of α as a sum of elements from Σ λ,θ is a refinement of this decomposition.
As is apparent from the results stated in the introduction, indivisible roots in Σ λ,θ play an important role in this paper.Occasionally it is useful to compare this with the condition of being Σ-indivisible, i.e., being indivisible in Σ λ,θ : Theorem 2.2.If α ∈ Σ λ,θ is imaginary, with α = mβ for some indivisible root β, then one of the following hold: ∈ Σ λ,θ and m > 1 can be chosen arbitrarily.
The following converse to (b) holds: if Proof.Once again, choose once again ν ∈ C Q 0 such that R + λ,θ = R + ν and let F ν be the "relative fundamental domain", as defined in [16, §7].Then Theorem 2.2 follows from [16,Theorem 8.1] provided that α ∈ F λ,θ .Namely, there it is described precisely the set F ν \ Σ ν , which has a very special form, called types (I), (II), and (III).Type (I) is the isotropic case: namely the multiples by positive integers m ≥ 2 of the imaginary root of an affine Dynkin subquiver.They are divisible.Types (II) and (III) are indivisible, and anisotropic.
If α is not in F ν then, by definition, there is a sequence of admissible reflections (whose product is w say) mapping α to w(α) ∈ F w(ν) (where w(ν) uses the action of dual reflections rather than reflections).Moreover, by [16, Lemma 5.2], w(α) also belongs to Σ w(ν) .Thus, it suffices to note that if trichotomy of the theorem holds for w(α), then it also holds for the root α.
The final statement follows from [17, Proposition 1.2 (3)].For the convenience of the reader we recall the proof, since it is closely related to the above.As we mentioned, the anisotropic cases (II) and (III) mentioned above are both indivisible.Thus every divisible anisotropic element of F ν is in Σ ν .So the above reductions imply the statement.
Corollary 2.3.In the canonical decomposition (2), Proof.This follows immediately from the final statement of Theorem 2.2, by the definition of the canonical decomposition.
Notice that Theorem 2.2 says that if β is an indivisible anisotropic root such that some multiple of β belongs to Σ λ,θ , then every proper multiple of β belongs to Σ λ,θ .However, in some cases β itself need not belong to Σ λ,θ .
2.4.Stability.Let θ ∈ Z Q 0 be a stability condition.Given a representation M of Q (e.g., a module over Π λ (Q)), let θ(M ) := θ • dim M .Note that a representation M of Π λ (Q) is the same as a point in the zero fiber µ −1 (λ).Recall that a Π λ (Q)-representation M (hence also a point in µ −1 (λ)) such that θ(M ) = 0, is said to be θ-stable, respectively θ-semistable, if θ(M ′ ) < 0, respectively We define a partial order on Z Q 0 by setting Proof.By definition, we have a G(α)-equivariant embedding µ −1 (λ) θ ′ ֒→ µ −1 (λ) θ .This induces a morphism between geometric quotients.We need to show that this morphism is projective.This is local on M λ (α, θ).Therefore we may choose n ≫ 0 and a nθ-semi-invariant f and consider the open subsets It is clear that this morphism is Poisson.
It follows from the proof of Lemma 2.4 that if θ ′′ ≥ θ ′ ≥ θ then the projective morphism We will frequently use the fact that for each point x ∈ M λ (α, θ), there is a unique closed G(α)orbit in the fibre over x of the quotient map ξ : µ −1 (λ) θ → M λ (α, θ).Recall that this closed orbit is denoted T (x).

Canonical Decompositions of the Quiver Variety
In this section we recall the canonical decomposition of quiver varieties described in [17], and show that it holds in slightly greater generality than stated there.
Therefore y decomposes into a direct sum y e 1 1 ⊕ • • • ⊕ y e k k of θ-stable representations, with multiplicity.Let β (i) = dim y i .The point x is said to have representation type τ = (e 1 , β (1) ; . . .; e k , β (k) ).Associated to this is the stabilizer group G τ = G(α) y , which is independent of the choice of y up to conjugation in G(α).Even though µ −1 (λ) θ is not generally affine, the fact that a nonzero morphism between θ-stable representations is an isomorphism implies: Lemma 3.2.The group G τ is reductive.
In fact, it is isomorphic to k i=1 GL e i (C).We denote the conjugacy class of a closed subgroup H of G(α) by (H).Given a reductive subgroup H of G(α), let M λ (α, θ) (H) denote the set of points x such that the stabilizer of any y ∈ T (x) belongs to (H).We order the conjugacy classes of reductive subgroups of G(α) by (H) ≤ (L) if and only if L is conjugate to a subgroup of H.

3.2.
Étale local structure.In this section, we recall the étale local structure of M λ (α, θ), as described in [18, Section 4].Since it is assumed in op.cit.that θ = 0, we provide some details to ensure the results are still applicable in this more general setting.Let x, y, y 1 , . . ., y k , β (1) , . . ., β (k) , and τ be as in Section 3.1.Let Q ′ be the quiver with k vertices whose double has 2p(β (i) ) loops at vertex i and −(β (i) , β (j) ) arrows from vertex i to j.The proof of Theorem 3.3 is given in section 3.3 below.By taking the completion M λ (α, θ) x of M λ (α, θ) at x and the completion M 0 (e, 0) 0 of M 0 (e, 0) at 0, the formal analogue of Theorem 3.3 is: There is an isomorphism of formal Poisson schemes M λ (α, θ) x ≃ M 0 (e, 0) 0 .Remark 3.5.An easy calculation shows that p(α) = p(e).It can also be deduced from the fact that dim M λ (α, θ) x = dim M 0 (e, 0) 0 .This fact will be useful later.Lemma 3.6.The group H is isomorphic to G(e) and θ| H is the trivial character.
Proof.The isomorphism H ≃ G(e) follows from the fact that Hom is the trivial stability condition.
As in [18], define ν : Theorem 3.3 follows from the following more precise result.
(b) the morphisms φ and ψ induce étale Poisson maps (c) There is an isomorphism of Poisson varieties, If we assume that y ∈ µ −1 (λ) then for k ∈ h and l ∈ L, Then δ is H-invariant.We let C δ denote the non-vanishing locus of δ.Then We recall that a G-morphism φ : X → Y is said to be excellent if Proof.This is a direct consequence of Luna's Fundamental Lemma [41], together with the fact that In particular, [27,Lemma 3.7] says that the corresponding étale morphism of Hamiltonian Proposition 3.9.There exist H-saturated open subsets Z of ν −1 (0) and U of W such that the morphism Proof.Let ω = ω| W .As in [18,Lemma 4.3], ω is a H-invariant symplectic form on W , with corresponding moment map μ.Write p : C → W for the projection map along C ⊥ and p : ν −1 (0) → W for the restriction of p to ν −1 (0).We claim that p * ω = ω| ν −1 (0) and p This follows, by definition, from p * ω = ω| C and p * μ = µ H | C .The latter two equalities can be checked by a direct computation.
By [18,Lemma 4.5], the map ν is smooth at 0 and ω| ν −1 (0) is non-degenerate at 0 with moment map µ H | ν −1 (0) .Moreover, loc.cit.shows that the kernel of d 0 ν is W , thus d 0 p : T 0 ν −1 (0) → T 0 W is the identity map.This implies that p : ν −1 (0) → W is étale at 0. Applying Luna's Fundamental Lemma once again, we deduce that there are H-saturated affine open subset Z ⊂ ν −1 (0) and Shrinking Z if necessary, we may assume that p * ω = ω| ν −1 (0) is non-degenerate on Z. Since We will show later that this isomorphism is Poisson.Let M (H) be the set of points Lemma 3.10.The set M (H) is a smooth locally closed subset of M θ with Proof.To show that M (H) is locally closed, it suffices to prove that, for each m ∈ M (H) , there exists some  In order to prove identity (6), we apply Luna's slice theorem [41].There exists an excellent map φ : G× H S → U , where S is a slice to the G-orbit at m.Then, Thus, Thus, as required.
Lemma 3.11.The variety µ −1 (λ) θ ∩ M (H) is smooth, with . By Lemma 3.10, we have ⊥ is isotropic.Therefore, we just need to show that the dimension of µ −1 (λ) θ ∩M (H) , as a reduced variety, is also equal to dim((M H ∩ (g • y) ⊥ ) ⊕ (g • y) H ). We have Set-theoretically, this equals G/H × ν −1 (0) H (which is smooth) and there is an étale map from this space to G/H × W H . Thus, we just need to show that Theorem 3.12.There exists a unique symplectic form ω H on M λ (α, θ) (H) such that where π : We claim that we have a commutative diagram of linear maps where the vertical map on the left is just projection.
Since φ is excellent, we have an identification commutes, with φ and φ/G being étale.Under the identification T 0 ν −1 (0) H = W H , the differential map dη : We deduce that π is a smooth morphism on Y .Hence → Ω 2 Y is an embedding, with image Ω 2 Y G .Thus, there is a unique (closed) 2-form ω H on (M ∩ V ) (H) , whose pull-back along π equals ω| Y .
Finally, to prove that ω H is symplectic it suffices to prove that the radical of ω| Y at m equals g/h.Clearly the latter is contained in the former.Since T m Y = W H ⊕ (g/h), it suffices to show that ω| W H is non-degenerate.Recall that ω = ω| W is non-degenerate.Then W H is a symplectic subspace since ω is H-invariant.
Next, we show that the symplectic forms ω H come from the Poisson structure on µ −1 (λ) θ //G.
Choose a function f defined on (V ∩ M) (H) and denote by the same symbol an arbitrary lift to V ∩ M. Since the form ω H is non-degenerate on (V ∩ M) (H) there exists a Hamiltonian vector field Since the quotient map π : Y → (V ∩ M) (H) is smooth, we can choose a lift of η.In fact, if we ask that the lift be G-invariant, it is unique, and so we will denote it by η too.If f is a lift of f to Finally, if we choose an arbitrary lift η ′ of η to V , then Finally, we complete the proof of Theorem 3.7.
Proof of Theorem 3.7.All claims, except for the final one, follow from Lemma 3.8 and Proposition 3.9.Thus, it suffices to note that the isomorphism Ψ of ( 5) is Poisson.Choose a generic point n in Then there exists some K ⊂ H such that n ∈ ((Z ∩ν −1 (0)∩µ −1 H (0))/ /H) (K) .Both Poisson structures on this open stratum are non-degenerate.Therefore, it suffices to show that the corresponding symplectic 2-forms agree via Ψ.Recall that the symplectic form on ((Z ∩ ν −1 (0) ∩ µ −1 H (0))/ /H) (K) is the unique form such that its pull-back to Similarly, the symplectic form on ((φ * µ) −1 (λ) θ / /G) (K) is the unique symplectic form whose pullback to Therefore, since the map Ψ is induced by the closed embedding j, it suffices to show that . But this follows from the fact that and φ • j is the map c → c + m, so that j * φ * ω = ω| ν −1 (0)∩C δ , since ω is invariant under translation.
The following result is an important consequence of Theorem 3.14.

Hyperkähler twisting. Let
be the canonical decomposition of α with respect to Σ λ .It is shown in [17] that Theorem 3.16.[17] There is an isomorphism of varieties We now adapt Crawley-Boevey's result to the case where θ = 0: be the canonical decomposition of α with respect to Σ λ,θ .Then, there is an isomorphism of Poisson varieties The proof of Theorem 3.17 is given at the end of section 3.5.In order to deduce Theorem 3.17 for all classes (H).In particular, the homeomorphism maps stable representations to stable (= simple) representations.
Proof.We follow the setup described in the proof of [15,Lemma 3].We have moment maps As shown in [34, Corollary 6.2], the Kempf-Ness Theorem says that the embedding Since the embedding is clearly continuous and the topology on the quotients µ −1 C (λ)∩ µ −1 R (iθ)/U (α) and M λ (α, θ) is the quotient topology (for the latter space, see [54, Corollary 1.6 and Remark 1.7]), the bijection ( 7) is continuous.
) (which exists by Proposition 3.1).Then Lemma 3.6 says that G x = G(e) and [34,Proposition 6.5] implies that U (α) x = U (e).Hence G(α) x = U (α) C x .Therefore the homeomorphism (7) restricts to a bijection Let the quaternions H = R ⊕ Ri ⊕ Rj ⊕ Rk act on Rep(Q, α) by extending the usual complex structure so that j Here the dagger denotes the Hermitian adjoint.In general, This action commutes with the action of U (α) and satisfies Let h = (i − j)/ √ 2. Then multiplication by h defines a homeomorphism Since multiplication by h commutes with the action of U (α), this homeomorphism descends to a homeomorphism which preserves the stratification by stabilizer type.
Thus, the map Ψ is the composition of three homeomorphisms, each of which preserves the stratification.
Remark 3.19.Our general assumption that λ ∈ R Q 0 if θ = 0 is required in the proof of Proposition 3.18 to ensure that multiplication by h lands in µ −1 R (0).Equation (8) implies that it would suffice to assume more generally that there exists z ∈ C such that |z| = 1 and zλ ∈ R Q 0 .It is natural to expect that Theorem 3.17 holds without the assumption λ ∈ R Q 0 .Remark 3.20.Using the notion of smooth structures on stratified symplectic spaces, as defined in [59], one can presumably strengthen Proposition 3.18 to the statement that there is a diffeomorphism of stratified symplectic spaces M λ (α, θ) Proof.We begin by showing that the variety M λ (α, θ) is connected.Proposition 3.18 implies that M λ (α, θ) is connected if and only if M ν (α, 0) is connected.The latter is known to be connected (and nonempty) by [17,Corollary 1.4].We can now prove Theorem 1.3.

thought of as a subgroup of G(α). There is a natural H(α)-equivariant inclusion
. This is an inclusion of symplectic vector spaces.Since the moment map for the action of H(α) on T * Rep(Q, α) is the composition of the moment map for G(α) followed by projection from the Lie algebra of G(α) to the Lie algebra of H(α), the above inclusion restricts to an inclusion i (µ −1 This map, which sends a tuple of representations (M i,j ) to the direct sum i,j M i,j clearly factors through i S n i M λ (σ (i) , θ) .It is this map that we call φ.
Passing to the analytic topology, Proposition 3.18 implies that we get a commutative diagram where both vertical arrows are homeomorphisms and the bottom horizontal arrow is an isomorphism by Theorem 3.16.Therefore, we conclude that φ is bijective.Since we are working over the complex numbers, and we have shown in Proposition 3.21 that M λ (α, θ) is normal, we conclude by Zariski's main theorem that φ is an isomorphism.
As a consequence, we can compute the dimension of M λ (α, θ), which in the case θ = 0 is [16,Corollary 1.4].We begin with the following basic lemma: Proof.Let U be the subset of M λ (α, θ) consisting of θ-stable representations.Since α is assumed to be in Σ λ,θ , Corollary 3.22 implies that U is a dense open subset of M λ (α, θ).Let V be the open subset of Rep(Q, α) consisting of θ-stable representations.Then U is the image of µ −1 (λ)∩ V under the quotient map and hence V is non-empty.The group PG(α) acts freely on V and µ is smooth when restricted to V .Thus, as required.For the second statement, observe that dim(V ∩ µ −1 (λ)) = dim U + dim PG(α) since PG(α) acts freely on V .
Then we immediately conclude Corollary 3.24.
) is defined similarly.Now the point is that under the embedding This implies that the two-form on the open leaf of i S n i M λ (σ (i) , θ) is the pull-back, under φ, of the symplectic two-form on the open leaf of M λ (α, θ).
Using Proposition 3.21, we can now show that each stratum M λ (α, θ) τ is connected.
Remark 3.26.The statement of Corollary 3.25 (at least in the case θ = 0) appears as Theorem 1.3 of [43].However, it is falsely claimed in Proposition 4.5 of that paper that the irreducibility of the stratum M λ (α, 0) τ follows from a result of G. Schwarz.
Proposition 3.28.The restricted moment map µ : In particular, if α ∈ Σ λ,θ , then the variety µ −1 (λ) θ is a complete intersection in the open subset Proof.By Lemma 3.23 and Theorem 3.27, all of the fibers µ −1 (λ) for λ ∈ B α,θ have the same dimension, α•α+2p(α)−1.Since this equals the difference of dimensions dim Rep(Q, α)−dim pg(α), it follows that the subset of the base where the fiber has this dimension is open, call it U .Then, since B α,θ is smooth, and µ −1 (U ) is open (hence smooth and therefore Cohen-Macaulay), it follows that the moment map is flat as stated, and therefore that every fiber is a complete intersection.

Smooth vs. stable points
As usual, choose a deformation parameter λ ∈ R Q 0 , a stability parameter θ ∈ Z Q 0 , and a dimension vector α ∈ NR + λ,θ .The main goal of this section is to prove Theorem 1.15, which says that x ∈ M λ (α, θ) is canonically θ-polystable if and only if it is in the smooth locus of M λ (α, θ).
4.1.Isotropic roots.In this section, we briefly consider quiver varieties associated to isotropic roots.The subgroup of GL(Z Q 0 ) generated by the reflection at loop free vertices is denoted W (Q). Lemma 4.1.Let α ∈ Σ λ,θ be an isotropic root.Then there exists w ∈ W (Q) such that δ = wα is in the fundamental domain, Q ′ = Supp δ is an affine Dynkin quiver, δ| Q ′ is the minimal imaginary root and M λ (α, θ) ≃ M wλ (δ, wθ).
Proof.As the name implies, the fundamental domain F(Q) is a fundamental domain for the action of the reflection group W (Q) of Q on the set of imaginary roots.Therefore there exists w such that wα ∈ F(Q).The fact that Q ′ is affine Dynkin and δ| Q ′ is the minimal imaginary root follows from [28, Lemma 1.9 (d)].
Thus, we show that δ ∈ Σ wλ,wθ and M λ (α, θ) ≃ M wλ (δ, wθ).The Lusztig-Maffei-Nakajima reflection isomorphisms of quiver varieties (see in particular [42,Theorem 26]) shows that if either λ i or θ i is non-zero (equivalently, as explained in example 1, if . Moreover, the fact that s i permutes the set As in the proof of [13,Proposition 16.10], the key thing to note is that δ is specified by the fact that it is the unique element of minimal height in the orbit W (Q) • α.
Thus, if α / ∈ F(Q), then there exists i ∈ Q 0 such that (α, e i ) > 0. This implies that e i / ∈ Σ λ,θ and ht(s i α) < ht(α).Since every element in the orbit W (Q) • α is a positive root (and hence has positive height) this cannot continue forever, and the result follows.
In particular, we note that Lemma 4.1 implies that if α ∈ Σ λ,θ is an isotropic root, then M λ (α, θ) is the partial resolution of a partial deformation of a Kleinian singularity.Moreover, the type of the Kleinian singularity is specified by the support of wα ∈ F. are valid in our setting.First, notice that, under the isomorphism of Theorem 3.17, the open subset of canonically θ-polystable points in M λ (α, θ) is the product of the canonically θ-polystable points in the spaces S n i M λ (σ (i) , θ).Therefore it suffices to show that the set of canonically θ- Val singularity.In particular, it is a 2-dimensional (quasi-projective) variety.This implies that the smooth locus of S n i M λ (σ (i) , θ) equals On the other hand, the set of canonically θ-polystable points in ) is the set of canonically θ-polystable points.Therefore, in this case it suffices to show that M λ (σ (i) , θ) sm equals U .Finally, in the case where σ (i) is an anisotropic root, we have Thus, we are reduced to considering the situation where α ∈ Σ λ,θ is an imaginary root.In this case, a point x is canonically θ-polystable if and only if it is θ-stable.As in the proof of Corollary 3.24, it is clear from the definition of M λ (α, θ) that the set of θ-stable points is contained in the smooth locus.Therefore it suffices to show that if x is not θ-stable then it is a singular point.As in section 3.2, let x be the image of a θ-polystable representation y = y e 1 1 ⊕ • • • ⊕ y e ℓ ℓ (with the y i θ-stable).Let β (i) = dim y i .Let Q ′ be the quiver with ℓ vertices whose double has 2p(β (i) ) loops at vertex i and −(β (i) , β (j) ) arrows between vertex i and j.The ℓ-tuple e = (e 1 , . . ., e ℓ ) defines a dimension vector for the quiver Q ′ .By Theorem 3.3, it suffices to show that 0 is contained in the singular locus of M 0 (e, 0).
Returning to the proof of Theorem 1.15, with Proposition 4.2 in hand, the argument given in the proof of [38,Theorem 3.2] goes through verbatim.This completes the proof of Theorem 1.15.

4.3.
The proof of Corollary 1.17.By Theorem 1.15, M λ (α, θ) is smooth if and only if every point is canonically θ-polystable.As in the reduction argument given at the start of the proof of Theorem 1.15, this means that n i must be 1 when σ (i) is an isotropic root.Moreover, it is clear that M λ (σ (i) , θ) consists only of θ-stable points if and only if σ (i) is minimal.
Since α is anisotropic, 2α is also an anisotropic root.Choose a generic stability parameter 4 is a partial projective resolution.In fact it is birational, since it is an isomorphism over the θ-stable locus (see the proof of Theorem 6.13 for more details).Thus, if Y → M λ (2α, θ ′ ) is a projective symplectic resolution, then so is the composite Y → M λ (2α, θ), i.e., it is enough to show that we can resolve M λ (2α, θ ′ ) symplectically.Fix X = M λ (2α, θ ′ ).
Then X = X 2 ⊔ X 1 ⊔ X 0 , where, by Theorem 1.15, X 0 is the smooth locus consisting of θ ′ -stable representations, X 1 parameterizes representations Let X denote the blowup of X the along the sheaf of ideals of the reduced singular locus X 1 ⊔ X 0 .
The corollary will follow from the following claim: X → X is a projective symplectic resolution.
Clearly, X → X is a projective birational morphism, therefore we just need to show that X is smooth and the symplectic 2-form on X 0 extends to a symplectic 2-form on X.We check this in a neighborhood of x ∈ X 2 and of y ∈ X 1 .First consider x ∈ X 2 .Replacing X by some affine open neighborhood of x, Theorem 3.3 says that there is an affine Z with where π and ρ are étale.Let X(2, 2) → X(2, 2), resp.Z → Z, denote the blowup along the reduced singular locus.Then As noted in [31,Remark 4.6], X(2, 2) → X(2, 2) is a projective symplectic resolution.Now Lemma 5.2 below and (12) imply that X → X is a projective symplectic resolution.
For y ∈ X 1 , Theorem 3.3 shows that there is an étale equivalence between a neighborhood of y and a neighborhood of the origin in a certain quiver variety, independent of the choice of y ∈ X 1 .In particular such a neighborhood is also étale equivalent to a neighborhood of a point of X 1 inside the neighborhood of x ∈ X 2 used above, so the result follows from the previous statement.(One can also compute explicitly: the quiver needed is the one with two vertices, one arrow in each direction between the two vertices, and also two loops at each vertex, so the quiver variety is isomorphic to which is an A 1 singularity and hence blowing up the reduced ideal sheaf of the singular locus gives a projective symplectic resolution).
It remains to prove the following standard lemma: Lemma 5.2.Let X be a symplectic singularity and π : X → X a proper morphism.Then π is a symplectic resolution if and only if it is so after a surjective étale base change i.e. being a symplectic resolution is an étale local property.
Notice that we are not making the (false) claim that X admits a symplectic resolution if and only if it does so étale locally.
Proof.Passing to the generic points of X and X, the fact that a surjective étale morphism is faithfully flat implies that π is birational if and only if it is so after base change.Therefore it suffices to check that the extension ω ′ of the pullback π * ω is non-degenerate.If b : Z → X is a surjective étale morphism, then so too is b : Z = X × X Z → X.The form ω ′ will be non-degenerate if and only if b * ω ′ is non-degenerate.

Factoriality of quiver varieties
In this section, which is the technical heart of the paper, we consider the case of a divisible anisotropic root.Fix α ∈ Σ λ,θ to be an indivisible anisotropic root, and let n ≥ 2 such that such that (p(α), n) = (2, 2).We prove the key result, Corollary 6.9, which says that if θ is generic then M λ (nα, θ) is a locally factorial variety.
For the third part, we use again (14), noticing the following points: the RHS of ( 14) is increasing in d; the RHS is increased if we replace (ℓ i , n i ) by (ℓ i − 1, n i ); ( For the final claim, observe that each stratum of M λ (nα, θ) consists of representations of the where the x i are pairwise non-isomorphic θ-stable representations of fixed dimension vectors α i ∈ Σ λ,θ .Under the assumptions given, each α i must be a multiple of α.
Therefore the representation type is of the form να for some weighted partition ν of n.
Proof.Since the stratum of representation type ρ = (n, α) is contained in the closure of all the other strata of type να, it suffices to show that there is no stratum β = (e 1 , β (1) ; . . .; e l , β (l) ) of any other type such that M λ (nα, θ) ρ ⊂ M λ (nα, θ) β .Assume otherwise.If G ρ ≃ GL n (C) is the stabilizer of some x ∈ M λ (nα, θ) ρ , then the Hilbert-Mumford criterion implies that there exists some y ∈ M λ (nα, θ) β whose stabilizer G β is contained in G ρ .Let V i be the nα i -dimensional vector space at the vertex i on which G(nα) acts.Then, for each g ∈ G(nα) and u ∈ C × , the u-eigenspace of g is the direct sum over the u-eigenspaces g| V i .In particular, it has a well-defined dimension vector.Now the elements g of G ρ all have the property that the dimension vector of the u-eigenspace of g is of the form rα for some r ∈ Z ≥0 .On the other hand, since β is not "of polystable representation M has dimension α • α − dim End(M ), which is maximized when M is stable.Thus if Z has codimension at least four (which is the case for us), then we can ignore the polystable part of ξ −1 (Z).
Next, if we consider a stratum Z of type (1, a; 1, b), note that every representation in this stratum is either polystable or an indecomposable extension of two non-isomorphic representations.The latter type is a brick, since there is a unique stable quotient and a unique stable subrepresentation and the two are nonisomorphic.Therefore applying the previous paragraph together with Lemma 6.1 (2) shows that we can ignore ξ −1 (Z) (the non-free locus has overall codimension at least four).This proves the final assertion.
Since µ is regular on the locus where PG(nα) acts freely, µ −1 (λ) θ free lies in the smooth locus of V .We conclude from the last assertion of the proposition that the singular locus of V has codimension at least 4 (i.e., property R 3 holds).Since V is a local complete intersection, and hence Cohen-Macaulay, it satisfies Serre's condition S 2 , so it is normal.
Finally, it follows from a theorem of Grothendieck, [32,Theorem 3.12], that since V is a complete intersection and satisfies R 3 , the local ring O V,x of any point x ∈ V is a unique factorization domain.
The result that allows us to descend local factoriality from V to the quotient U is the following theorem by Drezet.Since the version given in [20] concerns the moduli space of semistable sheaves on a smooth surface, we provide full details to ensure the arguments are applicable in our situation.
Let G be a connected reductive group.Lemma 6.6.Let V be a locally factorial normal affine G-variety and V s ⊂ V a dense open subset of V , whose complement has codimension at least two in V .Then every G-equivariant line bundle on V s extends to a G-equivariant line bundle on V .
Proof.The fact that V is normal and locally factorial implies that Hence if L 0 is a G-equivariant line bundle on V s , forgetting the equivariant structure, there is an extension L to V .To show that the extension L has a G-equivariant structure, one repeats the argument of [21, Lemme 5.2], which uses the fact that the codimension of V V s is at least two.(ii) For every line bundle M 0 on U s , there exists an open subset U 0 ⊂ U containing both x and U s such that M 0 extends to a line bundle M on U 0 .
(iii) For every G-equivariant line bundle L on V , the stabilizer of y acts trivially on the fiber L y .
Proof.Recall that O U,x is a unique factorization domain if and only if every height one prime is principal.Geometrically, this means that for every hypersurface Y of U , the sheaf of ideals I Y is free at x. (ii) implies (iii).Suppose that L is a G-equivariant line bundle on V .Since G acts freely on V s , the restriction L| Vs descends to the line bundle M 0 = (L| Vs )/G on U s .Let M be the extension of M 0 to U 0 .Then the G-equivariant line bundle ξ * M agrees with L on V s .This implies, as in the previous paragraph, that ξ * M = L on ξ −1 (U 0 ).In particular, since y ∈ ξ −1 (U 0 ), the stabilizer of y acts trivially on L y .
(iii) implies (ii).Let M 0 be a line bundle on U s .By Lemma 6.6, ξ * M 0 extends to a G-equivariant line bundle L on V .Recall by definition of lift that G • y is closed in V .Therefore Lemma 6.8 below says that there is an affine open neighborhood U ′ of x such that G y ′ acts trivially on L y ′ for all y Then, by descent [20, Theorem 1.1], there exists a line bundle M on U 0 such that ξ * M ≃ L. In particular, M extends M 0 .
Let Y be a variety admitting an algebraic action of a reductive group G. Assume that there exists a good quotient ξ : Y → X = Y / /G.The following result, which says that the descent locus of an equivariant line bundle is open, is presumably well-known, but we were unable to find it in the literature.
Proof.The proof of the lemma can be easily deduced from the proof of [21,Theorem 2.3].It is shown there that one can find a G-invariant section s ′ : O → L| O , which trivializes L| O .As explained in loc.cit., the fact that O is closed in Y implies that one can lift s ′ to a G-invariant section s ∈ Γ(ξ −1 (U ′ ), L), where U ′ is some affine open neighborhood of ξ(y).Let W be the (non-empty) open subset of ξ −1 (U ′ ) consisting of all points y ′ such that s(y ′ ) = 0 i.e. s trivializes L over W . Then it suffices to show that there is some affine neighborhood U of ξ(y) such that ξ −1 (U ) ⊂ W . Again, following [21, Theorem 2.3], the sets ξ −1 (U ′ ) W and O are G-stable closed subsets of ξ −1 (U ′ ).Therefore the fact that ξ is a good quotient implies that ξ(ξ −1 (U ′ ) W ) and ξ(O) = {ξ(y)} are closed, disjoint subsets of U ′ .Thus, there exists an affine neighborhood U of ξ(y) such that U ∩ ξ(ξ −1 (U ′ ) W ) = ∅, as required.Corollary 6.9.Assume that (p(α), n) = (2, 2).Then the variety U is locally factorial.
Proof.Let G = PG(α) and U s = M λ (nα, θ) s .Proposition 6.5 implies that V is normal and locally factorial.This implies that U is normal.Moreover, by Lemma 6.1, the codimension of the complement to U s in U has codimension at least two.Thus, assumptions (a) and (b) of Theorem 6.7 are satisfied.By Lemma 6.1 (2) and Corollary 6.4, the complement to µ −1 (λ) θ s in V has codimension at least 4. In particular, assumption (c) of Theorem 6.7 is also satisfied.
Next, recall from the proof of Lemma 6.2, the stratum of type ρ = (n, α) is contained in the closure of all other strata in U .If y is a lift in µ −1 (λ) θ of a point of M λ (nα, θ) ρ then y corresponds to a representation M ⊕n 0 , where M 0 ∈ M λ (α, θ) is a stable Π λ (Q)-module.Therefore PG(nα) y = P GL n has no non-trivial characters.In particular, PG(nα) y will act trivially on L y for any PG(nα)equivariant line bundle on V .Hence, we deduce from Theorem 6.7 that M λ (nα, θ) is factorial at every point of M λ (nα, θ) ρ .Now consider an arbitrary stratum M λ (nα, θ) να in U .If M λ (nα, θ) is factorial at one point of the stratum then it will be factorial at every point in the stratum (for a rigorous proof of this fact, repeat the argument given in the proof of [32,Theorem 5.3]).On the other hand, a theorem of Boissière, Gabber and Serman [9] says that the subset of factorial points of U is an open subset.
Since this open subset is a union of strata and contains the unique closed stratum, it must be the whole of U .Remark 6.10.Notice that if θ is generic then U = M λ (nα, θ).Hence Corollary 6.9 says that M λ (nα, θ) is a locally factorial variety.This is precisely the statement of Theorem 1.10.In the special case where Q has a single vertex, g loops and α = e 1 , the parameters λ = θ = 0 are generic.
6.3.The proof of Theorem 1.2.By Proposition 3.21, we know that M λ (α, θ) is irreducible and normal.Therefore, it suffices to show that it admits symplectic singularities.Since the isomorphism of Theorem 3.17 is Poisson, it suffices to show that the varieties S n i M λ (σ (i) , θ) admit symplectic singularities.If σ (i) is real there is nothing to check.0 to a resolution of the whole of M 0 (α, 0).For the converse statement, we restrict a symplectic resolution of M 0 (α, 0) to the formal neighborhood of zero.By Corollary 3.4, if one point in a stratum M λ (α, θ) τ ⊂ M λ (α, θ) is formally resolvable, then so too is every other point in the stratum.If τ = (e 1 , β (1) ; . . .; e k , β (k) ), then define the greatest common divisor gcd(τ ) of τ to be the greatest common divisor of the e i .If the greatest common divisor of τ is k, then each point in M λ (α, θ) τ corresponds to a representation of the form Y ⊕k for some θ-polystable representation Y .Let U fr ⊂ M λ (α, θ) be the union of all strata M λ (α, θ) τ such that gcd(τ ) = 1.Lemma 6.17.Let α ∈ Σ λ,θ .Then U fr is a dense open subset of M λ (α, θ).
Proof.The set U fr is dense because it contains the open stratum M λ (α, θ) (1,α) , consisting of stable representations.We will show that the complement to U fr is closed in M λ (α, θ).It suffices to show that if the greatest common divisor of ρ is greater than one and M λ (α, θ) τ ⊂ M λ (α, θ) ρ then gcd(τ ) > 1 too.The argument is similar to the proof of Lemma 6.2.Let x ∈ M λ (α, θ) ρ and G ρ ⊂ G(α) its stabilizer.By Proposition 3.15, there exists x ′ ∈ M λ (α, θ) τ such that its stabilizer G τ contains G ρ .Let gcd(ρ) = k, so that x corresponds to a representation Y ⊗ V for some θ-polystable representation Y , and k-dimensional vector space V .Notice that α = k dim Y .Then GL(V ) is a subgroup of G ρ , and hence of G τ too.An elementary argument shows that this implies that x ′ corresponds to a representation Y ′ ⊗ V for some θ-polystable representation Y ′ .Thus, gcd(τ ) > 1.
Remark 6.19.If U fr M λ (α, θ) then Corollary 1.11 implies that any open subset of U fr not contained in the smooth locus of M λ (α, θ) does not admit a symplectic resolution, i.e. the singular locus of U fr consists of points that cannot be resolved Zariski locally, but do admit a resolution in a formal neighborhood (in fact étale locally).

Namikawa's Weyl group
In the paper [51], Namikawa defined a finite group W associated to any conic affine symplectic singularity X such that the symplectic form on X has weight ℓ > 0 with respect to the torus action.The group W acts as a reflection group on H 2 (Y, R), where Y → X is any Q-factorial terminalization of X, whose existence is guaranteed by the minimal model program.The group W plays a key role in the birational geometry of X; see [53] and [3].
One computes W as follows: let L be a codimension 2 leaf of X and x ∈ L. Then the formal neighborhood of x in X is isomorphic to the formal neighborhood of 0 in C 2(n−1) × C 2 /Γ, where 2n = dim X and Γ ⊂ SL Thus, in order to compute W , it is essential to classify the codimension 2 leaves of X, and describe π 1 (L).This is the goal of this section.
Therefore the support of α on the quiver is connected.We can assume, up to replacing the quiver by the subquiver whose vertices are the support of α, and whose arrows are the ones with endpoints in the support, that α is sincere.Then, the quiver is connected.We may assume that α is imaginary, otherwise the statement is vacuous.
(c) The γ (i) are pairwise distinct real roots.
Let us now set α (i) := β (i) for 1 ≤ i ≤ s and α (i) = γ (i−s) for s + 1 ≤ i ≤ s + t.Let k i := n i for 1 ≤ i ≤ s and k i = m i−s for s + 1 ≤ i ≤ s + t; let k = (k 1 , . . ., k s+t ).By Theorem 3.3, at a point of this stratum, M λ (α, θ) is étale-equivalent to M Q ′ (0, k) 0 , where the notation means we use the quiver Q ′ instead of Q. Recall that Q ′ is the quiver with s + t vertices, 2p(α (i) ) loops at the ith vertex and −(α (i) , α (j) ) arrows between i and j.
Note that Q ′′ is obtained from Q ′ by discarding all loops at vertices.We will prove that, in the case that the stratum has codimension two, M Q ′′ (0, k) 0 étale-locally describes a transverse slice to the stratum.
Lemma 7.1.Suppose that τ is as in (16) and moreover n i = 1 for all i.Then at every point of the stratum there is an étale-local transverse slice isomorphic to a neighborhood of zero in M Q ′′ (0, k).
Proof.A neighborhood of a point of the stratum is étale-equivalent to a neighborhood of zero in M Q ′′ (0, k).Inside the latter, the stratum containing zero consists of the representations which are a direct sum of simple representations, one at each vertex.At the vertices 1, . . ., s, this representation has dimension one; at the other vertices there are no loops and hence the simple representations are the standard ones.The stratum has dimension 2 s i=1 p(β (i) ), where p(β (i) ) equals the number of loops at the vertex i.A transverse slice is thus given by the representations which assign zero to all of the loops, which obviously identifies with M Q ′′ (0, k).Lemma 7.2.Suppose that τ has codimension two.Then n i = 1 for all i.Moreover, the anisotropic β (i) are pairwise distinct, except in the case where Q ′′ is of affine type A 1 , so that s = 2, t = 0, and β := β (1) = β (2) has self pairing (β, β) = −2.
Proof.The codimension two condition can be written as: Remark 7.4.The lemma can be strengthened to prove: for any decomposition α = j α (j) with α (j) ∈ NR + λ,θ , we have j p(α (j) ) ≤ i n i p(σ (i) ).This generalizes an observation on [17, p. 3] (dealing with the case where the α (j) are roots).To prove this, for arbitrary α (j) , we can apply the lemma to each of the α (j) , and then we get that j p(α (j) ) ≤ j p(β (j) ) for some roots β (j) ∈ R + λ,θ with α = β (j) ; then we are back in the case of roots so that j n i p(σ (i) ) ≥ j p(β (j) ).
Remark 7.5.The arguments of [16,17] can be generalized to the context of the pair (λ, θ), which as we pointed out in §2.3 would eliminate the need of picking a λ ′ as in the proof of the lemma above.
(a) Varieties whose dimension vectors are indivisible roots; (b) Symmetric powers of deformations or partial resolutions of du Val singularities (C 2 /Γ for Γ < SL 2 (C));

1. 2 .
Symplectic leaves and the étale local structure.As Hamiltonian reductions, quiver varieties have a natural Poisson structure.The symplectic leaves of this Poisson structure are the maximal connected (analytic immersed) submanifolds on which the Poisson bracket is nondegenerate.Put differently, the reduction naturally is foliated by symplectic submanifolds.For example, the locus of stable representations of the doubled quiver inside µ −1 (λ) consists of free closed orbits under the group PG(α) := G(α)/C × , hence its Hamiltonian reduction here is well known to be symplectic.If nonempty, this forms an open dense symplectic leaf of the quiver variety.

1. 9 .
Acknowledgments.The first author was partially supported by EPSRC grant EP/N005058/1.The second author was partially supported by NSF Grant DMS-1406553.The authors are grateful to the University of Glasgow for the hospitality provided during the workshop "Symplectic representation theory", where part of this work was done, and the second author to the 2015 Park City Mathematics Institute as well as to the Max Planck Institute for Mathematics for excellent working environments.The proof of the theorems and corollaries stated in the introduction can be found in the following sections denotes the vector space of representations of Q of dimension α.The group G(α) := i∈Q 0 GL α i (C) acts on Rep(Q, α); write g(α) = Lie G(α).The torus C × in G(α) of diagonal matrices acts trivially on Rep(Q, α).Thus, the action factors through PG(α) := G(α)/C × .Let pg(α) := Lie PG(α) = g(α)/C.
The k-tuple e = (e 1 , . . ., e k ) defines a dimension vector for the quiver Q ′ .If X and Y are Poisson varieties, then we say that there is a étale Poisson isomorphism between a neighborhood of x ∈ X and y ∈ Y if there exists a Poisson variety Z and Poisson morphisms Y ψ ←− Z φ −→ X and z ∈ Z such that φ(z) = x, ψ(z) = y and both φ and ψ are étale at z. Theorem 3.3.There is an étale Poisson isomorphism between a neighborhood of 0 in µ −1 Q ′ (0)/ /G(e) and a neighborhood of x ∈ M λ (α, θ).

3. 3 .
The proof of Theorem 3.3.Fix M = Rep(Q, α) and G = G(α).Recall that M has a canonical G-invariant symplectic form ω. Since y ∈ M θ , there exists some n > 0 and nθ-semiinvariant function γ such that γ(y) = 0. We fix such a γ, and let M γ ⊂ M θ be the affine open subset of M defined by the non-vanishing of γ.Let H := G(α) y be the stabilizer of y in G(α) and h the Lie algebra of H. Since h is reductive we can fix a h-stable complement L to h in g.By [18, Lemma 4.1], the H-submodule g • y ⊂ M is isotropic, and by [18, Corollary 2.3], there exists a coisotropic H-module complement C to g • y in M .Let W = (g • y) ⊥ ∩ C. The composition of µ : M → g * with the restriction map g * → h * is denoted µ H . Notice that µ H is simply the moment map for the action of H on M .The restriction of µ H to W is denoted μ.There is a natural identification of W with Rep(Q ′ , e) such that μ = µ Q ′ .

Theorem 3 . 7 .
There exists a G-saturated affine open set V ⊂ M θ , and H-saturated affine open sets Z ⊂ C and U ⊂ Rep(Q ′ , e) such that (a) there are étale Poisson morphisms

Lemma 3 . 8 .
(a) φ is étale.(b) The induced map φ/G : X/ /G → Y / /G is étale.(c) The morphism X → Y × Y / /G X/ /G is an isomorphism.There exists an affine, H-saturated open neighbourhood Z of 0 in C δ , such that φ restricts to an excellent Poisson morphism where V is the affine open set of Lemma 3.8, and set M = M λ (α, θ).Abusing notation, we will also write M ∩ V for the affine open subset

Lemma 3 . 13 .Theorem 3 . 14 .
For each f ∈ C[V ] G , the Hamiltonian vector field ζ f is tangent to M (H) .Proof.By Lemma 3.10, M (H) is smooth, therefore it suffices to show that (ζ f ) y ∈ T y M (H) for all y ∈ (M (H) ) H . Recall from Lemma 3.10 that T y M (H) = M H ⊕ (g/h) H .The canonical map Der(V ) → T y M (H) is H-equivariant.Since {−, −} is G-invariant, and f ∈ C[V ] G , the Hamiltonian vector field ζ f belongs to Der(V ) G ⊂ Der(V ) H . Hence (ζ f ) y ∈ (T y M ) H = M H ⊂ T y M (H) ,as required.The space M λ (α, θ) (H) is a locally closed Poisson subvariety, such that the restriction {−, −}| M λ (α,θ) (H) of the Poisson bracket on M λ (α, θ) equals the Poisson structure induced by ω H .In particular, it is non-degenerate.Proof.Again, let M = M λ (α, θ).First we show that it is a Poisson subvariety.It suffices to show that each Hamiltonian vector field ζ respect to its natural Poisson bracket.Proof.It is well-known that the stratification of Rep(Q, α) θ / /G(α) by stabilizer type is finite, with smooth locally closed strata.Therefore the stratification {M λ (α, θ) τ } of M λ (α, θ) is finite with locally closed strata.Thus it suffices to show that (a) each stratum is smooth, and (b) the Poisson structure is non-degenerate on each stratum.In fact, (b) implies (a), and both statements are implied by Theorem 3.14.
x is a domain.This ring embeds into the complete local ring of x in M λ (α, θ).By Corollary 3.4, the complete local ring of x in M λ (α, θ) is isomorphic to the complete local ring of 0 in M 0 (e, 0).By [17, Corollary 1.4], this is a domain.Finally, normality is an étale local property, [45, Remark 2.24 and Proposition 3.17].Therefore, as in the previous paragraph this follows from Theorem 3.3 and [18, Theorem 1.1].

Finally, we need
to check that the morphism φ is Poisson.Since both varieties are normal by Proposition 3.21, it suffices to show that φ induces an isomorphism of smooth symplectic varieties between the open leaf of M λ (α, θ) and the open leaf of i S n i M λ (σ (i) , θ) .By Proposition 3.15, the symplectic leaves of M λ (α, θ) are connected components of the strata given by stabilizer type.The explicit description of φ given at the start of this section shows that φ restricts to an isomorphism between strata.In particular, φ restricts to an isomorphism between the open leaves.The symplectic structure on the open leaf of M λ (α, θ) comes from the symplectic structure on T * Rep(Q, α).More specifically, the non-degenerate closed form on the latter space restricts to a degenerate G(α)-invariant two-form on µ −1 (λ) θ .Hence it descends to a closed two-form on M λ (α, θ).The restriction of this two-form to the open leaf is non-degenerate.The two-form on the open leaf of

4. 2 .
The proof of Theorem 1.15.The proof of Theorem 1.15 follows closely the arguments given in[38, Theorem 3.2].We provide the necessary details that show that the arguments of loc.cit.

Theorem 6 . 7 (
[20], Theorem A).Let V be a locally factorial, normal G-variety, with good quotient ξ : V → U := V / /G.Assume that there exists an open subset U s ⊂ U such that (a) the complement to U s has codimension at least two in U ,(b) V s := ξ −1 (U s ) → U s is a principal G-bundle; and(c) the complement to V s has codimension at least two in V .Let x ∈ U and y ∈ T (x) a lift in V (so that G • y is closed in V ).The following are equivalent:(i) The local ring O U,x is a unique factorization domain.
(i) implies (ii).It suffices to assume that M 0 = I Y , where Y is a hypersurface in U s .If Y is the closure of Y in U , then M = I Y ∩U 0 is the required extension.(ii) implies (i).Let Y be a hypersurface in U .We wish to show that I Y is free at x. Let M be the extension of I Y | Us to U 0 .The line bundle M corresponds to a Cartier divisor D on U 0 ; M = O U 0 (D).Then, I Y | Us = O Us (D ∩ U s ), and the divisors Y and −D ∩ U s are linearly equivalent.Since, by assumption, the codimension of the complement to U s in U has codimension at least two and U is normal, Y ≃ −D.Hence M = I Y is free at x.

Lemma 6 . 8 .
Let L be a G-equivariant line bundle on Y and y ∈ Y a closed point such that the orbit O = G • y is closed and the stabilizer G y of y acts trivially on the fiber L y .Then there exists an affine open neighborhood U of ξ(y) such that the stabilizer G y ′ acts trivially on L y ′ for all
2 (C) is a finite group; see [52, Lemma 1.3].Associated to Γ, via the McKay correspondence, is a Weyl group W L of type A, D or E. The fundamental group π 1 (L) acts on W L via Dynkin automorphisms.Let W ′ L denote the centralizer of π 1 (L) in W L .Then By a result of Richardson, [57, Proposition 3.3], the fact that all stabilizers are connected implies that there is a G-stable open set U such that the stabilizer G u of each u ∈ U is conjugate to a subgroup of H.In particular, we see that if n This is closed by [10, Lemma 2.2].It will follow that M (H) is smooth if we can prove identity (6), since M H is smooth by [61, Corollary 6.5].