SHEAVES ON ALE SPACES AND QUIVER VARIETIES Introduction ...

MOSCOW MATHEMATICAL JOURNAL Volume 7, Number 4, October–December 2007, Pages 699–722

SHEAVES ON ALE SPACES AND QUIVER VARIETIES HIRAKU NAKAJIMA Dedicated to Victor Ginzburg on his fiftieth birthday

Abstract. We identify a quiver variety of an affine type with a framed moduli space of torsion free sheaves on an ALE space, a fiber of a simultaneous resolution of the semi-universal deformation of C2 /Γ. This result is an analog of a similar identification for a framed moduli space of anti-self-dual connections on an ALE space, given by Kronheimer and the author. It simultaneously generalizes the description on C2 given in Chapter 2 of my“Lectures on Hilbert schemes of points on surfaces” to arbitrary ALE space, and also the description of the Hilbert schemes of points on the ALE space by Kuznetsov to arbitrary torsion free sheaves. 2000 Math. Subj. Class. Primary 14D21; Secondary 53C26, 16G20. Key words and phrases. ALE space, ADHM description, coherent sheaf, framed moduli space.

Introduction This paper is a sequel to [9], where a framed moduli space of anti-self-dual connections on an ALE space (see [8]) is described as a space of equivalence classes of solutions of a quadratic equation for certain finite-dimensional matrices. It is similar to the celebrated ADHM (Atiyah–Drinfeld–Hitchin–Manin) description for anti-self-dual connections on S 4 (or rather R4 in our context) [1]. The latter space is a hyperk¨ ahler quotient of a linear space by a product of unitary groups. By the general theory relating a symplectic quotient and a geometric invariant theory quotient developed by Kempf–Ness, Kirwan and others (see [13, Ch. 8]), it is isomorphic to a holomorphic symplectic quotient of the linear space by the product of general linear groups. On the other hand, by the Hitchin–Kobayashi correspondence, proved in [3] in this setting, the framed moduli space of anti-self-dual connections is isomorphic to the framed moduli space of locally free sheaves. In this paper, we give a similar description for the framed moduli space of torsion free sheaves on the ALE space. It is also described as a holomorphic symplectic quotient of the same linear spaces, but the stability condition used to define quotients is slightly changed. If we view the spaces as a hyperk¨ ahler quotient, the change is given by a perturbation Received January 7, 2007. Supported by the Grant-in-aid for Scientific Research (No.17340005), JSPS. c

2007 Independent University of Moscow

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of the value of the real part of moment map. This modification of the stability condition was already appeared, in a simpler form, when the ALE space is R4 ∼ = C2 [18, Ch. 2]. Our description also generalizes the same one of the Hilbert scheme of points on an ALE space by Kuznetsov [10] to arbitrary torsion free sheaves. The argument also gives a new proof for the description of locally free sheaves, which does not depend on the Hitchin–Kobayashi correspondence nor the hyperk¨ahler quotient. In [15] the author introduced a quiver variety, as a holomorphic symplectic quotient of a similar linear space, and studied their geometric properties, as well as its relation to the representation theory of a Kac–Moody Lie algebra. In this setting, the space of stability conditions has a chamber structure. The stability condition for anti-self-dual connections is on a wall, called the level 0 hyperplane, while one for torsion free sheaves is in an adjacent chamber. There is another chamber, corresponding to a framed moduli space of Γ-equivariant torsion free sheaves on C2 by [18, Ch. 2] if the value of the complex moment map is 0 (see also [12], [23]). Here Γ is a finite subgroup of SL2 (C), and the fundamental group of the end of the ALE space. (The ALE space is diffeomorphic to the minimal resolution of C2 /Γ.) As the underlying C ∞ -structures of quiver varieties are independent of the choice of the stability condition outside walls [15, Cor. 4.2], we can choose any stability condition for topological questions. But the Hecke correspondences, used to define representations of the Kac–Moody Lie algebra, have been best understood for the stability condition corresponding to Γ-equivariant sheaves, and it was the reason why this stability condition was used in the later study (e.g. [17]). The quiver variety depends also on the choice of the value of complex moment map. In this paper, we choose it from the complexified level 0 hyperplane, as there is no corresponding ALE space otherwise. If we choose it outside the complexified level 0 hyperplane, the corresponding quiver variety is the framed moduli space of Γ-equivariant torsion free sheaves on a noncommutative deformation of P2 [4]. As the walls on the stability condition are root hyperplanes which kill the value of the complex moment map, all walls disappear and any stability condition is automatically satisfied in this case. Hence the problem, studied in this paper, does not make sense in the noncommutative setting. The main result of this paper was obtained some years ago, and has been explained to some people, including Vasserot, Wang, Haiman. I had not written down the detail, as the proof is just an exercise after [9] and [18, Ch. 2]. But I change my mind, as K. Nagao, under my supervision, uses this result to show that the representation of an affine Lie algebra on the direct sum of homology groups of moduli spaces of rank 1 torsion free sheaves on an ALE space, constructed from the Heisenberg algebra representation via the Frenkel–Kac construction (see [18, Ch. 9]), coincides with the representation constructed on homology groups of moduli spaces of Γ-equivariant sheaves [17] via the natural diffeomorphisms [14]. The paper is organized as follows. In Section 1 we recall the definition of quiver varieties and the ADHM description of anti-self-dual connections on an ALE space. In Section 2 we state the main result. In Section 3 we introduce a compactification of an ALE space, which will be used to replace the L2 -kernel of Dolbeault operators used in [9] by cohomology groups. In Section 4 we study the stability condition

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corresponding to torsion free sheaves on an ALE space. This is a technical heart of this paper, which was not appeared neither in [9] nor [18]. The tools used here are the Harder–Narasimhan filtration and Jordan–H¨older filtration. In Section 5 we complete the proof of the main result. Once the stability condition is understood, the rests of arguments follow closely those in [9] and [18]. We will work on the complex-analytic category with the ordinary topology in this paper. When the ALE space is biholomorphic to the minimal resolution of a simple singularity C2 /Γ, we can work on the complex algebraic category, but it is not clear whether this is possible in general. Acknowledgments. The author is grateful to Kentaro Nagao for discussion on results of this paper and their application to the representation theory of affine Lie algebras. 1. Preliminaries 1(i). Quiver varieties. We fix notations for quiver varieties in this subsection. Let (I, E) be a finite graph of an affine type, where I is the set of vertices and E the set of edges. Let A be the adjacency matrix of the graph. Then C = 2I − A is a (symmetric) Cartan matrix of an affine type. Let H be the set of pairs consisting of an edge together with its orientation. For h ∈ H, we denote by in(h) (resp. ¯ the out(h)) the incoming (resp. outgoing) vertex of h. For h ∈ H we denote h same edge as h with the reverse orientation. An orientation Ω of the graph is a ¯ ∪ Ω = H, Ω ¯ ∩ Ω = ∅. The orientation defines a function subset Ω ⊂ H such that Ω ¯ ε : H → {±1} Lgiven by ε(h) = 1 if h ∈ Ω and = −1 if h ∈ Ω. Let V = i∈I Vi be an I-graded vector space. We define its dimension vector def

by dim V = (dim Vi )i∈I ∈ ZI>0 . If V 1 , V 2 are I-graded vector spaces, we introduce following vector spaces M def L(V 1 , V 2 ) = Hom(Vi1 , Vi2 ), i∈I 1

2

def

E(V , V ) =

M

1 2 Hom(Vout(h) , Vin(h) ).

h∈H

For B = (Bh ) ∈ E(V 1 , V 2 ), C = (Ch ) ∈ E(V 2 , V 3 ), we define a multiplication of B and C by  X  def CB = Ch Bh¯ ∈ L(V 1 , V 3 ). in(h)=i 1

i

2

Multiplications ba, Ba of a ∈ L(V , V ), b ∈ L(V 2 , V 3 ), B ∈ E(V 2 , V 3P ) are defined in obvious manner. If a ∈ L(V 1 , V 1 ), its trace tr(a) is understood as k tr(ak ). Let V , W be I-graded vector spaces. We define def

M (V, W ) = E(V, V ) ⊕ L(W, V ) ⊕ L(V, W ). If the dimension vectors of V , W are only relevant in the context, we simply denote this by M (v, w) where v = dim V , w = dim W . The components of an element

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in M (V, W ) will be denoted by B, a, b respectively. This space has a holomorphic symplectic form given by def

ω((B, a, b), (B 0 , a0 , b0 )) = tr(εBB 0 ) + tr(ab0 − a0 b), where εB is an element of E(V, V ) definedQby (εB)h = ε(h)Bh . Let G ≡ GV ≡ Gv be the Lie group i GL(Vi ). It acts on M (V, W ) in an obvious manner, preserving the symplectic form. The moment map vanishing at the origin is given by µ(B, a, b) = εBB + ab ∈ L(V, V ), where the dual of the Lie algebra of G is identified with L(V, V ) via the trace. I Let L ζC = (ζC,i ) ∈ C . We define a corresponding element in the center of Lie G by i ζC,i idVi , where we delete the summand corresponding to i if Vi = 0. Let µ−1 (ζC ) be an affine algebraic variety (not necessarily irreducible) defined as the zero set of µ − ζC . The group G acts on µ−1 (ζC ). We now define stability conditions. def P For ζR = (ζR,i )i∈I ∈ RI , let ζR · dim V = i∈I ζR,i dim Vi . Definition 1.1. A point (B, a, b) ∈ M is ζR -semistable if the following two conditions are satisfied: (1) If an I-graded subspace S of V is contained in Ker b and B-invariant, then ζR · dim S 6 0. (2) If an I-graded subspace T of V contains Im a and is B-invariant, then ζR · dim T 6 ζR · dim V . We say (B, a, b) is ζR -stable if the strict inequalities hold in (1),(2) unless S = 0, T = V respectively. Note that (B, a, b) is ζR -(semi)stable if and only if (B ∗ , b∗ , a∗ ) is (−ζR )-(semi) stable. This will be used later frequently. If ζR,i > 0 for all i, the condition (2) is superfluous, and the condition (1) turns out to be the nonexistence of nonzero B-invariant I-graded subspaces S = (Sk ) contained in Ker b (and in this case ζR -stability and ζR -semistability are equivalent.) This is the stability condition used in [17, 3.9]. The case when ζR,i < 0 for all i is also important. The condition (1) is superfluous and the condition (2) turns out to be the nonexistence of proper I-graded B-invariant subspace T containing Im a. This coincides with the natural condition for the description of Hilbert schemes of points on C2 [18, Sec. 1]. We also need the stability condition for B ∈ E(V, V ), i. e., when W = 0. Definition 1.2. Suppose that ζR ·dim V = 0. A point B ∈ E(V, V ) is ζR -semistable if the following is satisfied: • If an I-graded subspace S of V is B-invariant, then ζR · dim S 6 0. A point B is ζR -stable if the strict inequality holds unless S = 0 or S = V . (The definition is the same as W 6= 0 case for the ζR -semistability, but different for the ζR -stability.)

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s ss Let H(ζ (resp. H(ζ ) be the set of ζR -stable (resp. ζR -semistable) points R ,ζC ) R ,ζC ) −1 in µC (ζC ). We say two ζR -semistable points (B, a, b), (B 0 , a0 , b0 ) are S-equivalent when the ss closures of GV -orbits intersect in H(ζ . We denote the pair (ζR , ζC ) by ζ for R ,ζC ) brevity. We define def

ss /∼, Mζ ≡ Mζ (V, W ) ≡ Mζ (v, w) = H(ζ R ,ζC ) def

reg reg s Mreg ζ ≡ Mζ (V, W ) ≡ Mζ (v, w) = H(ζR ,ζC ) /GV ,

where ∼ denotes the S-equivalence relation. These can be defined as quotients in the geometric invariant theory. (See [7].) 1(ii). The chamber structure. Fix a dimension vector v. Let def

R+ = {θ = (θi ) ∈ ZI>0 | t θCθ 6 2} \ {0}, def

R+ (v) = {θ ∈ R+ | θi 6 dimC Vi for all i}, def

Dθ = {x = (xi ) ∈ RI | x · θ = 0} for θ ∈ R+ . When the graph is of affine type, R+ is the set of positive roots, and Dθ is the wall defined by the root θ. Though R+ is an infinite set, but R+ (v) is always finite. Proposition 1.3 [16, 2.8]. Suppose [

ζ = (ζR , ζC ) ∈ (R ⊕ C)I \

(R ⊕ C) ⊗ Dθ .

θ∈R+ (v)

Then every semistable point is stable, so that the regular locus Mreg coincides with ζ Mζ . Fix a complex parameter ζC and move a real parameter ζR . A connected component of [ RI \ Dθ θ∈R+ (v) ζC ·θ=0

is called a chamber. Lemma 1.4. Take two real parameters ζR and ζR0 so that both (ζR , ζC ) and (ζR0 , ζC ) are contained in the set in Proposition 1.3. If ζR and ζR0 are in the same chamber, the corresponding stability conditions are equivalent. Proof. Suppose that corresponding stability conditions are not equivalent, i. e., there exists a point (B, a, b) ∈ µ−1 (ζC ) which is ζR -stable, but not ζR0 -stable. Therefore there exists a B-invariant I-graded subspace S or T violating the inequalities in Definition 1.1 for ζR0 , but not for ζR . If we connect ζR and ζR0 by a path, there is a point ζR00 on the path such that the subspace S (or T ) satisfies the equality in Definition 1.1 for ζR00 . In particular, the stability condition and the semistability condition are not equivalent for ζR00 . Therefore by Proposition 1.3 this cannot happen if ζR and ζR0 are in the same chamber. 

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1(iii). A review of the ADHM description [9]. (See also [16, Sec. 2].) Take and fix an affine Dynkin graph. Let I, H be as before, and let us choose an orientation Ω. Let 0 ∈ I be the vertex corresponding to the simple root, which is the negative of the highest weight root of the corresponding simple Lie algebra. Let δ be the vector in the kernel of the affine Cartan matrix whose 0-component is equal to 1. Such a vector is uniquely determined. Let Gδ be the complex Lie group corresponding to δ as in Section 1(i). Choose the parameter ζ ◦ = (ζR◦ , ζC◦ ) ∈ R3 ⊗Z, where Z ⊂ R[I] is the level 0 hyperplane {x ∈ R[I] | x · δ = 0}. We further assume that ζ ◦ is generic, i. e., it is not contained in any R3 ⊗ Dθ where Dθ is a real root hyperplane. Let def Xζ ◦ = {ξ ∈ M (δ, 0) | µ(ξ) = −ζC◦ }//(−ζR◦ ) (Gδ /C∗ ) , (1.5) where ‘//(−ζR◦ ) ’ means the GIT quotient ‘/∼’ with respect to the parameter (−ζR◦ ) defined as in Section 1(i). As we assume ζ ◦ is generic, we have s ∗ Xζ ◦ = H(−ζ ◦ ,−ζ ◦ ) / (Gδ /C ) , R

C

◦ where is the set of (−ζR )-stable points in µ−1 C (−ζC ) as before. Note that both the value of the moment map and the parameter for the stability condition are the negative of those used in the definition of the quiver variety Mζ . Note also that the group C∗ of scalars in Gδ acts trivially on M (δ, 0), so we can consider the action of the quotient group Gδ /C∗ . Kronheimer [8] showed that if ζ ◦ is generic, then (a) Xζ ◦ is a smooth complex surface (in fact, it is a 4-dimensional hyperk¨ahler manifold), (b) Xζ ◦ is diffeomorphic to the minimal resolution of C2 /Γ, where Γ is the finite subgroup of SL2 (C) associated to the affine Dynkin graph, (c) Xζ ◦ has a K¨ ahler metric (in fact, hyperk¨ahler), which is ALE (asymptotically locally Euclidean) of order 4, i. e., there is
a compact subset K ⊂ Xζ ◦ and a diffeomorphism Xζ ◦ \ K → C2 \ Br (0) /Γ under which the metric is approximated by the standard Euclidean metric on C2 /Γ; it is written in the Euclidean coordinate (xi )4i=1 s H(−ζ ◦ ◦ R ,−ζC )

gij = δij + aij P4 2 with ∂ p aij = O(r−4−p ), p > 0, where r2 = i=1 xi and ∂ denotes the differentiation with respect to the coordinates xi . Here Br (0) is the ball of radius r centered at 0. (The domain Xζ ◦ \ K will be referred to as the end of Xζ ◦ .) Recall also
that X0 is isometric to C2 /Γ and the diffeomorphism Xζ ◦ \ K → C2 \ Br (0) /Γ was given as a restriction of a certain map Xζ ◦ → X0 [8, Cor. 3.2, Proof of Cor. 3.12]. s ∗ By the construction, H(−ζ ◦ ◦ can be considered as a principal Gδ /C -bundle R ,−ζC ) over Xζ ◦ . It has a natural holomorphic structure. (In fact, the hyperk¨ahler quotient construction of Xζ ◦ also gives a reduction of the structure group of the principal bundle to the maximal compact group of Gδ /C∗ and a natural connection on the

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reduced principal bundle as a by-product. This is anti-self-dual [6] and has fith ∗ nite Q action [9, 2.2].) Let δi be the i -component of δ. We identify Gδ /C with ◦ GL(δ ). For each i ∈ I, we have the associated vector bundle over X i ζ : i6=0 def

Ri = µ−1 (−ζC◦ ) ×Gδ /C∗ Cδi , Q where Gδ /C∗ acts on Cδi through the projection Gδ /C∗ = i6=0 GL(δi ) → GL(δi ). When i = 0, we understand R0 as the trivial line bundle. We call Ri a tautological bundle. It also has a compatible hermitian metric, whose associated hermitian connection A (which is, in fact, anti-self-dual) approximates an irreducible flat unitary connection A0 at the end, i. e., A − A0 = O(r−3 ),

∇A − ∇A0 = O(r−4 ),

...,

(1.6)

where ∇ denotes the covariant derivative with respect to A0 . Here ‘compatible’ means that the holomorphic structure of Ri is given by the (0, 1)-part ∂¯¯A of the connection A. In fact, the flat connection A0 was the canonical connection on X0 ∼ = C2 /Γ, constructed exactly in the same way as A. (See [9, Prop. 2.2(i)].) From the construction of the isometry X0 ∼ = C2 /Γ [8, Cor.3.2], this irreducible flat connection corresponds to an irreducible Γ-module Ri which corresponds to the vertex i by the McKay correspondence. By the construction, there exists a bundle homomorphism ξh : Rout(h) → Rin(h) for each h ∈ H. This homomorphism is called a tautological homomorphism. Suppose I-graded vector spaces V , W and data (B, a, b) ∈ M (V, W ) satisfying µ(B, a, b) = ζC◦ and the ζR◦ -stability condition are given. (ζ ◦ is the same as above.) In our context, (B, a, b) is called an ADHM data. Let us consider vector bundles E(R∗ , V ),

L(R∗ , W ),

where (1) V and W are (collections of) trivial vector bundles, (2) E( , ), L( , ) are defined exactly as before by replacing vector spaces by vector bundles. We define a complex of vector bundles σ

τ

L(R∗ , V ) − → E(R∗ , V ) ⊕ L(R∗ , W ) − → L(R∗ , V ),

(1.7)

by σ(η) = (Bη − ηξ ∗ ) ⊕ bη,

τ (η 0 ⊕ η 00 ) = εBη 0 + εη 0 ξ ∗ + aη 00 .

This is a complex thanks to the conditions µ(ξ ∗ ) = ζC = µ(B, a, b). Also thanks to the ζR◦ -stability conditions for ξ ∗ and (B, a, b), σ is injective and τ is surjective on the fiber at any point in Xζ ◦ (as linear maps). (See Proposition 4.1(3).) Therefore Ker τ / Im σ is a vector bundle over Xζ ◦ , which has a natural holomorphic structure induced from E(R∗ , V ), L(R∗ , W ). (In fact, if we give hermitian metrics on V , W , we have a natural anti-self-dual connection compatible with the holomorphic structure.) The inverse map is constructed as follows. Suppose a holomorphic vector bundle E on Xζ ◦ is given. We put a framing on E. This means the following: we fix a C ∞ vector bundle E∞ over the end of Xζ ◦ and put a hermitian metric together with a flat unitary connection corresponding to a representation of Γ. Then a framing

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Φ of E is an isomorphism between the underlying C ∞ -vector bundle of E and E∞ over the end of Xζ ◦ such that the Dolbeault operator ∂¯¯A of E is approximated by the (0, 1)-part ∂¯¯A0 of the connection A0 on E∞ as ∂¯¯A = ∂¯¯A + α 0

−3

with α = O(r ) with a similar decay in the derivatives with respect to the connection A0 . If we put a hermitian metric on E which is equal to (or more generally, is approximated in order O(r−2 ) by) the given one on E∞ via Φ at the end, then it satisfies (1.6). We say two holomorphic vector bundles E, E 0 together with framings Φ, Φ0 are isomorphic, if there exists an isomorphism ϕ : E → E 0 of holomorphic vector bundles such that Φ0 ϕΦ−1 is approximated by the identity homomorphism in O(r−2 ) with a similar decay in the derivatives. A holomorphic vector bundle obtained from the ADHM data (B, a, b) has such a framing, canonical up to an isomorphism. This follows from the finiteness of the action in [9, Prop. 4.1] with Uhlenbeck’s removableL singularities theorem. The representation of Γ corresponding to E∞ is given by i Wi ⊗ Ri , where Ri is the irreducible representation of Γ corresponding to the vertex i via the McKay correspondence. We define I-graded vector spaces V , W by def Vi = L2 -kernel of ∂¯¯A ⊕ ∂¯¯A∗ : Ω0,1 (E ⊗ R∗i ) → Ω0,2 (E ⊗ R∗i ) ⊕ Ω0,0 (E ⊗ R∗i ), def

Wi = bounded harmonic sections of E ⊗ R∗i .

(1.8)

Here ∂¯¯A∗ is the formal adjoint of ∂¯¯A defined by using the connection A given by a hermitian metric on E which is approximated by the one on E∞ via Φ at the end, and the Laplacian is defined as ∂¯¯A∗ ∂¯¯A . We define linear maps Bh : Vout(h) → Vin(h) , ai : Wi → Vi by Bh (vout(h) ) = L2 -projection of (1E ⊗ ξh∗ )vout(h) , √ ¯¯ i . ai (wi ) = 2 ∂w The definition of b : V → W is more involved. Roughly speaking, we identify Wi with the fiber of E at infinity and b(v) is given by the ‘limiting value’ of v. We refer the original paper for the precise statement. There is also a definition via a double complex. Then (B, a, b) ∈ M (V, W ) satisfies µ(B, a, b) = ζC◦ and the ζR◦ -stability condition. Moreover, vectors v, w correspond to the data for the vector bundle E as follows. As we explained, we have a flat connection and the corresponding representation L ⊕wi of Γ. Then w is given by its decomposition to irreducibles: . Here the i∈I ρi set of isomorphism classes of irreducible representations is identified with I via the McKay correspondence. This correspondence is an obvious consequence of the fact that W is identified with the fiber at the end. (See [9, Lem. 5.2].) The vector v is the dimension of cohomology groups, and is given by Chern classes of E. The formulas are X c1 (E) = ui c1 (Ri ), where u = w − Cv, i6=0

ch2 (E) =

X i

(1.9) ui ch2 (Ri ) + 2v · δ ch2 (O(`∞ )).

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Here C is the affine Cartan matrix, and ch2 is the H 4 -part of the Chern character, and O(`∞ ) is a line bundle on a compactification of Xζ ◦ , explained later. In fact, we can interpret the second equality as an equality for the curvature integral over Xζ ◦ . (See [9, the first and fourth displayed equations on p. 302].) These formulas are consequences of the construction of E as a cohomology of the complex (1.7) (or rather the complex (3.1) on the compactification). As explained in [9, p. 302], these formulas determine the vector v, once w is determined by the flat connection at the end. The main result of [9] (more precisely, the holomorphic version of the proof) says that Mreg ζ ◦ (v, w) is bijective to the isomorphism classes of framed holomorphic vector bundle over Xζ ◦ , i. e., the framed moduli space of holomorphic vector bundles. The following result was also proved in [9, Prop. 9.2] (see also Corollary 4.3 about a comment on its proof): Proposition 1.10. We have Mζ ◦ (v, w) =

G

k ◦ Mreg ζ ◦ (v − kδ, w) × S Xζ ,

k>0 k

where S Xζ ◦ is the k

th

symmetric product of Xζ ◦ .

◦ As Mreg ζ ◦ (v−kδ, w) is the framed moduli space of locally free sheaves on Xζ with a smaller second Chern number, the above description means that M(ζR◦ ,ζC◦ ) (v, w) is the Uhlenbeck (partial) compactification of Mreg (ζR◦ ,ζC◦ ) (v, w). Our main result is an identification of the Gieseker–Maruyama (partial) compactification with the space M(ζR ,ζC◦ ) (v, w) for a suitable choice of ζR .

2. Main Result We fix dimension vectors v, w. 2(i). A stability parameter ζR . Let ζR◦ ∈ R[I] be the parameter for the stability condition, in the level 0 hyperplane ζR◦ · δ = 0 as in Section 1(iii). We take a parameter ζR from the chamber containing ζR◦ in its closure with ζR · δ < 0. These conditions uniquely determine the chamber containing ζR . As ζR is not contained in any root hyperplane, the stability and semistability are equivalent for ζR . Note that the chamber structure depends on the choice of v. The relevant chamber becomes smaller and smaller if we increase v in general. The following figure describes (1) chambers when ζC = 0 for type A1 , i. e., I = {0, 1}, A01 = 2, where A is the adjacency matrix as in Section 1(i). The followings are obvious from the definition and our choice: Lemma 2.1. (1) If (B, a, b) is ζR -stable, then it is ζR◦ -semistable. (2) If (B, a, b) is ζR◦ -stable, then it is ζR -stable. Therefore we have a projective morphism M(ζR ,ζC◦ ) (v, w) → M(ζR◦ ,ζC◦ ) (v, w).

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ζR,1 ζR ·α0 =0

ζR,0 ζR ·α1 =0 ζR ·(α1 +δ)=0 ζR◦

ζR

ζR ·δ=ζR,0 +ζR,1 =0 ζR ·(α0 +δ)=0 Figure 1. Chambers It is an isomorphism over Mreg (ζR◦ ,ζC◦ ) (v, w). This is the morphism from the GiesekerMaruyama compactification to the Uhlenbeck compactification, constructed by J. Li [11] in the setting of moduli spaces on a projective surface. (See also [18, Ch. 3] for the case C2 .) 2(ii). Statement. Let E be a torsion free sheaf on Xζ ◦ . Its double dual E ∨∨ is a locally free sheaf, i. e., a holomorphic vector bundle on Xζ ◦ . We have the canonical exact sequence 0 → E → E ∨∨ → Q → 0. The quotient sheaf Q := E ∨∨ /E is supported at finitely many points in Xζ ◦ . Conversely the torsion free sheaf E can be recovered from the locally free sheaf E ∨∨ together with the surjection E ∨∨ → Q as the kernel. We define a framing of E as a framing of E ∨∨ . In fact, a framing of the locally free sheaf E ∨∨ can be defined in several ways. As in Section 1(iii), we may consider it as an isomorphism between E ∨∨ and a fixed C ∞ -vector bundle over the end of Xζ ◦ . But for the proof ¯¯ζ ◦ of the following theorem, it is more appropriate to take a compactification X ¯¯ζ ◦ together with an isomorphism of Xζ ◦ and consider a torsion free sheave E on X between the restriction of E and a given bundle on the boundary divisor `∞ . (More ¯¯ζ ◦ as an orbifold. See Section 3.) precisely, we need to consider X Theorem 2.2. There is a bijection between M(ζR ,ζC◦ ) (v, w) and the framed moduli space of torsion free sheaves (E, Φ) on Xζ ◦ , where w corresponding to the representation of Γ corresponding to the flat connection at the end, and v is given by Chern classes of E by the formulas (1.9).

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The proof is a mixture of that for vector bundles in [9] and that for framed torsion free sheaves on C2 (i. e., a special case Γ = {1}) in [18, Ch. 2]. The construction of a torsion free sheaf from the ADHM data is exactly the same, once we notice that the complex (1.7) gives a torsion free sheaf, even if we allow σ is injective except finitely many points. This was already observed for C2 in [18, Ch. 2], but the proof that the ζR -stability condition is equivalent to the injectivity of σ outside a finite set is more complicated. (See Proposition 4.1.) To get an ADHM data from a framed ¯¯ζ ◦ instead torsion-free sheaf, we use cohomology groups on the compactification X of L2 -harmonic forms in (1.8). This part will be the same as C2 -case, where the compactification is P2 , the complex (1.7) naturally arises from the Beilinson spectral sequence, and the result was well-known. We do not try to give a complex-analytic structure on the framed moduli space of torsion free sheaves directly in this paper, though it should be probably possible. But from the construction below, it is clear that we have the universal family on M(ζR ,ζC◦ ) (v, w) × Xζ ◦ and M(ζR ,ζC◦ ) (v, w) is a fine moduli space of the functor for framed torsion free sheaves on Xζ ◦ . Thus the above theorem can be regarded as a construction of the framed moduli space via M(ζR ,ζC◦ ) (v, w). ¯¯ζ ◦ 3. A Compactification of Xζ ◦ to an Orbifold X We introduce a compactification of Xζ ◦ , which will be used to replace the space of L2 -harmonic forms by the cohomology group, as is explained at the end of the previous section. 3(i). It is easiest to describe the compactification when Xζ ◦ is the minimal resolution of C2 /Γ. We consider P2 /Γ and resolve the singularity at the origin, but keep the singularity on the line at infinity `∞ untouched . For general Xζ ◦ , we have a coordinate system at the end Xζ ◦ \ K → (C2 \ Br (0))/Γ such that the complex structure is approximated by the standard one on C2 /Γ up to order O(r−4 ). Let ¯¯ζ ◦ def X = Xζ ◦ ∪ `∞ , where `∞ = P1 /Γ. We endow a structure of a differential orbifold so that (Xζ ◦ \ K) ∪ `∞ is identified with (P2 \ Br (0))/Γ via the coordinate system at the end. Here the ‘orbifold’ means that we remember the action of Γ on the ˜ = P2 \ Br (0) of `∞ . For example, a smooth function on X ¯¯ζ ◦ is a neighborhood U ˜ ˜ smooth function on Xζ ◦ and a Γ-invariant function on U , patched along (U /Γ) \ `∞ in an obvious sense. ¯¯ζ ◦ : Let (z1 , z2 ) be the Let us give a structure of a complex analytic orbifold on X 2 complex Euclidean coordinate system on C . The almost complex structure J on Xζ ◦ is approximated by the standard almost complex structure J0 on (C2 \Br (0))/Γ ¯¯ζ ◦ as J − J0 = O(r−4 ) with r2 = |z1 |2 + |z2 |2 . Other coordinate systems on X 2 (associated with the corresponding homogeneous coordinate system on P ) are given by (v, w) = (1/z1 , z2 /z1 ), (v 0 , w0 ) = (z1 /z2 , 1/z2 ). Then we have J − J0 = O(|v|3 ), and = O(|w0 |3 ) (with a similar decay in the derivatives). Thus J extends to an ¯¯ζ ◦ as of class C 2,1 , which coincides with the almost complex structure J¯ on X 2 standard one on P /Γ on `∞ . As the Nijenhuis tensor vanishes on Xζ ◦ , it also ¯¯ζ ◦ by the continuity. By the Newlander-Nirenberg theorem, extended vanishes on X

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by [19] under a less differentiable condition, J¯ is integrable. It is also clear that `∞ ¯¯ζ ◦ . is the complex submanifold (divisor, as it is of codimension 1) of X In fact, this compactification can be done more explicitly by using the weighted projective space as follows (see [22] for detail): The ALE space Xζ ◦ is a fiber of a simultaneous resolution of the semi-universal deformation of C2 /Γ [8, Sec. 4]. Therefore, Xζ ◦ , or its singular model obtained by collapsing (−2)-curves, is written as a hypersurface in C3 , e. g., x4 + y 3 + z 2 + a1 yx2 + a2 yx + a3 x2 + a4 y + a5 x + a6 = 0 for E6 . When Xζ ◦ is the resolution of C2 /Γ, the singular model is given by x4 + y 3 + z 2 = 0, and x, y, z correspond to generators of C[z1 , z2 ]Γ . We compactify the hypersurface in C3 to a hypersurface in the weight projective space P(3, 4, 6, 1) given by {x4 + y 3 + z 2 + a1 w2 yx2 + a2 w5 yx + a3 w6 x2 + a4 w8 y + a5 w9 x + a6 w12 = 0}, where C∗ acts on C4 by (x, y, z, w) 7→ (t3 x, t4 y, t6 z, tw) and P(3, 4, 6, 1) = C4 \ {0}/C∗ . The line at infinity `∞ is identified with w = 0, and we have three singular points of the hypersurface which are on `∞ (if a1 , . . . , a6 is generic, all singular points are on w = 0) of type C2 /(Z/p), where p = 3, 4, 6 and Z/p acts by (z1 , z2 ) 7→ (ζz1 , ζz2 ) with ζ = exp(2πi/p). These points correspond to Γ-orbits in P1 which have stabilizers order greater than 2. (Any point is stabilized by diag(−1, −1) ∈ Γ.) Now Xζ ◦ is obtained by resolving singular points outside `∞ (if any). Strictly speaking, the case An with n even was not covered by [22]. In this case, we can only describe the compactification of (C2 /Γ)/{±1}, where (−1) acts by (z0 , z1 ) 7→ (−z0 , −z1 ). Then we can compactify the {±1}-quotient of the hypersurface {xn+1 + yz = a1 + a2 x + · · · + an xn−1 } (where (−1) acts by (x, y, z) 7→ (x, −y, −z)) by a hypersurface in P(2, l + 1, l + 1, 2) as above. ¯¯ζ ◦ (for example, it shows that X ¯¯ζ ◦ is algebraic) will not This explicit form of X ¯¯ be used later, and we just need the fact that Xζ ◦ is approximated by P2 /Γ near `∞ . Note we still need to put an orbifold structure even in this explicit form. For ¯¯ζ ◦ is either P1 × P1 (if ζ ◦ 6= 0) or P(T ∗ P1 ⊕ C) (if example, if Γ = {±1}, then X C ◦ ζC = 0), hence it is a smooth complex surface. But we need to remember the trivial action of Γ. Next suppose that a holomorphic vector bundle E over Xζ ◦ together with a framing Φ : E|Xζ◦ \K → E∞ in the sense of Section 1(iii) is given. We then have a hermitian metric and a connection A on E which is approximated by the flat connection A0 on E∞ as A − A0 = O(r−3 ). Note that E∞ together with A0 is identified with a trivial vector bundle on the Γ-covering C2 \ Br (0) → Xζ ◦ \ K together with the Γ-equivariant structure. Since E∞ extends to the orbifold (P2 \ ¯¯ζ ◦ by gluing via Φ. As Br (0))/Γ across `∞ , we can construct an extension of E to X 2 0 2 ¯¯ζ ◦ A−A0 = O(|v| ), = O(|w | ) in the other coordinate systems (v, w), (v 0 , w0 ) of X as before, the connection A (and the associated holomorphic structure) extends to a holomorphic vector bundle (of class C 1,1 , though it is not a problem by [19]) on ¯¯ζ ◦ . In particular, the tautological bundle Ri extends to X ¯¯ζ ◦ . X By the same argument as in [8, Prop. 3.14] and [9, Prop. 2.2(i)], the tautological homomorphism ξh is approximated by the coordinates z1 , z2 on C2 /Γ, pulled back to Xζ ◦ via the coordinate system at the end Xζ ◦ \ K → (C2 \ Br (0))/Γ as ξh − (z1 , z2 ) = O(|(z1 , z2 )|−1 ). Here (z1 , z2 ) is identified with the tautological

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homomorphism for the tautological bundles on X0 ∼ = C2 /Γ via [8, Cor.3.2]. As ¯¯ζ ◦ as (z1 , z2 ) extends to a bundle homomorphism over P2 /Γ, ξh also extends to X ξh : Rout(h) → Rin(h) (`∞ ) where Rin(h) (`∞ ) = Rin(h) ⊗ O(`∞ ) and O(`∞ ) is the line bundle corresponding to the divisor `∞ in the orbifold sense, constructed by gluing O on Xζ ◦ and O(1) on P2 /Γ. Here the Γ-equivariant structure on O(1) is defined so that z0 : O → O(1) is Γ-equivariant. ¯¯ζ ◦ as an isoHereafter we understand a framing of a torsion free sheaf E on X morphism between E|`∞ and (ρ ⊗ OP1 )/Γ, where ρ is a representation of Γ. 3(ii). Extension of the complex. Next we show that the complex (1.7) extends ¯¯ζ ◦ as to X σ

τ

L(R∗ , V )(−`∞ ) − → E(R∗ , V ) ⊕ L(R∗ , W ) − → L(R∗ , V )(`∞ ).

(3.1)

In order to check this assertion, we need to introduce two copies of Γ, denoted by Γ and Γleft , and rewrite above in terms of Γ, Γleft . In fact, such description was originally used in [9]. We identify an I-graded vector space V with a Γ-module M Vi ⊗ Ri , i

where Ri is the irreducible Γ-module corresponding to the vertex i via the McKay correspondence. We denote this Γ-module also by V for brevity. Under this identification, the imaginary root vector δ corresponds to the regular representation R of Γ. The vector space L(V, W ) is identified with HomΓ (V, W ) the space of Γ-homomorphisms. The same is true for L(V, V ), L(W, V ). The vector space E(V 1 , V 2 ) is identified with HomΓ (V 1 , Q ⊗ V 2 ), where Q is the natural 2-dimensional representation of Γ given by the inclusion Γ ⊂ SL2 (C). This is a rephrasing of the McKay correspondence; dim HomΓ (Ri , Q ⊗ Rj ) is the adjacency matrix of the affine Dynkin diagram. The group GV is identified with GLΓ (V ), the group of Γ-module automorphisms of V . Now the ALE space Xζ ◦ is considered as a quotient by GLΓ (R)/C∗ , where R is the regular representation of Γ. Recall that the regular representation of Γ is the set of complex valued functions on Γ, where Γ acts by the right translation. Therefore we have a commuting action of another copy of Γ, denoted by Γleft , by the left translation. Then the regular representation decomposes into M R= Rileft ⊗ Ri ∗ i

as a Γ × Γ-module. Here the vertex i∗ is defined so that Ri∗ is the dual representation of Ri . We consider the corresponding sum of the tautological bundles Ri ’s M R i∗ ⊗ R i . (3.2) left

i

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L We denote the above bundle also by R, though it was i Ri before. Then we have L(R∗ , V ) = HomΓ (R∗ , V ). The tautological homomorphism ξ is a section of HomΓ (R, Q ⊗ R). Now it becomes clear that the complex (1.7) extends to the compactification as in (3.1). In a neighborhood of `∞ , it is approximated (is exactly equal outside the exceptional locus if ζC◦ = 0) by the following Γleft -equivariant complex on P2 : σ

HomΓ (R∗ , V ) ⊗ OP2 (−`∞ ) − → HomΓ (R∗ , Q ⊗ V ) ⊗ OP2 ⊕ HomΓ (R∗ , W ) ⊗ OP2 τ

− → HomΓ (R∗ , V ) ⊗ OP2 (`∞ ), 

(3.3)



z0 B 1 − z1 σ =  z0 B 2 − z2  , z0 b


τ = −(z0 B2 − z2 ), z0 B1 − z1 , z0 a ,

where [z0 : z1 : z2 ] is the homogeneous coordinate of P2 , and Γleft acts on P2 in the natural way. We have taken a basis of Q ∼ = C2 and write B ∈ HomΓ (V, Q ⊗ V ) ⊂ Hom(V, Q ⊗ V ) as a pair (B1 , B2 ) of endmorphisms of V . Let us consider the restriction of this complex to {z0 = 0} and check that we have a framing of Ker τ / Im σ. We have Ker σ = Coker τ . The middle cohomology group is equal L0 =left to HomΓ (R∗ , W ) ⊗ OP1 = R ⊗ Wi ⊗ OP1 . Therefore it gives the correct i i Γ∼ = Γleft -equivariant locally free sheaf on P1 . 3(iii). A resolution of the diagonal. In [9, §3], a resolution of a skyscraper sheaf at a point in Xζ ◦ was constructed. In fact, in our setting, it is more appropriate to consider it as a resolution of the diagonal in Xζ ◦ × Xζ ◦ as V2 ∨
Γ Γ Γ 0 → R  R∗ ⊗ Q → (R  R∗ ⊗ Q∨ ) → (R  R∗ ) → O∆ → 0, where the first and second differentials are defined by the inner products i(ξ), i(ξ ∗ ) of tautological homomorphisms as i(ξ)  1 − 1  i(ξ ∗ ), and the last differential is given by the restriction to the diagonal and taking the contraction. Over the end of Xζ ◦ , it is approximated by the complex over C2 /Γ × C2 /Γ, obtained by replacing R by (R ⊗ OC2 )Γ the regular representation considered as a flat bundle over C2 /Γ. The differentials are given by replacing ξ by (z1 , z2 ). In order to distinguish two ‘Γ’ in C2 /Γ × C2 /Γ with ‘Γ’ already appeared in the complex, we denote the former by Γleft,1 , Γleft,2 . The complex lifts to a Γleft,1 × Γleft,2 -equivariant complex on C2 × C 2 : Γ

Γ

0 → (R ⊗ R∗ ) ⊗ OC2 ×C2 → (R ⊗ R∗ ⊗ Q∨ ) ⊗ OC2 ×C2 Γ

→ (R ⊗ R∗ ) ⊗ OC2 ×C2 →

M

Oγ∆ → 0,

(3.4)

γ∈Γleft,1

where γ∆ = {(z, z 0 ) ∈ C2 ×C2 | z 0 = γz}, and the last differential is given by taking the character values at (γ, 1) ∈ Γleft,1 ×Γleft,2 . If we forget the Γleft,1 ×Γleft,2 -action, this is just the direct sum of #Γ-copies of the usual Koszul complex 0 → OC2 ×C2 → C2 ⊗ OC2 ×C2 → OC2 ×C2 → O∆ → 0. This Koszul complex extends to a resolution of the diagonal in P2 × P2 as V2 ∨ 0 → OP2 (−2`∞ )  Q → O(−`∞ )  Q∨ → OP2 ×P2 → O∆ → 0,

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where Q is given by the exact sequence 0 → O(−`∞ ) → OP⊕3 2 → Q → 0. (See [18, ⊕3 (2.3)].) If we put a Γ-equivariant structure on OP2 by considering it as (Q ⊕ R0 ) ⊗ OP2 , we have a natural Γ-equivariant structure on Q, and its restriction to C2 is naturally isomorphic to Q ⊗ OC2 ∼ = Q∨ ⊗ OC2 . Therefore (3.4) extends to P2 × P2 as V2 ∨
Γ Γ 0 → R ⊗ O(−2`∞ )  R∗ ⊗ Q → (R ⊗ O(−`∞ )  R∗ ⊗ Q∨ ) M Γ Oγ∆ → 0. (3.5) → (R ⊗ R∗ ) ⊗ OP2 ×P2 → γ∈Γleft,1

¯¯ζ ◦ by gluing the trivial bundle Q ⊗ OX ◦ and We construct a vector bundle Q on X ζ the restriction of Q to a neighborhood of P1 in P2 . Then we have the resolution of ¯¯ζ ◦ : the diagonal of X V2 ∨
Γ Γ 0 → R(−2`∞ )  R∗ ⊗ Q → (R(−`∞ )  R∗ ⊗ Q∨ ) Γ

→ (R  R∗ ) → O∆ → 0. (3.6) 4. Study of the Stability Condition The purpose of this section is to prove the following result: Proposition 4.1. Let (B, a, b) ∈ µ−1 (ζC◦ ) and consider the complex (1.7). We consider σ, τ as linear maps on the fiber at a point in Xζ ◦ . Then (1) (B, a, b) is ζR -stable if and only if σ is injective possibly except finitely many points and τ is surjective at any point. (2) (B, a, b) is ζR◦ -semistable if and only if σ is injective and τ is surjective possibly except finitely many points. (3) (B, a, b) is ζR◦ -stable if and only if σ is injective and τ is surjective at any point. Remark 4.2. The ‘if’ part of (3) was originally proved in [16, Th. 3.7]. The proof depends on the Hitchin–Kobayashi correspondence for ALE spaces proved by Bando [3]. Our proof here is more algebraic, in particular avoids the use of a nonlinear PDE. 4(i). Proof of Proposition 4.1 for the ‘only if ’ part. (1) We first show that τ is surjective if (B, a, b) is ζR -stable. The assertion for σ will be proved during the proof of (2). Take ξ ∈ Xζ ◦ and suppose η is in the kernel of τξ∗ . We consider Ker η ⊂ V . As ∗ τξ η = 0, it is B-invariant and contains Im a. By the ζR -stability of (B, a, b), we have ζR · dim Ker η 6 ζR dim V. This implies ζR◦ · dim (V / Ker η) > 0 from our choice of ζR . On the other hand, V / Ker η ∼ = Im η ⊂ R∗ξ is ξ ∗ -invariant. From the (−ζR◦ )stability condition of ξ, we have ζR◦ · dim (V / Ker η) 6 0, and the inequality is strict unless V / Ker η is R∗ξ or 0. Combining with the above inequality, we find that the equality must hold, hence V / Ker η is R∗ξ or 0. If V / Ker η = R∗ξ , then we have ζR · dim (V / Ker η) = ζR · δ < 0

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from our choice of ζR . But it contradicts with the ζR -stability of (B, a, b) as it violates the inequality above. Therefore we must have V / Ker η = 0, i. e., η = 0. Therefore Ker τξ∗ = 0, i. e., τ is surjective everywhere. (2) We show that σ is injective except finitely many points if (B, a, b) is ζR◦ semistable. We consider L(R∗ , V ) = L(V ∗ , R). Suppose Ker σξ 6= 0 for ξ ∈ Xζ ◦ and take 0 6= η ∈ Ker σ. Then Ker η ⊂ V ∗ is B ∗ -invariant and contains Im b∗ . Then (−ζR◦ )semistability of (B ∗ , a∗ , b∗ ) implies ζR◦ ·dim Ker η > ζR◦ ·dim V, i. e., ζR◦ ·dim Im η 6 0. On the other hand, Im η ⊂ Rξ is ξ-invariant. Then the (−ζR◦ )-stability of ξ implies ζR◦ ·dim Im η > 0. This is the opposite of the above inequality, hence we have the equality. Therefore from the stability condition, we must have either Im η = 0 or Im η = Rξ . But the former possibility is killed thanks to our assumption η 6= 0. Therefore η is surjective and η ∗ ∈ L(R∗ξ , V ) is injective. Take a point ξ 0 ∈ Xζ ◦ different from ξ and consider an exact sequence 0 → L(R∗ξ0 , R∗ξ ) → L(R∗ξ0 , V ) → L(R∗ξ0 , V / Im η ∗ ) → 0. By [9, Lem. 3.8] the restriction of σξ0 to L(R∗ξ0 , R∗ξ ) is injective. Therefore if σ is not injective at ξ 0 , then σ corresponding to the data V / Im η ∗ is not injective at ξ 0 either. We next claim that (B, a, b) on the quotient V / Im η ∗ is ζR◦ -semistable. Then we repeat the above argument for V / Im η ∗ if Ker σξ0 6= 0 for ξ 0 ∈ Xζ ◦ , and continue. As the dimension decreases, this procedure eventually stops after finite number of steps. Suppose that S ⊂ V / Im η ∗ is B-invariant and is contained in Ker b. We consider its inverse image Se in V . It is also B-invariant and is contained in Ker b. The ζR◦ -semistability of (B, a, b) on V implies ζR◦ · dim Se 6 0. As ζR◦ · Im η ∗ = ζR◦ · δ = 0, we have ζR◦ · dim S 6 0. The case of a B-invariant subspace T containing Im a can be proved in the same way. Therefore (B, a, b) on the quotient V / Im η ∗ is ζR◦ -semistable. The assertion for τ can be proved in the same way by studying what happens with V / Ker η = R∗ξ in the proof of (1). (3) As in the proof of (1), we take η ∈ Ker τξ∗ . By the ζR◦ -stability of (B, a, b), we have ζR◦ · dim Ker η 6 ζR◦ dim V . But we also have the opposite inequality, as in the proof of (1). Therefore the equality holds. By the ζR◦ -stability again, we have Ker η = V , i. e., Ker τξ∗ = 0. The assertion for σ can be proved in the same way. Thus we complete the proof of the ‘only if’ part of Proposition 4.1.  Recall that the ζR◦ -stability for M (v, 0) was slightly modified from that for M (v, w). The analog of the above (3) is the following: Corollary 4.3. Suppose 0 6= v satisfies ζ · v = 0. Suppose B ∈ µ−1 (ζC◦ ) ⊂ M (v, 0) is ζR◦ -stable. Then v = δ and B corresponds to a point ξ in Xζ ◦ . Moreover, (1) σ is injective and τ is surjective at ξ 0 6= ξ (2) σ has the 1-dimensional kernel consisting of scalar endmorphisms at ξ. (3) The image of τ is the codimension 1 subspace of trace-free endmorphisms at ξ.

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Here we choose the isomorphism R∗ξ ∼ = V so that a representative of ξ ∗ is equal to B in (2), (3), hence scalar endmorphisms and trace-free endmorphisms make sense. Remark 4.4. This corollary was stated in [9, Prop. 9.2(ii)], but the proof was rather sketchy. Proof. Consider the complex (1.7) with W = 0. As ζR◦ -stability implies the ζR stability, τ is surjective thanks to Proposition 4.1(1). We assume σ is also injective everywhere. Then the complex is exact as the alternating sum of dimensions of terms is 0. Note also that the complex (1.7) extends to the compactification as in (3.1), and is also exact at infinity (this is true, regardless of the stability condition). Then the alternating sum of the Chern character vanishes, that is 0 = ch(L(R∗ , V )(−`∞ )) + ch(L(R∗ , V )(`∞ )) − ch(E(R∗ , V )) X X = C ij dim Vi ch(Rj ) + 2 rank Ri dim Vi ch2 (O(`∞ )), i,j

i

where C ij is the affine Cartan matrix, and ch2 is the H 4 -part of the Chern character. If we restrict this to the un-compactified part Xζ ◦ , ch(Rj ) gives a basis of H ∗ (XP ζ ◦ , Q) (which follows from [9, (2.2)] together with c1 (R0 ) = 0), hence we have i C ij dim Vi = 0. Therefore dim V is a multiple of δ. But if we substitute this back to the above equality, the first term vanishes, so the second term must vanish. We claim ch2 (O(`∞ )) = 1/(2#Γ), in particular, it is nonzero: Note first that we can make the compactification on the universal family over R3 ⊗ Z from the construction of the coordinate system at the end in [8, Proof of Cor. 3.12]. ¯¯ζ ◦ is the minimal resolution Therefore, by the continuity, it is enough to assume X 2 of singularity of P /Γ at the origin. Then O(`∞ ) is the pull-back from the line bundle O(1)/Γ over P2 /Γ and we get the assertion from ch2 (O(1)) = 1/2. Now the vanishing of the second term implies V = 0. This is a contradiction. Therefore σ has a kernel at a point ξ ∈ Xζ ◦ . We now argue as in the proof of Proposition 4.1(2). We have an injective homomorphism η ∗ ∈ L(R∗ξ , V ). But we have ζR◦ · Im η ∗ = 0 as before, hence we must have Im η ∗ = V by the ζR◦ -stability of B. Once we know v = δ and B corresponding to a point ξ ∈ Xζ ◦ , then the proofs of (1), (2), (3) were given in [9, Sec. 3]. But they are also clear from the above consideration.  4(ii). Modifications of definitions. For the proof of the ‘if’ part of Proposition 4.1, we shall use the Harder–Narasimhan filtration and Jordan–H¨older filtration. For this purpose we need a modification of the definition of a data (B, a, b) following [5] and its stability condition following [21]. This will be done in the this subsection. b E) b be the finite graph obtained from We fix dimension vectors v, w. Let (I, (I, E) by adding a new vertex ∞ and wi edges between ∞ and i for each i ∈ I. We choose orientations of new edges as ∞ to i. (See [5, Remarks starting p. 260].) b ∈ Z[Ib] by setting v b∞ = 1 and v bi = v i for i ∈ I. We define a new vector v

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We take and fix a base of Wi for each i. Then a point (B, a, b) ∈ M (v, w) is b For a complex parameter identified with a point in M (b v ). We denote it by B. ζC ∈ C[I] and a stability parameter ζR ∈ R[I], we define ζbC ∈ C[Ib], ζbR ∈ R[Ib] so that (ζbR or C )i = (ζR or C )i for i ∈ I and b = 0. ζbR or C · v These conditions uniquely fix ζbR or C . Then MζR ,ζC (v, w) can be identified with MζbR ,ζbC (b v ). b ∈ M (b Let (B, a, b) ∈ M (v, w) and let B v ) be the corresponding data. Let b b acts. Let S be an b on which B Vb be the I-graded vector space of dimension v b I-graded vector subspace S of Vb with dim S∞ = 0 or 1, which is invariant under b When dim S∞ = 0, it means that we have the I-graded subspace L Si such B. i∈I that Si ⊂ L Ker bi for each i. When dim S∞ = 1, it means that we have the I-graded subspace i∈I Si such that Si ⊃ Im ai for each i. Thus two cases appeared in the original definition of the stability condition can be expressed in a single way, i. e., b ∈ M (b (B, a, b) is ζR -semistable if and only if B v ) satisfies the following: if S is an b b b we have ζbR · dim S 6 0. I-graded subspace of V invariant under B, We modify the definition of the stability condition to a slightly different form, b vector space. (We only need to consider following [21]. Let Vb 0 be another I-graded 0 dim V∞ = 0 or 1, but the definition makes sense without this condition.) Assume Vb 0 6= 0, we define the slope of Vb 0 by b b0 def ζR · dim V θ(Vb 0 ) = P . b0 i∈Ib dim Vi b ∈ M (Vb 0 ) is ζR -semistable if we have We say a point B b 6 θ(Vb 0 ) θ(S) b b If the inequality is strict for any nonzero I-graded subspace Sb invariant under B. 0 b b b unless S = V , we say B is ζR -stable. If we apply this definition to the original Vb , it is equivalent to the original definition as ζbR · dim Vb = 0. When ζbR · dim Vb 6= 0, we define a new parameter ζbR0 by ζbR − θ(Vb 0 )(1, 1, . . . , 1) ∈ R[Ib], then the above stability is equivalent to the original stability condition for the parameter ζbR0 , which satisfies ζbR0 · dim Vb 0 = 0 by its definition. Therefore this definition does not yield anything new for a fixed Vb 0 , however it is convenient when we discuss several Vb 0 ’s at the same time, in particular, when we discuss the Harder–Narasimhan filtration, as pointed out by [21]. b as a representation of the quiver (I, b H) b (with obvious H), b the We consider B b Harder–Narasimhan filtration and the Jordan–H¨older filtration exists on B thanks to [21]. In the subsequent subsections, we shall use this stability condition for the parameter ζR◦ .

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b corresponding to 4(iii). Proof of the ‘if ’ part of Proposition 4.1(2). Take B (B, a, b) ∈ µ−1 (ζC◦ ). We take the Harder–Narasimhan filtration with respect to the stability parameter ζR◦ : Vb = Vb 0 ⊃ Vb 1 ⊃ · · · ⊃ Vb m ⊃ Vb m+1 = 0 b is ζ ◦ -semistable for k = 0, . . . , m and θ(Vb 0 /Vb 1 ) < θ(Vb 1 /Vb 2 ) < such that grVb k (B) R b is the representation of the new quiver defined · · · < θ(Vb m /Vb m+1 ). Here grVb k (B) k k+1 b b We assume B b is not ζ ◦ -semistable, on the I-vector space Vb /Vb induced from B. R and hence m > 0, and will lead to a contradiction. We have exactly one k such that dim(Vb k /Vb k+1 )∞ = 1 and dim(Vb l /Vb l+1 )∞ = 0 for l 6= k. If θ(Vb 0 /Vb 1 ) > 0, then all other θ(Vb k /Vb k+1 ) > 0, and we have a contradiction b b V b0 b1 with θ(Vb ) = PζR ·dim b = 0. Therefore we must have θ(V /V ) < 0. Similarly we dim V i

i∈Ib

have θ(Vb m ) > 0. We consider the complex (1.7) for the data corresponding to Vb 0 /Vb 1 . By the assumption that τ is surjective except possibly finitely many points for the data for Vb 0 , the same is true for Vb 0 /Vb 1 . Let us study σ, and suppose that σ has a nonzero kernel at ξ ∈ Xζ ◦ . We take 0 6= η ∗ ∈ Ker σ ⊂ L((V 0 /V 1 )∗ , Rξ ). We have ξη = ηB ∗ , ηb∗ = 0. The (−ζR◦ )-stability condition for ξ implies ζR◦ · dim Im η > 0. b ∗ implies θ(Ker d η) > θ(Vb 0 /Vb 1 ), On the other hand, the (−ζR◦ )-semistability for B 0 1 d ηi = Ker ηi for i ∈ I and Ker d η∞ = (Vb /Vb )∞ . Therefore we get where Ker

=

d η) = θ(Ker

ζbR◦ · dim(Vb 0 /Vb 1 ) − ζR◦ · dim Im η P d i∈Ib dim Ker ηi

P

b 0 /Vb 1 )i

i∈Ib dim(V

P

i∈Ib dim Ker ηi

d

ζ ◦ · dim Im η θ(Vb 0 /Vb 1 ) − P R . d i∈Ib dim Ker ηi

Combining with two inequalities above, we have P b0 b1 i∈Ib dim(V /V )i 0 b1 b θ(V /V ) 6 P θ(Vb 0 /Vb 1 ). d ηi dim Ker b i∈I As η 6= 0, the factor

P

b 0 /V b 1 )i

b dim(V

i∈I P

i∈Ib dim Ker ηi

is greater than 1, hence this inequality con-

d

tradicts with θ(Vb 0 /Vb 1 ) < 0, observed above. Therefore σ is injective everywhere. Next consider the complex (1.7) for the data corresponding to Vb m . By the assumption, σ is injective except possibly finitely many points. Let us study τ , and suppose that τ is not surjective at ξ ∈ Xζ ◦ . We take 0 6= η 0 ∈ Ker τ ∗ ⊂ L(V, R∗ξ ). We have ξ ∗ η 0 = −η 0 B, η 0 a = 0. The ζR◦ -stability condition for ξ ∗ implies b implies ζR◦ · dim Im η 0 6 0. On the other hand, the ζR◦ -stability condition for B 0 m d η 0 ) 6 θ(Vb m ), where Ker d η 0 = Ker η 0 for i ∈ I and Ker d η∞ θ(Ker = Vb∞ . Therefore i i we get P bm ζ ◦ · dim Im η b dim Vi 0 d θ(Ker η ) = P i∈I θ(Vb m ) − P R . d d 0 b dim Ker ηi b dim Ker η i∈I

i∈I

i

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H. NAKAJIMA

Combining with two inequalities above, we have P bm b dim Vi m b θ(Vb m ). θ(V ) > P i∈I 0 d dim Ker η i i∈Ib As η 0 6= 0, the factor

P P

bm i∈Ib dim Vi

i∈Ib dim Ker ηi

is greater than 1, hence this inequality contra-

d

dicts with θ(Vb m ) > 0, observed above. Therefore τ is surjective everywhere. Summarizing two considerations, we find that σ is injective and τ is surjective except finitely many points for both Vb 0 /Vb 1 and Vb m . As we remarked at the m beginning of the discussion, either of the ∞-component (Vb 0 /Vb 1 )∞ or Vb∞ is zero. m b Suppose V∞ = 0. The other case will lead to a contradiction exactly in the same m way. As Vb∞ = 0, the complex (1.7) for Vb m has the Euler characteristic 0. Hence it is exact outside finite number of points. Therefore the alternating sum of the first Chern classes of terms is zero. By the same argument as in the proof of Corollary 4.3, we get X 0= C ij dim Vim c1 (Rj ). i,j 2

As P {c1 (Rj )}j6=m0 is a basis of H (Xζ ◦ , Q)P(see [9, (2.2)]) and c1 (R0 ) = 0, we have = 0 for j 6= 0. But as j δj C ij = 0 and δ0 = 1 6= 0, the same i C ij dim Vi is true for j = 0. Therefore dim V m is in the kernel of C, hence is a multiple of δ. But it implies ζR◦ dim V m = 0 and θ(Vb m ) = 0. This contradicts with the b is ζ ◦ -semistable, and this completes the proof of inequality θ(Vb m ) > 0. Hence B R Proposition 4.1(2). b corresponding 4(iv). Proof of the ‘if ’ part of Proposition 4.1(3), (1). Take B to (B, a, b). We assume that σ, τ satisfy the condition in either (1) or (3). Then b the condition for (2) is satisfied in either cases, hence we already know that B ◦ is ζR -semistable, thanks to the proof in the previous subsection. We consider the Jordan-H¨ older filtration with respect to the stability parameter ζR◦ : Vb = Vb 0 ⊃ Vb 1 ⊃ · · · ⊃ Vb m ⊃ Vb m+1 = 0 b is ζ ◦ -stable for k = 0, . . . , m and θ(Vb 0 /Vb 1 ) = θ(Vb 1 /Vb 2 ) = such that grVb k (B) R · · · = θ(Vb m /Vb m+1 ) = 0. Note that the ζR◦ -stability condition is equivalent to the original one, as θ(Vb ) is zero. We have exactly one k such that dim(Vb k /Vb k+1 )∞ = 1 and others are dim(Vb l /Vb l+1 )∞ = 0 (l 6= k). By Corollary 4.3, such Vb l /Vb l+1 corresponds to a point in Xζ ◦ . Let us consider Vb 0 /Vb 1 . As τ is surjective for the data for Vb 0 , the same is true for 0 Vb /Vb 1 . But by Corollary 4.3(3), this is not possible if (Vb 0 /Vb 1 )∞ = 0. Therefore the above k must be 0, and all others correspond to points in Xζ ◦ . Suppose further that σ is injective for the data Vb 0 . The same is true for Vb m . m By Corollary 4.3(2), this is not possible if Vb∞ = 0. This means that we must ◦ b have m = 0, i. e., B is ζR -stable. This completes the proof of the ‘if’ part of Proposition 4.1(3).

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We now start the proof of the ‘if’ part of Proposition 4.1(1). Let S be an b b b b I-graded subspace of V , which is invariant under B. We consider S ∩ V k /S ∩ Vb k+1 . b hence we have This is invariant under B,
(4.5) ζR◦ · dim S ∩ Vb k /S ∩ Vb k+1 6 0 b by the ζR◦ -stability of grVb k (B). Taking the sum over all k = 0, . . . , m, we get ζR◦ · dim S 6 0. If the inequality in (4.5) is strict for some k, then ζR◦ · dim S < 0. As ζR is arbitrary close to ζR , we have ζR · dim S < 0. Thus the inequality for the ζR -stability holds. Hence it is enough to assume that the inequality in (4.5) is the b we have S ∩ Vb k /S ∩ Vb k+1 is equality for any k. By the ζR◦ -stability of grVb k (B), k b k+1 b either 0 or V /V . 0 We first consider the case S∞ = 0. As Vb∞ = C, we have S ∩ Vb 0 /S ∩ Vb 1 = 0. For k 6= 0, we have
ζR · dim S ∩ Vb k /S ∩ Vb k+1 = 0 or ζR · δ. Summing up over k = 0, 1, . . . , m, we have ζR · dim S ∈ Z>0 ζR · δ ⊂ R60 by our choice ζR · δ < 0. Therefore the inequality for the ζR -stability holds in this case. We finally suppose S∞ = C. As (S ∩ Vb 1 )∞ = 0, we have S ∩ Vb 0 /S ∩ Vb 1 = Vb 0 /Vb 1 . We consider the following exact sequence 0 → L(V k /S∩V k +V k+1 , R∗ ) → L(V k /S∩V k , R∗ ) → L(V k+1 /S∩V k+1 , R∗ ) → 0, where V k is the I-graded part of Vb k . We have either V k /S ∩V k +V k+1 = V k /V k+1 or = 0. We assume S 6= Vb and take the first k > 0 such that S ∩ Vb k /S ∩ Vb k+1 = 0. As V k /S ∩ V k + V k+1 = V k /V k+1 , we have 0 6= Ker τ ∗ | L(V k /S ∩ V k + V k+1 , R∗ ) by Corollary 4.3. By the above exact sequence, we have 0 6= Ker τ ∗ | L(V k /S ∩V k , R∗ ). Then we consider the above exact sequence with k replaced by k − 1. As V k−1 /S ∩ V k−1 + V k = 0 by our choice of k, we have 0 6= Ker τ ∗ | L(V k−1 /S ∩ V k−1 , R∗ ). Repeating this argument, we get 0 6= Ker τ ∗ | L(V /S, R∗ ). As L(V /S, R∗ ) ⊂ L(V, R∗ ), this contradicts with the assumption that τ is surjective. Hence we must have S = Vb in this case. Therefore the proof of the ζR -stability is completed. 5. Proof of the Main Theorem 5(i). From ADHM data to a torsion free sheaf. Suppose I-graded vector spaces V , W and data (B, a, b) ∈ M (V, W ) satisfying µ(B, a, b) = ζC◦ and the ζR stability condition are given. We consider the complex (1.7). By Proposition 4.1(1), τ is surjective and σ is injective outside a finite set. Therefore E = Ker τ / Im σ is a torsion-free sheaf on Xζ ◦ . Indeed, suppose we have e ∈ Ker τ with f e ∈ Im σ for some 0 6= f ∈ O. As E is locally free outside a finite set, e = σ(ξ) for some ξ possibly defined only outside a finite set. But ξ extends by Hartog’s theorem. We have two exact sequences 0 → L(R∗ , V ) → Ker τ → E → 0, 0 → Ker τ → E(R∗ , V ) ⊕ L(R∗ , W ) → L(R∗ , V ) → 0

720

H. NAKAJIMA

¯¯ζ ◦ of sheaves. We have already checked that it extends to the compactification X and has the correct Γ-module w in Section 3. The above two exact sequences can ¯¯ζ ◦ if we twist terms by O(`∞ ) appropriately. Then it is clear that be extended to X E satisfies the correct Chern class (1.9). 5(ii). From a torsion free sheaf to ADHM data. First note that standard facts from the theory of coherent sheaves on a nonsingular surface, e.g., the Serre duality, the vanishing of H p with p > 2, the fact that the double dual of a coherent ¯¯ζ ◦ , as it is locally sheaf is a vector bundle, etc, extend to the case of our orbifold X a nonsigular surface with a finite group action. For example, the Serre duality for orbifolds is proved just as it is proved for complex manifolds using the Hodge decomposition. The canonical sheaf is in the sense of orbifolds; see e.g., [2]. ¯¯ζ ◦ together with a framing Φ : E|` ∼ Let E be a torsion free sheaf on X ∞ = L ⊕wi ⊗ O . We then define R ` ∞ i i ¯¯ζ ◦ , E ⊗ R∗ (−2`∞ )), V = H 1 (X ¯¯ζ ◦ , E ⊗ R∗ (−`∞ )), V 0 = H 1 (X e = H 1 (X ¯¯ζ ◦ , E ⊗ Q∗ ⊗ R∗ (−`∞ )). W These are Γ-modules, where the action comes from the factor R∗ or Q∗ ⊗ R∗ . These cohomology groups are generalization of ones used for P2 in [18, Ch. 2]. By a similar argument as in [18, Lemma 2.4], we have vanishing of other cohomology groups H q for q = 0, 2 corresponding to above ones. The only difference is that ¯¯ζ ◦ , E ⊗ R∗ (−k`∞ )) = 0 for sufficiently large k (instead of the Serre we use H 0 (X vanishing theorem), and then the Serre duality ¯¯ζ ◦ , E ⊗ R∗ (k`∞ )) ∼ ¯¯ζ ◦ , E ∗ ⊗ R((−3 − k)`∞ ))∨ H 2 (X = H 0 (X if E is locally free (in the sense of orbifolds). From the cohomology exact sequence from 0 → E → E ∨∨ → E ∨∨ /E → 0, we can reduce the vanishing of H 2 to the case when E is locally free. Also by the same argument in [18, Ch. 2], we have the identification V ∼ = V 0, ∼ e W = Q ⊗ V ⊕ W , where W is the fiber at a point in `∞ . We use the framing Φ at this stage. Then from the resolution of the diagonal (3.6), we find that E is the middle cohomology group of the complex (3.1). (More precisely, we consider p∗2 E(−`∞ ) ⊗ O∆ and calculate its p1∗ -image. See [18, §2.1].) The ADHM data (B, a, b) are read from σ, τ , thus they are endmorphisms on cohomology groups as in Section 1(iii). The equation µ(B, a, b) = ζC is the consequence of τ σ = 0. The ζR -stability is the consequence of Proposition 4.1(1) and the torsion freeness of E: τ is surjective and σ is injective outside the support of E ∨∨ /E. From the above description, it is clear that the composite of two maps torsion free sheaf E → ADHM data (B, a, b) → torsion free sheaf E 0 is identity. Thus we have the completeness of the ADHM description. (This proof of the completeness is similar to the proof in [9, §6.b)], but is slightly different as we use a different cohomology groups.)

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5(iii). Uniqueness. We prove the composite of two maps ADHM data (B, a, b) → torsion free sheaf E → ADHM data (B 0 , a0 , b0 ) is identity. This means that the monad description of E as Ker τ / Im σ in (3.1) is unique. By a general result for monads (see [20, Ch. II, Lem. 4.1.3]), this assertion follows from the following vanishing result: ¯¯ζ ◦ , Ri ⊗ R∗ (−q`∞ )) = 0 for q = 1, 2 and arbitrary Lemma 5.1. We have H p (X j p, i, j. Proof. The proof is just the adaptation of one in [9, Lem. 7.1] in our current setting. From the definition of the inverse transform in Section 5(ii), the assertion means the uniqueness for the special case E = Ri . In fact, Ri is the middle cohomology of (3.1) if we put V = 0, W = Ri . Then from the formula for V ∼ = V 0 in Section 5(ii), the uniqueness of the description implies the required vanishing. e for E = Ri as in Section 5(ii). We already know that Let us define V , V 0 , W 0 ∼ e V = V and W = Q ⊗ V ⊕ W and W is the fiber at a point in `∞ , hence = Ri . Let v = dim V , w = dim W . By Section 5(ii) we know that E is the middle cohomology of (3.1) with these V , W . Therefore the Chern classes are given by the formula (1.9). Substituting E = Ri in the left hand side of (1.9), we get u = w and v = 0. (A similar argument was already used in the proof of Corollary 4.3.)  References [1] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562 [2] W. L. Baily, Jr., The decomposition theorem for V -manifolds, Amer. J. Math. 78 (1956), 862–888. MR 0100103 [3] S. Bando, Einstein–Hermitian metrics on noncompact K¨ ahler manifolds, Einstein metrics and Yang-Mills connections (Sanda, 1990), Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 27–33. MR 1215276 [4] V. Baranovsky, V. Ginzburg, and A. Kuznetsov, Quiver varieties and a noncommutative P2 , Compositio Math. 134 (2002), no. 3, 283–318. MR 1943905 [5] W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257–293. MR 1834739 [6] T. Gocho and H. Nakajima, Einstein–Hermitian connections on hyper-K¨ ahler quotients, J. Math. Soc. Japan 44 (1992), no. 1, 43–51. MR 1139657 [7] A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461 [8] P. B. Kronheimer, The construction of ALE spaces as hyper-K¨ ahler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. MR 992334 [9] P. B. Kronheimer and H. Nakajima, Yang–Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263–307. MR 1075769 [10] A. Kuznetsov, Quiver varieties and Hilbert schemes, Preprint arXiv:math/0111092 [math.AG]. [11] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993), no. 2, 417–466. MR 1205451 [12] G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141–182. MR 1623674 [13] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 [14] K. Nagao, Quiver varieties and Frenkel–Kac construction, Preprint arXiv:math/0703107 [math.RT].

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