On McKay correspondence for a finite small subgroup of GL(2, ℂ)

Akira Ishii

1 Introduction

In this paper, we shall generalize a result of Ito and Nakamura [IN], McKay correspondence for simple singularities via Hilbert scheme of G-orbits(see below), to the case of general quotient surface singularities without case-by-case studies. Our results, which were conjectured by Riemenschneider [R2], will be obtained by using results of Wunram [W] which we shall reformulate in terms of derived functors discussed by Bridgeland, King and Reid [BKR].

Before going into details, let us review the history of understanding of twodimensional McKay correspondence. McKay correspondence [M] for a finite subgroup G of SL(2, ℂ) gives a bijective correspondence of non-trivial irreducible representations of G with irreducible exceptional curves of the minimal resolution of the simple singularity ℂ^{2}/G. Since this observation depends on explicit computations on group representations, efforts were made to understand this phenomenon geometrically. Gonzalez-Sprinberg and Verdier [GV] described McKay correspondence as an isomorphism of Grothendieck groups with case-by-case checking. Then Artin and Verdier [AV] gave a short proof using properties of reflexive modules over rational surface singularities. Hilbert scheme of G-orbits (G-Hilb), introduced by Nakamura, allows us a moduli-theoretic approach to the problem. Ito and Nakamura [IN], noting that G-Hilb is the minimal resolution, gave another way to associate a representation to an irreducible component of the exceptional locus. They studied I/mI as a representation of G for exceptional [I] ∈ G-Hilb with case-by-case analysis. Now McKay correspondence has been understood as an equivalence of derived categories by Kapranov and Vasserot [KV] and generalized to higher dimensions including all the cases of finite subgroups of SL(3, ℂ) by Bridgeland, King and Reid [BKR].

In this paper, we shall generalize the original situation to the case of a finite small subgroup G of GL(2, ℂ). In this case, there are less irreducible exceptional curves than non-trivial irreducible representations in general and we cannot expect equivalence of derived categories nor isomorphism of Grothendieck groups. But Wunram [W] obtained a bijective correspondence of non-trivial special irreducible representations (see Section 4) with irreducible components of the exceptional locus, by generalizing the arguments on reflexive modules of Artin and Verdier [AV], and of Esnault [E]. On G-Hilb side, when G is a cyclic group, Kidoh [K] gave a concrete description of G-Hilb in terms of continued fraction expansions. In particular, she proved that G-Hilb is the minimal resolution in this case. Riemenschneider [R2], looking at her result, pointed out that we can obtain Ito-Nakamura type correspondence of irreducible exceptional curves with non-trivial special representations in the case of cyclic groups. Moreover, he conjectured that these should be true for general finite small subgroups. In this paper, we shall prove his conjecture by using the adjointness of two derived functors discussed by Bridgeland, King and Reid [BKR], and the description of a functor via results of Wunram [W], with basic properties of reflexive modules over rational surface singularities.

Let us state the results more precisely. Let G ⊂ GL(2, ℂ) be a finite small subgroup, that is, a finite subgroup acting freely on ℂ^{2} \ {0}. Recall that a Gcluster Z is a finite subscheme of ℂ^{2} such that Z is G-invariant and such that H^{0}(O_{Z}) is the regular representation of G. Let G- Hilb(ℂ^{2}) be the moduli space of G-clusters on ℂ^{2}, which is a finite union of connected components of the G-fixed point locus (Hilb^{#G}(ℂ^{2}))^{G} of Hilb^{#G}(ℂ^{2}). Then G- Hilb(ℂ^{2}) is smooth since so is Hilb^{#G}(ℂ^{2}) (Fogarty [F]) and the G-action can be locally linearized. We don’t know the connectedness of G- Hilb(ℂ^{2}) (which will be proved in section 7) at this stage and so let Y denote the connected component dominating ℂ^{2}/G. In the case G ⊂ SL(2, ℂ), Ito and Nakamura proved the following. (Crawley-Boevey [C] gave a proof of this result without case-by-case checking, by using the theory of preprojective algebras.)

Theorem (Ito-Nakamura, [IN]). When G ⊂ SL(2, ℂ), the following hold.

1. Y is the minimal resolution of ℂ^{2}/G.

2. There is a bijection of irreducible exceptional curves {E_{1}, …, E_{n}} with nontrivial irreducible representations {ρ_{1}, …, ρ_{n}} such that

I_{y}/(mI_{y} + 𝔫) ≅ \begin{cases}ρ_{i} & if y ∈ E_{i} and y ∈ E_{j}(j ≠ i) \\ ρ_{i} ⊕ ρ_{j} & if y ∈ E_{i} ∩ E_{j} \end{cases}

where 𝔫 = (m^{G})O_{ℂ^{2}}. Moreover this bijection coincides with original McKay’s correspondence.

For general finite small G ⊂ GL(2, ℂ), Riemenschneider [R2] conjectured that both the assertions hold, where in 2 we should choose special representations introduced by Wunram in [W] and noted that the conjecture is true in the case of cyclic groups due to the work of Kidoh [K].

Both parts of his conjecture will be proved in this paper. We shall prove that Y is the minimal resolution in Theorem 3.1 where we use a basic property of a reflexive module. The correspondence as in 2 of irreducible components with special representations will be given in Theorem 6.2. We shall also determine the socle of O_{Z} in Theorem 6.3. For these, we shall consider the derived categories D_{c}(Y) of coherent sheaves with compact support on Y and D_{c}^{G}(ℂ^{2}) of G-equivariant coherent sheaves with compact support on ℂ^{2}. By using the universal family of G-clusters, we can define an adjoint pair of functors between these derived categories as in

[KV] and [BKR]. The most essential part is Theorem 5.1, which is a reformulation of Wunram’s ‘multiplication formula’ on reflexive modules. It can be seen to be McKay correspondence on derived category level (though not an equivalence) and in fact is a generalization of a result in [KV]. The connectedness of G-Hilb will also be proved by using the adjointness of these functors in Theorem 7.1.

Acknowledgements The author is grateful to Oswald Riemenschneider, whose lecture in Kyoto led him to the problem, for comments and interest. This work is based on invaluable discussions with Yukari Ito, Rie Kidoh and Hiraku Nakajima in a seminar at Tokyo Metropolitan University where the author introduced [BKR] to which he owes a lot. He also expresses his thanks to Alastair King for useful suggestions.

2 Representations, reflexive modules and full sheaves

Let G ⊂ GL(2, ℂ) be a finite small subgroup, where finite subgroup is said to be small if it acts freely on ℂ^{2} \ {0}. X = ℂ^{2}/G denotes the associated quotient singularity with quotient map

π : ℂ^{2} → X

and we denote by

ϕ : X̃ → X

the minimal resolution with fundamental cycle F. In this section, we review the correspondence among representations of G, reflexive modules on X and full sheaves on X̃. To establish these correspondences as bijections without inessential arguments, we should consider the germ of X at 0 and its fiber instead of X and X̃ respectively. But by abuse of terminology, we say ‘reflexive modules on X’ and ‘full sheaves on X̃’ even when we should consider the germs.

Let ρ be a (finite dimensional) representation of G. We define the associated O_{X}-module as

M := (O_{ℂ^{2}} ⊗_{ℂ} ρ^{∗})^{G},

where we use the contragradient ρ^{∗} of ρ according to [GV]. Then M is a reflexive module on X. The reflexivity means that the canonical map of M to its double dual M^{∨∨} is an isomorphism. This is equivalent to the condition M ≅ H^{0}(X \ {0}; M|_{X\{0}}) since X is normal and 2-dimensional. In the above way, we have a bijective correspondence of representations of G with reflexive modules on X.

For a reflexive module M on X, we define a coherent sheaf M̃ = ϕ^{∗}M/torsion on X̃. Then we have the following lemma (see [E]).

Lemma 2.1. M̃ has the following properties.

1. M̃ is a locally free sheaf,

2. M̃ is generated by global sections,

3. R^{1}ϕ_{∗}((M̃)^{∨} ⊗ KX̃) = 0.

This motivates the following definition.

Definition 2.2. ([E]) A locally free sheaf M on X̃ with the above properties is called a full sheaf.

M ↦ M̃ gives a bijective correspondence of reflexive modules with full sheaves on X̃ (see [E]).

Thus we have bijective correspondences of representations, reflexive modules and full sheaves. In this correspondence, an irreducible representation corresponds to an indecomposable reflexive module and to an indecomposable full sheaf.

3 Minimality of Y

Let G ⊂ GL(2, ℂ) be a small finite subgroup and Y ⊆ G- Hilb(ℂ^{2}) be as in the introduction. First we show the following.

Theorem 3.1. Y is the minimal resolution of X = ℂ^{2}/G.

Proof. Let π : ℂ^{2} → X = ℂ^{2}/G be the quotient map and let ϕ : X̃ → X be the minimal resolution. If we identify ℂ^{2} with the graph of π in ℂ^{2} ×X, then the second projection makes π_{∗}O_{ℂ^{2}} a quotient of the O_{X}-algebra O_{X} ⊗_{ℂ} H^{0}(O_{ℂ^{2}}) ≅ O_{X}[x_{1}, x_{2}] by its G-invariant ideal (sheaf). Since ϕ^{∗} is right exact, ϕ^{∗}π_{∗}O_{ℂ^{2}} is a quotient of OX̃[x_{1}, x_{2}] by its ideal sheaf. The torsion part (as an OX̃-module) of ϕ^{∗}π_{∗}O_{ℂ^{2}} must be preserved by its local endomorphisms as an OX̃-module and in particular by multiplications by local sections of OX̃[x_{1}, x_{2}]. This shows that the torsion part is an OX̃[x_{1}, x_{2}]-submodule and hence { \atop π_{∗}O_{ℂ^{2}}} is again a quotient of OX̃[x_{1}, x_{2}] by its G-invariant ideal sheaf. { \atop π_{∗}O_{ℂ^{2}}} is locally free by Lemma 2.1 and therefore it gives rise to a flat family of G-clusters parameterized by X̃. Thus we have a morphism X̃ → G- Hilb(ℂ^{2}) which induces an isomorphism away from the fibers over 0 ∈ ℂ^{2}/G. Since X̃ is the mininal resolution and Y is non-singular, we obtain X̃ ≅ Y.

If we denote by Z ⊂ Y × ℂ^{2} the universal subscheme, we have the following commutative diagram.

\begin{matrix}Z & ⟶^{q} & ℂ^{2} \\ ^{p}↓ & & ↓^{π} \\ Y & ⟶^{f} & X = ℂ^{2}/G \end{matrix}  (3.1)

We put R = p_{∗}O_{Z}. Then R is a locally free sheaf on Y since p is flat and finite.

Corollary 3.2. R is a full sheaf. Moreover, if ρ is a representation of G, then (p_{∗}(q^{∗}(O_{ℂ^{2}} ⊗ ρ^{∗})))^{G} is also a full sheaf.

Proof. In the proof of the theorem, we saw that R ≅ { \atop π_{∗}O_{ℂ^{2}}} and hence is a full sheaf. We can decompose R with respect to the G-action as

R = ⨁_{ρ} R_{ρ} ⊗ ρ

where ρ runs through all the irreducible representations. If ρ is an irreducible representation, then (p_{∗}(q^{∗}(O_{ℂ^{2}} ⊗ ρ^{∗})))^{G} ≅ R_{ρ} is a direct summand of R and hence is a full sheaf.

4 Wunram’s special representations and the multiplication formula

Generalizing the original McKay correspondence to the case of finite small G ⊂ GL(2, ℂ), Wunram [W] associated to each irreducible exceptional curve an indecomposable reflexive module, which are said to be special by Riemenschneider [R1], [R3]. Moreover, Wunram generalized the ‘multiplication formula’ by Esnault and Knörrer [EK] which describes the tensor property of the original McKay correspondence in terms of reflexive modules and full sheaves. In this section, we recall these Wunram’s results which are essential to our argument. Let f : Y → X = ℂ^{2}/G be the minimal resolution and denote by F the fundamental cycle.

Definition 4.1 ([R1]). A full sheaf M on Y is said to be special if

R^{1}f_{∗}(M^{∨})(= H^{1}(M^{∨})) = 0.

A reflexive module on X (or representation of G) is said to be special if the corresponding full sheaf is special.

Note that if G ⊂ SL(2, ℂ), then all the reflexive modules are special (see Lemma 2.1).

Theorem 4.2 (Wunram, [W]). There is a bijective correspondence of (isomorphism classes of) non-trivial indecomposable special reflexive modules M_{i} with irreducible exceptional curves E_{i} such that c_{1}(M̃_{i}).E_{j} = δ_{ij}. The rank of M_{i} is r_{i} = c_{1}(M̃_{i}).F, the coefficient of E_{i} in the fundamental cycle F.

Here we recall the construction of M_{i} in the above theorem. Let D_{i} be a curve on Y which intersects transversely to the exceptional locus at one point in E_{i}. Take a minimal generator φ_{1}, …, φ_{r} of f_{∗}O_{D} as an O_{X}-module and let N_{i} be the kernel of the surjection O_{Y}^{⊕r} → O_{D} given by (φ_{1}, …, φ_{r}). Then it is easy to see that M_{i} := N_{i}^{∨} is the desired special full sheaf. As a consequence, M_{i} is constructed via an exact sequence

0 → (M̃_{i})^{∨} → O_{Y}^{⊕r} → O_{D_{i}} → 0.  (4.1)

Next we shall review the ‘multiplication formula’, Theorem 1.3 in [W]. Let

0 → K_{ℂ^{2}} → Ω_{ℂ^{2}}^{1} → O_{ℂ^{2}} → O_{0} → 0  (4.2)

be the Koszul complex of m = m_{0,ℂ^{2}}, which is G-equivariant. For an irreducible representation ρ of G and the corresponding reflexive module M, tensor the Koszul complex (4.2) with ρ^{∗} and take the G-invariant part. Then we obtain a complex(the Auslander-Reiten sequence)

0 → τ(M) → N_{M} → M → 0  (4.3)

of reflexive modules on X, where τ(M) = (M ⊗ ω_{X})^{∨∨} and N_{M} = (M ⊗ Ω_{X}^{1})^{∨∨}. Note that (4.3) is exact unless M ≅ O_{X} and if M = O_{X}, then we have an exact sequence

0 → τ(O_{X})(= ω_{X}) → N_{O_{X}} → O_{X} → O_{0} → 0.  (4.4)

The multiplication formula is stated as follows.

Theorem 4.3 (Wunram, [W]). For an indecomposable reflexive module M, we have

c_{1}(Ñ_{M}) − c_{1}(M̃) − c_{1}({ \atop τ(M)}) = \begin{cases}−F & if M ≅ O_{X} \\ E_{i} & if M = M_{i} is special with c_{1}(M̃_{i}).E_{i}=1 \\ 0 & if M is non-special \end{cases}

We shall give the proof of this after restating the theorem in terms of a derived functor in the next section.

5 Multiplication formula in terms of a derived functor

In this section, we translate the multiplication formula into derived category language and moreover give a proof of it because it is essential to our arguments. Let D_{c}(Y) denote the derived category of coherent sheaves with compact support on Y and let D_{c}^{G}(ℂ^{2}) denote the derived category of G-equivariant coherent sheaves with compact support on ℂ^{2} (see [BKR].) We define a functor Ψ : D_{c}^{G}(ℂ^{2}) → D_{c}(Y) by

Ψ(−) := [p_{∗}𝕃q^{∗}(−)]^{G}.

Note p is finite and hence p_{∗} is exact. Then the refined version of multiplication formula can be stated as below. This is a generalization of [KV] Theorem 2.3.

Theorem 5.1. For an irreducible representation ρ of G,

Ψ(O_{0} ⊗ ρ^{∗}) ≅ \begin{cases}O_{E_{i}}(−1)[1] & if ρ = ρ_{i} is special and non-trivial, \\ O_{F} & if ρ is trivial, \\ 0 & if ρ is non-special, \end{cases}

where O_{0} is the structure sheaf of reduced {0} ⊂ ℂ^{2}, ρ_{i} denotes the special representation corresponding to an irreducible exceptional divisor E_{i} and F is the fundamental cycle.

Proof. 𝕃q^{∗}(O_{0} ⊗ ρ^{∗}) is defined by replacing O_{0} ⊗ ρ^{∗} by its (equivariant) locally free resolution. We can calculate this by taking the Koszul complex (4.2) tensored with ρ^{∗}: Ψ(O_{o} ⊗ ρ^{∗}) is given by the complex

0 → (p_{∗}q^{∗}(K_{ℂ^{2}} ⊗ ρ^{∗}))^{G} → (p_{∗}q^{∗}(Ω_{ℂ^{2}}^{1} ⊗ ρ^{∗}))^{G} → (p_{∗}q^{∗}(O_{ℂ^{2}} ⊗ ρ^{∗}))^{G} → 0.

By virtue of Corollary 3.2, this is a complex of full sheaves and hence this coincides with the complex

0 → { \atop τ(M)} →^{φ} Ñ_{M} →^{ψ} M̃ → 0,  (5.1)

obtained as the torsion-free pull back of (4.3). Since φ is always injective, we have

Ψ^{i}(O_{0} ⊗ ρ^{∗}) = 0 (i ≠ 0, −1)  (5.2)

for arbitrary representation ρ, where Ψ^{i}(−) denotes the i-the cohomology sheaf of the complex Ψ(−). From the exactness of (4.3) (if ρ is non-trivial) and (4.4) (if ρ is trivial), we deduce

Ψ^{0}(O_{0} ⊗ ρ^{∗})(= coker ψ) ≅ \begin{cases}0 & if ρ is non-trivial \\ O_{F} & if ρ is trivial. \end{cases}  (5.3)

Putting

F_{ρ} = Ψ^{(−1)}(O_{0} ⊗ ρ^{∗})(= ker ψ/ Im φ),

we shall determine these sheaves. Note that

supp(F_{ρ}) ⊆ F ( set-theoritically),  (5.4)
H^{0}(F_{ρ}) = H^{1}(F_{ρ}) = 0,  (5.5)
F_{ρ} is purely of dimension 1 (if it is non-zero).  (5.6)

The exactness of (4.3)(or (4.4)) and the rationality of the singularity imply (5.4) and (5.5). (5.6) follows from either (5.5) or the definition of F_{ρ} as a quotient of a locally free sheaf by its locally free subsheaf of the same rank.

As a first step, we shall show the statement of Theorem 4.3:

c_{1}(F_{ρ}) = \begin{cases}E_{i} & if ρ = ρ_{i} is special corresponding to E_{i} \\ 0 & otherwise \end{cases}  (5.7)

To do this along Wunram’s idea, we calculate p_{∗}𝕃q^{∗}(O_{0}) by using the Koszul complex (4.2). Then it is given by

0 → R ⊗_{ℂ} det(ρ_{nat}^{∗}) → R ⊗_{ℂ} ρ_{nat}^{∗} → R → 0,  (5.8)

where R = p_{∗}O_{Z}. (5.8) can be decomposed as

p_{∗}𝕃q^{∗}(O_{0}) = ⨁_{ρ : irreducible representations} Ψ(O_{0} ⊗ ρ^{∗}) ⊗_{ℂ} ρ  (5.9)

with respect to the G-action. Since the alternating sum of the first Chern classes of sheaves in (5.8) is obviously 0, (5.9) with (5.2) and (5.3) deduces

∑_{ρ : irreducible representations}(dim ρ)c_{1}(F_{ρ}) = c_{1}(Ψ^{0}(O_{0} ⊗ ρ_{0})) = F = ∑ r_{i}E_{i}.

In this equation, each c_{1}(F_{ρ}) is represented by an effective divisor supported by the exceptional locus by virtue of (5.4) and r_{i} = dim ρ_{i} as in Theorem 4.2. Thus (5.7) is proved if we show supp(F_{ρ_{i}}) ⊇ E_{i} for non-trivial special representations ρ_{i} corresponding to E_{i}.

We note that (4.3) is non-split by the non-splitness of the Koszul complex (4.2). On the other hand, for the special reflexive module M_{i} (corresponding to ρ_{i}), we have

Ext_{O_{Y}}^{1} (M̃_{i}, { \atop τ(M_{i})}) = 0

since H^{1}(M̃_{i}^{∨}) = 0 (speciality) and since { \atop τ(M_{i})} is generated by global sections. It follows that (5.1) for M_{i} is non-exact and we have

0 ≠ Ext_{O_{Y}}^{1} (M̃_{i}, ker ψ) ≅ Ext_{O_{Y}}^{1} (M̃_{i}, F_{ρ_{i}}).

From the construction (4.1) of M_{i}, we obtain an exact sequence

0 → M̃_{i}^{∨} ⊗ F_{ρ_{i}} → F_{ρ_{i}}^{⊕r} → F_{ρ_{i}} ⊗ O_{D_{i}} → 0.

We have H^{1}(F_{ρ_{i}}) = 0((5.5)), H^{1}(M̃_{i}^{∨}⊗F_{ρ_{i}}) ≠ 0 and hence H^{0}(F_{ρ_{i}} ⊗O_{D_{i}}) ≠ 0 by the above exact sequence. This implies supp(F_{ρ_{i}}) ∩ D_{i} ≠ ∅ and hence supp(F_{ρ_{i}}) ⊇ E_{i}. (see (5.4) and (5.6).) Thus we have proved (5.7).

Now (5.5), (5.6) and (5.7) imply

F_{ρ} ≅ \begin{cases}O_{E_{i}}(−1) & if ρ = ρ_{i} \\ 0 & otherwise. \end{cases}

This together with (5.2) and (5.3) proves our assertion.

6 Representations constructed from ideals

In this section, we shall establish the Ito-Nakamura type correspondence using the results in the previous section. We have the commutative diagram

Y × ℂ^{2} ⟶^{π_{ℂ^{2}}}  ℂ^{2}
π_{Y} ↓  ↓^{π}
Y ⟶^{f} X = ℂ^{2}/G.

We defined the functor Ψ : D_{c}^{G}(ℂ^{2}) → D_{c}(Y) so that

Ψ(−) := [p_{∗}𝕃q^{∗}(−)]^{G} = [ℝπ_{Y} _{∗}(O_{Z} ⊗^{𝕃}_{O_{Y ×ℂ^{2}}} π_{ℂ^{2}}^{∗}(−))]^{G}.

Let Φ : D_{c}(Y) → D_{c}^{G}(ℂ^{2}) denote the functor in the other direction defined as

Φ(−) := ℝπ_{ℂ^{2}}_{∗}(O_{Z}^{∨} ⊗^{𝕃} π_{Y}^{∗} (− ⊗ ρ_{0}) ⊗ (π_{ℂ^{2}}^{∗}K_{ℂ^{2}}))[2],

where

O_{Z}^{∨} = ℝHom_{O_{Y ×ℂ^{2}}}(O_{Z}, O_{Y ×ℂ^{2}})

is the derived dual of O_{Z} and where − ⊗ ρ_{0} denotes the functor

D_{c}(Y) → D_{c}^{G}(Y)

attaching the trivial G-action. Then we have

Lemma 6.1. Ψ is a left adjoint of Φ.

Proof. This can be proved as in [BKR]:

Hom_{D_{c}(Y)}(Ψ(♣), ♠) ≅ Hom_{D_{c}(Y)}([ℝπ_{Y} _{∗}(O_{Z} ⊗^{𝕃}_{O_{Y ×ℂ^{2}}} π_{ℂ^{2}}^{∗}(♣))]^{G}, ♠)

≅ Hom_{D_{c}^{G}(Y)}(ℝπ_{Y} _{∗}(O_{Z} ⊗^{𝕃}_{O_{Y ×ℂ^{2}}} π_{ℂ^{2}}^{∗}(♣)), ♠ ⊗ ρ_{0})

≅ Hom_{D^{G}(Y ×ℂ^{2})}(O_{Z} ⊗^{𝕃}_{O_{Y ×ℂ^{2}}} π_{ℂ^{2}}^{∗}(♣), π_{Y}^{∗} (♠ ⊗ ρ_{0}) ⊗ (π_{ℂ^{2}}^{∗}K_{ℂ^{2}})[2])

≅ Hom_{D^{G}(Y ×ℂ^{2})}(π_{ℂ^{2}}^{∗}(♣), O_{Z}^{∨} ⊗^{𝕃} π_{Y}^{∗} (♠ ⊗ ρ_{0}) ⊗ (π_{ℂ^{2}}^{∗}K_{ℂ^{2}})[2])

≅ Hom_{D_{c}^{G}(ℂ^{2})}(♣, ℝπ_{ℂ^{2}}_{∗}(O_{Z}^{∨} ⊗^{𝕃} π_{Y}^{∗} (♠ ⊗ ρ_{0}) ⊗ (π_{ℂ^{2}}^{∗}K_{ℂ^{2}})[2]))

≅ Hom_{D_{c}^{G}(ℂ^{2})}(♣, Φ(♠))

Here, the third isomorphism is the equivariant Grothendieck duality (see [BKR]) for

π_{Y}, which can be applied since O_{Z} ⊗^{𝕃}_{O_{Y ×ℂ^{2}}} π_{ℂ^{2}}^{∗}(♣) has proper support over Y. The other adjunctions are rather standard ([BKR]).

We use this lemma in the form

Hom_{D_{c}(Y)}^{k}(Ψ(O_{0} ⊗ ρ^{∗}), O_{y}) ≅ Hom_{D_{c}^{G}(ℂ^{2})}^{k}(O_{0} ⊗ ρ^{∗}, Φ(O_{y})),  (6.1)

where ρ is an irreducible representation of G, O_{0} and O_{y} are the structure sheaves of {0} ⊂ ℂ^{2} and {y} ⊂ Y respectively.

We have calculated Ψ(O_{0}⊗ρ^{∗}) in Theorem 5.1. On the other hand, the definition of Φ implies

Φ(O_{y}) = O_{Z_{y}}^{∨} ⊗ K_{ℂ^{2}}[2],  (6.2)

where Z_{y} is the subscheme of ℂ^{2} corresponding to y and O_{Z_{y}}^{∨} = ℝHom_{O_{ℂ^{2}}}(O_{Z_{y}}, O_{ℂ^{2}}) is the derived dual on ℂ^{2}.

From (6.1), Theorem 5.1 and (6.2), if ρ = ρ_{i} is special, then we deduce that

Ext_{O_{Y}}^{k−1}(O_{E_{i}}(−1), O_{y}) ≅ Hom_{D_{c}^{G}(ℂ^{2})}^{k}(O_{0} ⊗_{ℂ} ρ_{i}^{∗}, O_{Z_{y}}^{∨} ⊗ K_{ℂ^{2}}[2])
≅ Hom_{D_{c}^{G}(ℂ^{2})}^{k}(O_{Z_{y}}, (O_{0} ⊗_{ℂ} ρ_{i}^{∗})^{∨} ⊗ K_{ℂ^{2}}[2])
≅ Hom_{D_{c}^{G}(ℂ^{2})}^{k}(O_{Z_{y}}, O_{0} ⊗_{ℂ} ρ_{i}))
= G- Ext_{O_{ℂ^{2}}}^{k}(O_{Z_{y}}, O_{0} ⊗_{ℂ} ρ_{i})  (6.3)

where (O_{0} ⊗_{ℂ} ρ_{i}^{∗})^{∨} is also the derived dual on ℂ^{2}. Note that the third isomorphism above is given by the duality for {0} ↪ ℂ^{2} (which can easily be verified by using the Koszul complex (4.2)). Similarly, we have

Ext_{O_{Y}}^{k} (O_{F} , O_{y}) ≅ G- Ext_{O_{ℂ^{2}}}^{k}(O_{Z_{y}}, O_{0} ⊗_{ℂ} ρ_{0})  (6.4)

for the trivial representation ρ_{0} and

0 ≅ G- Ext_{O_{ℂ^{2}}}^{k}(O_{Z_{y}}, O_{0} ⊗_{ℂ} ρ)  (6.5)

for non-special irreducible representation ρ. If we put k = 1 in (6.3), (6.4) and (6.5), then we deduce that G- Ext_{O_{ℂ^{2}}}^{1}(O_{Z_{y}}, O_{0} ⊗_{ℂ} ρ) is non-zero(then is 1-dimensional) if and only if

(ρ = ρ_{0} and y ∈ F) or (ρ = ρ_{i} and y ∈ E_{i}).

For a non-trivial G-equivariant extension

0 → O_{0} ⊗_{ℂ} ρ → F → O_{Z_{y}} → 0

with ρ irreducible, we can lift 1 ∈ O_{Z_{y}} to an invariant section s of F. Then the non-triviality of the extension and the irreducibility of ρ induce that F is generated by s as an O_{ℂ^{2}}-module. Therefore F is of the form

F ≅ O_{ℂ^{2}}/J

with mI_{y} ⊆ J ⊆ I_{y} and I_{y}/J ≅ O_{0} ⊗ ρ. Thus we obtain Theorem 6.2. For y ∈ Y, denote by I_{y} the ideal corresponding to y and let m be the maximal ideal of O_{ℂ^{2}} corresponding to the origin 0. If y ∈ Y is in the exceptional locus, then, as representations of G, we have

I_{y}/mI_{y} ≅ \begin{cases}ρ_{i} ⊕ ρ_{0} & if y ∈ E_{i} and y ∉ E_{j} for j ≠ i, \\ ρ_{i} ⊕ ρ_{j} ⊕ ρ_{0} & if y ∈ E_{i} ∩ E_{j}, \end{cases}

where ρ_{i} is the special representation associated with the irreducible exceptional curve E_{i}.

We can also calculate the socle of O_{Z_{y}}. By Serre duality we have

G- Ext_{O_{ℂ^{2}}}^{k}(O_{Z_{y}}, O_{0} ⊗ ρ) ≅ G- Ext_{O_{ℂ^{2}}}^{2−k}(O_{0} ⊗ ρ ⊗ K_{ℂ^{2}}^{∗}, O_{Z_{y}})
≅ G- Ext_{O_{ℂ^{2}}}^{2−k}(O_{0} ⊗ ρ ⊗ det ρ_{nat}, O_{Z_{y}})

Thus by putting k = 2 in (6.3), (6.4) and (6.5) we obtain

Theorem 6.3. Under the same notation as in the previous theorem, the socle of O_{Z_{y}} is

(I_{y} : m)/I_{y} ≅ \begin{cases}ρ_{i} ⊗ det ρ_{nat} & if y ∈ E_{i} and y ∉ E_{j} for j ≠ i, \\ (ρ_{i} ⊕ ρ_{j}) ⊗ det ρ_{nat} & if y ∈ E_{i} ∩ E_{j}. \end{cases}

7 Connectedness of G-Hilb(ℂ^{2})

In this section, we prove the following theorem.

Theorem 7.1. G-Hilb(ℂ^{2}) is connected.

Proof. We shall show that Y = G-Hilb. Assume to the contrary that there exists a G-cluster [Z] ∈ G-Hilb(ℂ^{2}) \ Y. We consider Ψ(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]) under the notation of the previous sections. Since O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2] ≅ Hom_{ℂ}(O_{Z}, ℂ) is represented by a sheaf, Ψ(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]) is given by a complex

0 → Ñ^{−2} →^{φ} Ñ^{−1} →^{ψ} Ñ^{0} → 0  (7.1)

of full sheaves on Y.

From the assumption [Z] ∉ Y, we deduce that for any y ∈ Y,

Hom_{D_{c}(Y)}(Ψ(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]), O_{y}) ≅ Hom_{D_{c}^{G}(ℂ^{2})}(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2], Φ(O_{y}))

≅ Hom_{D_{c}^{G}(ℂ^{2})}(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2], O_{Z_{y}}^{∨} ⊗ K_{ℂ^{2}}[2])

≅ Hom_{D_{c}^{G}(ℂ^{2})}(O_{Z_{y}}, O_{Z}) = 0.

It follows that ψ is surjective and hence if we put

F := ker ψ/ im φ

then

Ψ(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]) = F[1].

We shall calculate H^{1}(F). Since (7.1) is a complex of acyclic sheaves, we can calculate the cohomology of F by taking global sections of (7.1):

H^{1}(F) ≅ coker f_{∗}ψ : N^{−1} → N^{0}.

Since H^{0}(O_{Z}) is a regular representation of G, so is H^{0}(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]) ≅ H^{0}(O_{Z})^{∗} and we see coker f_{∗}ψ ≅ (H^{0}(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]))^{G} = ℂ. As a consequence, we have H^{1}(F) ≠ 0 and hence dim supp(F) = 1. Therefore c_{1}(F) ≠ 0 in the Picard group.

On the other hand, the first Chern class (in the Picard group) of a sheaf (or a complex) can be determined by its class in the Grothendieck group and hence we have

c_{1}(F) = −c_{1}(Ψ(O_{Z}^{∨} ⊗ K_{ℂ^{2}}[2]))
= − ∑_{ρ: irreducible representations}(dim ρ)c_{1}(Ψ(ρ^{∗} ⊗ O_{0})) = 0

by Theorem 5.1. Thus we have met a contradiction.

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Department of Civil Engineering Systems, Kyoto University, Kyoto, 606-8501, Japan

email: akira@kusm.kyoto-u.ac.jp