Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.


Introduction
In recent years, the term McKay correspondence has come to describe any relation between the group theory of a finite subgroup G of SL(n, C) and the topology of a crepant resolution of C n /G. Its starting point is Du Val's observation [DuV34] that for a finite subgroup G of SL(2, C) the exceptional divisor E of the unique crepant resolution C 2 /G of C 2 /G is a tree of rational curves with self-intersection −2 and that these trees are in one-to-one correspondence with the simply-laced Coxeter-Dynkin diagrams. Brieskorn [Bri68] further observed that the the intersection matrix of the irreducible components of E is exactly the negative Cartan matrix of the simple Lie algebra associated to the Coxeter-Dynkin tree. This way a correspondence between the finite subgroups of SL(2, C) and the simply-laced Coxeter-Dynkin diagrams was established. This correspondence was rediscovered from a combinatorial point of view by McKay [McK80], who noticed that for finite subgroups of SL(2, C) there exists a bijection between the set Irr(G) of irreducible representations and the set of vertices of the extended Coxeter-Dynkin diagram. To show this, he relates the matrix A = (a ρσ ) ρ,σ∈Irr(G) corresponding to the decomposition of the tensor products to the Cartan matrixC of the extended Coxeter-Dynkin diagram via A = 2I −C. These observations together give a complete description of the topology of C 2 /G in terms of G and its embedding in SL(2, C), which is now known under the name of the (classical) McKay correspondence: The irreducible components Σ ρ of E are labelled by the non-trivial irreducible representations ρ of G and Σ ρ · Σ σ = −c ρσ .
(1.1) However, the role of the irreducible representations of G in the above description is still elusive. Hence, it is desirable to find a geometrical interpretation for them.
The first geometric realization of this classical McKay correspondence was given by Gonzalez-Sprinberg and Verdier [GV83] at the level of K-theory. They showed that the irreducible representation ρ of G ⊂ SL(2, C) give rise to holomorphic bundles R ρ on C 2 /G, called tautological bundles, and these form a basis in K-theory. This way they obtained an isomorphism between the representation ring of G, the G-equivariant K-theory of C 2 and the K-theory of the crepant resolution. Moreover, they showed that the first Chern classes of the tautological bundles form a basis in cohomology, dual to the basis given by the irreducible components of the exceptional divisor. Using a concrete description of the crepant resolution, Kapranov and Vasserot [KV00] lifted this correspondence to the derived category and showed that it is given by a Fourier-Mukai transform. Yet another interpretation was given by Kronheimer and Nakajima [KN90]. Using the index theorem, they obtained the Cartan matrix as the multiplicative matrix of the first Chern classes of the tautological bundles X c 1 (R ρ )c 1 (R σ ) = −(C −1 ) ρσ , (1.2) see [KN90,Theorem A.7].
The string theory insight of Dixon, Harvey, Vafa and Witten [DHVW86] suggested that such McKay correspondences between the topology of a crepant resolution and the representation theory of the finite group should also hold in higher dimension. They introduced the orbifold Euler number, a number expressed in terms of the group G and its action on C n , and conjectured that it is equal to the Euler characteristic of any crepant resolution of C n /G, whenever such a resolution exists. This has sparked an enormous body of work in mathematics including among other results the case-by-case proof of this conjecture for finite subgroups of SL(3, C) of Ito [Ito95], Markushevich [Mar97] and Roan [Roa96], its refinement for Betti and Hodge numbers of Batyrev and Dais [BD96], the notion of orbifold cohomology of Chen and Ruan [CR04], and the new versions of the McKay correspondence of Ito and Reid [IR96].
Recall that the existence and uniqueness of a crepant resolution depends crucially on the dimension: In dimension two and three a crepant resolution always exists; however, it is unique only in dimension two, as in dimension three any flop gives another one. In higher dimensions crepant resolutions need not exist at all. Moreover, above dimension two the link between finite groups and Coxeter-Dynkin diagrams is missing. Hence, the insight gained from the geometrical interpretation of the classical McKay correspondence is of key importance to understanding its extensions to higher dimensions. In the case of finite subgroups of SL(3, C), the Ktheory interpretation was carried out by Ito and Nakajima [IN00], while the derived category equivalence was established by Bridgeland, King and Reid [BKR01] and Craw and Ishii [CI04]. All these give additive information about the cohomology of the crepant resolution and they are invariant under its choice. In this work, we obtain the counterpart of Kronheimer and Nakajima's result and make the first steps towards understanding the multiplicative structure of the cohomology of crepant resolutions of C 3 /G.
We consider the case of a finite subgroup G of SL(3, C) which acts freely on C 3 \ {0}. This forces G to be cyclic of prime order. Let X be a projective crepant resolution of C 3 /G. From the work of Craw and Ishii, we know that X is a moduli space M θ of G-constelations (a sheaf theoretic generalisation of the notion of Gorbit) which are stable with respect the parameter θ ∈ Θ Q , see Definition 2.3. As such, X naturally comes with a collection of holomorphic bundles R ρ , one for each representation ρ of G. The bundles corresponding to the irreducible representations of G form a basis in K-theory, and hence their Chern characters form a basis in H * (X, R). Using the natural action of G on C 3 , one can define a matrix C, see (7.5), which is the higher dimensional analogue of the Cartan matrix associated to the finite subgroups of SL(2, C) from the matrix A. Theorem 1.3. Let G be a finite subgroup of SL(3, C) which acts with an isolated fixed point on C 3 . Let X be a projective crepant resolution of C 3 /G. Then for all nontrivial ρ, σ ∈ Irr(G). Here ch := ch − rk is the reduced Chern character.
Remark 1.5. This result has first appeared in a preprint of the first named author [Deg03], but it had a gap in its proof.
Formula (1.4) should be viewed as an analogue of Kronheimer's and Nakajima's formula (1.2). It determines a part of the multiplicative structure in cohomology of all the projective crepant resolutions of C 3 /G that only depends on the finite group G and its embedding in SL(3, C). Thus we have found a new McKay correspondence. More precisely, note that the left-hand side of (1.4) can be written as Using the Chern-Weil theory, c 1 (R ρ ) = 1 2π F A θ,ρ where F A θ,ρ is the curvature of the natural (1, 1)-connection induced on R ρ . By Theorem 3.9, F A θ,ρ defines a class in the weighted L 2 -cohomology of X, which can be identified, using a result of Hausel, Hunsicker and Mazzeo [HHM04,Corollary 4], with H k c (X, C). Moreover, since on the crepant resolution X, we have H 2 c (X, C) ∼ = H 1,1 (X, C), with the last generated by the Poincaré duals of the exceptional divisors, the class corresponding to c 1 (R ρ ) can be represented by compactly supported (1, 1)-forms. Hence, (1.4) describes the part of triple pairing X : H 1,1 (X, C) × H 1,1 (X, C) × H 1,1 (X, C) → C, (1.7) mapping (α, β, γ) to X α ∧ β ∧ γ where γ = α − β and α, β ∈ {c 1 (R ρ ) | ρ ∈ Irr(G)}. The issues of completely determinig the multiplicative structure in cohomology as well as of describing the entire part which is invariant under the choice of the crepant resolution are still open and will be investigated in future work.
We prove Theorem 1.3 in same manner as Kronheimer and Nakajima by studying the index formula for certain Dirac operators whose index we show to be zero. The condition that the group G acts with an isolated fixed point on C 3 ensures that any crepant resolution of the orbifold C 3 /G is an ALE space, in which case the index of the Dirac operator is given by the Atiyah-Patodi-Singer (APS) index theorem [APS75]. For all the other finite subgroups of SL(3, C), the geometry of the crepant resolution is that of a QALE manifold, as introduced by Joyce [Joy00]. The generalization of the APS index theorem to QALE manifolds is work in progress by the first named author [DM12].
In the course of proving Theorem 1.3, we also gain more insight into the geometry of the crepant resolutions X = M θ . Since M θ is constructed via geometric invariant theory (GIT), it follows that the complex manifold M θ underlying M θ can be constructed as a Kähler quotient. This construction, which we present in Section 3, is the generalisation to higher dimension of Kronheimer's construction of ALE gravitational instantons [Kro89] and was first carried out by Sardo-Infirri [SI96]. One of its consequences is that M θ carries a natural Kähler metric g θ . Our second main result establishes the existence of special metrics on M θ and its tautological bundles R ρ .
Theorem 1.8. Let G be a finite subgroup of SL(3, C) acting with an isolated fixed point on C 3 . Let θ ∈ Θ Q be a generic rational stability parameter. Then 1. M θ carries a ALE Ricci-flat Kähler metric g θ,RF , which is in the same Kähler class as g θ .
The existence of the Ricci-flat Kähler metric on M θ is a consequence of Joyce's proof of the Calabi conjecture for ALE crepant resolutions [Joy00, Section 8], while the existence of HYM metric is a consequence of the properties of the Laplace operator on ALE manifolds. The difficulty lies in proving the infinitesimal rigidity statement. In fact, the key ingredient for proving both Theorem 1.3 and the rigidity of the HYM metric in Theorem 1.8 is the vanishing result in Lemma 5.1.
The results here are of interest in the context of higher dimensional gauge theory. For example, one can use them to extend the second named author's construction of G 2 -instantons on generalized Kummer constructions [Wal11] to G 2 -manifolds arising from G 2 -orbifolds with codimension 6 singularities.
The paper is organised as follows. In Section 2 we briefly recall the construction of crepant resolutions as moduli spaces of G-constelations, introduce the Fourier-Mukai transform and collect the results of Bridgeland, King and Reid [BKR01] and Craw and Ishii [CI04] that are relevant for our work. In Section 3 we present the Kähler counterpart of the construction of crepant resolutions and use it to describe its geometry. In Section 4 we prove the existence part of Thereom 1.8, while in Section 5 we prove the rigidity statement. Section 6 introduces the Dirac operator on crepant resolutions and establishes its properties. We finish with the proof of our main Theorem 1.3 in Section 7.

Moduli spaces of G-constellations
Let G be a finite subgroup of SL(3, C). We denote by Irr(G) the set of irreducible representations, by Rep(G) the representation ring, and by R the regular representation of G.
Definition 2.1. A G-sheaf on C 3 is a coherent sheaf F together with an action of G which is equivariant with respect to the action of G on C 3 . A G-constellation on C 3 is a G-sheaf F such that H 0 (C 3 , F ) ∼ = R as G-modules. An isomorphism of G-constellations is an isomorphism of sheaves intertwining the G-actions.
Remark 2.2. The (set theoretic) support of a G-constellation is a G-orbit in C 3 and can thus be thought of as a point in C 3 /G. From this point of view one can think of G-constellations as a sheaf-theoretic generalisation of the notion of G-orbit.
Theorem 2.4 (Craw and Ishii [CI04, Section 2.1]). If θ ∈ Θ Q is generic, then there exists a smooth fine moduli space M θ of θ-stable G-constellations on C 3 . Moreover, associated to each representation ρ of G there is a locally free sheaf R ρ on M θ . If ρ and σ are two representations of G, Sketch of the Proof. The construction of M θ uses GIT and is based on ideas of King [Kin94] and Sardo-Infirri [SI96].
In fact, every G-constellation arises this way. Furthermore, two points in N yield isomorphic G-constellations if and only if they are related by a G-equivariant automorphism of R, i.e., an element of GL(R) Since the diagonal C * ⊂ GL(R) G acts trivially on N , the action of GL(R) G descends to an action of PGL(R) G . An integral stability parameter θ ∈ Θ thus determines a character χ θ : PGL(R) G → C * defined by King [Kin94, Proposition 3.1] proved that an element of N is stable (resp. semistable) in the sense of GIT with respect to χ θ if and only if the corresponding G-constellation is θ-stable (resp. θ-semi-stable). Let N s θ (resp. N ss θ ) be the set GIT (semi-)stable points with respect to χ θ in N and let be the corresponding GIT quotient. As schemes, M kθ = M θ for any k ∈ N and therefore, the above construction extends to rational stability parameters θ ∈ Θ Q as well.
To see that M θ is indeed a fine moduli space, we construct a universal Gconstellation U θ on M θ × C 3 . For this, we identify where Irr 0 (G) is the set of non-trivial irreducible representations of G. In this way, PGL(R) G acts on R. This makes Let now ρ : G → Aut(R ρ ) be a representation of G. Since by the above identification PGL(R) G acts on R ρ , the construction that associates to R the sheaf R can be carried on for R ρ , giving rise to the sheaf R ρ . It is then clear that To obtain further insight into the spaces M θ , it is helpful to make use of the language of derived categories. We first recall the bounded derived category D(A ) associated with an abelian category A . For details we refer the reader to Bühler's notes [Büh07], as well as Thomas' article [Tho01] and Huybrechts' book [Huy06], both of which underline the importance of derived categories of coherent sheaves in algebraic geometry. Roughly speaking, D(A ) is obtained from the category of bounded chain complexes in A by formally inverting quasi-isomorphisms. If A, B ∈ A are considered as bounded chain complexes concentrated in degree zero, then Hom D(A ) (A, B) is a complex whose cohomology computes Ext • (A, B), that is, If B is another abelian category and f : A → B is a left-exact functor, then one associates to it a right derived functor Rf : A similar construction holds for g : A → B a right-exact functor, producing a left derived functor Lg. When working with derived categories, it is customary to write f and g instead of Rf and Lg. We follow this custom in this article.
An important example of such a derived category is D(Coh(X)), the bounded derived category of coherent sheaves D(Coh(X)) over a scheme X. If X and Y are two schemes and K ∈ Coh(X × Y ) is a coherent sheaf, then the Fourier-Mukai transform with kernel K is the functor Φ K : D(Coh(X)) → D(Coh(Y )) defined by Here p * 1 , (p 2 ) * and ⊗ are taken in the derived sense, with p 1 and p 2 denoting the projections from X × Y to X and Y respectively.
In our context, we denote by D(M θ ) the bounded derived category of coherent sheaves on M θ and by D G (C 3 ) the bounded derived category of G-equivariant coherent sheaves on C 3 , which is the same as the bounded derived category D([C 3 /G]) of coherent sheaves on the stack [C 3 /G]. One of the key ideas of Bridgeland, King and Reid [BKR01] is to introduce the Fourier-Mukai transform Φ θ : to study crepant resolutions. Here p : M θ × C 3 → M θ and q : M θ × C 3 → C 3 are the two projections. Using it we have the following descriptions of the projective crepant resolutions of C 3 /G: Theorem 2.11 (Craw and Ishii [CI04, Proposition 2.2 and Theorem 2.5]). For each θ ∈ Θ Q , there exists a morphism π θ : M θ → C 3 /G which associates to each isomorphism class of G-constellations its support. If θ ∈ Θ Q is generic, then π θ is a projective crepant resolution and the Fourier-Mukai transform Φ θ is an equivalence of derived categories.
Remark 2.12. Bridgeland, King and Reid [BKR01] first proved this result for Nakamura's G-Hilbert scheme. Craw and Ishii observed that the proof carries over to the more general moduli spaces of G-constellations. The fact that π θ is a crepant resolution is essentially a consequence of Φ θ being an equivalence of derived category, in which case there exists a categorical criterion for a resolution to be crepant [BKR01, Lemma 3.1].
Theorem 2.13 (Craw and Ishii [CI04, Theorem 1.1]). If G is an abelian subgroup of SL(3, C), then every projective crepant resolution of C 3 /G is a moduli space of θ-stable G-constellations for some generic θ ∈ Θ Q .

M θ via Kähler reduction
We now approach the previous discussion from the Kähler point of view. There is no loss in assuming that the finite group G ⊂ SL(3, C) preserves the standard Hermitian metric on C 3 , that is, G ⊂ SU (3). Moreover, we fix a G-invariant Hermitian metric on R. In this set-up, N defined in (2.5) is a cone in the Hermitian Here we identify B ∈ (End(R) ⊗ C 3 ) G with a triple (B 1 , B 2 , B 3 ) of endomorphisms of R. Proof. It is enough to prove this for the action of U (R) G . If ξ ∈ u(R) G , then the corresponding vector field X ξ on (End For θ ∈ Θ R , we define ζ θ ∈ pu(R) G * by for all ξ ∈ pu(R) G . Here π ρ : R → C dim ρ ⊗R ρ is the projection onto the ρ-isotypical component of the the regular representation, and ξ · π ρ is thought of as an element in End(R). Note that if θ is integral, then ζ θ = −idχ θ ∈ pu(R) G * , with χ θ the character associated to θ as defined in (2.6). Moreover, since the centre of u(R) G is spanned by {iπ ρ | ρ ∈ Irr(G)}, we can identify Θ R with the centre of (pu(R) G ) * via θ → ζ θ .
With this identification, it follows that θ ∈ Θ R is generic if and only if for all proper subrepresentations 0 S R we have ζ θ (iπ S ) = 0, with π S : R → S denoting the orthogonal projection onto S.
For each θ ∈ Θ R , we denote by corresponding Kähler quotient. Moreover, for each representation ρ : G → GL(R ρ ) of G, PU(R) G acts on R ρ and gives rise to the bundle This bundle is holomorphic, since the holomorphic structure on µ −1 (θ) × V is PU(R) G -equivariant and thus passes down to R ρ . We call R ρ the tautological holomorphic bundle associated to the representation ρ of G.
We describe now the relation between the algebraic objects M θ and R ρ defined in Theorem 2.4 and the holomorphic objects M θ and R ρ defined above. Since µ(B), iπ S = θ(iπ S ) = θ(S), it follows that θ(H 0 (E )) ≥ 0.
King [Kin94,Theorem 6. 1] shows that the Kempf-Ness theorem holds in this case: If θ ∈ Θ, then each PGL(R) G -orbit which is closed in N ss θ meets µ −1 (ζ θ ) in precisely one PU(R) G -orbit, and meets no other orbit. From this we have the following result: Proposition 3.5. Suppose θ ∈ Θ Q is generic. Then the inclusion µ −1 (ζ θ ) → N s θ = N ss θ induces a biholomorphic map from M θ to the analytification of M θ . This map identifies the holomorphic bundle R ρ with the analytification of the locally free sheaf R ρ . Now, for each θ ∈ Θ R , let g θ and ω θ be the metric and the Kähler form on M θ induced by the Kähler quotient construction. We also have a natural PU(R) Gconnection A θ whose horizontal space is the orthogonal complement of the tangent space to the orbit in T B (µ −1 (θ)). To describe the geometry of M θ and the behaviour of g θ and A θ , we first need the following definitions: Definition 3.6. Let G be a finite subgroup of SU(3) acting freely on C 3 \ {0}. A noncompact Riemannian manifold (X, g) of real dimension 6 is called an ALE manifold asymptotic to C 3 /G to order τ > 0, if there exists a compact subset K ⊂ X and a diffeomorphism π : Here r := |x| denotes the radius function on C 3 and g 0 denotes the standard metric on C 3 . The pair (X \ K, π) is called an ALE end and the function r, extended smoothly to X so that r : X → [1, ∞), is a radius function on X.
Definition 3.7. Similarly, a connection A on a complex vector bundle E of rank k on the ALE manifold (X, g) is called asymptotically flat of order τ > 0 if there exists a flat connection A 0 on the ALE end of X so that  2. If θ ∈ Θ Q generic, then M θ is smooth and the induced Kähler metric g θ is ALE of order 4.

The PU(R) G -connection A θ is an
(1, 1)-connection which is asymptotically flat of order 2. In particular, its curvature decays like r −4 .
In the case of finite subgroups of SU(2), the analogous theorem was proven by Kronheimer [Kro89] and by Gocho and Nakajima [GN92]. For the above theorem, the smoothness of the Kähler quotient M θ for θ ∈ Θ Q generic follows from the identification with the algebraic quotient M θ provided by Proposition 3.5 and the result of Theorem 2.11. The first statement and the remaining of the second were proved by Sardo-Infirri [SI96] by generalising Kronheimer's proof. The proof of the third statement is a direct generalization of Gocho and Nakajima's argument.
Remark 3.10. It seems reasonable to expect that the second statement in Theorem 3.9 holds for all generic θ ∈ Θ R . Because of the homogeneity of the moment map, the statement follows for all tθ with θ ∈ Θ Q generic and t a strictly positive real number. On the other hand, when θ ∈ Θ R generic, we can show that PU(R) G acts freely on µ −1 (θ). However, to conclude that M θ is smooth, one still needs to show that µ −1 (θ) is contained in the smooth locus of N .

Ricci-flat metrics on M θ and Hermitian-Yang-Mills metrics on R ρ
In contrast to Kronheimer's and Nakajima's work [Kro89,KN90], the metric g θ on M θ obtained from the Kähler reduction is not necessarily Ricci-flat and the connections A ρ on R ρ are not necessarily Hermitian-Yang-Mills (HYM  Corollary 4.5. Let θ ∈ Θ Q generic. Then for each ρ ∈ Irr(G) the tautological bundle R ρ on M θ carries an asymptotically flat HYM metric with respect to g θ,RF .
Remark 4.6. Using some of the results derived in Section 5, one can show that the HYM connection associated with h in Proposition 4.4 is asymptotically flat of order 5. Thus the Hermitian metric h is asymptotically flat of order 4.
Remark 4.7. Using heat flow methods, Bando [Ban93] proved that every holomorphic bundle E over an ALE Kähler manifold which admits a Hermitian metric h 0 with |F h0 | = O r −2−ε does in fact carry a HYM metric.
The case of line bundles is much simpler and follows from the Laplace operator being an isomorphism between certain weighted Sobolev spaces. For k a nonegative integer and δ ∈ R, we denote by W k,2 δ (X) the completion of C ∞ 0 (X) with respect the norm Here m denotes the real dimension of X. Let ∆ δ : W k+2,2 δ (X) → W k,2 δ−2 (X) denote the corresponding completion of the Laplacian ∆.
Proof. The weighted Laplacian ∆ δ : W k+2,2 δ (X) → W k,2 δ−2 (X) is a Fredholm operator if and only if the weight parameter δ is not in its set of indicial roots at infinity. This is a discrete set that does not intersect the interval (−m + 2, 0), see Bartnik [Bar86, Sections 1 and 2] for details. Moreover, for δ < 0, the kernel of ∆ δ is trivial by the maximum principle. On the other hand, the cokernel of ∆ δ is isomorphic to the kernel of its formal adjoint ∆ m−2−δ . Therefore, for δ ∈ (−m + 2, 0), ∆ δ is an isomorphism.

Rigidity of HYM metrics on the holomorphic tautological bundles
In this section we prove the rigidity statement in Theorem 1.8. This will be an immediate consequence of Lemma 5.1, which is the main vanishing result of this paper. If the metric h is HYM, then H 1 A is the space of infinitesimal deformations. Hence, the HYM metrics on R ρ constructed in the first part of Theorem 1.8 are infinitesimally rigid for all ρ ∈ Irr(G), thus completing the proof of Theorem 1.8.
The strategy for proving Lemma 5.1 is as follows: We first reduce to a problem in complex geometry, see Proposition 5.8. Using GAGA, we translate this into an algebraic geometry problem, see (5.9), which we then solve using the results of Bridgeland, King and Reid [BKR01] and Craw and Ishii [CI04] discussed in Section 2.
It is a useful heuristic to think of bundles with decaying connections as bundles on a compactification whose restrictions to the "divisor at infinity" satisfy certain conditions, like being flat for example. Accordingly, we compactify M θ at infinity by gluing M θ and (P 3 \ {[0 : 0 : 0 : 1]})/G along M θ \ π −1 θ (0) = (C 3 \ {0})/G. The resulting spaceM θ is not a complex manifold, but rather a complex orbifold. We denote its divisor at infinity by D. This is a smooth orbifold divisor, i.e., it lifts to a smooth divisors in covers of the uniformising charts. The bundle R extends over D to a bundleR onM θ . The following result reduces the proof of Lemma 5.1 to a problem in complex geometry.
The proof of Proposition 5.2 requires two preparatory results. Proof. Since the i * and∂ commute, A • forms a complex. Moreover, it is clear that E(−D) is the kernel of A 0 ∂ → A 1 . The proof that A • is a resolution uses two ingredients: the Grothendieck-Dolbeault Lemma and the fact that if U is a sufficiently small open set, then holomorphic sections on D ∩ U extend to U . We show that these assertions hold also for orbifolds. Let U be a small open set which is covered by a uniformising chart U /Γ. Lifting everything up toŨ , E corresponds to a Γ-equivariant holomorphic bundleẼ and D to a Γ-equivariant smooth divisorD. If α ∈ Ω (0,k) (U, E) satisfies ∂α = 0, then so does its liftα ∈ Ω (0,k) (Ũ ,Ẽ) Γ . If U (and thusŨ ) is sufficiently small, then the usual Grothendieck-Dolbeault Lemma yieldsβ ∈ Ω (0,k−1) (Ũ ,Ẽ) satisfying∂β =α. There is no loss in assuming thatβ is Γ-invariant and thus pushes down to the desired primitive β ∈ Ω (0,k−1) (U, E) of α. We thus obtain the Grothendieck-Dolbeault Lemma for orbifolds. Now, if s is a holomorphic section of E over D ∩U , we lift it to the uniformising chartŨ , where, provided U is sufficiently small, we find a Γ-equivariant extension. We then push this extension down to U . Hence, the proof of the second assertion in orbifold set-up.
Let now U be a small open set of Z and let α ∈ Ω 0,k (U, E) with∂α = 0. By the Grothendieck-Dolbeault Lemma after possibly shrinking U , we can find β ∈ Ω 0,k (U, E) satisfying∂β = α. If k ≥ 2, we apply the Grothendieck-Dolbeault Lemma once more to obtain γ ∈ Ω 0,k−2 (U ∩ D, E) such that∂γ = i * β. We extend γ smoothly to all of U . Then β −∂γ ∈ A k−1 (U ) yields the desired primitive of α on U . When k = 1, we know that β restricts to a holomorphic section β| D of E| U∩D , which can be extended to a holomorphic section δ on U . Hence, β − δ ∈ A 0 (U ) is the desired primitive of α.
Finally, (A • , d) is an acyclic resolution of E(−D), since the sheaves A • are C ∞modules and therefore soft.
Remark 5.4. In the definition of A k it is not strictly necessary to require that α be smooth. In fact, a simple application of elliptic regularity shows that it suffices that elements of A k be in the Hölder space C n−k,α , where n denote the complex dimension of Z.
Proposition 5.5. If a ∈ H 1 A , then Proof. First observe that using simple scaling considerations and standard elliptic theory, (5.6) for k > 0 follows from the case k = 0.
It is rather straightforward to obtain a = O r −4 using the maximum principle. To obtain the stronger decay estimate it is customary to make use of a refined Kato inequality, see, e.g., Bando, Kasue and Nakajima [BKN89]. The Kato inequality is a consequence of the following application of the Cauchy-Schwarz inequality: | ∇ A a, a | ≤ |∇a| |a|. But, the equation∂a =∂ * A a = 0 imposes a linear constraint on ∇ A a, which is incompatible with equality in the previous estimate unless ∇ A a = 0. Hence, there exists a constant γ < 1, such that |d|a|| ≤ γ|∇ A a| on the set U := {x ∈ M θ : a(x) = 0}. A more detailed analysis shows that γ can be chosen to be 5/6. For a systematic treatment of refined Kato inequalities we refer to the work of Calderbank, Gauduchon and Herzlich [CGH00].
We set γ = 5/6 and let σ = 2 − 1/γ 2 = 4/5. Using the refined Kato inequality for a, we have The Weitzenböck formula for ∇ * A ∇ A a gives with R the Riemannian curvature operator and F A the curvature of the connection A. Since ∆∂ a = 0, and since by Theorem 3.9 the metric on M θ is ALE of order 4 and the curvature F A decays like r −4 , it follows that there exist positive constants c, τ > 0 so that on U we have We show that f = O r −4 , which is equivalent to the desired decay estimate for a. Note that on U , Since f is bounded, using [Joy00, Theorem 8.3.6(a)], we find g = O (r −τ ), such that Here (−) + denotes taking the positive part. Then f − g is a subharmonic function on M θ and must achieve its maximum at the boundary boundary of U or at infinity. Hence f ≤ g = O (r −τ ). Then by (5.7), ∆f = O r −2−2τ and the above procedure yields f = O r −2τ . Reiterating this argument k-times gives f = O r −kτ for all k < (n − 2)/τ . For the biggest k with this property, we have 2 + (k + 1)τ > n. Then by [Joy00, Theorem 8.3.6(b)], we can chose g above such that g = O r −4 . Therefore, f = O r −4 as desired.
Proof of Proposition 5.2. Given a ∈ H 1 A , we extend it to a 1-form onM θ vanishing along D. From Proposition 5.5 it follows that a vanishes to third order along D. Hence, a is in C 2,α and we can regard it as an element of A 1 M θ . Since∂a = 0, by Proposition 5.3 it gives an element [a] ∈ H 1 M θ , End R (−D) . This defines a linear map i : . We show now that i is injective. For this, assume that there exists b ∈ A 0 M θ so that a =∂b. Since b vanishes along D, its restriction to M θ decays like r −1 . Using this together with a = O(r −5 ), we can integrate by parts to obtain It follows that a vanishes, and thus i is injective.
To prove Lemma 5.1 it now sufficies to establish the following result: It follows, essentially from the definition of π θ , that Using (5.12) as well as the push-pull formula ( This concludes the proof. Before we embark on the proof of (5.9), it is useful to recall some basic properties of local cohomology, see, e.g., [Har77, Chapter III, Exercise 2.3]. Let D be a closed subset of X and let E be a sheaf on X. Denote by Γ D (X, E ) the subspace of Γ(X, E ) consisting of sections whose support is contained in D. The functor Γ D (X, −) is left-exact. Then H • D (X, E ) := R • Γ D (X, E ) is called the local cohomology of E with respect to D. Local cohomology is related to the usual cohomology of E by the following long exact sequence Moreover, it satisfies excision, that is, if U is an open subset in X containing D, then there is natural isomorphism Proof of Proposition 5.8. We already reduced the proof of this to the proof of the vanishing (5.9). Since by Proposition 5.10 H 1 (M θ , E nd(R)) = 0, the long exact sequence associated to the local cohomology yields We show that the first map in this sequence is an isomorphism. This gives the desired vanishing, H 1 (M θ , E nd R (−D)) = 0.
Let H denote the hyperplane section in P 3 . By excision, we have Theorem 6.1. For δ ∈ (−2n − 1, 0), D ± E,δ is Fredholm and its index is given by Here η E (s) := λ =0 sign(λ)|λ| −s is the eta-function of the spectrum of the Dirac operator restricted to the boundary at infinity S 2n−1 /G of the ALE manifold X, and η E (0) is the eta-invariant. AlsoÂ(X) is the HirzebruchÂ-polynomial applied to the Pontrjagin forms p i (X) of the ALE metric on X.
Proof. The fact that D ± E,δ is Fredholm is proved as in Proposition 4.9 by noting that the set of indicial roots does not intersect (−2n − 1, 0). This can be seen, for example, by realising that the indicial roots correspond to the eigenvalues of the Dirac operator on S 2n−1 /G shifted by − 2n−1 2 . The index formula follows from Atiyah-Patodi-Singer index theorem [APS75].
We consider the case when X is a Calabi-Yau manifold and E underlies a holomorphic vector bundle E with a Hermitian metric h whose induced Chern connection is A. Then there exists a canonical spin structure on X with S + = Λ 0,even T * C X and S − = Λ 0,odd T * C X.
The corresponding twisted Dirac operator is Now, we take X to be M θ for θ ∈ Θ Q generic and E to be R. Using Theorem 1.8, we equip M θ with an ALE Calabi-Yau metric g θ,RF and R with an HYM metric.

Geometrical McKay correspondence
In this section we prove Theorem 1.3 and derive its consequences. The proof uses the Atiyah-Patodi-Singer index theorem for ALE manifolds (6.2). In order to apply it, we need to compute the eta-invariant term that appears in this formula.
Proposition 7.1. Let G be a finite subgroup of SL(n, C) acting freely on C n \ {0}. Assume that X is a smooth ALE spin manifold assymptotic to C n /G and let (E, A) be a asymptotically flat bundle on X whose fiber at infinity is E ∞ . Then, the etainvariant for the Dirac operator D E,δ on X is given by provided −2n + 1 < δ < 0. In this formula χ E∞ denotes the character of the representation corresponding to the action of G on the vector space E ∞ . Remark 7.3. Note that for any g ∈ SL(n, C), n i=0 (−1) i χ Λ i C n (g) = det(id − g). Since G is chosen to act freely on C n , det(I n − g) = 0 for all g ∈ G \ {e}, and thus all the denominators in formula (7.2) are non-zero.
This proposition is a consequence of the Lefschetz fixed-point formula, in the sense that η E is the contribution from the fixed locus under the action of G on C n . It can be also proved using the definition of the eta-invariant as the analytic continuation at 0 of the eta-series corresponding to the spectrum of the Dirac operator on the boundary at infinity of the orbifold C n /G. This last approach gives the generalization of the above formula to the case of non-isolated singularities [Deg01].
Consider the virtual representation n i=0 (−1) i Λ i C n of G. For each ρ ∈ Irr(G) we have the decomposition into irreducibles Λ i C n ⊗ ρ = σ∈Irr(G) a (i) ρσ σ. LetC be the matrix with entries c ρσ for ρ, σ ∈ Irr(G), and let C be the principal submatrix ofC obtained by erasing the line and column corresponding to the trivial representation.
When G is a finite subgroup of SL(2, C), the matrix C is the Cartan matrix of the unique simple Lie algebra corresponding to G, whileC is the extended version. This is the essence of the McKay correspondence [McK80]. Note that for n ≥ 3 this matrix is not the Cartan matrix associated to a Lie algebra, nor is it a generalised Cartan matrix as appearing in the context of Lie algebras.
Remark 7.12. Note that formula (7.11) gives that the matrix C is invertible. In the case of a finite subgroup of SL(2, C), the invertibility of C was a direct consequence of the McKay Correspondence, given that C is the Cartan matrix associated to a simply-laced Dynkin diagram [McK80].