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Partial Resolutions of Orbifold Singularities via Moduli

Spaces of HYM-type Bundles

Alexander V. Sardo Infirri

Abstract. Let be a finite group acting linearly on Cn, freelyoutside the origin, and let N be the number of conjugacy classesof minus one.

A construction of Kronheimer [Kro89] of moduli spaces X oftranslation-invariant -equivariant instantons on C2 is generalisedto Cn.

The moduli spaces X depend on a parameter QN . Thefollowing results are proved: for = 0, X0 is isomorphic to Cn/;if 6= 0, the natural maps X X0 are partial resolutions. Themoduli X are furthermore shown to admit Kahler metrics whichare Asymptotically Locally Euclidean (ALE).

A description of the singularities of X using deformation com-plexes is given, and is applied in particular to the case SU(3).It is conjectured that for general and generic that the singu-larities of X are at most quadratic. When SU(3) a natu-ral holomorphic 3-form is constructed on the smooth locus of X,which is conjectured to be non-vanishing. The morphims X X0are expected to be crepant resolutions and X to be smooth forgeneric choices of the parameter . Related open problems inhigher-dimensional complex geometry are also mentioned.

The paper has a companion paper [SI96b] which identifies themoduli X with representation moduli of McKay quivers, and de-scribes them completely in the case of abelian groups.

Contents

0. Introduction 20.1. Background 20.2. Main Results 30.3. Two Conjectures 30.4. Related Questions 40.5. Methods 50.6. Outline 5

Date: 2 October 1996.MATHS SUBJECT CLASSIFICATION (1991): 32S45 (Primary) 32L07,

53C07, 32M05 (Secondary)1

http://arXiv.org/abs/alg-geom/9610004v1

2 Alexander V. Sardo Infirri

0.7. Acknowledgments 51. Geometric Invariant Theory 61.1. Linearisations and GIT quotients 61.2. Stability and Extended G-equivalence 71.3. Quotients for non-trivial linearisations 72. Moduli of Hermitian-Yang-Mills Connections 82.1. Connections over Symplectic Manifolds 82.2. Connections over Kahler manifolds, Holomorphic Structures

and the Hermitian-Yang-Mills condition 93. Construction of X 113.1. Invariant HYM Connections 113.2. The Vector Space M 123.3. Symplectic and Kahler structure. 133.4. The Variety N of Commuting Matrices 143.5. Moment Map 144. Variation of Quotients and Partial Resolutions 154.1. The Universal Quotient 164.2. The Zero-Momentum Quotient 174.3. Non-zero Momentum and Partial Resolutions 195. ALE Metrics 196. Deformation Complexes 216.1. Differential Forms and Graded Lie Algebras 216.2. Local Description of X 236.3. Kuranishi Germs and Formality 247. Subgroups of SU(3) and Cubic Forms 257.1. Example 27References 28

0. Introduction

This paper is concerned with affine orbifold singularities, namelywith singularities of the type X = Cn/ for a finite group actinglinearly on Cn.

More precisely, this paper gives a method for constructing partialresolutions of X, namely birational morphisms Y X which are iso-morphisms over the smooth locus of X.

0.1. Background. The method in question was first introduced byKronheimer [Kro89]. It can be described in various ways, dependingon ones point of view. One description (although maybe not the moststraight-forward) is to construct moduli spaces X of instantons on the

Orbifold Singularities and Moduli of Bundles 3

trivial bundle CnR Cn. Here R denotes the regular representationspace for the group , and is a linearisation of the bundle action. Theinstantons are required to satisfy Hermitian-Yang-Mills-type equations,as well as additional -equivariance and translation-invariance proper-ties.

In Kronheimers case, SU(2), and the moduli spaces X can infact be viewed as hyper-Kahler quotients. Kronheimer shows that X0is isomorphic to C2/, that the natural maps X X0 are partial res-olutions for 6= 0, and that indeed X coincides with the minimal reso-lution of C2/ for generic choices of . Furthermore, X inherit naturalhyper-Kahler metrics on their non-singular locus, which are shown tobe Asymptotically Locally Euclidean (ALE): they asymptotically ap-proximate the Euclidean metric at infinity (up to terms vanishing withthe inverse of the fourth power of the radial coordinate).

0.2. Main Results. In the present case n is any integer greater thanor equal to 2, and U(n) is assumed to act on Cn freely outsidethe origin for any n 2, which means that X = Cn/ has an iso-lated singularity.1 As a result, the moduli X are only Kahler ratherthan hyper-Kahler quotients (in actual fact they are more convenientlydescribed in term of geometric invariant theory). The main result is

Theorem (c.f. Thms. 4.2, 4.7 and 5.1 in the text). Let act linearlyon Cn and freely outside the origin and let X be the moduli spaces con-structed in Section 3.5.

Then X0 is isomorphic to X = Cn/, and for 6= 0, the naturalmorphisms X X0 are partial resolutions. Furthermore, the inher-ited Kahler metrics on the smooth loci of X are Asymptotically LocallyEuclidean in the sense of [Kro89].

0.3. Two Conjectures. The final sections of the paper discuss anddevelop two conjectures.

Conjecture (c.f. Conj. 1). The singularities ofX (for generic , say)are at most quadratic algebraic.

This is a common occurrence for moduli spaces of this kind [Nad88,GM87, GM90]. Its proof can be reduced to proving the formalityof a certain differential graded Lie algebra (DGLA) by the methodsof [GM90]. This is done in Section 6, where the singularities of Xare described in terms of deformation complexes [AHS77, DK90]. Theconcept of formality is explained, and it is suggested that the complex

1This is for the purpose of simplicity the method would seem to be applicableto the general case with some modifications

4 Alexander V. Sardo Infirri

relevant to X may be formal, in a way similar to Tian [Tia87] andTodorovs [Tod89] work. This would imply that the singularities of Xat most quadratic algebraic. This conjecture is also checked by com-puter for low order abelian groups in U(3) using the methods in thecompanion paper [SI96b].

Conjecture (c.f. Conj. 2 in the main text). If SU(3), the mor-phisms X X0 are crepant, and if is generic, X is smooth and itsEuler number is equal to the orbifold Euler number of X0 as definedin [DHVW85].

The fact that X has at most quadratic singularities has been verifiedfor the abelian subgroups of order less than 11. The smoothness of Xhas been verified in the abelian cases 1

3(1, 1, 1), 1

6(1, 2, 3), 1

7(1, 2, 4),

18(1, 2, 5), 1

9(1, 2, 6), 1

10(1, 2, 7) and 1

11(1, 2, 8). Both these verifications

were done by a brute-force listing of singularities of X for all possible , using the methods given in the companion paper [SI96b].

The cases SU(n) present a particular interest. The problemof constructing a crepant resolution of C3/ with the same orbifoldEuler number was only recently completed [MOP87, Roa91, Mar93,Roa93, Ito94, Roa90]. For the case C4/, one can obtain some inter-esting analogous results if one considers terminalisations rather thanresolutions [SI96a].

In Section 7, a natural holomorphic 3-form is constructed on thesmooth locus of X : this is conjectured to be non-vanishing. Its normis shown to be constant if and only if the induced metric on X isRicci-flat (which does not usually turn out to be the case, however).

0.4. Related Questions. This paper has a companion paper [SI96b]in which the moduli X are identified with representation moduli ofMcKay quivers. This allows one to explicitly describe the case ofAbelian in terms of flows on the McKay quiver. Explicit com-putations are carried out for groups of low order, and the conjecturesabout the smoothness and the triviality of the canonical bundle arechecked (by brute-force computer calculations) for abelian subgroupsof SU(3) of order less than or equal to 11.

Many questions are left open by the present work, besides the conjec-tures already mentioned. For instance, do the birational models X ofCn/ possess any special properties with regards to their singularities(are they terminal?) Are all terminal models for a given 3-fold singu-larity obtained by this construction? What is the relationship betweenthe different X? Are they related by flips/flops? Is it possible, bychoosing very special values of , to produce blowups X X0 which

Orbifold Singularities and Moduli of Bundles 5

are interesting from the point of view of higher dimensional geometry,for instance, for the construction of flips in dimensions 4 and greater?

0.5. Methods. The methods used include geometric invariant theory,Kahler quotients, and elementary theory of the moduli of bundles, thenecessary aspects of which are reviewed in Sections 1, 2 and 3. Further-more, the same construction is presented under different angles withthe intention that the reader who is familiar with one of them (or withKronheimers work [Kro89]) will be able to follow the discussion easily.

The later sections devoted to the various conjectures raised by themain results touch on the theory of deformation complexes, and con-cepts of Kuranishi germs, formality, and so on. Some background isalso provided, although not as extensive as to be able to describe it asself-contained, given the conjectural nature of the material.

0.6. Outline. The outline of this paper is as follows.Section 1 reviews material regarding geometric invariant theory quo-

tients which is necessary to define the moduli X . No essentially newmaterial is involved, although the formulation of some of the resultsmay be un-familiar to non-specialists.

Section 2 review material concerning moduli of Hermitian-Yang-Millsconnections. This is not essential to the understanding of the mainresults, although some familiarity is desirable for the understanding ofSection 6.

Section 3 deals with the definition and construction of the moduliX.

Section 4 gives the proof that X0 is isomorphic to X and that X X0 are partial resolutions.

Section 5 proves that the induced metrics on X are ALE.Section 6 contains the discussion of the singularities of X in the

language of deformation complexes. This includes the conjecture thatthe singularities of X are at most quadratic algebraic.

Section 7 deals with the case SU(3), the construction of theholomorphic three-form on the non-singular locus of X and conjec-ture 2.

0.7. Acknowledgments. The present paper and its companion pa-per [SI96b] consist mostly2 of excerpts of my D.Phil. thesis [SI94], andI wish to acknowledge the University of Oxford and Wolfson College fortheir hospitality during its preparation. I am grateful to the RhodesTrust for financial support during my first three years, and to Wolfson

2Minor portions have been rewritten to include references to advances in thefield made since then.

6 Alexander V. Sardo Infirri

College for a loan in my final year. The conversion from thesis to articleformat was done while I was a Research Assistant in RIMS, Kyoto.

I also take the opportunity to thank my supervisors Peter Kron-heimer and Sir Michael Atiyah who provided me with constant advice,encouragement and support and whose mathematical insight has beenan inspiration. I also wish to thank William Crawley-Boevey, MichelBrion, Gavin Brown, Jack Evans, Partha Guha, Katrina Hicks, FrancesKirwan, Alistair Mees, Alvise Munari, Martyn Quick, David Reed,Miles Reid, Michael Thaddeus, and, last but not least, my parents andfamily.

1. Geometric Invariant Theory

This section recalls the geometric invariant theory of affine varietiesand proves some results which shall be needed in the sequel. Theseresults have been included here, because, although they are well-knownto the experts, no elementary treatment exists.3

1.1. Linearisations and GIT quotients. LetG be a reductive groupacting linearly on a complex affine variety X. In this situation,4 a(G-)linearisation is a lifting of the G-action to the trivial line bundleL X. Such a linearisation is determined completely by the actionof G on the fibres of L, namely by a character : G C. For everycharacter , denote by L the trivial bundle endowed with the cor-responding linearisation. The space of G-invariant sections of L isdenoted by H0(L)

G.The geometric invariant theory (GIT) quotient of X by G with re-

spect to is defined by

X// G := Proj

kNH0(kL)

G.

Example 1.1. If = 0 is the trivial character, then the correspondingquotient X//0G coincides with the usual affine GIT quotient X//G. Infact, suppose that X = SpecR for a finitely generated ring R and letz0 be a coordinate in the fibre of L0. Then kNH0(kL0)G = R[z0]G =RG[z0], so taking Proj gives:

X//0G = ProjRG[z0] = SpecR

G = X//G,

which is the usual affine GIT quotient.

3For the generalisation of these results to arbitrary quasi-projective varieties,see [Tha94, DH94].

4When X is only quasi-projective the definition of a linearisation involves spec-ifying an ample bundle over X as well as a lift of the action to the bundle.

Orbifold Singularities and Moduli of Bundles 7

1.2. Stability and Extended G-equivalence. The GIT quotientcan be obtained by first restricting attention to the open setXss() Xof so called semi-stable points. A point x in X is called semi-stable(with respect to ) if there exists a G-invariant section of kL (forsome k in N) which is non-vanishing at x. As a set, X// G is the quo-tient of Xss() by the extended G-equivalence relation induced by theclosure of the G-orbits:

x y Gx Gy 6= .Thus the G-invariant quotient mapXss() X// G for the equivalencerelation can map several G-orbits to the same point.

For most of the points, this does not happen, however. This isbecause the closure of an open orbit is obtained by adding orbitsof smaller dimension, so since the dimension of the orbit is a lowersemi-continuous function on X, it follows that there is an open subsetXs() Xss() of points which have full-dimensional closed G-orbits the so-called stable points and there is a one-one correspondencebetween the orbits of G in Xs() and their images in the GIT quo-tient. In other words, the GIT quotient contains, as an open set, thegeometric quotient Xs()/G.

1.3. Quotients for non-trivial linearisations. Example 1.1 showedthat the GIT quotient for the trivial linearisation coincides with theaffine GIT quotient. The following theorem uses the notion of stabilityto show that the quotients for non-trivial are closely related to theaffine quotient.

Theorem 1.2. The GIT quotients admit projective morphisms

: X// G X//0Gwhich are isomorphisms over

1 (Xs(0)) Xs(0).

Proof. Let X = SpecR for some finitely generated ring R, and let zbe a complex coordinate in the fibre of L . The GIT quotient X// G isgiven by taking Proj of the G-invariant part of R[z ], where R[z ] is tobe considered as an algebra graded by the powers of z . The previousexample showed that when is zero, ProjRG[z0] = SpecR

G.For a general non-zero , R[z ]

G 6= RG[z ]. However, the degree-zeropart of R[z ]

G is always RG, and this shows [Har77, Example II.4.8.1and Cor. II.5.16] that X//G is projective over X//G = SpecR

G.Finally, the map X// G X//0G comes from the descent of the

composition Xss() Xss(0) X//0G and this is one-one wheneverXss(0) X//0G is, so the last statement of the theorem follows.

8 Alexander V. Sardo Infirri

Remark 1.3. If X contains a 0-stable point, then Xs(0) is open andnon-empty, so dense in X. Its image Xs(0)//0G is open and densein X//0G, and therefore, : X// G X//0G is an isomorphism on adense open subset, i.e. is birational.

2. Moduli of Hermitian-Yang-Mills Connections

This section recalls basic background concerning Hermitian-Yang-Mills connections and the construction of their moduli, following [DK90].The construction of X will appear as a specialisation of the materialin this section. However, the reader wishing to go straight to the pointcan skip this section and find a self-contained construction of X insection 3.

2.1. Connections over Symplectic Manifolds. Suppose X is acompact symplectic manifold (X,) of dimension 2n and let E be acomplex vector bundle over X. The bundle of infinitesimal automor-phisms of E will be denoted gl(E) or EndE.

2.1.1. Connections. Consider connections on E, namely linear maps : 0X(E) 1X(E) which satisfy the Leibniz condition. Any connec-tion on E can be expressed in a local neighbourhood U X as

d = d+ , (2.1)

for 1U(EndE). On the other hand, if one considers the differenceof two connections, one gets a global one-form with values in EndE.

2.1.2. Hermitian Structure. Let h be a positive definite Hermitian in-ner product on the fibres of E and denote by u(E) gl(E) the realsub-bundle of unitary automorphisms determined by h. The connec-tion is said to be compatible with the Hermitian structure if

dh(s, t) = h(s, t) + h(s,t).Such a connection will have local one-form representatives 1U(u(E));the space A of all such connections is an infinite-dimensional affinespace modeled on 1X(u(E)).

Remark 2.1. Why does one fix a Hermitian structure on E? One rea-son is because fixing a Hermitian structure on E amounts, by Chern-Weil theory, to fixing the topological invariants of the connections: forany Hermitian connection d on E, the Chern polynomials c1(E) andc2(E)c1(E)2 are represented respectively by i2 traceF and traceF 2.

Orbifold Singularities and Moduli of Bundles 9

2.1.3. Symplectic Structure and Gauge Group. The space A has a sym-plectic structure defined by

(a, b) :=

X

trace(a b) n1,

for a, b 1X(u(E)) tangent vectors to A.The gauge group G is the group of automorphisms of E which respect

the Hermitian structure in the fibres and cover the identity map of X.It acts on A by

g := gg1(the condition that g is unitary ensures that the new connection iscompatible with h) and preserves the symplectic form .

2.1.4. Moment Map. The Lie algebra of the gauge group is LieG =0X(u(E)). The moment map for the action of the gauge group is givenby [DK90, Prop. 6.5.8]

: A 0X(u(E)) 7 s 7

Xtrace sF n1 (2.2)

Note that the G-equivariance follows because the curvature transformsas a tensor under gauge transformations.

2.2. Connections over Kahler manifolds, Holomorphic Struc-tures and the Hermitian-Yang-Mills condition. Now supposethat X is in fact a Kahler manifold.

2.2.1. Complex Structure and the Decomposition of Curvature. Thereis a natural decomposition 1X(EndE) = (

1,0X 0,1X )0(EndE), and

if a connection is expressed in a local holomorphic frame {zi} accordingto (2.1) it takes the form

=

i

idzi i dzi,

for i smooth sections of EndE.The local connection d splits into a sum of (1, 0) and (0, 1) parts

+ given by:

= +

i

idzi (2.3)

=

i

i dzi. (2.4)

The space A becomes a (flat) Kahler manifold when tangent vectors areidentified with their (0, 1) parts, or in other words, when connections

10 Alexander V. Sardo Infirri

are represented by their operators. The holomorphic tangent spaceto A is of course isomorphic to 0,1X (EndE).

The (1, 1) and (2, 0) parts of the curvature F of d are given by

F 1,1 =

i,j

(

jzi

j

zi [i, j ]

)

dzi dzj , (2.5)

and

F 2,0 =

i,j

(

izj

+1

2[i, j]

)

dzi dzj , (2.6)

with F 0,2 equal to minus the Hermitian adjoint of F2,0 .

2.2.2. The Hermitian-Yang-Mills condition. A connection on E is calledHermitian-Yang-Mills (HYM ) if the inner product of its curvature withthe Kahler form is a central element of 1X(u(E)). This is in fact amoment map condition: using the identity

F n1 =1

nF, n =:

1

n(F)

n,

the map in equation (2.2) becomes

()(s) =Vol(X)

(n 1)! trace(sF),

so, embedding the Lie algebra of G its dual in the usual way, we seethat the moment map becomes a constant multiple of

: A 0X(u(E)) 7 F.

The moduli space of HYM connections is the Kahler quotient 1(0)/G.

2.2.3. Holomorphic Bundles. Suppose instead that the Hermitian con-nection d is required to induce a holomorphic structure on E. By theNewlander-Nirenberg theorem, prescribing such a structure is exactlyequivalent to specifying a connection d which is integrable, i.e. whose(0, 1)-part is such that = 0. This is equivalent to the conditionthat F is of type (1, 1) and gives a Kahler subvariety A1,1 A. Thisvariety parametrises all the possible holomorphic structures which canbe put on the C bundle E X.

The action of G on A extends to an action of its complexificationGC, which can be thought of naturally as the group of all general linear

Orbifold Singularities and Moduli of Bundles 11

automorphisms of E covering the identity map on X. Put g := (g)1

and let

g := gg1, (2.7)g := gg1. (2.8)

This action of GC preserves the space A1,1, and its orbits are equiv-alence classes of holomorphic bundles. To get a nice moduli space (aquasi-projective variety), one must restrict to the so-called semi-stablebundles, or in other words, consider a GIT quotient A1,1//GC. A theo-rem of Uhlenbeck and Yau [UY86] states that the moduli space of stablebundles with the same topological type as E X coincides with themoduli space of Hermitian-Yang-Mills connections on E. This is aninfinite-dimensional version of the correspondence between symplecticand algebro-geometric quotients.

Remark 2.2. One should really use the quotient of GC by the scalarautomorphisms, since they act trivially on A. The resulting groupthen has a trivial centre so there is only one linearisation of the action;it essentially determined by the degree and rank of E.

3. Construction of X

A construction of X from scratch will be given in this section.Let Q be an n-dimensional complex representation of a finite group

. Average over the group elements to get a positive definite Hermitianinner product on Q such that U(Q). Let R be the regular represen-tation of , i.e. the free -module which is generated over C by a basis{e| }, and on which acts via the morphism : AutCRdefined by:

e := ()e := e.

3.1. Invariant HYM Connections.

Note . The reader who has not read Section 2 or is not interested inthe moduli of bundles point of view can jump directly to 3.2.

The construction of X is based on a variation on the construction inthe previous section. It consists, roughly speaking, in applying the con-struction to the case where the compact Kahler manifold X is replacedby the germ of the singularity Q/.

More precisely, start with Q, the dual vector-space to Q, and E =Q R Q the trivial vector bundle with fibre R. Consider theconnections on E which are invariant under all translations in Q.These connections are determined by their value at one point, and

12 Alexander V. Sardo Infirri

they form a finite-dimensional vectorspace which can be identified withM = Q C EndCR by choosing an isomorphism 1Q = Q. Theconstructions in the previous section are now valid, because transla-tion invariance eliminates any problems one might have with the non-compactness of base space Q. Most aspects are indeed a lot simpler:it suffices to set all the derivatives of the i equal to zero, and to ignoreany integrals over the base space and all the formulas remain valid.

An unusual feature of these invariant connections is that there areseveral moduli spaces: the usual Hermitian-Yang-Mills condition for aconnection states that the contraction of the curvature with the Kahlerform should be a central element of the Lie algebra of the gauge group.In case of general connections, there is only one gauge-invariant mo-mentum level set because the centre of the gauge group G is trivial.In the case of invariant connections the gauge group that is relevant isa much smaller group K, consisting of gauge transformations whichare invariant with respect to all translations and the action of . Thisgroup consists of unitary endomorphisms of R which commute withthe action of , and has a non-trivial centre (consisting of the traceless-endomorphisms : R R). The non-triviality of the centre meansthat there are several gauge invariant momentum level sets, and henceseveral possible moduli X.

For the sake of the readers unfamiliar with the material in section 2,the construction of X is given in detail without making any referenceto the bundle construction.

3.2. The Vector Space M. Let M = Q C EndC R and let acton EndC R by conjugation via

AutR: T := ()T ()1 (3.1)

This makes M into a -module. Its -invariant part is denoted M:

M := (Q EndR) . (3.2)The spaces M and M can be described explicitly by choosing a basis{ql}nl=1 for Q, and defining the components of M by:

=n

l=1

ql l. (3.3)

In this way the elements M are identified with n-tuples of linearmaps i : R R. The elements of M correspond to those n-tupleswhich satisfy the following equivariance condition

l

kll = ()k()1, k, , (3.4)

Orbifold Singularities and Moduli of Bundles 13

where = (kl) is the matrix corresponding to the action of the element on Q with respect to the basis {ql}nl=1.3.3. Symplectic and Kahler structure. Endow R with a fixed pos-itive definite Hermitian inner product , which makes the standardbasis e orthonormal. The inner product on R also defines a real struc-ture on the space EndC R of C-linear endomorphisms of R by the Her-mitian adjoint operation in the usual way:

T x, y := x, Ty. x, y R.Define a positive definite Hermitian inner product h on M by

h : M M C(, ) 7 i trace(ii )

(3.5)

The definition of h is independent of the basis of Q up to a unitarytransformations, and restricts to an inner product on M. As usual, hinduces two forms on the underlying real vector-space to M :

a non-degenerate symmetric bilinear form g = (h) called theRiemannian metric associated to h

g : M M R(, ) 7 1

2

i trace(ii + i

i )

a non-degenerate skew-symmetric bilinear form = (h) calledthe Kahler form associated to h

: M M R(, ) 7 1

21

i trace(ii ii )

This gives a Riemannian metric g and a Kahler form of type (1, 1)on M and M, related as usual by

(, ) = g(, i). (3.6)

This makes M,M and all their complex subvarieties into Kahler va-rieties.

The group GL(R) of automorphisms of R acts on M by conjugationon EndR:

i 7 gig1, g GL(R). (3.7)In fact, the scalars act trivially, and the action descends to an actionof G := PGL(R) := GL(R)/GL(1).

The subgroup GLR of endomorphisms which commute with theaction of acts on M and, in the same way, there is a free action ofG = PGL(R) on M. A maximal compact subgroup of G (resp. G)is given by K = PU(R) (resp. K := PU(R)). The compact group K(resp. K) leaves the Kahler structure on M (resp. M) invariant.

14 Alexander V. Sardo Infirri

3.4. The Variety N of Commuting Matrices. Define the follow-ing natural map:

: Q EndR 2Q EndR

k qk k 7

k,l qk ql[k, l]This definition is independent of the basis ofQ (in terms of connections,it corresponds to calculating the (0, 2) part of the curvature). Denotethe restriction of to M by the same letter. Define

N := 1(0) M ; (3.8)it is a cone (i.e. it is invariant under multiplication by non-zero scalars)which is an intersection of quadrics in M given by the coordinate func-tions of . In the representation of equation (3.3), its points consist ofn-tuples of commuting r r matrices:

N = {(1, . . . , n) : i Matr(C), [i, j] = 0.}Its -invariant part N consists of those commuting matrices satisfyingthe equivariance condition (3.4).

3.5. Moment Map. Consider the vector space M with the hermitianinner product h and the action ofK. The Lie algebra ofK is isomorphicto suR, which consists of traceless skew-Hermitian endomorphisms ofR. Using the invariant inner product

a, b := trace(ab) = trace ab,identify suR with its dual in the usual way. Then the moment map isfor the action of K is

: M suR 7 k[k, k].

(3.9)

(In the language of of connections, the map corresponds to con-tracting the curvature of the connection with the Kahler form =

i dqi dqi 1,1Q .) The moment map for the action of K on M isobtained simply by restriction.

The Kahler quotients are defined by

X :=1() N

K, Centre(suR). (3.10)

As was remarked in section 1, to make this definition rigorous, oneneeds to make sense of the Kahler structure on X . One way to do thisis by restricting to take on integral values. Then, by the correspon-dence between Kahler and GIT quotients, one has

X = N // G, (3.11)

Orbifold Singularities and Moduli of Bundles 15

where on the right-hand side specifies the linearisation of the actionof G on the trivial line bundle N C.

Remark 3.1. In the case SU(2), one has 2Q = R0 the trivialrepresentation. Identifying C2 with the quaternions, the vector spaceM becomes a hyper-Kahler manifold with 3 distinct complex struc-tures I, J,K and corresponding associated Kahler forms I , J , K .The map is then a moment map for the complex symplectic formC = J + iK which is itself holomorphic with respect to I and thequotients X are quotients of M

with respect to the hyper-Kahler mo-ment map given by (, ), where the second (complex) variable is setto zero. Kronheimer [Kro86, Kro89] exploits this fact to show that Xare the minimal resolutions of C2/ for generic values of . Further-more, by varying the level set of , he obtains universal deformations.If dimM > 2 however, 1() is not G-invariant for non-zero .

Remark 3.2. In fact, there is a further action on M by GLQ (actingon Q on the left). If GLQ, then it is easy to see that this preservesM, and indeed N . Furthermore, the centre Z() of is a subgroupof GLQ which acts trivially because of (3.4), so there is an actionof G := GLQ/Z() which the quotients X inherit. The compactsubgroup K := U(Q)/Z() acts in a Hamiltonian fashion, and themoment map for this is given by

: X (LieUQ)[] 7 b 7ij bij traceij .

(3.12)

In general this does not provide one with very much information: forinstance, if Q is irreducible, G = C and () is the identity endo-morphism of Q times the sum of the norm squared of the is. In thecase where is abelian, however, G contains an algebraic torus of di-mension n acting freely. The components of are the norm squaredof the matrices i. This is exploited in the companion paper [SI96b] toobtain a complete description of X by the methods of toric geometry.

4. Variation of Quotients and Partial Resolutions

The zero momentum quotient X0 is better understood if viewed asa two-stage construction: first construct a universal quotient N0 byignoring the -equivariance condition (i.e. perform the same construc-tion with replaced by the trivial group) and then obtain X0 as its-invariant part.

16 Alexander V. Sardo Infirri

4.1. The Universal Quotient. Consider taking symplectic quotientsof N with respect to the action of K. Since the centre of its Lie algebrais trivial, there is only one quotient, with momentum zero:

N0 =N 1(0)

K= N //G.

Lemma 4.1. The reduction N0 is isomorphic to configuration spaceof r = || points in Q:

N0 = Symr(Q) := Qr/r,where r denotes the permutation group on r letters acting component-wise on the Cartesian product Qr.

Proof. The proof simply adapts Kronheimers [Kro86, Lemma 5.2.1]. Itis shown that the K-orbits in N 1(0) can be identified in a one-oneway with the r-orbits in Q

r. Let N 1(0) have components iwith respect to a basis qi. The conditions () = () = 0 give

[i, j] = 0, for all i, j

i

[i , i] = 0.

If one denotes by Ai the operator ad(i), one has, using the Jacobiidentity and the above equations:

i

AiAi(j ) =

i

[[i , i], j ] + [[

j ,

i ], i] = 0.

The positivity of AiAi implies AiAi(

j ) = 0 for all i and j, and hence

Aj(j ) = [

j , j] = 0. Thus N0 is the variety of n-tuples of nor-

mal commuting endomorphisms of R modulo simultaneous conjuga-tion. Any such n-tuple can simultaneously diagonalised by conjuga-tion by a unitary matrix. This means that the orbit of an n-tuple isdetermined by the eigenvalues of its components; more precisely, thereare orthonormal vectors v R indexed by the elements andcorresponding eigenvalues i C such that:

i(v) = iv, for all i, . (4.1)

This gives r elements

:=

i

iqi Q,

which could be called the eigenvalues of . These are defined up toa permutation, because one can always conjugate by an elementarymatrix which permutes the rows of the is.

Orbifold Singularities and Moduli of Bundles 17

In geometrical language, denote by M the subspace of n-tuplesof matrices which are diagonal with respect to the standard basis eof R. The unitary automorphism of R which maps e to v moves into . The slice can be identified with Qr by mapping to itsr eigenvalues (listed in some specified order). In this way inheritsan action of r, and the U(R)-orbit of 1(0) N intersects ina single r-orbit.

4.2. The Zero-Momentum Quotient.

Theorem 4.2. If acts freely outside the origin, then X0 = Q/ asvarieties.

Proof. The proof consists in showing that the K-orbits in N 1(0)can be identified in a one-one way with the -orbits in Q. Let N 1(0) and let v1 be an eigenvector of with eigenvalue 1 :=

i i1qi Q:

i(v1) = i1v1, i = 1, . . . , n.

By equivariance of ,

i(R()v) = (Q()1)i(R()v), for all , and all i,

so the eigenvectors and eigenvalues of are given by

:= Q()1 and v := R()v1, for all

i.e. they lie in orbits of . Since acts freely outside the origin,the eigenvalues are either all zero or all distinct and non-zero. In thelatter case, the eigenvectors therefore form a basis of R. The unitaryautomorphism of R defined by e 7 v commutes with the action of, so defines an element of K. If M denotes the n-tuples ofendomorphisms of R which are diagonal with respect to the standardbasis {e} then the automorphism carries into . The map 7

i i1qi identifies

with Q in a manner that is compatible with the-action on both sides. Furthermore, the K-orbit of intersects

in precisely one -orbit. Thus X0 = / = Q/.The following lemma will be useful in the section on ALE metrics

(cf. [Kro86]).

Lemma 4.3. If acts freely outside the origin the map

1(0) N /K /is an isometry when is given the metric it inherits as a subspace ofM, namely the Euclidean metric. Furthermore, the bundle 1(0) N X0 is flat.

18 Alexander V. Sardo Infirri

Proof. The key point is that the subspace is everywhere orthogonalto the orbits of K: a tangent vector to the orbits consists of an n-tupleof matrices of the form [, i] for some su(R), and these matricesare always zero on the diagonal, so orthogonal to .

This shows that the bundle 1(0) N X0 is flat, and thedefinition of the quotient metric on X0 implies that the map X0 =1(0) N /K / is an isometry.

4.2.1. Case when does not act freely outside the origin. Let N 1(0) have an eigenvalue Q. If the stabiliser of istrivial, then has r distinct eigenvalues, corresponding to the elementsof the orbit . This determines the components i completely on thewhole of R.

On the other hand, if has a non-trivial stabiliser , then thisdetermines on the sub-representation W := span E R, whereE is the eigenspace corresponding to . In fact, E is a representationof the stabiliser subgroup and W is simply the representation of induced by E:

W = IndE.

If dimE < ||, then W 6= R and restricts to an endomorphismof W . Let

be an eigenvalue of the restriction; the equivariancecondition then determines on the factor W . Continuing in this way,one obtains a decomposition of R:

R = W W . . . .From this discussion, one obtains the following description of the

quotient X0:

Theorem 4.4. There is an inclusion Q/ X0; this inclusion is anisomorphism if and only if acts freely on Q outside the origin.

Proof. The first statement follows because, for any orbit in Q, onecan construct an n-tuple of diagonal matrices whose -eigenspace hasdimension equal to the stabiliser . The orbit of such an under K

consists of commuting matrices with eigenvalue with multiplicity||, for all .

For the second statement, note that the if direction is theorem 4.2.The only if direction follows because if is an eigenvalue of withnon-trivial stabiliser and with multiplicity one, one can set to be zeroon W (since 0 is a fixed point of , the equivariance condition (3.4)does not imply the existence of other eigenvalues).

In general, X0 corresponds configurations of r = || points of Qwhich are unions of orbits of , and hence give rise to a decomposition

Orbifold Singularities and Moduli of Bundles 19

of R into induced representations

R =

i

IndiEi ,

where Ei denote the -eigenspace of an element .

Remark 4.5. When doesnt act freely outside the origin, the quotientX0 may end up containing all sorts of things. For instance, for the groupaction 1

5(0, 1,1), the quotient X0 contains a copy of C3/Z5, a copy of

C5, eight copies of C2, etc...

4.3. Non-zero Momentum and Partial Resolutions. By theo-rem 1.1.2, in the case where is integral, there are projective mor-phisms : X X0 which are isomorphisms over the set of pointswhich have finite K-stabilisers.

Proposition 4.6. If acts freely outside the origin, The stabilisers ofK on N 1(0) are trivial everywhere except at = 0.Proof. An automorphism T of R which fixes N 1(0) mustpreserve the (simultaneous) eigenspaces of . If T also commutes withthe action of , its action on an eigenvector v R determines its actionon the linear span of the -orbit of v. In the case where acts freelyand is non-zero this means that T is only allowed to multiply eacheigenvector by the same non-zero constant and this constant mustbe of modulus one if T is unitary. Such a T thus corresponds to theidentity element in the quotient group K = PU(R).

Applying the theorem about Kahler quotients, one gets the followingtheorem:

Theorem 4.7. If acts on Q freely outside the origin, and is in-tegral, there are projective morphisms : X X0 = Q/ which areisomorphisms outside the set 1 (0).

Remark 4.8. Even in the case that does not act freely outside theorigin, it is likely that there are still birational maps from X to thecomponent of X0 which is isomorphic to Q/ and which are isomor-phisms outside the singular set.

5. ALE Metrics

The quotients X inherit a metric g from the metric g on the am-bient space M. This section shows that these are ALE metrics.

A metric g on a real m-dimensional Riemannian manifold X is calledasymptotically locally Euclidean (ALE ) if there exists a compact subset

20 Alexander V. Sardo Infirri

C X whose complement X \ C has a finite covering X \ C which isdiffeomorphic to the complement of a ball in Rm, and such that, in the

pulled-back coordinates x1, . . . , xm on X \ C, g takes the formgij = ij + aij , (5.1)

where |paij | = O(r4p) for p 0, where r =

i x2i denotes the

radial distance in Rm and denotes the differentiation with respect tothe coordinates x1, . . . , xm.

Theorem 5.1. The metrics on X are ALE: for any , there is anexpansion in powers of r

g = +

k2hk()r

2k, (5.2)

where (r, ) denote polar coordinates in R2n = Q. This expansion isanalytic and may be differentiated term by term.

Proof. Kronheimers proof [Kro86, Prop.5.5.1] goes through with theappropriate modifications. The metric g restricted to the unit ballr = 1 is an analytic function of , so admits an expansion

g|r=1 =

f

where are multi-indices in the coordinates of . The moment mapbeing quadratic homogeneous implies that

g(r, ) = gr2(1, ).

Hence the expansion for g takes the form

k0hk()r

2k,

where the hk =

||=k f are analytic functions of the radial coordi-

nates.It remains to show that h0 = and that h1 = 0. The first state-

ment is equivalent to showing that the identification X0 Q/ is anisometry. This was done in Lemma 4.3.

For the second statement, one must show that the variation of gwith is zero at = 0 in every direction ((LieK))K. Themetric g is determined entirely by the Kahler form and the inducedcomplex structure J . Since the latter is the same for all , it is sufficientto prove that

|=0 = 0

Orbifold Singularities and Moduli of Bundles 21

for all . A general formula for the variation of the induced symplecticform is given by Duistermaat and Heckman in [DH82]. Away fromthe singularities, the projection 1() N X is a principal K-bundle whose connection is given by the Levi-Civita connection for theinduced metric on X . If denotes the curvature, regarded locally asan element of 2X su(R), then the formula for the variation of isgiven by

= ,.In the present case, lemma 4.3 tells us that 0 = 0, so the variation iszero for = 0, and this concludes the proof.

6. Deformation Complexes

The question of the local geometry of the moduli spaces X can bestudied using the tools of deformation complexes.

6.1. Differential Forms and Graded Lie Algebras. Define thevectorspaces

Mp,q := p,qQ EndR,for p, q N, whose typical element is of the form

= IJdqI dqJ ,where the summation convention is used for the multi-indices I, J andwhere, as usual, dqI = iIdqi, and dqJ = jJdqj. Define the degreeof to be deg := |I|+ |J | and write (1) for (1)deg . The productof two elements , is defined to be

:= IJI

J dqI dqJ dqI dqJ

,

and the bracket of any two elements Mp,q and Mp,q is definedby

[, ] := [IJ , I

J ]dqI dqJ dqI dqJ

= (1). (6.1)In the equation above and elsewhere, (1) means (1)deg deg andnot (1)deg (1)deg . Writing M r := p+q=r Mp,q, the algebra Minherits the structure of a graded Lie algebra, namely a graded algebrawith a bracket satisfying

[M r,Ms] M r+s,(graded) skew-commutativity

[, ] = (1) [, ],and the (graded) Jacobi identity:

(1) [, [, ]] + (1)[, [, ]] + (1)[, [, ]] = 0.

22 Alexander V. Sardo Infirri

There are two sub-algebras M,0 and M0, of M. Defining the adjointof = I

JdqI dqJ to be

:= (IJ)dqI dqJ ,

then () = (1) and [, ] = (1) [, ].The Jacobi identity implies that if or has odd degree then

[, [, ]] =1

2[[, ], ]. (6.2)

If = idqi M0,1, and one defines

: Mp,q Mp,q+1 7 [, ],

then, using (6.2), one sees that the sequences

Mp, : Mp,00Mp,1

1Mp,2 . . . (6.3)

of vectorspaces are complexes precisely when [, ] = 0, i.e. when N . Write Hp,q for the cohomology groups Hq(Mp,, ). If oneintroduces a metric on Mp,q by using the standard inner product onEndR and making

12p+q{dqI dqJ}|I|=p,

|J |=q

orthonormal [GH78, p.80], one can define the adjoint operator , andthe Laplacian 2 :=

+

. Their kernels give harmonic repre-

sentatives for the cohomology groups in the usual way

Hp,q := ker 2p,q Mp,q.

If : Mp,q Mp1,q1 denotes the operation of contraction with theKahler form = dqi dqi, then the definition of the adjoint and theinvariance of the trace under cyclic permutations give

= [, ], (6.4)

or, in coordinates,

()IJ = [

j ,

IjJ ]. (6.5)

Writing := dq1 dqn, the n-th power of is

n = n!(1)(n1)(n2)/2 in

2n . (6.6)

Orbifold Singularities and Moduli of Bundles 23

6.2. Local Description of X. Under the identification M = M0,1,

the derivative of the action (3.7) of GL(R) on M is given by

0 : M0,0 M0,1.

On the other hand, the derivative of 7 F 0,2 is (twice)

1 : M0,1 M0,2,

so the Zariski tangent space to N // GL(R) at an element is givenby the first cohomology group of the Atiyah-Hitchin-Singer [AHS77]deformation complex

M0,00M0,1

1 M0,2,

i.e. by H0,1 . From the point of view of the Kahler quotient, one cansee this as follows: the Zariski tangent space to X at [] is givenby ker d() ker d(). By definition, the derivative of is dual tothe action of K, so d() = 2 = 2[, ], as can be verifiedby remarking that () = [, ]. Hence ker d() = Im , so thetangent spaces indeed coincide.

A local model forN // GL(R) in a neighbourhood of a point [] where N has trivial stabiliser is given by solving the equation F 0,2a+ = 0for in the slice

{ M0,1 : = 0, small}.

This comes down to solving the system of equations

= 0 (6.7)

+1

2[, ] = 0, (6.8)

in a neighbourhood of the origin. Kuranishis argument [Kur62, Kur65]shows that the solution set is given by the zero set of a map : H0,1 H0,2 whose two-jet at the origin is given by

(2) : H0,1 H0,2 7 H([, ]).

where

H : M0,2 H0,2denotes the orthogonal projection to the harmonic subspace. Similarstatements hold for X and M

0,,.

24 Alexander V. Sardo Infirri

6.3. Kuranishi Germs and Formality. This whole discussion canbe phrased in more abstract language of deformation functors and dif-ferential graded Lie algebras. Additional details and background canbe found in [GM90, GM88] and [DGMS75]. The algebra M0, is actu-ally a differential graded Lie algebra (DGLA) when endowed with thedifferential . The metric on M

p,q makes it into an analytic DGLA,namely a DGLA which possesses a norm compatible with its differentialand bracket, and which induces what is essentially a Hodge decomposi-tion of its graded pieces with finite dimensional topological summandsHi which are the analogues of the harmonic forms. When has trivialstabiliser, X is locally analytically isomorphic in the neighbourhoodof [] to the Kuranishi germ KM associated to M

0,,. The results sofar are stated in the following theorem.

Theorem 6.1. The sequence of vector spaces (M0,, ) is a complex(and therefore a differential graded Lie algebra) if and only if N .Furthermore, if N , the Zariski tangent space to X at [] isisomorphic to the first cohomology group of its -invariant part

H0,1, := H1(M0,,, )

and if has trivial K-stabilisers, then X is locally isomorphic to itsKuranishi germ

KM0,, = { M0,1,| = +1

2[, ] = 0}.

In general the Kuranishi germ KL of an analytic DGLA (L, d) is(Banach analytically) isomorphic to the germ at 0 of

{ (Im d) L1|d + 12[, ] = 0},

where (Im d) is a fixed complement of the image of d in L1. Goldmanand Millson [GM90] prove that KL is invariant under quasi-isomorphisms,namely chains of homomorphisms of DGLAs5

L L L L

which induce isomorphisms in cohomology. When L is quasi-isomorphicto its cohomology (which is a DGLA when endowed with the zero dif-ferential), L is called formal and it follows that KL is analyticallyisomorphic to the quadratic cone

QL := { H1 : [, ] = 0}.5An important point is that the intermediate L, L, . . . need not have any an-

alytic structure and that the intermediate arrows need not preserve any splittings.

Orbifold Singularities and Moduli of Bundles 25

One way in which this can happen is if the bracket of two har-monic elements of degree 1 is harmonic. This is the case, for instancefor the moduli space of flat Hermitian-Yang-Mills connections over acompact Kahler manifold [Nad88, GM87, GM90]. If in addition, thecup-product on H1 is zero, then KL = H1 and the deformation spaceis a smooth manifold (even if H2 6= 0). This is the case, for instancefor the moduli space of complex structures over a Calabi-Yau n-fold,namely a compact Kahler manifold with a nowhere vanishing holomor-phic (n, 0)-form [GM90]. These moduli were studied by F. Bogomolov.The key fact which implies the formality of the DGLA and the van-ishing of the cup-product in this case was proved by Tian [Tia87] andTodorov [Tod89].

In the case of the algebra M0,, formula (6.2) with and inter-changed shows that the bracket of two harmonic elements in H0,1 is-closed. However, it does not follow that

([, ]) = 0; indeed this

is easily seen to be false, since [, ] = 2. Nevertheless, it doesnot seem unreasonable to expect that M0,, can also be proved to beformal for generic , maybe by imitating Tian and Todorovs method.

Conjecture 1. The differential graded Lie algebra (M0,,, ) is for-mal for all N 1() and generic , and therefore X has, forthese , at worst quadratic algebraic singularities.

Another conjecture is the following:

Conjecture 2. If SU(3), can one imitate the Tian-Todorov proofand show that the Kuranishi germ of (M0,,, ) is isomorphic toH0,1,for generic , i.e. that X is smooth?

The fact that X has at most quadratic singularities has been verifiedfor the abelian subgroups of order less than 11. The smoothness of Xhas been verified in the abelian cases 1

3(1, 1, 1), 1

6(1, 2, 3), 1

7(1, 2, 4),

18(1, 2, 5), 1

9(1, 2, 6), 1

10(1, 2, 7) and 1

11(1, 2, 8). Both these verifications

were done by exhaustive listing of singularities of X for all possible ,using the methods given in the companion paper [SI96b].

A different approach is available in the specific case of SU(3); this ispresented next.

7. Subgroups of SU(3) and Cubic Forms

Suppose that SU(3). If 1() and , H0,1, , then, asremarked in the previous section,

[, ] = 0,

26 Alexander V. Sardo Infirri

but [, ] is not in H0,2, . However, considerations of type show that itdiffers from its harmonic projection by a term , for some M0,1,.For H0,1,

trace([, ]) trace(H([, ])) = trace([, ])= trace([, ])

= 0, since H0,1, . (7.1)

This shows that the tensor

H0,1, H0,1, H0,1, C(, , ) 7 trace(H([, ])),

is totally symmetric on H0,1, (the isomorphism 3,3Q= C has been

used). An easy polarisation argument shows that it is completely de-termined by the corresponding cubic form

C : H0,1, C 7 trace(([, ])).

Proposition 7.1. The singularity of X has no quadratic part if andonly if C() = 0 for all H0,1, and all 1().

Proof. Suppose X has no quadratic part at []. Then (2)() =H([, ]) = 0. But this implies that C() = 0 by equation (7.1).

Conversely, if C() = 0 for all H0,1, then the correspondingtotally symmetric tensor vanishes on all triples (, , ) for all , H0,1, . Since this is true for all , it must be that H([, ]) Im ,i.e. (2)() = 0 in H0,2, .

There is a natural 3-vector whose value on three elements of H0,1,of is given by

(, , ) := trace(). (7.2)

This is symmetric under cyclic permutations of the entries, so decom-poses into a totally skew-symmetric part skew and a totally symmetricpart, which is nothing but the totally symmetric tensor correspondingto C. The proposition above implies the

Corollary 7.2. IfX is smooth, then defines an element of 3,0(X).

Conjecture 3. The canonical sheaf OX (KX ) is locally free, and isgenerated by the non-vanishing (3, 0)-form when X is smooth.

Orbifold Singularities and Moduli of Bundles 27

Taking the wedge of with its complex conjugate gives

(, , , , , ) = (ijk trace ijk)(k trace k) ,

(7.3)

=

ijk

trace ijk

2

, (7.4)

where

ijk denotes the sum over distinct i, j and k. On the otherhand, the symplectic form on X is simply the restriction of thesymplectic form defined in (3.5), and equation (6.6) gives

=3i

4 . (7.5)

Suppose that (, , ) are an orthonormal triple in T 1,0 X . Then thevalue of the coefficient of in (7.4) is equal to 2 2.Hence X has trivial canonical bundle if this coefficient is never zerofor all 1().Lemma 7.3. The Kahler manifold X is Ricci-flat if and only if thecoefficient of in (7.4) is constant for all orthonormal triples(, , ) in H0,1, and all 1() N .Proof. This follows because if X is Ricci-flat, there exists a holomor-phic (3, 0)-form which is covariant constant on X. Hence willdiffer from by a holomorphic function f . Now Liouvilles theoremimplies that f is either constant or unbounded. Since is clearlybounded (by 6 ), f must be constant.7.1. Example. Let us work out a specific example. Consider thegroup = 3 of order 3 acting on C3 with weights (1, 1, 1). Thefollowing configuration of matrices is easily seen to define a point of1() N , where = (|A|2, |A|2 |B|2, |B|2), (A,B R):

1 =

0 A 00 0 B0 0 0

dq1, 2 = 3 = 0. (7.6)

The tangent space is three-dimensional and is generated by the follow-ing orthonormal elements (recall that dqi2 = 2)

1 =12

0 0 00 0 01 0 0

dq1, i =1

2(A2 +B2)

0 A 00 0 B0 0 0

dqi,

(7.7)

28 Alexander V. Sardo Infirri

for i = 2, 3, so this defines a smooth point of X . The value of 2 atthis point is

1.A.B + 1.B.A

2

2(A2 +B2)

2

= 12

(

AB

A2 +B2

)2

, (7.8)

and so this is non-zero away from AB = 0 (which correspond to non-generic values of ).

At the point

1 =

0 A + C 00 0 B + CC 0 0

dq1, 2 = 3 = 0, (7.9)

in 1()N , the tangent space is still three-dimensional, with ortho-normal generators

i =16

0 1 00 0 11 0 0

dqi, i = 1, 2, 3, (7.10)

The value of 2 however is now

6

(

16

)3

2

= 16 . (7.11)

In fact, all the points of 1() N are of the form (7.6) or (7.9)(modulo permutations of the indices 1, 2, 3)6. Thus it has been shown,in a rather laborious way, that away from certain degenerate valuesof , is non-vanishing on X and KX is therefore trivial. In fact,X = OP2(3).

Since the coefficient of in (7.8) is always smaller than 1/8, onealso deduces that is not a constant multiple of onany of the quotients X , and therefore by lemma 7.3 that the inducedmetric is never Ricci-flat.

Remark 7.4. The space OP2(3) does have a standard Ricci-flat met-ric, as was first noted by Calabi [Cal79].

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Orbifold Singularities and Moduli of Bundles 29

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[Kro89] P. B. Kronheimer, The construction of ALE spaces as hyper-Kahlerquotients, J. Differential Geom. 29 (1989), no. 3, 665683.

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[Kur65] M. Kuranishi, New proof for the existence of locally complete families ofcomplex structures, Proc. Conf. Complex Analysis (Minneapolis, 1964),Springer, Berlin, 1965, pp. 142154.

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[MOP87] D. G. Markushevich, M. A. Olshanetsky, and A. M. Perelomov, De-scription of a class of superstring compactifications related to semisim-ple Lie algebras, Comm. Math. Phys. 111 (1987), no. 2, 247274.

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30 Alexander V. Sardo Infirri

[SI94] Alexander V. Sardo Infirri, Resolutions of orbifold singularities andrepresentation moduli of McKay quivers, D. Phil. thesis, University ofOxford, 1994, Available as Pre-Print RIMS-984.

[SI96a] Alexander V. Sardo Infirri, Crepant terminalisations and orbifold Eulernumber for SL(4) singularities, to appear, 1996.

[SI96b] Alexander V. Sardo Infirri, Resolutions of orbifold singularities and thetransportation problem on the McKay quiver, to appear, 1996.

[Tha94] M. Thaddeus, Geometric invariant theory and flips, eprint alg-geom/9405004, 1994.

[Tia87] Gang Tian, Smoothness of the universal deformation space of compactCalabi-Yau manifolds and its Petersson-Weil metric, Mathematical as-pects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys.,vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629646.

[Tod89] Andrey N. Todorov, The Weil-Petersson geometry of the moduli spaceof SU(n) (n 3) (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126(1989), no. 2, 325346.

[UY86] K. Uhlenbeck and S. T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math.39 (1986), 257.

E-mail address : sacha@kurims.kyoto-u.ac.jp

Research Institute for Mathematical Sciences, Kyoto University,

Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-01, Japan

http://arXiv.org/abs/alg-geom/9405004http://arXiv.org/abs/alg-geom/9405004