Orbifolds and Finite Group Representations

We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We shall concern only the quotient singularity of hypersurface type. The abelian group $A_r(n)$ for $A$-type hypersurface quotient singularity of dimension $n$ is introduced. For $n=4$, the structure of Hilbert scheme of group orbits and crepant resolutions of $A_r(4)$-singularity are obtained. The flop procedure of 4-folds is explicitly constructed through the process.


Introduction
It is well known that the theory of "minimal" resolutions of singularity of algebraic (or analytical) varieties differs significantly when the (complex) dimension of the variety is larger than two. As the prime achievement in algebraic geometry of the 1980-s, the minimal model program in the 3-dimensional birational geometry carried out by S. Mori and others has provided an effective tool for the study of algebraic 3-folds, (see [16] and references therein). Meanwhile, Gorenstein quotient singularities in dimension 3 has attracted considerable interests among geometers due to the development of string theory, by which the orbifold Euler characteristic of an orbifold was proposed as the vacuum description of models built upon the quotient of a manifold [5]. The consistency of physical theory then demanded the existence of crepant resolutions which are compatible with the orbifold Euler characteristic. The complete mathematical justification of the conjecture was obtained in the mid-90s (see [24] and references therein). However, due to the computational nature of methods in the proof, the qualitative understanding of the these crepant resolutions has still been lacking on certain aspects from a mathematical viewpoint. Until very recently, by the development of Hilbert scheme of a finite group G-orbits, initiated by Nakamura and others with the result obtained [1,6,10,11,17], it strongly indicates a promising role of the finite group in problems of resolutions of quotient singularities. In particular a plausible method has been suggested on the study of geometry of orbifolds through the group representation theory. It has been known that McKay correspondence [15] between representations of Kleinian groups and affine A-D-E root diagrams has revealed a profound geometrical structure on the minimal resolution of the quotient surface singularity (see e.g., [7]). A similar connection between the finite group and general quotient singularity theories would be expected. Yet, the interest of this interplay of geometry and group representations would not only aim on the research of crepant resolutions, but also on its own right, due to possible implications on understanding some certain special kind of group representations by engaging the rich algebraic geometry techniques.
In this article, we shall study problems related to the crepant resolutions of quotient singularities of higher dimension n, (mainly for n ≥ 4). Due to the many complicated exceptional cases of the problem, we shall restrict ourselves here only on those of the hypersurface singularity type. The purpose of this paper is to present certain primitive results of our first attempt on the study of the higher dimensional hypersurface orbifolds under the principle of "geometrization" of finite group representations. We shall give a brief account of the progress recently made. The main issue we deal with in this work will be the higher dimensional generalization of the A-type Kleinian surface singularity, the A r (n)-hypersurface singularity of dimension n (see (4) of the content). For n = 4, we are able to determine the detailed structure of A r (4)-Hilbert scheme, and its relation with crepant resolutions of C 4 /A r (4). In the process, an explicit "flop" construction of 4-folds among different crepant resolutions is found. In this article, we shall only sketch the main ideas behind the proof of these results, referring the reader to our forecoming paper [2] for a more complete description of methods and arguments used. This paper is organized as follows. In Sect. 2, we shall give a brief introduction of the general scheme of engaging finite group representations in the birational geometry of orbifolds. Its connection with the Hilbert scheme of G-orbits for a finite linear group G on C n , Hilb G (C n ), introduced in [11], will be explained in Sect. 3. In Sect. 4, we first review certain basic facts in toric geometry, which will be presented in the form most suitable for our goal, then focusing the case on A r (n)-singularity. For n = 3, we will give a thorough discussion on the explicit toric structure of Hilb Ar(3) (C 3 ) as an illustration of the general result obtained by Nakamura on abelian group G in [17] 1 . In Sect. 5, we deal with a special case of 4-dimensional orbifold with G = A 1 (4), and derive the detailed structure of Hilb G (C 4 ). Its relation with the crepant resolutions of C 4 /G is given, so is the "flop" relation among crepant resolutions. In Sect. 6, we describe the result of G = A r (4) for the arbitrary r , then end with some concluding remarks.
Notations. To present our work, we prepare some notations. In this paper, by an orbifold we shall always mean the orbit space for a finite group action on a smooth complex manifold. For a finite group G, we denote The trivial representation of G will be denoted by 1. For a G-module W , i.e., a G-linear representation on a vector space W , one has the canonical irreducible decomposition: The vector space W ρ will be called the ρ-factor of the G-module W .
For an analytic variety X, we shall not distinct the notions of vector bundle and locally free O Xsheaf over X. For a vector bundle V over X, an automorphisms of V means a linear automorphism with the identity on X. If the bundle V is acted by a group G as bundle automorphisms, we shall call V a G-bundle.

Representation Theory in Algebraic Geometry of Orbifolds
In this paper, G will always denote a finite (non-trivial) subgroup of GL n (C) for n ≥ 2, and S G := C n /G with the canonical projection, and o := π G (0) ∈ S G . When G is a subgroup of SL n (C), which will be our main concern later in this paper, G acts on C n freely outside a finite collection of linear subspaces with codimension ≥ 2. Then the orbifold S G has a non-empty singular set, Sing(S G ), of codimension ≥ 2, in fact, o ∈ Sing(S G ).
For G in GL n (C), S G is a singular variety in general. By a birational morphism of a variety over S G , we shall always mean a proper birational morphism σ from variety X to S G which defines a biregular map between X \ σ −1 (Sing(S G )) and S G \ Sing(S G ), One has the commutative diagram, Denote F X the coherent O X -sheaf over X obtained by the push-forward of the structure sheaf of X × S G C n , The sheaf F X has the following functorial property, namely for X, X ′ birational over S G with the commutative diagram, Furthermore, all the morphisms in the above are compatible with the natural G-structure on F X induced from the G-action on C n via (2). One has the canonical G-decomposition of F X , where (F X ) ρ is the ρ-factor of F X , and it is a coherent O X -sheaf over X. The geometrical fiber of F X , (F X ) ρ over an element x of X are defined by , F X is a vector bundle of rank |G| with the regular G-representation on each geometric fiber. Hence (F X ) ρ is a vector bundle over X − σ −1 (Sing(S G )) of the rank equal to dim.V ρ . For x ∈ X, there exists a G-invariant ideal I(x) in C[Z](:= C[Z 1 , · · · , Z n ]) such that the following relation holds, We have (F X ) ρ, In particular, for s = o, we have where the subscript 0 means the maximal ideal with polynomials vanishing at the origin. For a birational variety X over S G via σ in (1), the following relations of G-invariant ideals of C[Z] hold: A certain connection exists between algebraic geometry and G-modules through the variety X. For x ∈ X, there is a direct sum G-decomposition of C[Z], Here such that the following relations hold for the finite-dimensional G-modules, Consider the canonical G-decomposition of I(x) ⊥ , Note that I(x) ⊥ ρ is isomorphic to a positive finite copies of V ρ . Then the affine structure of X near x is determined by the C-algebra generated by all the G-invariant rational function f (Z) such that f (Z)I(x) ⊥ ρ ⊂ I(x) for some ρ.

Hilbert Scheme of Finite Group Orbits
Among the varieties X birational over S G with F X a vector bundle, there exists a minimal object, called the G-Hilbert scheme in [11,17], For another X, the map (1) can be factored through a birational morphism λ from X onto Hilb G (C n ) via σ Hilb , λ : X −→ Hilb G (C n ) .
In fact, the ideal I(x),x ∈ X, of (3) are with the co-length |G|, which gives rise to the above map λ of X to Hilb G (C n ). We shall denote X G as the normal variety over S G defined by By the fact that every biregular automorphism of S G can always be lifted to one on Hilb G (C n ), hence on X G , one has the following result.
Lemma 1 Denote Aut(S G ) the group of biregular automorphisms of S G . Then Hilb G (C n ) and X G are varieties over S G with the Aut(S G )-equivariant covering morphisms.

By the definition of Hilb
The fiber of the vector bundle F Hilb G (C n ) over p can be identified with he regular G-representation space C[Z]/I(p). Our study will mainly concern on the relation of crepant resolutions of S G and Hilb G (C n ). For this purpose we will assume for the rest of the paper that the group G is a subgroup of SL n (C): which is the same to say that S G has the Gorenstein quotient singularity. For n = 2, these groups were classified by F. Klein into A-D-E types [13], the singularities are called Kleinian singularities. The minimal resolution S G of S G has the trivial canonical bundle ( i.e., crepant), by [9]. In [11,17], Y. Ito and I. Nakamura showed that Hilb G (C 2 ) is equal to the minimal resolution S G . For n = 3, it has been known that there exist crepant resolutions for a 3-dimensional Gorenstein orbifold (see [24] and references therein). Two different crepant resolutions of the same orbifolds are connected by a sequences of flop processes (see e.g., [20]). It was expected that Hilb G (C 3 ) is one of those crepant resolutions. The assertion has been confirmed in the abelian group case in [17], and in general by [1]. For the motivation of our later study on the higher dimensional singularities, we now illustrate the relation between G-Hilbert scheme and the minimal resolution in dimension 2, i.e., surface singularities. For the rest of this section, we are going to describe the structure of Hilb G (C 2 ) for the A-type Kleinian group, The affine ring of ) and G-invariant polynomials is the algebra generated by , for some k. With the method of continued fraction [9], it is known that the minimal resolution S G of S G has the trivial canonical bundle with an open affine cover .
The configuration can be realized in the following tree diagram: P P P P ✏ ✏ ✏ ✏ P P P P ✏ ✏ ✏ ✏ P P P P ✏ ✏ ✏ ✏ P P P P ✏ ✏ ✏ ✏ P P P P Fig. 1] Exceptional curve configuration in the minimal resolution of C 2 /A r .
It is easy to see that the ideal I(ô k ) is given by hence the G-module C[Z]/I(ô k ) is the regular representation isomorphic the following one, One can represent monomials in the above expression as the ones with • in the following picture: For x ∈ U k , the ideal I(x) has the expression The classes in C[Z]/I(x) represented by monomials in (5) still form a basis, hence give rise to a local frame of the vector bundle F S G over U k . The divisor E k+1 is defined by β = 0, and its element approaches toô k+1 as α tends to infinity.

Abelian Orbifolds and Toric Geometry
In this section we discuss the abelian group case of G in the previous section using methods in toric geometry. We shall consider G as a subgroup of the diagonal group of GL n (C), denoted by T 0 := C * n , and regard C n as the partial compactification of T 0 , Define the n-torus T with the toric embedding in S G (= C n /G) by Techniques in toric geometry rely on lattices of one-parameter subgroups, characters of T 0 , T , For our convenience, we shall make the following identification of N 0 , N with lattices in R n . An element x in R n has the coordinates x i with respective to the standard basis (e 1 , · · · , e n ): x i e i ∈ R n .
The dual lattice M 0 of N 0 is the standard one in the dual space R n * , and we shall identify it with the group of monomials of Z 1 , . . . , Z n via the correspondence: The dual lattice M of N is the sublattice of M 0 corresponding to the set of G-invariant monomials.
Over the T -space S G , we now consider only those varieties X which is normal and birational over S G with a T -structure, hence as it has been known, are presented by certain combinatorial data by the toric method [4,12,18]. Note that by Lemma (1), X G is a toric variety over S G . In general, a toric variety over S G is described by a fan Σ = {σ α | σ ∈ I} whose support equals to the first quadrant of R n , i.e., a rational convex cone decomposition of the first quadrant of R n . Equivalently, it is determined by the intersection of the fan and the simplex △ where The data in △ is given by Λ = {△ α | α ∈ I}, where △ α := σ α △. The △ α s form a decomposition of △ by convex subsets, having the vertices in △ ∩ Q n . Note that for σ α = { 0}, we have △ α = ∅. We shall call Λ a rational polytope decomposition of △, and denote the corresponding toric variety by X Λ . We call Λ an integral polytope decomposition of △ if all the vertices of Λ are in N . For a rational polytope decomposition Λ of △, we define Λ(i) : Then T -orbits in X Λ are parameterized by n−1 i=−1 Λ(i). In fact, for each △ α ∈ Λ(i), there associates a T -orbit of the dimension n − 1 − i, denoted by orb(△ α ). A toric divisor in X Λ is the closure of a n − 1 dimensional orbit, denoted by D v = orb(v) for v ∈ Λ(0). The canonical sheaf of X Λ has the expression in terms of toric divisors (see, e.g. [12]), where m v is the positive integer such that m v v is a primitive element of N . In particular, the crepant property of X Λ , i.e. ω X Λ = O X Λ , is given by the integral condition of Λ. The non-singular criterion of X Λ is the simplicial decomposition of Λ together with the multiplicity one property, i.e., for each Λ α ∈ △(n − 1), the elements m v v, v ∈ Λ α ∩ Λ(0), form a Z-basis of N . The following results are known for toric variety over S G (see e.g. [19] and references therein): (1) The Euler number of X Λ is given by χ(X Λ ) = |Λ(n − 1)|.
(2) For a rational polytope decomposition Λ of △, any two of the following three conditions implies the third one: It is easy to see that the following result holds for the sheaf F X Λ .
Lemma 2 Let Λ be a rational polytope decomposition of △, and x 0 be the zero-dimensional toric orbit in X Λ corresponding to an element △ α 0 in Λ(n − 1). Let Z I (j) , 1 ≤ j ≤ N , be a finite collection of monomials whose classes generate the G-module C[Z]/I(x 0 ). Then the classes of Z I (j) s also generate C[Z]/I(y) for y ∈ orb(△ β ) with △ β ⊆ △ α 0 .
Note that the above group for n = 2 is the same as A r in (4). For a general n, A r (n)-invariant polynomials in C[Z] are generated by the following (n + 1) ones: This implies that S Ar(n) is the singular hypersurface in C n+1 , S Ar(n) = {(x, y 1 , · · · , y n ) ∈ C n+1 | x r+1 = y 1 · · · y n } .
For the rest of this paper, we will conduct the discussion of abelian orbifolds mainly on the group A r (n). The ideal I(o) of C[Z] associated to the element o ∈ S Ar(n) is given by For 1 = ρ ∈ Irr(A n (r)), the dimension of I(o) ⊥ ρ is always greater than one. In fact, one can describe explicitly a set of monomial generators of I(o) ⊥ ρ . For example, say I(o) ⊥ ρ containing an element Z I with I = (i 1 , . . . , i n ), i 1 = 0 and i s ≤ i s+1 , then I(o) ⊥ ρ is generated by Z K s with K = (k 1 , . . . , k n ) given by here j runs through 1 to n. In particular for r = 1, the dimension of I(o) ⊥ ρ is equal to 2 for ρ = 1, with a basis consisting of Z I , Z I ′ whose indices satisfy the relations, 0 ≤ i s , i s ′ ≤ 1, i s + i s ′ = 1 for 1 ≤ s ≤ n.
For n = 3, by the general result of Nakamura on an abelian group G (Theorem 4.2 in [17]), Hilb Ar(3) (C 3 ) is a crepant toric variety. To illustrate this fact, we give here a direct derivation of the result by working on the explicitly described toric variety.
Example. It is easy to see that △ ∩ N consists of the following elements, For y in X Ξ with the value U j equal to α j , by computation the ideal I(y) has the following generators, hence the description of ideals for y in X Ξ with V j = β j , Therefore we have shown that X Ξ is birational over Hilb G (C 3 ). Now we are going to show that they are in fact the same. Let x be an element in Hilb G (C 3 ) represented by an monomial ideal J = I(x), (i.e, with a set of generators composed of monomials). Then the regular G-module J ⊥ is generated by |G| monomials, and x lies over the element o of S G , equivalently, J contains the ideal C[Z] G 0 . Denote l i the smallest non-negative integer such that Z l i i ∈ J, l i,j the smallest non-negative integer with (Z i Z j ) l i,j ∈ J for i = j. Hence 1 ≤ l i ≤ r + 1, and In particular, Z l 1 −1 1 ∈ J ⊥ and (Z 2 Z 3 ) r+2−l 1 ∈ J. By the description (8) for I(o) ⊥ , Z l 1 1 is the only monomial in the basis of I(o) ⊥ for the corresponding character of G, the same for (Z 2 Z 3 ) r+1−l 1 . Hence (Z 2 Z 3 ) r+2−l 1 ∈ J ⊥ , which implies l 1 + l 2,3 = r + 2. Similarly, we have l 2 + l 1,3 = l 3 + l 1,2 = r + 2. Again by (8), Z l 1 −1 If 3 j=1 l j = 2(r + 1) + 1, the ideal J corresponds to the ideal of x m 1 ,m 2 ,m 3 u in X Ξ with For 3 j=1 l j = 2(r + 1) + 2, the ideal J corresponds to the ideal for the element   1, 2). An "•" indicates a monomial in I ⊥ while an "×" means one in I. The difference between two graphs are marked by broken segments.

A 1 (4)-Singularity and Flop of 4-folds
We now study the A r (n)-singularity with n ≥ 4. For simplicity, let us consider the case r = 1, i.e., G = A 1 (n), (indeed, no conceptual difficulties arise for higher values of r). The N -integral elements in △ are as follows: where v i,j := 1 2 (e i + e j ) for i = j. Other than the whole simplex △, there is only one integral polytope decomposition of △ invariant under permutations of coordinates, denoted by Ξ, which we now describe as follows. There are n + 1 elements in Ξ(n − 1): △ i , 1 ≤ i ≤ n, together with ✸, where △ i is the simplex generated by e i and v i,j for j = i, and ✸ = the closure of △ \ n i=1 △ i . In fact, ✸ is the convex hull spanned by all the v i,j for i = j. The lower dimensional polytopes of Ξ are given by the faces of those in Ξ(n − 1). Then X Ξ has the trivial canonical sheaf. However only for n = 2, 3, X Ξ is a crepant resolution of S A 1 (n) (see, e.g., [19]). In fact, one has the following result for higher n.
Lemma 3 For n = 4, the toric variety X Ξ is smooth except one isolated singularity, which is the 0-dimensional T-orbit corresponding to ✸.
Proof. In general, for n ≥ 4, it is easy to see that for each i, the vertices of △ i form a Z-basis of N , for example, say i = 1, it which follows from |A 1 (n)| = 2 n−1 , and det(e 1 , v 1,2 , · · · , v 1,n ) = 1 2 n−1 .
Hence X Ξ is nonsingular near the T -orbits associated to simplices in ∆ i . As ✸ is not a simplex, orb(✸) is always a singular point of X Ξ . For n = 4, the statement of smoothness of X Ξ except orb(✸) follows from the fact that for 1 ≤ i ≤ 4, the vertices v i,j (j = i) of ✸ together with (1/2) 4 j=1 e j , from an N -basis. ✷ Remark.
(1) Denote x j := orb(△ j ) ∈ X Ξ for 1 ≤ j ≤ n. The inverse of the matrix of vertices of △ j , (v 1,j , · · · , v j−1,j , e j , v j+1,j , · · · , v n,j ) −1 , gives rise to affine coordinates (U 1 , . . . , U n ) around x j : Hence (2) We shall denote x ✸ the element orb(✸) in X Ξ , x ✸ := orb(✸). The singular structure of x ✸ is determined by those A 1 (n)-invariant polynomials, corresponding to M -integral elements in the cone dual to the one generated by ✸ in N R . It is easy to see that these polynomials are generated by the following ones: Hence we have I(x ✸ ) = Z 1 · · ·Ž j · · · Z n 1≤j≤n + I A 1 (n) .
Note that for n = 3, Y j s form the minimal generators for the invariant polynomials, which implies the smoothness of X Ξ . For n ≥ 4, x ✸ is an isolated singularity, but not of the hypersurface type. For n = 4, the X j , Y j (1 ≤ j ≤ 4) form a minimal set of generators of invariant polynomials, hence the structure near x ✸ in X Ξ is the 4-dimensional affine variety in C 8 defined by the relations: where i = j and {i ′ , j ′ } is the complimentary pair of {i, j}. ✷ For the rest of this section, we shall consider only the case n = 4. We shall discuss the crepant resolutions of S A 1 (4) , and its relation with Hilb A 1 (4) (C 4 ). Now the simplex △ is a tetrahedron, and ✸ is an octahedron, on which the symmetric group S 4 acts as the standard representation. The dual polygon of ✸ is the cubic. Faces of the octahedron ✸ are labeled by F j , F ′ j for 1 ≤ j ≤ 4, where x j e j ∈ ✸ | x j = 0} .
The dual of F j , F ′ j in the cubic are vertex, denoted by α j , α ′ j as in [Fig. 5]. Consider the rational simplicial decomposition Ξ * of △, which is a refinement of Ξ by adding the center [ Fig. 6] The rational simplicial decomposition Ξ * of △ for n = 4, r = 1.
which is non-singular with the canonical bundle ω = O X Ξ * (E), where E is an irreducible divisor isomorphic to the triple product of P 1 , Furthermore for {i, j, k} = {1, 2, 3}, the normal bundle of E when restricted on the P 1 -fiber, P 1 k , for the projection on P 1 × P 1 via the (i, j)-th factor, is the (−1)-hyperplane bundle: Proof. By Lemma 3 and Remark (1) after that, one can see the smoothness of X Ξ * on the affine chart corresponding to △ j , also its relation with Hilb G (C 4 ). For the rest of simplexes, the octahedron ✸ of Ξ is decomposed into eight simplexes corresponding to the faces F j , F ′ j of ✸. Denote C j (C ′ j ) the simplex of Ξ * spanned by c and F j (F ′ j respectively), and x C j , x C ′ j the elements in X Ξ * of the corresponding T -orbit. First we show that for It is easy to see that the vertices of F j together with 2c form a integral basis of N , the same for the vertices of F ′ j . For the convenience of notation, we can set j = 1 without loss of generality. Then we have the integral basis of M for the cones, dual to C 1 , C ′ 1 as follows: Therefore, the following 4 functions form a smooth coordinate of X Ξ * near x C j for j = 1, and one has Similarly the coordinates near x C ′ j for j = 1 are given by and we have It is easy to see that the G-modules, C[Z]/I(x C 1 ), C[Z]/I(x C 1 ), are both equivalent to the regular representation. Therefore the ideals give rise to distinct elements in Hilb A 1 (4) (C 4 ). In fact, one can show that X Ξ * = Hilb A 1 (4) (C 4 ) ( for the details, see [2]). By (6), the canonical bundle of X Ξ * is given by where E is the toric divisor D c . It is known that E is a 3-dimensional complete toric variety arisen from the star of c in Ξ * , which is given by the octahedron in [Fig. 5]; in fact, the cube in [Fig. 5] represents the toric orbits' structure. Therefore E is isomorphic to the triple product of P 1 as in (11). The conclusion of the normal bundle of E restricting on each P 1 -fiber will follow from techniques in toric geometry. For example, for fibers over the projection of E onto the (P 1 ) 2 corresponding to the 2-convex set spanned by v 1,2 , v 1,3 , v 3,4 , v 2,4 , one can perform the computation as follows. Let (U 1 , U 2 , U 3 , U 4 ) be the local coordinates near x C ′ 4 dual to the N -basis (2c, v 1,2 , v 1,3 , v 2,3 ), similarly the local coordinate (W 1 , W 2 , W 3 , W 4 ) near x C 1 dual to (2c, v 1,2 , v 1,3 , v 1,4 ). By 2c = v 1,4 + v 2,3 , one has the relations, This shows that the restriction of the normal bundle of E on a fiber P 1 over (U 2 , U 3 )-plane is the (−1)-hyperplane bundle. ✷ The sheaf F X Ξ * for X Ξ * in Theorem 1 is a vector bundle with the regular G-module on each fiber. The local frame of the vector bundle is provided by the structure of C[Z]/I(x) for x being the zero-dimensional toric orbit of X Ξ * . One can have a pictorial realization of monomial basis of these G-representations as follows. We start with the element x △ 1 , and the identification, The eigen-basis of the G-module I(x △ 1 ) ⊥ is given by monomials in the diagram of [ Fig. 7].  Fig. 8] The corresponding I ⊥ -graph for the simplex ∆ 2 , ∆ 3 and ∆ 4 in X Ξ * s. An "•" means a monomial in I(x △ i ) ⊥ , while an "×" means one in I(x △ i ). Fig. 9] The corresponding I ⊥ -graph for the simplex C 1 , C 2 , C 3 and C 4 Fig. 10] The corresponding I ⊥ -graph for the simplex By the standard blowing-down criterion of an exceptional divisor, the property (13) ensures the existence of a smooth 4-fold (X Ξ * ) k by blowing down the family of P 1 s along the projection p k (12) for each k. In fact, (X Ξ * ) k is also a toric variety X Ξ k where Ξ k is the refinement of Ξ by adding the segment connecting v k, 4 and v i,j to divide the central polygon ✸ into 4 simplexes, where {i, j, k} = {1, 2, 3}. Each X Ξ k is a crepant resolution of X Ξ (= S A 1 (4) ). We have the relation of refinements: Ξ ≺ Ξ k ≺ Ξ * for k = 1, 2, 3. The polyhedral decomposition in the central part ✸ appeared in the refinement relation are denoted by of which the pictorial realization is given in [Fig. 11]. The connection of smooth 4-folds for different ✸ k can be regarded as a "flop" of 4-folds suggested by the similar procedure in the theory of 3dimensional birational geometry. Each one is a "small" 2 resolution of a 4-dimensional isolated singularity defined by the equation (10). Hence we have shown the following result.
Theorem 2 For G = A 1 (4), there are crepant resolutions of S G obtained by blowing down the divisor E of Hilb G (C 4 ) along (12) in Theorem 1. Any two such resolutions differ by a "flop" procedure of 4-folds.

A r (4)-Singularity and Conclusion Remarks
For G = A r (n), n ≥ 4, the structure of Hilb G (C n ) and its relation with possible crepant resolutions of S G has been an on-going program under investigation. We have discussed the simplest case 14 and qualitative relation would still be interesting for the possible study of some other simple groups G in higher dimension. A such program is under consideration with initial progress being made. Even for the abelian group G in the dimension n = 3, the conclusion on the trivial canonical bundle of Hilb G (C 3 ) would raise a subtle question in the mirror problem of Calabi-Yau 3-folds in string theory. As an example, a standard well known one is the Fermat quintic in P 4 with the special marginal deformation family: X : 5 j=1 z 5 i + λz 1 z 2 z 3 z 4 z 5 = 0 , λ ∈ C .
With the maximal diagonal group SD of z i s preserving the family X, the mirror X * is constructed by "the" crepant resolution of X/SD, X * = X/SD (see e.g., [8,21]), by which the roles of H 1,1 , H 2,1 are interchangeable in the "quantum" sense. When working on the one-dimensional space H 1,1 (X) ∼ H 2,1 (X * ), the choice of crepant resolution X/SD makes no difference on the conclusion. While on the part of H 2,1 (X) ∼ H 1,1 (X * ), it has been known that many topological invariants, like Euler characteristic, Hodge numbers, elliptic genus, are independent of the choices of crepant resolutions, hence one obtains the same invariants for different choices of crepant resolutions as the model for X * . However, the topological triple intersection of cohomologies does differ for two crepant resolutions (see, e.g., [22]), hence the choice of crepant resolution as the mirror X * = X/SD will lead to the different effect on the topological cubic form of H 1,1 (X * ), upon which as the "classical" level, the quantum triple product of the physical mirror theory would be built (see, e.g., articles in [25]). The question of the "good" model for X * has rarely been raised in the past, partly due to the lack of mathematical knowledge on the issue. However, with the G-Hilbert scheme now given in Sect. 3, 4 as the mirror X * , it seems to have left some fundamental open problems on its formalism of mirror Calabi-Yau spaces and the question of the arbitrariness of the choice of crepant resolutions remains a mathematical question to be completely understood concerning its applicable physical theory. For the role of G-Hilbert scheme in the study of crepant resolution of S G , the conclusion we have obtained for G = A r (4) has indicated that Hilb G (C n ) couldn't be a crepant resolution of S G in general when the dimension n is greater than 3. Nevertheless the structure of Hilb G (C n ) is worthwhile for further study on its own right due to the interplay of geometry and group representations. Its understanding could still lead to the construction of crepant resolutions of S G in case such one does exist. It would be a promising direction of the geometrical study of orbifolds.