6_matsumura.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 4, 2012, 492-510

ISSN 1307-5543 – www.ejpam.com

Stringy and Orbiforld Cohomology of Wreath Product Orbifolds

Tomoo Matsumura

Department of Mathematical Sciences, ASARC, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon

305-701, South Korea

Abstract. Let [X/G] be an orbifold which is a global quotient of a compact almost complex manifold

X by a finite group G. Let Σn be the symmetric group on n letters. Their semidirect product Gn ⋊Σn

is called the wreath product of G and it naturally acts on the n-fold product X n, yielding the orbifold

[X n/(Gn⋊Σn)]. Let H (X n,Gn⋊Σn) be the stringy cohomology [7, 10] of the (Gn⋊Σn)-space X n. We

prove that the space Gn-invariants of H (X n,Gn ⋊Σn) is isomorphic to the algebra Hor b([X/G]){Σn}
introduced by Lehn and Sorger [14], where Hor b([X/G]) is the Chen-Ruan orbifold cohomology of

[X/G]. We also prove that, if X is a projective surface with trivial canonical class and Y is a crepant

resolution of X/G, then the Hilbert scheme of n points on Y , denoted by Y [n], is a crepant resolu-

tion of X n/(Gn ⋊ Σn). Furthermore, if H∗(Y ) is isomorphic to Hor b([X/G]) as Frobenius algebras,

then H∗(Y [n]) is isomorphic to H∗
or b
([X n/(Gn ⋊ Σn)]) as rings. Thus we verify a special case of the

cohomological hyper-Kähler resolution conjecture due to Ruan [22].

2010 Mathematics Subject Classifications: 14N35 14A20 14E15 14J81 14L30

Key Words and Phrases: Orbifold, Gromov-Witten Theory, Frobenius Algebra, Symmetric Product,

Wreath Product

1. Introduction

The stringy cohomology H (X ,G) of an almost complex manifold X with an action of a

finite group G was first introduced by Fantechi-Göttsche [7] and studied further by Jarvis-

Kaufmann-Kimura [10, 11]. It is a G-Frobenius algebra [23, 12] which is a G-equivariant

generalization of Frobenius algebras and the space of its G-invariants is the Chen-Ruan orb-

ifold cohomology H∗
or b
([X/G]) introduced in [4].

Let W be an orbifold and π : Y → W be a hyper-Kähler resolution of the coarse moduli

space W of W . Ruan’s cohomological hyper-Kähler resolution conjecture [22] predicts that

the ordinary cohomology ring of Y is isomorphic to the orbifold cohomology ring of W over

C-coefficients. This is a special case of the crepant resolution conjecture of Ruan [22] and

Bryan-Graber [3].

Email address: tomoomatsumura�kaist.a
.kr

http://www.ejpam.com 492 c© 2012 EJPAM All rights reserved.



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 493

Among the examples which support the cohomological hyper-Kähler resolution conjecture,

the symmetric product is perhaps the most fascinating. The symmetric group on n-letters, Σn,

naturally acts on the n-fold product Y n of a manifold Y , yielding the symmetric product orbifold

[Y n/Σn]. If Y is a projective surface with trivial canonical class, then the Hilbert scheme of n

points on Y , denoted by Y [n], is a hyper-Kähler resolution of the quotient space Y n/Σn [1].

Fantechi and Göttsche [7] showed that the ring of Σn-invariants of H (Y n,Σn) is isomorphic

to H∗(Y [n]) over C. Their proof proceeds by showing that H (Y n,Σn) is isomorphic to the

algebra H∗(X ){Sn} defined by Lehn and Sorger [14], i.e.

H (Y n,Σn)
∼= H∗(Y ){Σn} =⇒ H∗or b([Y

n/Σn])
∼= H∗(Y ){Σn}

Σn ∼= H∗(Y [n])

where the last isomorphism is due to [14] (see also [24, 18, 16]).

In this paper, we consider a generalization of the algebra isomorphism on the left-hand

side of the arrow above, namely, replace Y by an orbifold [X/G] and H∗(Y ) by H∗
or b
([X/G]).

The symmetric group Σn naturally acts on the n-fold product Gn and their semidirect product

Gn ⋊Σn is called the wreath product of G. It naturally acts on the n-fold product X n, yielding

the orbifold [X n/(Gn⋊Σn)]. This orbifold is called the wreath product orbifold of a G-space X .

The linear structure of the orbifold cohomology of a wreath product orbifold has been studied

in a sequence of papers by Qin, Wang and Zhou, cf. [18, 25, 26] through a careful analysis of

the fixed point loci. However, one of the goals of this paper is to analyze the multiplication

in stringy cohomology and in Chen-Ruan orbifold cohomology of a wreath product orbifold.

The multiplication in the special case when X = C2 and G is a finite subgroup of SL2(C) has

been studied in [6, 19].

The main result of this paper is Theorem 4 which proves that, when X is compact, there

is a canonical Σn-Frobenius algebra isomorphism

H (X n,Gn ⋊Σn)
Gn ∼= H∗or b([X/G]){Σn}.

When G is a trivial group, this isomorphism reduces to the isomorphism defined by Fantechi

and Göttsche [7]. This result means that H (X n,Gn ⋊ Σn)
Gn

gives a geometric construction

of the second quantization [Definition 8.16, 13] of an orbifold [X/G].

There are two results that play key roles in our proof of the main theorem. One is the

formula (1) proved in [11] for the obstruction bundle of the stringy cohomology. Since their

definition avoids any construction of complex curves, admissible covers, or moduli spaces, it

greatly simplifies the analysis of the obstruction bundle and allows us to write the obstruction

bundle of [X n/(Gn ⋊ Σn)] in terms of the ones of [X/G] and [X n/Σn]. The other result is

Theorem 6.5 of [13] which states that there is a unique product structure on a normalized,

special Σn-Frobenius algebra (reviewed in Appendix). Lemma 3 which computes the obstruc-

tion bundle in a certain case using the formula (1) is necessary to apply Theorem 6.5 of [13]

and prove our main theorem. The direct and geometric proof of Theorem 4 in the case of an

abelian group G is also available in [17].

In order to relate our result to Ruan’s conjecture, we prove that, if X/G is an even di-

mensional Gorenstein variety and Y is a crepant resolution of X/G, then the natural map

Y n/Σn −→ X n/(Gn ⋊Σn) is crepant (Theorem 5). This implies that, if Y is a projective sur-

face with the trivial canonical class, then Y [n] is a crepant resolution of X n/(Gn ⋊ Σn), i.e.



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 494

the composition Y [n] −→ Y n/Σn −→ X n/(Gn ⋊ Σn) is a crepant resolution (conjectured in

[25, p.20]). Together with Theorem 4 and the result in [14], we obtain a verification of the

cohomological hyper-Kähler resolution conjecture in a special case: if H∗(Y )∼= H∗
or b
([X/G]),

then

H∗or b([X
n/(Gn⋊Σn)])

∼= H∗or b([X/G]){Σn}
Σn ∼= H∗(Y ){Σn}

Σn ∼= H∗(Y [n]).

When X = C2 and G is a finite subgroup of SL2(C), it is proved in a completely different way

[6].

The structure of the rest of the paper is as follows. In Section 2, we review the defini-

tion of a G-Frobenius algebra and show that, if H is an (K ⋊ L)-Frobenius algebra, then the

K-invariants of H form an L-Frobenius algebra. Also we review the construction of stringy

and orbifold cohomology following [11]. In Section 3, we study wreath product orbifolds and

compute the obstruction bundles for the cases that we need to prove the main theorem. In

Section 4, we prove the main theorem. In Section 5, we prove the crepantness of the map

Y n/Σn −→ X n/(Gn ⋊Σn) and apply our main theorem to verify the spacial case of the coho-

mological hyper-Kähler resolution conjecture. In the Appendix, we review the construction of

Lehn-Sorger’s algebras and the uniqueness theorem of Kaufmann.

Unless otherwise specified, we assume throughout the paper that all groups are finite and

all group actions are left actions. Also, unless otherwise specified, all of the vector spaces are

finite dimensional and over Q, and all coefficient rings for cohomology and K-theory are Q.

2. G-Frobenius Algebras and Semidirect Products

Recall the definition of a G-Frobenius algebra for a group G from [11] Section 3.

Definition 1. Let G be a group. A G-graded G-module (H ,ρ) is a G-graded vector space

H :=
⊕

g∈GHg with the structure of a left G-module by isomorphisms ρg : H
≃
−→ H such

that ρg takes Hh to Hghg−1 for all g,h in G. We denote a vector in Hg by vg for any g ∈ G

Definition 2. A tuple (H ,ρ, ·,1,η) is said to be a G-(equivariant) Frobenius algebra provided

that the following properties hold:

i) (G-graded G-module) (H ,ρ) is a G-graded G-module.

ii) (Self-invariance) For all g in G, ρg : Hg →Hg is the identity map.

iii) (Metric) η is a symmetric non-degenerate bilinear form on H s.t. η(vg , vh) = 0 unless

gh= 1.

iv) (Associativity) (H , ·,1) is a unital associative algebra.

v) (G-graded Multiplication) vg · vh ∈Hgh for all g,h ∈ G.

vi) (Braided Commutativity) vg · vh = ρg(vh) · vg for all g,h ∈ G.

vii) (G-equivariance of the Multiplication) ρg(v) · ρg(w) = ρg(v · w) for all g in G, and all

v, w ∈H .



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 495

viii) (G-invariance of the Metric) η(ρg(v),ρg(w)) = η(v, w) for all g in G, and all v, w ∈H .

ix) (Invariance of the Metric) η(v1 · v2, v3) = η(v1, v2 · v3) for all v1, v2, v3 ∈H .

x) G-invariant Identity) ρg(1) = 1 for all g in G.

xi) (Trace Axiom) For all a, b in G and v in H[a,b], if Lv denotes the left multiplication by v,

then the following equation is satisfied: TrHa
(Lv ◦ρb) = TrHb

(ρa−1 ◦ Lv).

Remark 1.

1) The G-Frobenius algebras are introduced in [23, 12]. Definition 2.1.1 of [12] is slightly

more general than the above definition (see ii) and xi)). Our definition is obtained by

setting χg = 1 for all g ∈ G.

2) A G-Frobenius algebra when G = {1} is a Frobenius algebra in the usual sense.

3) We can also define a G-Frobenius superalgebra [12] by introducing Z/2Z-grading and by

introducing signs in the usual manner, c.f. Section 1.2 of [13].

Definition 3. A G-Frobenius algebra H is said to be Q-graded if each Hg comes with a Q-

grading Hg =
⊕

r∈QHg ,r and the G-action and the multiplication respect the Q-grading and

the metric η satisfies η(v, w) = 0 unless deg a + deg b = d ≥ 0, i.e. H has degree d. In this

paper, we assume that all G-Frobenius algebras are Q-graded.

2.1. K-Invariants of a K⋊ L-Frobenius Algebra

Let K and L be groups. Suppose that L acts on K from left where the action of l ∈ L on

k ∈ K is denoted by k
l
7→ kl−1

. Let K⋊ L be a semidirect of groups K and L with respect to

this action. We identify K with the normal subgroup K⋊ 1 and hence the left adjoint action

of L on K can be identified with the given action of L on K, namely, we have kl−1

= lkl−1, i.e.

lkl = kl or lk = kl−1

l.

Let (H ,ρ, ·,1,η) be a (K⋊L)-Frobenius algebra. Let πK : H →H be the averaging map

over K:

πK(v) :=
1

|K|

∑

k∈K

ρk(v).

The image πK(H ) is the space of K-invariants of H , which we denote by H
K. The direct

sum H[l] := ⊕k∈KHkl is a K-module and so denote its K-invariants also by H
K
[l]

:= πK(H[l]).

The following theorem is the starting point of this paper.

Theorem 1. If H is a (K⋊ L)-Frobenius algebra, then H
K is an L-Frobenius algebra.

Proof. All of the properties except the self-invariance property and the trace axiom follow

immediately from those properties of H . The self-invariance property of H
K is that, for all

l ∈ L, ρl : H
K
[l]
→H

K
[l]

is the identity map. This is true because of the self-invariance property



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 496

of H . Indeed, for all kl ∈ K ⋊ L, ρkl restricted to Hkl is the identity map so that ρk = ρl−1

on Hkl . Let v ∈Hk0 l , then

ρlπK(v) =
1

|K|

∑

k∈K

ρlρkv =
1

|K|

∑

k′∈K

ρk′ρl v =
1

|K|

∑

k′∈K

ρk′ρk−1
0

v =
1

|K|

∑

k′′∈K

ρk′′v = πK(v)

For the trace axiom for H
K, we need to show Tr

H K
[l1]
(Lvm

◦ ρl2
) = Tr

H K
[l2]
(ρl−1

1
◦ Lvm

), for

l1, l2 ∈ L and vm ∈H
K
[m]

where m = [l1, l2]. The left-hand side is

Tr
H K
[l1]
(Lvm

◦ρl2
) = TrH[l1]

(Lvm
◦ρl2

◦πK) =
1

|K|

∑

k1,k

TrHk1 l1
(Lvm

◦ρl2
◦ρk)

=
1

|K|

∑

k1,k2

TrHk1 l1
(Lvm

◦ρk2 l2
) =

1

|K|

∑

k1,k2

TrHk2 l2
(ρ(k1l1)

−1 ◦ Lvm
),

where the third equality is obtained by replacing the parameter kl−1
2 by k2 and the fourth

equality follows from the trace axiom for H . The right-hand side is

Tr
H K
[l2]
(ρl−1

1
◦ Lvm

) =
1

|K|

∑

k,k2

TrHk2 l2
(ρl−1

1
◦ Lvm

◦ρk) =
1

|K|

∑

k1,k2

TrHk2 l2
(ρl−1

1 k−1
1
◦ Lvm

),

where the second equality follows from the cyclicity of the trace and by replacing the param-

eter k
l−1
1

1 by k−1
1 . Thus, the trace axiom holds for the K-invariants H

K.

2.2. Stringy and Orbifold Cohomology

We review the definition of the stringy and Chen-Ruan orbifold cohomology, following

[7] and [11]. Let X be a compact almost complex manifold of complex dimension d with

an action ρ of a finite group G preserving the almost complex structure. Let X g1,··· ,gr be the

submanifold of points in X fixed by the subgroup generated by g1, · · · , gr ∈ G. Then

H (X ,G) :=
⊕

g∈G

H∗(X g)

is naturally a G-graded G-module where ρg : X h −→ X ghg−1

(x 7→ ρg x). The G-equivariant

multiplication requires the class of the obstruction bundle in rational K-theory K(X g ,h) ⊗ Q
[7, 11]:

R(g,h) = T X g ,h⊖ T X |X g,h ⊕Sg |X g,h ⊕Sh|X g,h ⊕S(gh)−1 |X g,h. (1)

Here, the class Sg in K(X g)⊗Q is given by

Sg :=

r−1
⊕

k=0

k

r
Wg ,k (S-bundle) (2)



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 497

where r is the order of g (i.e. g r = 1), and Wg ,k is the eigenbundle of Wg := T X |X g such that

g acts with the eigenvalue exp(2πki/r). Now the multiplication is defined by,

vg · vh := q∗
�

vg |X g,h ∪ vh|X g,h ∪ cg ,h

�

, cg ,h := ctop

�

R(g,h)
�

(3)

where q : X g ,h ,→ X gh is the obvious inclusion. The G-equivariance of this multiplication

follows from the G-equivariance of S and R:

ρ∗mSmgm−1 =Sg , ρ∗mR(mgm−1, mhm−1) =R(g,h). (4)

The metric η of H (X ,G) is defined by

η(vg , wg−1) :=

∫

X g

vg ∪ ι
∗wg−1 , and η(vg , wh) = 0 if gh 6= 1 (5)

where ι : X g −→ X g−1

is the identity map. The orbifold Q-grading is given by

degQ(vg) := |vg |+ 2 age(g), where age(g) := rkSg (6)

where |vg | is the ordinary degree of the cohomology class vg . The following summarizes the

algebraic structure of H (X ,G):

Theorem 2 ([7, 10, 11]). (H (X ,G), ·,1,η,ρ, degQ) is a Q-graded G-Frobenius (super-) alge-

bra of degree 2 dimC X . It is called the stringy cohomology of G-manifold X .

The G-invariants of the stringy cohomology is isomorphic as a Frobenius algebra to the

orbifold cohomology of Chen-Ruan [4], i.e. H (X ,G)G = H∗
or b
([X/G]). Here the metric ηCR

on H∗
or b
([X/G]) is given by

ηCR(v, w) :=
1

|G|
η(v, w) for v, w ∈H (X ,G)G. (7)

Remark 2. If G = K⋊ L, then we have an action of L on an orbifold [X/K]. For the action of

a group on an orbifold, see [15] or [21] for example. By Theorem 1, we have an L-Frobenius

algebra H (X ,K⋊L)K which should play the role of the stringy cohomology of L-orbifold [X/K].

In general, when L acts on an orbifold X , it should be possible to define its stringy cohomology

H (X ,L) analogously and to show that it is an L-Frobenius algebra. Then our main result in

this paper should be easily generalized for the symmetric product of a global quotient orbifold

[X/H] where H is a Lie group, namely H ([X/H]n,Σn)
∼= Hor b([X/H]){Σn} with the help of

the explicit formula for the obstruction bundle of [X/H] in [5], or if H is a torus, [2, 9].

Remark 3. In the case of the wreath product orbifold [X n/Gn ⋊ Σn] that we study, the main

result of this paper implies that H (X n,Gn ⋊Σn)
Gn gives a geometric construction of the second

quantization of an orbifold [X/G] (see [13]).

It is convenient to generalize the formula (3) to the multi-product:



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 498

Lemma 1. Let g1, · · · , gr ∈ G, q : Z := X g1,··· ,gr −→ X g1···gr . Then

vg1
· · · · · vgr

= q∗
�

vg1
|Z ∪ · · · ∪ vgr

|Z ∪ c(g1, · · · , gr)
�

,

where

R(g1, · · · , gr) :=T Z ⊖ T X |Z ⊕Sg1
⊕ · · · ⊕Sgr

⊕S(g1···gr)
−1

c(g1, · · · , gr) :=ctop(R(g1, · · · , gr)).

The proof of the associativity of the product in [11] can be easily generalized to the proof

of this lemma (c.f. Proposition 5.3 of [17]).

3. The Wreath Product Orbifold

In this section, we review the wreath product orbifold to fix the notation (cf. Section 1 of

[25]) and then compute the obstruction bundle in certain cases.

Notation 1. The set of conjugacy classes of G is denoted by Ḡ. For all α ∈ G, let ZG(α) be

the centralizer of α in G. The subgroup generated by the subset {αk}k=1,··· ,r of G is denoted by

〈α1, · · · ,αr〉. For a finite set J , let GJ be the set of maps, Map(J ,G) ∼= G|J | and let gi := g(i)

for g ∈ GJ and i ∈ J. If g ∈ GJ , then ḡ ∈ ḠJ is defined by ( ḡ)i := ḡi ∈ G. Let ∆J : G→ GJ be

the diagonal map and let ∆J
G

:= ∆J(G). The same notation is applied to any set, i.e. if X is a

manifold, then X J :=Map(J , X ), x i := x(i) for x ∈ X J and ∆J
X :=∆J where ∆J : X → X J is the

diagonal map.

Definition 4 (Wreath Product and Wreath Product Orbifold). Fix a finite set I of cardinality n

and let ΣI be the permutation group of the set I. For all σ,τ ∈ ΣI , let Iσ := I/〈σ〉 be the set of

orbits in I under the action of the subgroup 〈σ〉 and similarly let Iσ,τ := I/〈σ,τ〉. Let |σ| be the

minimum number of transpositions to express σ and then |σ|= n− |Iσ|.
The natural left action of ΣI on GI is given by σ : gi 7→ gσ−1(i) for all σ ∈ ΣI and g ∈ GI .

The semidirect product GI ⋊ ΣI is called the wreath product of G. Let X be a compact almost

complex manifold with a left action ρ of G. There is a natural left action of the wreath product

GI ⋊ΣI on X I , which we also denote by ρ. Namely, for gσ ∈ GI ⋊ΣI , ρgσ(x) ∈ X I is defined by

(ρgσ(x))i := ρgi
(xσ−1(i)).

Thus, we have an orbifold [X I/G I ⋊ΣI] which we call the wreath product orbifold associated

to [X/G].

Definition 5 (Cycle product). For each a ∈ Iσ, choose a representative ia ∈ a. For each σ ∈ ΣI ,

define a map θσ : GI −→ GIσ (g 7→ θσg ) where

(θσg )a := gσ|a|−1(ia)
gσ|a|−2(ia)

· · · gσ0(ia)
, ∀a ∈ Iσ. (8)

θσg is a cycle product of g with respect to σ. The map θσ depends on the choice of repre-

sentatives {ia}, but if we choose different representatives, then each (θσg )a is conjugated by some

element in G. Hence θ̄σg ∈ Ḡ
Iσ is independent of the choice of representatives {ia}.



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 499

Now we compute the orbits of the action of GI by conjugation on GI ⋊ΣI .

Proposition 1. Choose a representative ia in a for each a ∈ Iσ. For gσ ∈ GI ⋊ΣI , let

g := θσg ∈ G
Iσ . The orbit of gσ under the action of GI by conjugation is given by

Ogσ :=
n

g′σ ∈ GIσ
�

� θ̄σ
g ′
= ḡ

o

. (9)

Proof. Define εg ∈ G
I by (εg)i := ga if i = ia for some a ∈ Iσ and otherwise (εg)i = 1. Then

it is easy to check (νσg )
−1 · gσ · νσg = εgσ where νσg is given by

(νσg )σm(ia)
:= gσm(ia)

gσm−1(ia)
· · · gσ0(ia)

, m= 0, · · · , |a| − 1. (10)

Therefore gσ and εgσ are in the same orbit. On the other hand, for g,g′ in G Iσ , f ∈ GI

satisfies εgσ = f −1εg′σ f if and only if f ∈
∏

a∈Iσ
∆a

G and ga = f −1
ia

g′a fia
. Thus, gσ and g′σ

are in the same orbit if and only if θ̄σg = θ̄
σ
g ′

.

Remark 4. From the above proof, it is clear that ZGI (εgσ) =
∏

a∈Iσ
∆a
ZG(ga)

.

Lemma 2 ([Lemma 4 and 5, 25]). For gσ ∈ GI ⋊ ΣI , we have (X I)gσ = ρνσg

�

∏

a∈Iσ
∆a

Xga

�

where g := θσg .

Proof. When gσ = εgσ, it follows from [Lemma 4, 25]. In general, it follows from the

definition (10) of νσg . Indeed, (X I)gσ = (X I)
νσg ·εgσ·(ν

σ
g )
−1

= ρνσg

�

(X I)εgσ
�

.

3.1. The Obstruction Bundle of the Wreath Product Orbifold

Now we will compute the obstruction bundle of the wreath product orbifold in certain

cases. Theorem 3 is crucial because it roughly says that the S-bundles for the wreath product

[X I/GI⋊ΣI] can be written in terms of the S-bundles of [X/G] and [X I/ΣI]. We use Lemma 3

in the proof of our main theorem, in particular, in the proof of Proposition 3.

Theorem 3. Let σ ∈ ΣI ,g ∈ G
Iσ . Let εg ∈ G

I be the element defined in the proof of Proposition 1.

We have

Sεgσ =
∏

a∈Iσ

�

∆a
∗

�

Sga
⊕
|a| − 1

2
T X |Xga

��

where Sga
∈ K(X ga) is the S-bundle with respect to the action of G on X and ∆a : X ga ∼=∆a

Xga .

Proof. Without loss of generality, we can assume I = {1, · · · , n} and σ = (12 · · ·n). Let

εg = (g, 1, · · · , 1) ∈ GI . Let ρ be the natural left action of ΣI on V := Cn so that

ρσ(e j) = eσ−1( j) where {e1, · · · ,en} is the standard basis of V . As a 〈εgσ〉-equivariant vector

bundle, T X I |(X I )εgσ can be identified to (T∆X⊗V )|∆Xg
. Explicitly, εgσ acts on u⊗v ∈ Tp∆X⊗V

as ρεgσ(u⊗ v) = ρgu⊗ v1en+
∑n

j=2 u⊗ v je j−1 where v=
∑

j vjej . Let r be the order of g ∈ G

and let T∆X |∆Xg
=
⊕r−1

l=0 Ul be the eigenbundle decomposition of the diagonal action of g



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 500

where the eigenvalue on Ul is e2πi l

r . On the other hand, the eigenvalues of the action of σ on

V are e2πi k

n , k = 0, · · · , n− 1 and the corresponding eigenvectors are vk :=
∑n

j=1

�

e2πi k

n

� j

e j.

For each l = 0, · · · , r − 1, define vk,l =
∑n

j=1

�

e2πi
�

l

rn
+ k

n

�� j

e j and then {vk,l, k = 0, · · · , n− 1}

forms a basis of V . Thus we have the following decomposition

T X I |(X I )εgσ =

n−1
⊕

k=0

 

r−1
⊕

l=0

Ul ⊗ Vk,l

!

. (11)

where Vk,l is the 1-dimensional subspace spanned by vk,l . This turns out to be the eigenbundle

decomposition of the action of εgσ where the eigenvalue of Ul ⊗ Vk,l is e2πi
�

l

nr
+ k

n

�

. Indeed, if

ul ⊗ vk,l ∈ Ul ⊗ Vk,l ,

ρεgσ(ul ⊗ vk,l) = e2πi
�

l

rn
+ k

n

��

e2πi l

r

�

ul ⊗ en+

n
∑

j=2

�

e2πi
�

l

rn
+ k

n

�� j

ul ⊗ e j−1

= e2πi
�

l

rn
+ k

n

�







�

e2πi l

r

�

ul ⊗ en +

n
∑

j=2

�

e2πi
�

l

rn
+ k

n

�� j−1

ul ⊗ e j−1







= e2πi
�

l

rn
+ k

n

�

· ul ⊗ vk,l

Thus we have

Sεgσ =

n−1
⊕

k=0

r−1
⊕

l=0

�

l

nr
+

k

n

�

Ul ⊗ V l
k
=

r−1
⊕

l=0

l

r
Ul ⊕

n−1
⊕

k=0

k

n
T∆X |∆Xga

=∆∗

�

Sg ⊕
n− 1

2
T X |Xg

�

,

where the second equality follows from forgetting the group action and identifying Vk,l with

C.

Corollary 1. Theorem 3 leads to the following formula obtained in [26] through the direct

calculation:

age(gσ) := rkSεgσ =
dimC X · |σ|

2
+
∑

a∈Iσ

age(ga)

where age(ga) is the age of ga with respect to the action G on X .

We need the following lemma to compute the action of the untwisted sector of

H (X I ,GI ⋊ΣI)
GI

.

Lemma 3. For every h ∈ GI and g ∈ GIσ , let Za := X ga ,hi ,i∈a for a ∈ Iσ, then

(X I)h ∩ (X I)εgσ =
∏

a∈Iσ

∆a
Za

and R(h,εgσ) =
∏

a∈Iσ

∆a
∗R(hσ|a|−1(ia)

, · · · ,hσ0(ia)
,ga)



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 501

Proof. The first statement follows immediately from Lemma 2. For the second claim, with-

out loss of generality we can assume that I = {1, · · · , n}, σ = (12 · · ·n) and εg = (g, 1, · · · , 1).
Since (hεgσ)

−1 = (h−1
2

, · · · ,h−1
n ,g−1h−1

1
)σ−1, we have m · (hεgσ)

−1 ·m−1 = ε(hn···h2h1g)
−1σ−1

for some m ∈ 〈h1, · · · ,hn,g〉I . Now compute

R(h,εgσ) =T∆Z ⊖ T X I |∆Z
⊕∆∗

�

Sh1
|Z ⊕ · · · ⊕Shn

|Z
�

⊕∆∗

�

Sg|Z ⊕
n− 1

2
T X |Z

�

⊕ρ∗m∆∗

�

S(hn···h2h1g)
−1 |Z ⊕

n− 1

2
T X |Z

�

where Z := X g ,h1,··· ,hn . Since mi fixes Z , ρm|∆Z
= id and therefore

R(h,εgσ) =∆∗
�

T Z ⊖ T X |Z ⊕Sh1
|Z ⊕ · · · ⊕Shn

|Z ⊕Sg|Z ⊕S(hn···h2h1g)
−1 |Z
�

=∆∗R(hn, · · · ,h1,g).

The next lemma is only related to Proposition 4.

Lemma 4. Let σ ∈ Σ and let τ = (i j) be a transposition. Suppose that σ,τ are transversal, i.e.

|σ|+ |τ|= |στ| and let i ∈ a and j ∈ b for a, b ∈ Iσ. Then

(X I)σ ∩ (X I)gτg−1

=







∏

c∈Iσ\{a,b}

∆c
X






×ρg ′

�

∆a∪b
X

�

where g′ ∈ Ga∪b is given by g′
l
= gi if l ∈ a and g′

l
= g j if l ∈ b. Moreover, rkR(σ, gτg−1) = 0

and so c(σ, gτg−1) = 1.

Proof. The first statement is straightforward. Without loss of generality, we can assume

that Iσ,τ = {I}. Also we can assume that g = g′, since gτg−1 = g′τ(g′)−1. Since

gσg−1 = σ, we have R(σ, gτg−1) = ρg∗R(σ,τ). However, from Corollary 1 we can com-

pute that rkR(σ,τ) = 0 and thus c(σ, gτg−1)=1.

Remark 5. If G is abelian, the general computation of the obstruction bundle is available in

[17].

4. Lehn-Sorger’s Algebras and G
I -Invariants of Stringy Cohomology

In this section, we prove our main theorem. Since our G-Frobenius algebras are special

G-Frobenius algebras, we can use the structure theorems in [12] and [13]. For the summary

of definitions and theorems, please see the appendix. Let H (X I ,GI ⋊ ΣI ) be the stringy

cohomology of the (G I ⋊ΣI)-space X I reviewed in Section 2.2 and let H∗
or b
([X/G]){ΣI} be

the Lehn-Sorger algebra associated to H∗
or b
([X/G]) reviewed in Section 5. By Proposition 1,

the GI -invariants of H (X I ,GI ⋊ΣI) is

H (X I ,GI ⋊ΣI)
GI

=
⊕

σ∈ΣI

⊕

ḡ∈ḠIσ

Hḡ,σ, Hḡ,σ :=







⊕

gσ∈Ogσ

H∗((X I)gσ)







GI

.



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 502

On the other hand, the Lehn-Sorger algebra associated to H∗
or b
([X/G]) is

H∗or b([X/G]){ΣI}=
⊕

σ∈ΣI

⊕

ḡ∈ḠIσ

Aḡ,σ, Aḡ,σ :=







⊕

g′∈ḡ

H∗((X Iσ)g
′
)







GIσ

.

Proposition 2. There is a canonical isomorphism of graded ΣI -graded ΣI -modules which pre-

serves the metric:

Φ : H∗or b([X/G]){ΣI}
≃
−→H (X I ,GI ⋊ΣI)

GI

.

Proof. Choose {ia ∈ a}a∈Iσ
. Consider the following isomorphisms:

H∗((X Iσ)g)ZGIσ (g) ∼=Aḡ,σ, x 7→ Lx :=
∑

f∈GIσ

ρf∗(x);

H∗((X I)εgσ)ZGI (εgσ) ∼=Hḡ,σ, v 7→ Fv :=
∑

f ∈GI

ρf ∗(v).

From Lemma 2 and Remark 4, we also have ZGIσ (g) =
∏

a∈Iσ
ZG(ga)

∼= ZGI (εgσ) and

(X Iσ)g =
∏

a∈Iσ
X ga ∼= (X I)εgσ, which imply

H∗((X Iσ)g)ZGIσ (g) ∼= H∗((X I)εgσ)ZGI (εgσ)

where (x 7→ ∆x). Define Φ by Φ(Lx) := F∆x
. Because of the summations over G Iσ and GI , Φ

is independent of the choices we made. The ΣI -equivariance is clear from the commutative

diagram
∏

a∈Iσ
X ga

∼=
��

∼=
//
∏

a∈Iσ
∆a

Xga

∼=
��

∏

τ(a)∈I
τστ−1

X gτ(a)
∼=

//
∏

τ(a)∈I
τστ−1

∆
τ(a)

X
gτ(a)

It follows from Eq. (13), Corollary 1 and Eq. (6) that Φ preserves the Q-grading.

Since the component of H (X I ,GI ⋊ ΣI)
GI

graded by the identity permutation (the un-

twisted sector) is exactly the Frobenius algebra H∗(X I ,GI )G
I

= H∗
or b
([X/G])⊗I , this yields the

following proposition.

Proposition 3. The isomorphism Φ is an isomorphism as H∗
or b
([X/G])⊗I -modules. In particular,

H (X I ,GI ⋊ΣI)
GI

is a special ΣI -Frobenius algebra. Furthermore, Φ preserves the metric.

Proof. We will show that the actions of H∗
or b
([X/G])I on Aḡ,σ and on Hḡ,σ are identified

by Φ. Without loss of generality, we can assume I = {1, · · · , n} and σ = (12 · · ·n). Let

x g ∈ H∗(X g)ZG(g) for every g ∈ G and let Lxg
:=
∑

k∈Gρk(x g) ∈ A ḡ,σ and

Φ(Lxg
) =
∑

k∈GI ρk(∆xg
) ∈H ḡ,σ. We need to show that the Lehn-Sorger product

PLS :=







∑

f1,··· , fn∈G

ρf1
(xh1
)⊗ · · · ⊗ρfn

(xhn
)






·

 

∑

k∈G

ρk(x g)

!



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 503

corresponds via Φ to the stringy product

PST :=







∑

f1,··· , fn∈G

ρf1
(xh1
)⊗ · · · ⊗ρfn

(xhn
)






·





∑

k∈GI

ρk(∆xg
)



 .

Let Z := X g , f1h1 f −1
1 ,··· , fnhn f −1

n , q : Z ,→ X fnhn f −1
n ··· f1h1 f −1

1 g and ∆q :∆Z ,→∆
X

fnhn f−1
n ··· f1h1 f−1

1
g . Then

PLS is computed as follows.

PLS =
∑

f1,··· , fn,k∈G

�

ρf1
(xh1
) · · · · ·ρfn

(xhn
) ·ρk(x g)

�

=
∑

k∈G

ρk







∑

f1,··· , fn∈G

�

ρfn
(xhn
) · · · · ·ρf1

(xh1
) · x g

�







=
∑

k∈G

ρk







∑

f1,··· , fn∈G

q∗
�

ρfn
(xhn
)|Z ∪ · · · ∪ρf1

(xh1
)|Z ∪ x g |Z ∪ c( fnhn f −1

n , · · · , f1h1 f −1
1 , g)

�







where the first equality follows from the definition and the fact that gd(1,σ) = 0, the sec-

ond equality follows by the G-equivariance and the commutativity of the multiplication in

H (X ,G)G and replacing k−1 fi by fi and the third equality follows from Lemma 1. On the

other hand, PST is computed as follows.

PST =
∑

k∈GI

ρk







∑

f1,··· , fn∈G

�

ρf1
(xh1
)⊗ · · · ⊗ρfn

(xhn
)
�

·∆xg







=
∑

k∈GI

ρk







∑

f1,··· , fn∈G

∆q∗

�

ρf1
(xh1
)⊗ · · · ⊗ρfn

(xhn
)
�

�

∆Z
∪∆xg

|∆Z
∪∆∗c( fnhn f −1

n , · · · , f1h1 f −1
1 , g)













where the second equality follows from Lemma 3. Now it is clear that Φ(PLS) = PST .

Since the metric of a special G-Frobenius algebra is completely determined by the Frobe-

nius algebra structure on the untwisted sector and its action (see [Theorem 4.1, 12] or Ap-

pendix), the isomorphism also preserves the metric.

So far, we have proved that Φ is a G-equivariant isomorphism of special ΣI -reconstruction

data. Thus, by Theorem 4.1 of [12], if the associated graded cocycles coincide, then Φ is a

G-Frobenius algebra isomorphism. From Theorem 6.5 of [13] and the fact that the graded

cocycle of a Lehn-Sorger’s algebra is normalized [13, p.77,], it suffices to show:

Proposition 4. Let 1σ be the identity of the Frobenius algebra Hor b([X/G])
⊗Iσ . The graded

cocycle defined by the special ΣI -Frobenius algebra structure on H(X I ,GI ⋊ΣI)
GI

with the cyclic

generators {Φ(1σ)} is normalized.

Proof. The graded cocycle γ is defined by Φ(1σ) ·Φ(1τ) = γσ,τΦ(1στ). If 1σ is the identity

element of the ordinary cohomology ring H∗((X I)σ), then

Φ(1σ) =
1

|GIσ |

∑

f ∈GI

ρ f

�

1σ
�

.



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 504

Let σ,τ be transversal and |τ| = 1. Without loss of generality, we can assume Iσ,τ = {I} and

therefore, let τ = (i j) and Iσ = {a, b} where i ∈ a and j ∈ b. We compute

Φ(1σ) ·Φ(1τ) =







1

|GIσ |

∑

f ∈GI

ρf

�

1σ
�






·







1

|GIτ |

∑

g∈GI

ρg

�

1τ
�







=
1

|G||Iσ|+|Iτ|

∑

f ∈GI

ρf







∑

g∈GI

1σ ·ρg(1τ)







=
1

|G|n+1

∑

f ∈GI

ρf









|G|Iτ
∑

g∈GI/
�

∏

c∈Iτ
∆c

G

�

1σ ·ρg(1τ)









=
|G|n−1

|G|n+1

∑

f ∈GI

ρf







∑

g∈∆a
G
×∆b

G
/∆G

ρg(1σ · 1τ)







=
1

|G|

∑

f ∈GI

ρf

�

1στ
�

= Φ(1στ)

where the fourth equality follows from GI/
�

∏

c∈Iτ
∆c

G

�

∼=∆a
G
×∆b

G
/∆G and the GI -equivariance

of the stringy product. The last equality follows from 1σ · 1τ = 1στ which holds because of

transversality. Thus γσ,τ = 1.

Theorem 4. The canonical isomorphism Φ is an isomorphism of ΣI -Frobenius algebras.

Proof. Since the Lehn-Sorger algebra is always normalized (see Appendix), Propositions 2,

3 and 4 imply that Φ is an isomorphism of normalized special ΣI -reconstruction data. There-

fore by Theorem 4.1 of [12] and Theorem 6.5 of [13], we can conclude that Φ preserves the

ΣI -Frobenius algebra structures.

5. Hilbert Schemes and Wreath Products Orbifolds

In this section, we will relate the wreath product orbifold associated to a G-variety X to

the Hilbert scheme of n-points on Y when Y is a crepant resolution of X/G. Throughout the

section, all vector spaces are over C and we will work in the algebraic category.

Definition 6. Let W be a normal variety over C and let L be a rank 1, torsion free, coherent

sheaf of OW -module over W. L is called divisorial [20] if and only if any torsion free coherent

sheaf of OW -module, M , such that L ⊂M and Supp(M /L ) has codimension ≥ 2, coincides

with L .

Remark 6. Let L be divisorial. If W 0 ⊂W is a non-singular open subvariety such that W\W 0

has codimension ≥ 2, then L |X 0 is invertible and L = j∗(L |X 0) [20], where j : W 0 ,→ W



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 505

denotes the canonical inclusion. Let KW be the canonical divisor of W. By Proposition (7) in

[20], the canonical sheaf ωW := O(KW ) of W is divisorial. Hence, we have ωW = j∗ωW0 since

ωW |W0 =ωW 0 .

Definition 7. Let W and Y be normal varieties. A birational morphism π : Y → W is crepant

if ωY
∼= π∗ωW . A normal variety W is Gorenstein if and only if all of the local rings are

Cohen-Macaulay and KW is Cartier.

Lemma 5. Let W and Y be Gorenstein varieties. If π : Y → W is birational, then π∗KX is

divisorial.

Proof. Let dim W = dim Y = n. Since KW is Cartier, π∗KW is also Cartier. Hence, π∗ωW

is torsion-free and of rank 1. Let M be a torsion-free sheaf such that π∗ωW ⊂ M and the

dimension of SuppM /π∗ωW is at most n−2. Let L := KY−π
∗KW . L is Cartier and L := O(L)

is an invertible sheaf. It follows that π∗ωW ⊗L ∼=ωY ⊂M ⊗L and

dim(Supp((M ⊗L )/ωY ))≤ dim(SuppM /π∗ωW )≤ n− 2. Since Hn−1((M ⊗L )/ωY ) = 0,

we have Hn(M ⊗L ) ∼= Hn(ωY )
∼= C by Serre duality. Hence, there exists an element in

Hom(M ⊗L ,ωY ) which gives a splitting of the short exact sequence

0→ωY →M ⊗L → (M ⊗L )/ωY → 0. However, since M ⊗L is torsion-free,

(M ⊗L )/ωY = 0.

Theorem 5. Let W and Y be normal varieties with dimension ≥ 2. Suppose that W\W 0 has

codimension ≥ 2 and that Y n/Σn and W n/Σn are Gorenstein. If π : Y → W is a crepant

resolution, then the induced map π̃ : Y n/Σn→W n/Σn is crepant.

Proof. The smooth locus of W n/Σn is equal to (W n\∆⋆W )/Σn where ∆⋆W is the set of points

in W n with non-trivial isotropy. Let DY := π−1(∆⋆W ). Let π̄ : Y n\DY → W n\∆⋆W be the map

π×n restricted to Y n\DY . Since π×n : Y n → W n is crepant, KY n = (π×n)∗KW n . Consider the

commutative diagram
Y n ←−−− Y n\DY

π×n





y π̄





y

W n ←−−− W n\∆⋆W
where the horizontal arrows are the obvious inclusions. We have

KY n\DY
= KY n |Y n\DY

=
�

(π×n)∗KW n

�

|Y n\DY
= π̄∗(KWn |W n\∆⋆W

) = π̄∗KW n\∆⋆W
. (12)

Consider the following commutative diagram

Y n\DY

q
−−−→ (Y n\DY )/Σn

π̄





y π̃′





y

W n\∆⋆W
q′

−−−→ (W n\∆⋆W )/Σn



Tomoo Matsumura / Eur. J. Pure Appl. Math, 5 (2012), 492-510 506

where q and q′ are the canonical projections. Since the actions of ΣI on Y n\DY and W n\∆⋆W
are free, Equation (12) implies that K(Y n\DY )/Σn

= π̃′∗K(W n\∆⋆
W
)/Σn

. Hence

KY n/Σn
|(Y n\DY )/Σn

=
�

π̃∗KW n/Σn

�

|(Y n\DY )/Σn
.

Since both KY n/Σn
and π̃∗KW n/Σn

are divisorial (Remark 6, Lemma 5), we obtain

KY n/Σn
= π̃∗KW n/Σn

.

Remark 7. For a non-singular variety X with an action of a finite group G, the variety X/G is

Gorenstein if and only if the age of α on any connected component is an integer for all α ∈ G.

See Remark (3.2) in [20]. If dim X is even and X/G is Gorenstein, by Corollary 1, X n/Gn ⋊Σn

is Gorenstein. In particular, for a non-singular variety Y with even (complex) dimension, the age

of the symmetric product Y n/Σn is always an integer so that Y n/Σn is Gorenstein.

If Y is a smooth projective surface, then the Hilbert-Chow morphism Y [n] → Y n/Σn is a

resolution of singularities [8], which is also crepant [1]. Hence, together with Theorem 5 and

Remark 7, we obtain the following statement conjectured on p.20 of [25]:

Corollary 2. Let X be a smooth projective surface with an action of a finite group G. Suppose

that X/G is Gorenstein. If π : Y → X/G is a crepant resolution, then Y [n]→W n/Σn is a crepant

resolution.

Together with Theorem 4, we obtain the following result.

Theorem 6. Let Y be a smooth projective surface with trivial canonical class. Let X be a smooth

projective surface with an action of G such that X/G is Gorenstein. Suppose that π : Y → X/G is

a crepant resolution and that H∗(Y )∼= H∗
or b
([X/G]) as Frobenius algebras, then Y [n]→ X n/Σn

is a hyper-Kähler resolution and H∗(Y [n]) is isomorphic as a ring to H∗
or b
([X n/Gn ⋊Σn]).

Proof. We have H (Y n,Σn)
∼= H∗(Y ){Σn} ∼= H∗

or b
([X/G]){Σn} ∼=H (X n,Gn⋊Σn)

Gn

where

the first equality is due to [7] and the third is Theorem 4. Since H∗(Y [n]) ∼= H∗(Y ){Σn}
Σn

[14], we obtain the theorem by taking Σn-invariants everywhere in the above equality.

Theorem 6 is a special case of the following conjecture due to Ruan [22].

Conjecture 1 (Cohomological hyper-Kähler resolution conjecture). Suppose that Y → X be a

hyper-Kähler resolution of the coarse moduli space X of an orbifoldX . The ordinary cohomology

ring H∗(Y ) of Y is isomorphic to the Chen-Ruan orbifold cohomology ring H∗
or b
(X ) of X .

Remark 8. The conjecture in the special case of wreath product orbifolds has been verified when

X = C2 and G is a finite subgroup of SL2(C) in [6]. In particular, an explicit ring isomorphism

between H∗(Y [n]) and H∗
or b
([X n/Gn ⋊Σn]) has been established when X = C2 and G is a finite

cyclic subgroup of SL2(C) by using Fock space methods in [19].

Remark 9. In Theorem 6, the claim still holds if we replace the ordinary cohomology and orbifold

cohomology by ordinary K-theory and orbifold K-theory respectively by the results in [10].



REFERENCES 507

ACKNOWLEDGEMENTS The author is greatly indebted to his thesis advisor Takashi Kimura,

who has provided constant guidance throughout the course of this project. The author would

like to thank R. Kaufmann to point out his papers to improve significantly from my thesis

[17]. It is also pleasure to thank D. Abramovich, A. Craw, B. Fantechi, S. Okada, F. Perroni,

W. Wang for important advice and useful conversations. The author is supported by the Na-

tional Research Foundation of Korea (NRF) grants funded by the Korea government (MEST)

(No. 2012-0000795, 2011-0001181). He also would like to express his gratitude to the Alge-

braic Structure and its Application Research Institute at KAIST for providing him an excellent

research environment in 2011-2012.

References

[1] A. Beauville. Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ-

ential Geom., 18(4):755–782 (1984), 1983.

[2] L. A. Borisov, L. Chen, and G. G. Smith. The orbifold Chow ring of toric Deligne-Mumford

stacks. J. Amer. Math. Soc., 18(1):193–215 (electronic), 2005.

[3] J. Bryan and T. Graber. The crepant resolution conjecture. In Algebraic geometry—Seattle

2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 23–42. Amer. Math. Soc.,

Providence, RI, 2009.

[4] W. Chen and Y. Ruan. A new cohomology theory of orbifold. Comm. Math. Phys.,

248(1):1–31, 2004.

[5] D. Edidin, T. J. Jarvis, and T. Kimura. Logarithmic trace and orbifold products. Duke

Math. J., 153(3):427–473, 2010.

[6] P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and

deformed Harish-Chandra homomorphism. Invent. Math., 147(2):243–348, 2002.

[7] B. Fantechi and L. Göttsche. Orbifold cohomology for global quotients. Duke Math. J.,

117(2):197–227, 2003.

[8] J. Fogarty. Algebraic families on an algebraic surface. Amer. J. Math, 90:511–521, 1968.

[9] R. Goldin, T. S. Holm, and A. Knutson. Orbifold cohomology of torus quotients. Duke

Math. J., 139(1):89–139, 2007.

[10] T. J. Jarvis, R. Kaufmann, and T. Kimura. Pointed admissible G-covers and G-equivariant

cohomological field theories. Compos. Math., 141(4):926–978, 2005.

[11] T. J. Jarvis, R. Kaufmann, and T. Kimura. Stringy K-theory and the Chern character.

Invent. Math., 168(1):23–81, 2007.

[12] R. M. Kaufmann. Orbifolding Frobenius algebras. Internat. J. Math., 14(6):573–617,

2003.



REFERENCES 508

[13] R. M. Kaufmann. Second quantized Frobenius algebras. Comm. Math. Phys., 248(1):33–

83, 2004.

[14] M. Lehn and C. Sorger. The cup product of Hilbert schemes for K3 surfaces. Invent.

Math., 152(2):305–329, 2003.

[15] E. Lerman and A. Malkin. Hamiltonian group actions on symplectic Deligne-Mumford

stacks and toric orbifolds. Adv. Math., 229(2):984–1000, 2012.

[16] W. Li, Z. Qin, and W. Wang. Ideals of the cohomology rings of Hilbert schemes and their

applications. Trans. Amer. Math. Soc., 356(1):245–265 (electronic), 2004.

[17] T. Matsumura. Orbifold Cohomology of a Wreath Product Orbifold. PhD dissertation,

Boston University, Dept. Math., August 2007. arXiv:math/0610269.

[18] Z. Qin and W. Wang. Hilbert schemes and symmetric products: a dictionary. In Orbifolds

in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages

233–257. Amer. Math. Soc., Providence, RI, 2002.

[19] Z. Qin and W. Wang. Hilbert schemes of points on the minimal resolution and soliton

equations. In Lie algebras, vertex operator algebras and their applications, volume 442 of

Contemp. Math., pages 435–462. Amer. Math. Soc., Providence, RI, 2007.

[20] M. Reid. Canonical 3-folds. In Journées de Géometrie Algébrique d’Angers, Juillet 1979/Al-

gebraic Geometry, Angers, 1979, pages 273–310. Sijthoff & Noordhoff, Alphen aan den

Rijn, 1980.

[21] M. Romagny. Group actions on stacks and applications. Michigan Math. J., 53(1):209–

236, 2005.

[22] Y. Ruan. Stringy orbifolds. In Orbifolds in mathematics and physics (Madison, WI, 2001),

volume 310 of Contemp. Math., pages 259–299. Amer. Math. Soc., Providence, RI, 2002.

[23] V. Turaev. Homotopy field theory in dimension 2 and group-algebras.

arXiv:math/9910010, October 1999.

[24] B. Uribe. Orbifold cohomology of the symmetric product. Comm. Anal. Geom.,

13(1):113–128, 2005.

[25] W. Wang. Equivariant K-theory, wreath products, and Heisenberg algebra. Duke Math.

J., 103(1):1–23, 2000.

[26] W. Wang and J. Zhou. Orbifold Hodge numbers of the wreath product orbifolds. J.

Geom. Phys., 38(2):152–169, 2001.



REFERENCES 509

Appendix

Special ΣI -Frobenius Algebras

In this section, we recall definitions and results about special G-Frobenius algebras from

[12, 13] and the construction of Lehn-Sorger’s algebras [14].

Definition 8 ([Definition 4.1 and 4.2, 12]). A special G-Frobenius algebra (H ,ρ, ·, {1g},η)
is a G-Frobenius algebra (H ,ρ, ·,1e,η) with the choice of 1g ∈ Hg such that Hg = He · 1g

and ϕg(1h) = ϕg ,h1ghg−1 for some ϕg ,h ∈ k×. Let rg : He −→Hg be given by a 7→ a · 1g and

let Ig := ker g. Let ig be a section of rg . A special G-reconstruction datum is a collection of

Frobenius algebras (Hg , ·,ηg ,1g), g ∈ G with an action ρ of G on He and cyclic He-algebra

structures on (Hg , ·,1g) such that Hg and Hg−1 are isomorphic as He-algebras and

η(ρg(a),ρg(b)) = η(a, b).

Lemma 6 ([Proposition 4.1, 12]). A special G-Frobenius algebra (H ,ρ, ·, {1g},η) defines a

special G-reconstruction datum {(Hg , ·,ηg ,1g), g ∈ G,ρ}. The structure of a Frobenius algebra

in Hg is given by ag · bg := ig(ag) · ig(bg) · 1g and ηg(ag , bg) := η(ig(ag)1g , ig(bg)1g−1).

Definition 9 ([Definition 4.3 and 4.4, 12]). Let {(Hg , ·g ,ηg ,1g), g ∈ G,ρ} be a special G-

reconstruction datum. A graded cocycle is a map γ : G × G −→ He, (g,h) 7→ γg ,h such

that γg ,hγgh,k ≡ γg ,hkγh,k mod Ighk. A graded cocycle γ is compatible with the special G-

reconstruction datum if (Ig + Ih)γg ,h ⊂ Igh (section independence), γg ,g−1 = řg(1g) (metric

compatibility) and γe,h = 1e mod Ih where η
♯
g : ag 7→ ηg(ag , ) and řg := (η

♯
e)
−1 ◦ r∗g ◦η

♯
g . We

identify two cocycle γ and γ′ if γg ,h ≡ γ
′
g ,h

mod Igh.

A non-abelian cocycle is a map ϕ : G × G −→ k× satisfying ϕgh,k = ϕg ,hkh−1ϕh,k and

ϕe,g = ϕg ,e = 1. A graded cocycle γ and a non-abelian cocycle ϕ form a compatible pair if

ϕg ,hγghg−1,g = γg ,h and ϕk,gϕk,hγkgk−1,khk−1 = ϕk(γg ,h)ϕk,gh.

Theorem 7 (Reconstruction Theorem [Theorem 4.1, 12]). Let {(Hg , ·g ,ηg ,1g), g ∈ G,ρ} be

a special G-reconstruction datum. Then the structures of special G-Frobenius algebras inducing

the given reconstruction datum correspond bijectively to compatible pairs of a non-abelian cocycle

ϕ and a section independent cocycle γ compatible with the given special G-reconstruction datum,

such that ϕg ,g = 1 and TrHg
(Lc ◦ρh) = TrHh

(ρg−1 ◦ Lc) for any c ∈H[g ,h].

Remark 10. Given ϕg ,h and γg ,h, the multiplication on H = ⊕gHg is given by

ag · bh := rgh(ig(ag) ·e ih(bh) ·e γg ,h), the action ρg on Hh is defined by

ϕg(bh) := rghg−1(ϕg ,hρg(ih(bh))), and the metric is defined by

η(ag , bg−1) := ηe(ig(ag) ·e ig−1(bg−1) ·e γg ,g−1,1e) and ηe(ag , bh) = 0 if gh 6= e. Those definition

is independent from the choice of sections ig because of the section independence of γ. On the

other hand, given a G-Frobenius algebra, γ and ϕ are defined by 1g1h = γg ,h · 1gh (γg ,h ∈He)

and ϕg(1h) = ϕg ,h1ghg−1 (ϕg ,h ∈ k×).

Definition 10 (Normality). Let I be a finite set of cardinality |I | = n. Consider a special ΣI -

Frobenius algebra (H ,ρ, ·, {1σ},η). Two permutations σ,τ ∈ Σ are transversal if

|σ|+ |τ| = |στ|. The graded cocycle γ induced by a special ΣI -Frobenius algebra is normalized

if γσ,τ = 1e for all transversal pair σ,τ with |τ|= 1 [Definition 6.4, 13].



REFERENCES 510

Theorem 8 ([Theorem 6.5, 13]). If γ is normalized, then γ is completely determined by

γτ,τ = řτ(1τ) with |τ|= 1.

Lehn-Sorger’s Algebras

Let (A, ·, 1,η) be a commutative graded Frobenius algebra of degree 2d . If J is a finite set,

then the tensor product A⊗J over J is naturally a commutative graded Frobenius algebra. If

φ : J1 ։ J2 is a surjective map between finite sets, then there is an induced homomorphism

φ∗ : A⊗J1 ։ A⊗J2 defined by
⊗

i∈J1
ai 7→

⊗

j∈J2

�

∏

φ(i)= j ai

�

. By using the metric, we can

identify a Frobenius algebra with its vector space dual, and thus we also have an induced map

φ∗ : A⊗J2 −→ A⊗J1 .

Let I be a finite set of cardinality |I | = n. The underlying vector space for the Lehn-Sorger’s

algebra associated to A is

A{Σ} :=
⊕

σ∈ΣI

A⊗Iσ

where Iσ is the set of orbits in I by the action of the subgroup 〈σ〉 generated by σ. There is

an obvious ΣI -graded ΣI -module structure. The Q-grading is given by

degQ aσ := |aσ|+ d · |σ|, aσ ∈ A
⊗Iσ (13)

where |aσ| is the degree in A⊗Iσ and |σ| is the minimum transpositions to express σ.

The Euler class e of a Frobenius algebra A is defined as the image of 1 under the map

A −→ A ⊗ A −→ A where the first map is the comultiplication and the second map is the

multiplication. The graph defect gd(σ,τ) : Iσ,τ −→ Z≥0 is defined by

gd(σ,τ)c =
1

2
(|c|+ 2− |c/〈σ〉| − |c/〈τ〉| − |c/στ|) .

Using these, the Lehn-Sorger’s product is defined by

aσ · bτ = f ∗






fσ∗aσ · fτ∗bτ ·







⊗

c∈Iσ,τ

egd(σ,τ)c













where fσ : Iσ ։ Iσ,τ, fτ : Iτ։ Iσ,τ and f : Iστ։ Iσ,τ. Note that, if σ,τ are transversal, then

it is straightforward computation to see that gd(σ,τ) = 0. Thus it is also easy to see that the

induced graded cocycle γ is normalized.

Theorem 9 ([14, 13]). A{ΣI} is a graded special ΣI -Frobenius algebra and its graded cocycle γ

is normalized.