Table Of ContentADE
CHEN-RUAN COHOMOLOGY OF SINGULARITIES
FABIO PERRONI
Institut für Mathematik, Universität Züri
h
Winterthurerstrasse 190, CH-8057 Züri
h
fabio.perronimath.unizh.
h
7
0
0
Abstra
t
2
n WestudyRuan's
ohomAologi
al
repant resolution
onje
ture [41℄fororbifoldswithtraHns∗ve(r[sYa]l)ADE
a singularities. In the n-
ase we
ompuHte∗b(Zot)h(qt,h.e..,Cqh)en-Ruan
ohomology ring CR and
1 n
J the quantum
orre
ted
ohomology ring . The former is a
hieved in general, the
9 lHat∗er(u[Yp])to somHe∗a(dZd)i(tio1n)al, te
hnAi
al assumptions. We
onstru
t an expli
it isoAmorphism between
HC∗(RZ)(q1,a..n.,dqn) − in the q11-
=ase..,.v=erqifnyi=ng R1uan's
onje
ture. In the n-
ase, the family
] is not de(cid:28)ned for − . This implies that the
onje
ture should be
G An
slightly modi(cid:28)ed. We propose a new
onje
ture in the -
ase (Conj. 1.9). Finally, we prove Conj.
A
A 1.9 in the 2-
ase by
onstru
ting an expli
itisomorphism.
. 2000
h
Mathemati
s Subje
tClassi(cid:28)
ation : Primary14E15; Se
ondary 14N35; 14F45
t
a
m
0 Introdu
tion
[
2 TheChen-Ruan
ohomology was de(cid:28)nedbyChenandRuan[11℄foralmost
omplexorbifolds. This
v wasextendedtoanon-
ommutativeringbyFante
hiandGötts
he[18℄inthe
asewheretheorbifold
7 is a global quotient. Abramovi
h, Graber and Vistoli de(cid:28)ned the Chen-Ruan
ohomology in the
0 algebrai
ase [1℄.
[Y] Y
2 Let be a
omplexGorenstein orbifold su
h thatthe
oarse moduli spa
e admitsa
repant
ρ:Z Y Z
5 resolution → . Then, under some te
hni
al assumptions on , Ruan's
ohomologi
al
repant
0
06 mresoololugtyiornin
gonHjeC∗
Rtu(r[Ye],[4C1)℄ panreHddi∗t
(htZes,tsCho)e
eaxlliesdtenq
ueanotfuamn
isoormreo
trepdhis
mohobmetowloegeny trhinegCohfeZn-.RuTahne
loahteor-
is a deformation of the ring obtained using
ertain Gromov-Witten invariants of rational
/ Z ρ Z
h
urvesin whi
hare
ontra
tedundertheresolutionmap . Noti
ethatif
arriesanholomorphi
t symple
ti
stru
ture,thenthis
onje
turealsopredi
tstheexisten
eofanisomorphismbetweenthe
a [Y] Z
m Chen-Ruan
ohomology ring of and the
ohomology ring of . HilbrM r
: AninterestingtestinMg
aseforthe
onje
tureistheoneoftheHilberts
hemSeymrM of points
v onaproje
tivesurfa
e . Itisa
repantresolutionofthesymmetri
produ
t viatheChow
r=2 r
i morphism. Inthis
asethe
onje
turewasprovedbyW.-P.LiandZ.Qinfor [28℄,for general
X M
and with numeri
allyHt∗ri(vHiaillb
raMno)ni
al
lass by Fante
hi and Götts
he [18℄ (using the expli
it
r
omputation of the ring given by Lehn and Sorger [26℄), and independently by Uribe
a
[46℄. A di(cid:27)erent and self-
ontained proof of this result was given by Z. Qin and W. Wang [37℄. In
M
thesamesituation butwith quasi-proje
tive with a holomorphi
symple
ti
form,the
onje
ture
was proved by W.-P. Li, Z. Qin and W. Wang [29℄. In parti
ular this result generalizes the
ase
of the a(cid:30)ne plane obtained by Lehn and Sorger [27℄ and Vasserot [47℄ independently. The general
Y = V/G V G Sp(V)
ase where with
omplex symple
ti
ve
tor spa
e and ⊂ (cid:28)nite subgroup was
proved by Ginzburg and Kaledin [21℄. Let us point out that in the previous
ases (ex
ept [28℄) the
Z
resolution
arries a holomorphi
sympHle∗
(tZi
,sCt)ru
ture, hen
e the quantum
orre
ted
ohomology
ring
oin
ides withthe
ohomology riMng=P2 .rD=.3Edidin, W.-P.Li andZ. Qinpartially veri(cid:28)ed
Ruan's
onje
ture in the
ase where and , therequantum
orre
tions appeared [17℄.
ADE
TheaimofthispaperistostudyRuan's
onje
turefororbifoldswithtransversal singular-
[Y] ADE
ities (see Def.Y2.10). An orbifold has tRransCvkersal R singularities if,AéDtaEle lo
ally, the
oarse
modulispa
e isisomorphi
toaprodu
t × ,where isagermofan singularity. Noti
e
[Y] W Y 3
that for any Gorenstein orbifold , there exists a
losed subset ⊂ of
odimension ≥ su
h
1
Y W ADE
that \ hastransversal singularities. Thusthe
asewestudyisthegeneraloneifweignore
3
phenomenathato
ur in
odimension ≥ .
ADE
We des
ribe the twisted se
tors of orbifolds with singularities. After that, we
on
entrate
A
n
onthetransversal -
aseandweaddressRuan's
onje
tureby
omputingexpli
itlyboththeChen-
Ruan
ohomology (Th. 3.12) and the quantum
orre
tions (Prop. 5.4). The former is a
hieved in
general, regardingthelater weproposea
onje
tureonthevalueofsomeGromov-Witteninvariants
A A n 2
1 n
(Conj. 5.1)whi
hisprovedfullyinthe -
ase,andinthe -
ase( ≥ )underadditionalte
hni
al
assumptions. In a work in progress with B. Fante
hi we give a proof of Conj. 5.1 and
omputethe
D E
quantum
orre
tions in thetransversal and
ases. H∗ ([Y])
We
onstru
t an expli
it isomorphismHbe∗t(wZe)e(n1th)e Chen-Ruan
ohomAology ring CR and
1
the quantum
orre
ted
ohomology ring − in the transversal -
ase, verifying Ruan's
A 3
n
onje
ture (Se
. 6.1). Inthe -
ase, thequantum
orre
ted -point fun
tion
an notbe evaluated
q =...=q = 1
1 n
in − . This implies that Ruan's
onje
ture has to be slightly modi(cid:28)ed. We propose
A A
n 2
a modi(cid:28)
ation in the -
ase (Conj. 1.9) that we prove in the -
ase, by
onstru
ting an expli
it
isomorphism(see Prop. 6.2).
The stru
ture of the paper is the following. In Se
tion 1 we review the statement of the
oho-
ADE
mologi
al
repant resolution
onje
ture. Orbifolds with transversal singularities are de(cid:28)ned
in Se
tion 2. Then in Se
tion 3, we
ompute expli
itly the Chen-Ruan
ohomology ring of su
h
orbifolds. InSe
tion4 we provethatupto isomorphismthe
oarse modulispa
e ofanorbifold with
ADE Z
transversal singularities has a unique
repant resolution and we des
ribe the
ohomology
Z Z
ring of . In Se
tion 5, we state our
onje
ture about the Gromov-Witten invariants of (whose
proof in some parti
ular
ases is postponed to Se
tion 7). Using this, we
ompute the quantum
orre
ted
ohomology ring. Afterwards we put together these results to verify our modi(cid:28)
ation of
Ruan's
onje
ture.
Notation
C Y Z
We will work over the (cid:28)eld of
omplex numbers . Through out this paper, and will denote
d C Y S
proje
tivealgebrai
varietiesofdimension over . Thesingularlo
usof isdenotedby andthe
i:S Y
in
lusion by → .
[Y] Y
A
omplexorbifold meansa
omplexorbifold stru
tureover thetopologi
al spa
e . Inthis
Y
ontext, has the
omplex topology. Our referen
es for orbifolds are [10℄, [11℄, [33℄ and [36℄. In
parti
ular notations are taken from[10℄ and [36℄.
Wewill workwith
ohomologygroupswith
omplex
oe(cid:30)
ients, althoughmanyresultsarevalid
for rational
oe(cid:30)
ients.
1 The
ohomologi
al
repant resolution
onje
ture
In this se
tion we re
all the statement of the
ohomologi
al
repant resolution
onje
ture as given
by Y. Ruan in [41℄. The
onje
ture
laims a pre
ise relation between the Chen-Ruan
ohomology
[Y] Y
ring of a
omplex orbifold and the
ohomology ring of a
repant resolution of , when su
h a
resolution exists.
[Y] ι
(g)
De(cid:28)nition 1.1. A
omplexorbifold isGorensteinifthedegreeshiftingnumbers areintegers,
(g) T
for all ∈ .
[Y] Y
Noti
e that, if is Gorenstein, then the algebrai
variety is also Gorenstein and in parti
ular
K
Y
the
anoni
al sheaf is lo
ally free (see e.g. [38℄ and [39℄ for moredetails).
Y ρ : Z Y
Dreep(cid:28)annittiiofnρ∗1(K.2Y()[3∼=9℄K).Z.Let be a Gorenstein variety. A resolution of singularities → is
Crepant resolutions of Gorenstein varieties with quotient singularities are known to exist in di-
2 3 d = 2 Y
mensions and . In parti
ular, for a stronger result holds: every normal surfa
e admits
d = 3
a unique
repant resolution [2℄. In dimension the existen
e of a
repant resolution is proven
d 4
e.g. in [40℄ and in [9℄, however the uniqueness result does not hold. In dimension ≥
repant
resolutions not always exist.
We will work under thefollowing
[Y] ρ:Z Y
Assumption 1.3. Let beaGorensteinorbifoldand → a(cid:28)xed
repantresolution. Then
onsider theindu
edgroup homomorphism
ρ :H (Z,Q) H (Y,Q).
∗ 2 2
→ (1)
2
ρ n
Weassumethattheextremalrays
ontra
tedby aregeneratedby rational
urveswhosehomology
β ,...,β Q β ,...,β ρ
1 n 1 n ∗
lasses are linearly independentover . Then determine a basis of Ker
alled
integral basis [41℄.
ρ
ΓTh=ehonmolaogβy
lass ofanaye(cid:27)e
tive
urvethatis
ontra
tβedby
anbewritteninauniqquewaΓy
as l=1 lqal1, withqatnhe l's positive integers. F3or ea
h l we assign a formal variable l, so
orrespoPnds to 1 ··· n . The quantum
orre
ted -point fun
tion is
γ ,γ ,γ (q ,...,q ):= ΨZ(γ ,γ ,γ )qa1 qan,
h 1 2 3iqc 1 n Γ 1 2 3 1 ··· n (2)
a1,.X..,an>0
γ ,γ ,γ H∗(Z) Γ= n aβ ΨZ(γ ,γ ,γ )
where 1 2 3∈ are
ohomology
lasses, l=1 l l,and Γ 1 2 3 isthegenuszero
Z
Gromov-Witteninvariantof [41℄. P
q ,...,q
1 n
Assumption 1.4. We assume thaCtn(2) de(cid:28)nes an analyti
fuγn
,tγio,nγof the variables on
1 2 3 qc
some region of the
omplex spa
e . It will be denoted by h i . In the following, when
γ ,γ ,γ (q ,...,q )
1 2 3 qc 1 n
we evaluate h i on a point , we will impli
itly assume that it is de(cid:28)ned on su
h
a point.
q ,...,q
1 n
We now de(cid:28)nea family of rings dependingon theparameters .
γ ,γ ,γ (q ,...,q )
1 2 3 qc 1 n
De(cid:28)nition 1.5. The quantum
orre
ted triple interse
tion h i is de(cid:28)ned by
γ ,γ ,γ (q ,...,q ):= γ ,γ ,γ + γ ,γ ,γ (q ,...,q ),
1 2 3 ρ 1 n 1 2 3 1 2 3 qc 1 n
h i h i h i
γ ,γ ,γ := γ γ γ γ γ
whereh 1 2 3i Z 1∪ 2∪ 3. Thequantum
orre
ted
upprodu
t 1∗ρ 2 isde(cid:28)nedbyrequiring
that R
γ γ ,γ = γ ,γ ,γ (q ,...,q ) γ H∗(Z),
1 ρ 2 1 2 ρ 1 n
h ∗ i h i for all ∈
γ ,γ := γ γ
where h 1 2i Z 1∪ 2.
R
Remark 1.6. Our de(cid:28)nition of quantum
orre
ted triple interse
tion and of quantum
orre
ted
up produ
t is slightly di(cid:27)erent from the one given in [41℄. One
an re
over the original de(cid:28)nition
q = ... = q = 1
1 n
by giving to the parameters the value − , provided that this point belongs to the
3
domain of thequantum
orre
ted -pointfun
tion.
(q ,...,q ) 3
1 n
Proposition 1.7([13℄). Forany belongingtothedomainofthequantum
orre
ted -point
ρ
fun
tion, the quantum
orre
ted
up produ
t ∗ satis(cid:28)es the following properties.
H∗(Z)
Asso
iativity: it isasso
iative on , moreover it has a unit whi
h
oin
ides with the unit of the
Z
usual
up produ
t of .
γ1 ργ2 =( 1)deg γ1·deg γ2γ2 ργ1 γ1,γ2 H∗(Z)
Skewsymmetry: ∗ − ∗ , for any ∈ .
γ ,γ H∗(Z) deg (γ γ )=deg γ +deg γ
1 2 1 ρ 2 1 2
Homogeneity: for any ∈ , ∗ .
Z
De(cid:28)nition 1.8.HT∗h(eZ)quantum
orre
ted
ohomology ringHof∗(Z)i(sqth,.e..,faqm)ily of ring stru
tures on
theve
torspa
e givenby ∗ρ. It will be denotedby ρ 1 n .
We (cid:28)nally
ome to Ruan's
onje
ture, whose studyis thereason of this paper.
Cohomologi
al
repant resolution
onje
ture (Y. Ruan,[41℄)
Under the above hypothesis, there exists a ring isomorphism
H∗(Z)( 1,..., 1)=H∗ ([Y]).
ρ − − ∼ CR
A
n
As said, this
onje
ture needs to be slightly modi(cid:28)ed. Inthe -
ase we propose thefollowing
[Y] A
n
Conje
ture 1.9. Let be an orbifold with transversal -singularities and trivial monodromy
ρ:Z Y
(Def. 3.4), → be the
repant resolution (Prop. 4.2). Then the following map
Hρ∗(Z)(q1,...,qn)∼=HC∗R([Y]) (3)
n
E ζlk(ζk+ζ−k 2)1/2e
l k
7→ −
Xk=1
q =...=q =ζ (n+1) 1 E ,...,E
1 n 1 n
isaringisomorphismfor beaprimitive -throot of . Here arethe
e ,...,e
1 n
irredu
ible
omponents of the ex
eptional divisor (see Notation 4.5) and are the generators
ζ =exp 2πim
of the Chen-Ruan
ohomology (see Thm. 3.12). The square root in (3) means, for n+1 ,
“ ”
i(2 ζk ζ−k)1/2 0<m< n+1;
(ζk+ζ−k 2)1/2 = | − − | if 2
− ( i(2 ζk ζ−k)1/2 .
− | − − | otherwise
3
Remark 1.10. The isomorphism in theprevious
onje
ture is theone
onje
turedbyJ. Bryan,T.
A
n
Graber and R. Pandharipande [7℄ for the -
ase. It
oin
ides with the map found by W. Nahm
and K. Wendland [34℄. In a re
ent work, joint with S. Boissière and E. Mann [5℄, we prove that
(3)gives anisomorphismbetweentheChen-Ruan
ohomology ringoftheweightedproje
tive spa
e
[P(1,3,4,4)]
and the quantum
orre
ted
ohomology ring of its
repant resolution. We expe
t to
report on theveri(cid:28)
ation of Conj. 1.9 soon.
InChapter 2.2 we will see how to get (3) fromthe
lassi
al M
Kay
orresponden
e.
ADE
2 Orbifolds with singularities
ADE
In this Se
tion we de(cid:28)ne orbifolds with transversal singularities. They are generalizations of
Gorenstein orbifolds asso
iated to quotient surfa
e singularities, also
alled rational double points.
Therefore we (cid:28)rst re
all the de(cid:28)nition of su
h surfa
e singularities and
olle
t some properties. We
will follow [2℄, [15℄, [16℄.
2.1 Rational double points
DRe(cid:28)nCi3tion 2.1. A rational double poiCn2t/G(in shorGt RDP) is the germ oSfLa(2s,uCr)fa
e singularity
⊂ whi
h is isomorphi
to a quotient with a (cid:28)nite subgroup of .
Rational double points are Gorenstein. Indeed every variety with symple
ti
singularities is
Gorenstein [3℄.
SL(2,C)
Finitesubgroupsof are
lassi(cid:28)ed,upto
onjugation, andtheresultofthis
lassi(cid:28)
ation
is givenin thefollowing Theorem.
SL(2,C)
Theorem2.2 ([16℄). Any(cid:28)nite subgroup of is
onjugate to oneofthe followingsubgroups:
24 48
6 7
the binary tetrahedral group E of order ; the binary o
tahedral group E of order ; the binary
120 4(n 2) n 4
8 n
i
osahedral group E of order ; the binary dihedral group D of order − for ≥ ; the
n+1
n
y
li
group A of order .
It turns out that
onjugate subgroups give isomorphi
surfa
e singularities. Hen
e the above
lassi(cid:28)
ation indu
es a
lassi(cid:28)
ation of RDP's [16℄:
: xy zn+1 =0 n 1
n
A : x2+y−2z+zn−1 =0 for n≥4
n
D : x2+y3+z4 =0 for ≥
6
E : x2+y3+yz3=0 (4)
7
E : x2+y3+z5=0.
8
E
Resolution graph
R ρ:R˜ R
Anyrational doublepoint hasaunique
repantresolution → [2℄. Theex
eptionallo
usof
ρ E ,...,E 2
1 n
istheunionofrational
urves withself-interse
tionnumbers− . Moreover,itispossible
to asso
iate a graph to the
olle
tion of these
urves in the following way: there is a vertex for any
irredu
ible
omponentof theex
eptional lo
us; two verti
es are joined byanedge if and only if the
orresponding
omponents have non zero interse
tion. The list of the graphs obtained by resolving
rational doublepointsis givenin [15℄ and in [2℄. Ea
h of this graph is
alled resolution graph of the
orresponding singularity.
R C3
Notation 2.3. From now on, will denote a surfa
e0in C3de(cid:28)ned by one of the equatRions (4),
i.e. a surfa
ρe :wRi˜th aRrational double point at the origin ∈ . The
repant resolution of will be
denotedby → .
2.2 M
Kay
orresponden
e
R G SL(2,C) R Q=C2
Let be a RDP and ⊂ be a (cid:28)nitesubgroup
orresponding to . We denoteby
G SL(2,C) λ ,...,λ
0 m
therepresentationindu
edbythein
lusion ⊂ . Let bethe(isomorphism
lasses
G λ j =1,...,m
0
of)irredu
iblerepresentations of ,with beingthetrivialone. Then,forany we
an
Q λ
j
de
ompose ⊗ as follows
Q⊗λj =⊕mi=0aijλi, aij =dimCHomG(λi,Q⊗λj). (5)
G SL(2,C)
De(cid:28)nition 2.4. The M
Kay graph of ⊂a is the graph with one vΓ˜ertex for any irredu
ible
ij G
representation, two verti
es are joined by arrows. It will be denoted by . If we
onsider only
Γ
G
nontrivial representations, then we obtain the graph , whi
h will be
alled also M
Kay graph.
4
G
Remark2.5. In[32℄therepresentation graph of (i.e. whatwe
alltheM
Kaygraph)wasde(cid:28)ned
SL(2,C)
in a slightly di(cid:27)erent way. However it
an be shown that, for (cid:28)nite subgroups of , the two
de(cid:28)nitions
oin
ide.
Γ
G
The M
Kay
orresponden
e, in his original form, states that the graph
oin
ides with the
R
resolution graph of . The
orrespond[eRn]
e
an be obtaR˜ined geometri
ally by means of a map that
identi(cid:28)es the K-theory of the orbifold with that of , this is done in [22℄. We re
all brie(cid:29)y this
onstrGu
tion. C2 F C2
A -equivariant
oherent sheaf on is a
oherent sheaf on together with isomorphisms
α :g∗F F, g G
g
→ ∈
K([R])
whGi
hsatisfytheobvious
o
y
le
ondCiti2on. Let K(Rt˜h)eGrothendie
kringofisomorphism
lasses
of -equivariant
oherent sheaves on R˜. As usual, R(Gd)enotes the Grothendie
k ring of isomor-
phism
lasses of
oGherent sheaλvesRo(nG).λ∨Finally, set be the ring of isomorphism
lasses of
representations of . For any ∈ , denotesthe dual
lass.
We have thefollowing
λ G
Proposition 2.6 ([22℄). The map that asso
iates, to any representation of on the ve
tor spa
e
Vλ, the G-equivariant
oherent sheaf OC2⊗CVλ∨ indu
es a ring isomorphism
R(G) ∼= K([R]).
−→
We identify the two rings by means of this map.
Consider now theCartesian diagram
C˜2 pr2 R˜
−−−−−→
pr1 ρ
C??2 χ R??
y −−−−−→ y
χ
where is the quotientmap. The following result holds.
Theorem 2.7 ([22℄). Let π:R(G)=K([R]) K(R˜)
→
de(cid:28)ned by π:=Inv pr pr ∗,
◦ 2∗◦ 1
pr pr ∗ Inv
wGhere 2∗ and 1 are the
Manoni
R˜al morphisms aMndG is the appli
ation that asso
iates to any
-equivariant
oherent sheaf on the subsheaf of the invariants. Then
λ G E
λ
(i) for any irredu
ible representation of , there is a unique
omponent of the ex
eptional
E
divisor su
h that
rk(π(λ))=degλ c (π(λ))=c ( (E )).
and 1 1 OR˜ λ
λ E G
λ
The map 7→ is a bije
tion from the set of irredu
ible representations of to the set of
E λ = µ (E E ) = a a
λ µ λµ λµ
omponents of . For any 6 , · , where the 's are de(cid:28)ned in (5) and
( )
_·_ is the Poin
aré pairing.
π Z
(ii) is an isomorphism of -modules.
[R]
This Thm.
an be Ru˜sed to get a
orresponden
e between the Chen-Ruan
ohomology of
and the
ohomology of as follows (we refer to the next Chapter for the de(cid:28)nition of Chen-Ruan
ohomology). We have maps
Ch( ) Td(R˜):K(R˜) H∗(R˜)
_ · → (6)
h( ) d([R]):K([R]) H∗ ([R])
C _ ·T → CR (7)
Ch Td h d
where and aretheusualChen
hara
terandTodd
lassrespe
tively,C andT aretheChern
hara
terandTodd
lass fororbifolds as de(cid:28)nedbyToen[44℄, andthemultipli
ations aretheusual
π
upprodu
ts(nottheChen-Ruanonein these
ond
ase). Thenthemap of Thm.2.7, (6)and(7)
A
n
giveamapbetween
ohGomologZygroups. Weζw=orekxopu(t2tπhie)detCa∗ilsofthλis
omputationinthe -
ase.
Identifythegroup with n+1 and set n+1 ∈ . Let m be theirredu
ible represen-
Z V
tation of n+1 on λm whose
hara
ter is
l ζml.
7→
5
FromThm. 2.7 we have that
Ch(π(λ )) Td(R˜)=1+c ( (E )) H∗(R˜).
m · 1 OR˜ λm ∈ (8)
We
omputenow Ch OC2 ⊗CVλ∨m ·Td([R])∈HC∗R([R]). (9)
Foranyl∈Zn+1,
onsidertheres`tri
tion OC2´⊗CVλ∨m |(C2)l ofOC2⊗CVλ∨m tothe(cid:28)xedpointlo
us
(C2)l of l. Thea
tion of l on OC2⊗CVλ∨m` |(C2)l is give´nbythemultipli
ation byζ−lm. Hen
e
` ´
Ch(OC2⊗CVλ∨m)= ζ−lm·1H∗(R(l)),
l∈XZn+1
where 1H∗(R(l)) is the neutral element of the
ohomology ring of the twisted se
tor R(l), for any
l Z α K([R ]) [R ]
∈ n+1. Next we
ompute the
lass [R] ∈ 1 de(cid:28)ned in [44℄, where 1 is the inertia
C [R ] [R] [R ]
1 1
orbifold. Wedenoteby the
onormalsheafof withrespe
tto ,i.e. thesheafon whose
[R] l Z
n+1
restri
tiontoea
htwistedse
toristhe
onormalsheafofthetwistedse
torin . Forany ∈ ,
C C [R ] l = 0 C 0
l (l) l
set the restri
tion of to . Then, if , has rank . Otherwise it is given by the
λ λ Z
1 n n+1
representation ⊕ of . λ (C)=1 C+ 2C,
−1
− ∧
hen
e
1 l=0;
(α[R])|(C2)l =(2 ζl ζ−l if .
− − otherwise
Therefore n
1
Td([R])=1H∗(R(0))+ 2 ζl ζ−l ·1H∗(R(l)).
Xl=1 − −
Finally, we get
n ζ−lm
Ch(OC2⊗CVλm)·Td([R])=1H∗(R(0))+ 2 ζl ζ−l ·1H∗(R(l)).
Xl=1 − −
Remark 2.8. Theprevious pro
edure gives thefollowing map
H2(R˜) H2 ([R])
→ CR
n ζ−lm
E e,
m 7→ 2 ζl ζ−l l
Xl=1 − −
where we have used the same notation as in Conj. 1.9. It follows from Prop. 6.2 that this is not a
ring isomorphism. But it is
lear how to
hange thepro
edure to get the
orre
t map.
However the previous
omputation gives a way to get the isomorphism between the Chen-Ruan
ADE
ohomologyandthequantum
orre
ted
ohomologyofthe
repantresolutioninthe -
ase. This
will be obje
t of further investigations.
ADE
2.3 De(cid:28)nition of orbifolds with transversal singularities
We use the language of groupoids, and refer to [10℄ and to the referen
es there for a more detailed
dis
ussion of the relations between orbifolds and groupoids. To (cid:28)x notations, we re
all that an
Y
orbifold stru
ture on thefp:ara
omYpa
t Hausdor(cid:27) spa
e is d(e(cid:28),fn)ed to (be′,afn′)orbifold groupoid G
with a′homeomorphism |G|→ . Two orfbifoldfs′tru
tures G and G are equivalent i(cid:27) G
andG are Morita equivalentand themaps and are
ompatible under theequivalen
e relation.
[Y] Y
Then an orbifold is de(cid:28)ned to be a spa
e with an equivalent
lass of orbifold stru
tures. An
( ,f) [Y]
orbifoldstru
ture G insu
hanequivalen
e
lassisapresentation oftheorbifold . Theorbifold
[Y] TG
0
is
omplex if it is given in addition a
omplex stru
ture on the tangent bundle , whi
h is
equivariantunderthe G-a
tion.
Y V Y α
α
Anorbifold stru
ture over
an also be given byan open
overing { } of and, for any , a
U G χ :U /G V
α α α α α α
smoothvariety , a (cid:28)nitegroup a
ting onitu, andUanhomue′omUorphism y→ Y. This
α β
data must satis(cid:28)es the
ondition that, whenever ∈ and ∈ map to the same ∈ , then
6
W U u W′ U u′ ϕ:W W′
α β
thereeuxistun′eighborhoods ⊂ of and ⊂ of ,andanisomorphism → whi
h
sends in su
h thatthefollowing diagram
ommutes
W ϕ W′
−−−−−→
χα χβ
Y?? id Y??
y −−−−−→ y
Then,if we set
G := U ,
0 α α
⊔
G := (u,ϕ,u′)u u′ y Y, ϕ
1
{ | and mapto thesame ∈ and is a germof a lo
al isomorphism as above}
and the stru
ture maps de(cid:28)ned in the obvious way, we obtain a groupoid G whi
h is an orbifold
Y
stru
ture on .
Y ADE S
Wesaythattheva(rSie,tYy) hastransversal singularities ifthesingularlo(C
uks i0s
,oCnkne
tRed),
smooth, and the pair is lo
ally (in the
omplex topology) isomorphi
to ×{ } × .
We have thefollowing
Y ADE
Proposition 2.9. Let be a variety with transversal singularities. Then there is a unique
[Y] Y
omplex holomorphi
orbifold stru
ture on su
h that the (cid:28)xed point lo
us of the lo
al groups
2
has
odimension greater than .
Proof. This is a parti
ular
ase of the well known fa
t that every
omplex variety with quotient
singularities has a unique orbifold stru
ture su
h that the (cid:28)xed point lo
us of the lo
al groups has
2
odimension greater than (see e.g. [43℄).
ADE [Y]
De(cid:28)nition2.10. Anorbifold with transversal singularitiesistheorbifold asso
iated
Y ADE
to a variety with transversal singularities as in Prop. 2.9.
[Y] ADE
Notation 2.11. Let beanorbifoldwithtransversal singularities. Intherestofthepaper,
( ,f) [Y] y Y y / S V
α
we will usethepresentation G of de(cid:28)nedas follows. Let ∈ be a point. If ∈ ,take
y U := V χ := id y S V
to be a smooth open neighborhood of , α α and α Vα. If ∈ , then set α an open
neighborhood of theform Vα∼=Ck×R,
U :=Ck C2,
α
×
G :=G
α
and χ :U /G ∼= V ,
α α α α
→
G U := Ck C2 [Y] ( ,f)
α α
where a
ts on × only on the se
ond fa
tor. The presentation of , G , is
(U ,G ,χ )
α α α
onstru
ted as explained in the beginning of the Se
tion. The triple is
alled orbifold
y
hart at .
Y 3
Remark2.12. If isa -foldwith
anoni
alsingularities,thenwiththeex
eptionofatmosta(cid:28)nite
Y
numberofpoints,everypointin hasanopenneighborhoodwhi
hisnonsingular orisomorphi
to
C R
× [38℄.
3 Chen-Ruan
ohomology
A
n
InthisSe
tionwe
omputetheChen-Ruan
ohomologyoforbifoldswithtransversal singularities.
[Y]
As a ve
tor spa
e, theChen-Ruan
ohomology of is de(cid:28)nedby
H∗ ([Y]):= H∗−2ι(g)(Y ),
CR ⊕(g)∈T (g)
Y [Y ] [Y ] T
(g) (g) (1)
where is the
oarse moduli spa
e of the twisted (untwisted) se
tor ( ), is the set
[Y ] ι
1 (g)
of
onne
ted
omponents of theinertia orbifold , and His∗t(hYe ag)e (also
alled degree shifting)
(g)
[11℄. We work with
ohomology with
omplex
oe(cid:30)
ients, so denotessingular
ohomology
with
omplex
oe(cid:30)
ients.
[E]
CR
Theorbifold
upprodu
t∪ [Yi0s]de(cid:28)nedintermsofanobstru
tionbu3ndle ,whi
hisanorbifold
ve
torbundl(ego,vger,tghe) orbi3fold 3 ,thgesugb-ogrbi=fol1d oftheorbifold of -multise
tors
orresponding
to elements 1 2 3 ∈SG su
h that 1· 2· 3 , [10℄ [11℄.
There is an orbifold morphism
[τ]:[Y ] [Y]
1
→
whose underlying
ontinuous mapis
τ :Y Y
1
→
(y,(g) ) y.
y
7→
7
3.1 Inertia orbifold and monodromy
[Y] ADE
Westudysomepropertiesoftheinertiaorbifoldofanorbifold withtransversal singularities.
[Y]
Thepresentation of des
ribedin Not. 2.11 will be used.
[Y] S
Lemma 3.1. The orbifold indu
es a natural orbifold stru
ture on .
s t F :G Y
0
Proof. Let and bethesour
e andtargetmapsofG,anddenoteby → the
omposition
G f
0
of thequotientmap →|G| followed by . We de(cid:28)ne
H :=F−1(S) H :=t−1(H ).
0 1 0
and
t−1(H ) = s−1(H )
0 0
Sin
e , we obtain a groupoid H whose stru
ture maps are the restri
tion of the
H H f
0 1
stru
ture maps of G to and . The orbit spa
e |H| is
ontained in |G| and the restri
tion of
f S ( ,f) S
to|H|, |, is an homeomorphismfrom|H| to . Then H | is theorbifold stru
ture on .
[S] ( ,f) [S]
Notation 3.2. We denote by the orbifold given by the equivalen
e
lass of H | .
an be
[Y] [S] [Y] [N]
viewedas sub-orbifold of . The normalve
torbundleof in is denotedby .
τ :Y Y
1
Proposition 3.3. 1. The restri
tion of → to the
oarse moduli spa
e of the unionof the
Y
twisted se
tors, ⊔(g)6=(1) (g), is a topologi
al
overing
τ : Y S.
| ⊔(g)6=(1) (g) →
y S (τ)−1(y)
2. Foranypoint ∈G,t:h=e(cid:28)(sb,etr)−1|(y,y) is
anoni
allyidenti(cid:28)edwiththesetof
onjuga
y
lasses
y
of the lo
al group whi
h are di(cid:27)erent from the
lass of the neutral element
(1) G
y
, and hen
e with the set of the non trivial irredu
ible representations of .
y S [N] [S] [Y] 2
y
3. For ∈ , the (cid:28)ber of the normal bundle of in is a -dimensional representation
G Γ G [N]
of y, let Gy be the M
Kay graph of y with respe
t to y. Then, the monodromy of the
τ y Γ
overing | at takes values in the automorphism group of the M
Kay graph Gy.
π
G
Proof. 1. Following [10℄, we
onsider thefollowing Cartesian diagram whi
h de(cid:28)nes S and
G
G 1
S −−−−−→
π (s,t)
(10)
G??y0 −−−∆−−→ G0×??yG0
∆
G
where is the diagonal. S is a G-spa
e with a
tion givenby
G
1 s π G G
× S → S (11)
(a,b) aba−1
7→
⋉ [Y ]
G 1
and thea
tion-groupoid G S is a presentation of theinertia orbifold .
π : H π H G H
Let |H0 SG|H0 → 0 be the base
hange of with respe
t to the in
lusion 0 → 0 ( 0 is
[Y]
de(cid:28)nedintheproofofthepreviousLemma). Withrespe
ttoourpresentationof (seeNot. 2.11),
we have
SG|H0 ∼=H0×G.
H (G 1 )
G 0
Thea
tionofGonS restri
tstoana
tionon × −{ } ,whi
hunderthepreviousidenti(cid:28)
ation
is des
ribed as follows
(u,ϕ,u′),(u,g) (u′ =ϕ(u),ϕ g ϕ−1).
7→ ◦ ◦ (12)
` ⋉(H ´ (G 1 )) [Y ]
The asso
iate⋉d (aH
tion-(gGroupo1id),)G (U 0)G×α (−G{ }1,)is a presentation of ⊔(g)6=(1) (g) . The re-
0 α
stri
tion of G × −{ } to × −{ } is isomorphi
to the a
tion groupoid
G (U )G (G 1 ) ⇉(U )G (G 1 ),
α α
× × −{ } × −{ }
“ ”
[G (U )G (G 1 ) ⇉ (U )G (G 1 )]
α α
moreover the orbifolds × × −{ } × −{ } form an open
overing of
[Y ] (τ)−1((U )G) (U )G
⊔(g)6=(1) (g) . Thusweseeth`at | α isd´isjointunionof
opiesof α andtherestri
tion
τ
|
of on anyof these
omponentsis an homeomorphism. This proves thestatement.
2. It follows fromdiagram (10) and thea
tion (11) that
(τ)−1(y)=(π−1(y) id )/π−1(y)=(G id )/G
| −{ y} y−{ y} y
G τ
y |
where a
tsby
onjugation. Thisestablishthe
orresponden
ebetween(cid:28)bersof and
onjuga
y
lasses of lo
al groups.
8
3G. L=etGy∈S. Usinyg′the(
Uha)rGt (Uα,Gα,χα) aty we get anidenti(cid:28)
ation of thelo
al group Gy′ with
α α
, for any ∈ . It is
lear that these identi(cid:28)
ations respe
t the M
Kay graphs. It
remains to show that, if y ∈ Vα∩Vβ, the isomorphism Gα ∼= Gβ indu
ed by the orbifold stru
ture
respe
ts theM
Kay graphs. W U W′ U χ−1(y)
χ−1W(ye)re
all that, in thiϕs :siWtuatioWn,′if ⊂ α and ⊂ β arχe neiϕgh=boχrhoods of α and
β respe
tively, and → is an isomorphism su
h that β◦ α, thenthere exists a
λ:G G ϕ λ
α β
uniqueisomorphism → su
h that is -equivariant[33℄. We identifytherepresentations
G G λ
α β
of with that of by means of . In this way the irredu
ible representations
orrespond to
irredu
ible representations. Finally, thelinear map
T ϕ:T U T U
χ−α1(y) χ−β1(y) α → χ−β1(y) β
N G N G
gives an isomorphism between the representations UαG/Uα of α and UβG/Uβ of β. Now the
statement follows from the de(cid:28)nition of the M
Kay graph and of the monodromy of a topologi
al
over,see e.g. [31℄.
[Y] ADE y S
De(cid:28)nition3.4. Let beanorbifoldwithtransversal singularities, ∈ . Themonodromy
[Y] y y
of in is the monodromy, in , of the topologi
al
over
τ : Y S,
| ⊔(g)6=(1) (g) →
it is denoted by the group homomorphism
m :π (S,y) Aut(τ−1(y)).
y 1 → |
G=A , n 1, D n 4,E ,E ,E
n n 6 7 8
Remark 3.5. For ≥ ≥ (see Th. 2.2), theautomorphismgroup of
Γ
G
is givenas follows:
G (Γ )
G
Aut
A 1
1
{ }
A n 2 Z
n 2
≥
D S
4 3
D n 5 Z
n 2
≥
E Z
6 2
E 1
7
{ }
E 1
8
{ }
G (Γ )
G
where we have written ontheleft side thegroup and onthe rightAut .
Y (g) T
The previous
onsiderations give
onstraints on thetopology of the spa
es (g) for ∈ . The
following Corollary is an easy
onsequen
e of Prop. 3.3.
[Y] ADE
Corollary 3.6. Let be an orbifold with transversal singularities. Then, if the monodromy
S
is trivial, all the
oarse moduli spa
es of the twisted se
tors are
anoUni
alSly isomorphi
to . U˜
U IfthemonU˜odromyisnottrivial,thereexistsanopenneigAhDboErhood of anda
overingspa
e →
su
hthat has astru
ture oforbifoldwith transversal singularities andtrivial monodromy.
(g)=(1)
Proof. For any 6 , themap
τ :Y S
|Y(g) (g) →
[Y] τ
is a
onne
ted topologi
al
overing. If has trivial monodromy, then |Y(g) has also trivial mon-
τ
odromy. It follows that |Y(g) is anhomeomorphism.
U Y S
Assumenowthatthemonodromyisnottrivial. Let ⊂ be a tubular neighborhood of and
y Y
∈ a point. Then therepresentation
m :π (S,y) Aut(τ−1(y))
y 1 → |
U˜ U m U˜ U
y
guarantee the existen
eU˜of a
overing → with the same monodroAmDyE . Sin
e → is a
lo
al homeomorphism, is a
omplex analyAtiD
Espa
e with tran[Us˜v]ersal singular[iU˜ti]es, hen
e it
has a stru
ture of orbifold with transversal singularities . By
onstru
tion has trivial
monodromy.
[Y ] [Y] S
(g)
Remark 3.7. Noti
e that the twisted se
tors of depend only on a neighborhood of in
Y U Y S Y U ADE
. Indeed,let ⊂ beanopenneighborhoodof in ,then isavarietywithtransversal
[U ] [U] [Y ]
(g) (g)
singularities and the twisted se
tors of are
anoni
ally isomorphi
to . So,
[Y1]∼=[Y] [U(g)].
(g)∈TG,(g)6=(1)
9
[Y] A
n
Corollary 3.8. Let be an orbifold with transversal singularities and trivial monodromy. If
n 2 [N] [S] [Y]
[N≥]g , the[nN]tgh−e1norm[Sal] bundle of in is isomorphi
to the dire
t sum of two line bundles
and on ,
[N]=[N]g [N]g−1.
∼ ⊕
[N] N H G
Proof. A presentation of is given by the H-spa
e H0/G0 → 0, [10℄. The subset SG|H0 of 1
N H ( ,f)
(see (10))a
ts on H0/G0 → 0 (cid:28)xingthesour
e points. Be
ause of ourspe
ial presentation G
we have theidenti(cid:28)
ation
SG|H0 ∼=H0×G∼=H0×Zn+1,
then N =(N )g (N )g−1,
H0/G0 ∼ H0/G0 ⊕ H0/G0
g:Z C∗ Z Z
n+1 n+1 n+1
where → isageneratorofthegroupof
hara
tersof ,and a
tsonea
hfa
tor
bymultipli
ation(Nwith th)eg
orreHsponding(N
hara
te)rg.−1 H
In general, H0/G0 → 0 and H0/G0 → 0 are not H-spa
es. However, if the
G Z
y n+1
m(uo,ngo)droHmy isZtrivial, w(ue,ϕid,eun′t)ifyGthe lo
al groups with in su
h a way that, for any
0 n+1 1
∈ × and ∈ ,
ϕ g ϕ−1 =g.
◦ ◦
s,t : H H
1 0
Now, let → be sour
e and target maps of H. The previous
onsiderations imply that
themap
Φ:s∗(N )g t∗(N )g
H0/G0 → H0/G0
(u,ϕ,u′),v Tϕ(v)
7→
` H ´Φ
1
isanis(oNmorphi)sgmofve
torbundlesover . is[N
o]mg patiblewiththemu(Nltipli
at)igo−n1ofthegro[Nup]go−id1,
hen
e H0/G0 de(cid:28)nestheorbifoldlinebundle . Inthesameway, H0/G0 de(cid:28)nes .
3.2 Chen-Ruan
ohomology ring
[Y] A
n
Wenowdes
ribetheChen-Ruan
ohomologyringofanorbifold withtransversal singularities.
n= 1
We (cid:28)rst study the
ase . In this
ase, there is only one twisted se
tor whi
h is isomorphi
to
[S]
. Then, as a ve
torspa
e, theChen-Ruan
ohomology is givenby
H∗ ([Y])=H∗(Y) H∗−2(S) e .
CR ⊕ h i
1
Theobstru
tion bundlehas rankzero (see e.g. [18℄), so its top Chern
lass is . Then
1
(δ +α e) (δ +α e)=δ δ + i (α α )+(i∗(δ ) α +α i∗(δ ))e
1 1 ∪CR 2 2 1∪ 2 2 ∗ 1∪ 2 1 ∪ 2 1∪ 2
δ +α e,δ +α e H∗(Y) H∗−2(S) e
1 1 2 2
where ∈ ⊕ h i. This
an bededu
ede.g. fromtheDe
omposition
Lemma4.1.4 in [11℄.
A n 2
n
Case with ≥ and trivial monodromy.
We will use thefollowing
onvention.
G Z
y n+1
Convention 3.9. Sin
e the monodromyis trivial, we identify the lo
al groups with . We
useboth theadditive and multipli
ative notations for the groupoperation.
[N]g
Not[aNti]og−n13.10. The orbifold
up produ
t
an be des
ribed in terms of the Chern
lasses of
and . Butforlaterusewe(cid:28)ndmore
onvenienttodes
ribeitinadi(cid:27)erentway. Considerthe
morphism
f :[S] S
→
that,naively speaking, forgets theorbifold stru
ture. It is easy to see that
([N]g)⊗n+1 =f∗M, [N]g−1 ⊗n+1=f∗L and [N]g [N]g−1 =f∗K,
∼ ∼ ⊗ ∼ (13)
“ ”
M,L K S
for someline bundles and on . The orbifold
upprodu
t will be expressed in termsof the
M,L K
Chern
lasses of and .
10
