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ADE 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 isomorphism 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 Rãl 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 giveńbythemultipli 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) . Thusweseethàt | α isdísjointunionof 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 IfthemonUõdromyisnottrivial,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 eUõf 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