Enumerative Geometry of Calabi-Yau 5-Folds

Transcription

1 Enumrativ Gomtry of Calabi-Yau 5-Folds R. Pandharipand and A. Zingr 11 Fbruary 2008 Abstract Gromov-Wittn thory is usd to dfin an numrativ gomtry of curvs in Calabi-Yau 5-folds. W find rcursions for mting numbrs of gnus 0 curvs, and w dtrmin th contributions of moving multipl covrs of gnus 0 curvs to th gnus 1 Gromov-Wittn invariants. Th rsulting invariants, conjcturd to b intgral, ar analogous to th prviously dfind BPS counts for Calabi-Yau 3 and 4-folds. W commnt on th situation in highr dimnsions whr nw issus aris. Two main xampls ar considrd: th local Calabi-Yau P 2 with normal bundl 3 i=1 O 1 and th compact Calabi-Yau hyprsurfac X 7 P 6. In th formr cas, a closd form for our intgr invariants has bn conjcturd by G. Martin. In th lattr cas, w rcovr in low dgrs th classical numration of lliptic curvs by Ellingsrud and Strömm. Contnts 0 Introduction Ovrviw Elliptic invariants Exampls BPS stats Highr dimnsions Acknowldgmnts Gnus 0 invariants Configuration spacs of gnus 0 curvs Gnus 0 counts Justification of dgr rducing rcursions Ovrviw Chrn classs Th numbrs 2A Th numbrs 2B Th numbrs Gnus 1 counts Ovrviw Prliminaris Strata with ghost principal componnt I Strata with ghost principal componnt II

2 2.5 Strata with ffctiv principal componnt Local P Gromov-Wittn invariants Martin s conjctur Introduction 0.1 Ovrviw Lt X b a nonsingular projctiv varity ovr C. Lt M g,k X, β b th moduli spac of gnus g, k pointd stabl maps to X rprsnting th class β H 2 X, Z. Lt v i : M g,k X, β X b th valuation morphism at th i th marking. concrns th invariants N g,β γ 1,..., γ k = Th Gromov-Wittn thory of primary filds [M g,k X,β] vir i=1 k v i γ i Q, 0.1 whr γ i H X, Z. Th rlationship btwn th Gromov-Wittn invariants and th actual numrativ gomtry of curvs in X is subtl. An ovrviw of th subjct in low dimnsions can b found in th introduction of [10]. For Calabi-Yau 3-folds, th Aspinwall-Morrison formula [1] is conjcturd to produc intgr invariants in gnus 0. A full intgrality conjctur for th Gromov-Wittn thory of Calabi-Yau 3-folds was formulatd by Gopakumar and Vafa in [5, 6] in trms of BPS stats with gomtric motivation partially providd by [14]. Th Aspinwall-Morrison prdiction has bn xtndd to all Calabi-Yau n-folds in [10]: th numbrs n 0,β γ 1,..., γ k dfind by β 0 N 0,β γ 1,..., γ k q β = β 0 n 0,β γ 1,..., γ k d=1 1 d 3 k qdβ 0.2 ar conjcturd to b intgrs. Lt X b a Calabi-Yau of dimnsion n 4. Sinc Gromov-Wittn invariants of gnus g 2 of X vanish for dimnsional rasons, only intgrality prdictions for gnus 1 invariants of X rmain to b considrd. Th analogu of th gnus 1 Gopakumar-Vafa intgrality prdiction for Calabi- Yau 4-folds has bn formulatd in [10]. Hr, w find complt formulas in dimnsion 5 and rintrprt th dimnsion 4 prdictions. Th gomtry bcoms significantly mor complicatd in ach dimnsion. W discuss nw aspcts of th highr dimnsional cass. Th rlationship btwn Gromov-Wittn thory and numrativ gomtry in dimnsions gratr than 3 is simplst in th Calabi-Yau cas. Th Fano cas, vn in dimnsion 4, involvs complicatd highr gnus phnomna which hav not yt bn undrstood. 2

3 0.2 Elliptic invariants If X is Calabi-Yau, th virtual moduli cycl for M 1 X, β is of dimnsion 0. associatd Gromov-Wittn invariant by N 1,β, N 1,β = 1 Q. [M 1 X,β] vir W dnot th Intgrality prdictions for Calabi-Yau n-folds ar obtaind by rlating curv counts to Gromov- Wittn invariants in an idal Calabi-Yau X. All gnus 1 curvs in X ar assumd to b nonsingular, supr-rigid 1, and disjoint from othr curvs. Each gnus 1 dgr β curv thn contributs σd/d to N 1,dβ for vry d Z + via étal covrs, whr σd = i d i. Th gnus 1 to gnus 1 multipl covr contribution is indpndnt of dimnsion. If X is an idal Calabi-Yau 3-fold, th gnus 0 curvs in X ar also nonsingular, supr-rigid, and disjoint. Th contribution of a gnus 0 dgr β curv to N 1,dβ is thn th intgral of an Eulr class of an obstruction bundl on M 1 P 1, d, Obs = 1 [M 1 P 1,d] vir 12d, calculatd in [14]. Thus, if X is an idal Calabi-Yau 3-fold, β 0 N 1,β q β = β 0 n 1,β d=1 σd d qdβ 1 12 n 0,β log1 q β, 0.3 whr th numrativ invariant n 1,β is dfind by 0.3 and th gnus 0 invariant n 0,β is dfind by th Aspinwall-Morrison formula 0.2. Th invariants n 1,β ar thn conjcturd to b intgrs for all Calabi-Yau 3-folds. If X is an idal Calabi-Yau 4-fold, mbddd gnus 0 dgr β curvs in X form a nonsingular, compact, 1-dimnsional family M β. Th moving multipl covr calculation of Sction 2 of [10] shows that M β contributs χm β /24d to N 1,dβ for vry d Z +. Th calculation is don in two stps. First, th moving multipl covr intgral is don assuming vry gnus 0 dgr β curv is nonsingular. Scond, th contribution from th nodal curvs is dtrmind for a particular, but sufficintly rprsntativ, Calabi-Yau 4-fold X by localization. For an idal Calabi-Yau 4-fold X, β 0 N 1,β q β = β 0 n 1,β d=1 σd d qdβ 1 24 β 0 χm β log1 q β. 0.4 β 0 Th topological Eulr charactristic χm β is dtrmind by χm β = n 0,β c 2 X + +β 2 =β m β1,β 2, 1 A nonsingular curv E X with normal bundl N E is supr-rigid if, for vry dominant stabl map f : C E, th vanishing H 0 C, f N E = 0 holds. 3

4 whr m β1,β 2 is th numbr of ordrd pairs C 1, C 2 of rational curvs of classs and β 2 mting at point, s Sction 1.2 of [10]. Th mting numbrs m β1,β 2 can b xprssd in trms of th invariants n 0,β γ through a rcursion on th total dgr + β 2 by computing th xcss contribution to th topological Kunnth dcomposition of m β1 β 2, s Sctions 0.3 and 1.2 of [10]. Along with ths rcursions, rlations 0.2 and 0.4 ffctivly dtrmin th numbrs n 1,β in trms of th gnus 0 and gnus 1 Gromov-Wittn invariants of X. For arbitrary Calabi-Yau 4-folds, quation 0.4 is takn to b th dfinition of th numbrs n 1,β which ar conjcturd always to b intgrs. If X is an idal Calabi-Yau 5-fold, mbddd gnus 0 dgr β curvs in X form a nonsingular, compact, 2-dimnsional family M β. Howvr, as th nodal curvs ar mor complicatd, th localization stratgy of [10] dos not appar possibl. By viwing N 1,dβ as th numbr of solutions, countd with appropriat multiplicitis, of a prturbd -quation as in [4, 11], w show in Sction 2 that M β contributs 1 2c2 M β c 2 24d 1M β M β to N 1,dβ for vry d Z +. Thus, for an idal Calabi-Yau 5-fold X, β 0 N 1,β q β = β 0 n 1,β d=1 σd d qdβ 1 24 β 0 M β 2c2 M β c 2 1M β log1 q β. 0.5 Th last trm in 0.5 may b writtn in trms of various mting numbrs of total dgr β via a Grothndick-Rimann-Roch computation applid to th dformation charactrization of th tangnt bundl T M β. W pursu a mor fficint stratgy in Sctions 1 and 2. Dgr 1 maps from gnus 0 curvs to dgr β curvs in X ar rgular. Thus, quation 2.15 in [23] xprsss thir contribution to N 1,β in trms of counts of m-tupls of 1-markd curvs with cotangnt ψ- classs mting at th markd point. Th ψ-classs can b asily liminatd using th topological rcursion rlation at th cost of introducing counts of arbitrary mting configurations of rational curvs in X. Th lattr can b rcursivly dfind as in th cas of m β1,β2 in dimnsion 4. Rlations 0.2 and 0.5 thn rduc th numbrs n 1,β to functions of gnus 0 and gnus 1 Gromov-Wittn invariants. Lt X b an arbitrary Calabi-Yau 5-fold. Equation 0.5 togthr with th ruls providd in Sctions 1 and 2 for th calculation of M β 2c2 M β c 2 1M β in trms of th Gromov-Wittn invariants of X dfin th invariants n 1,β. W viw n 1,β as virtually numrating lliptic curvs in X. Conjctur 1 For all Calabi-Yau 5-folds X and curv classs β 0, th invariants n 1,β intgrs. ar 0.3 Exampls If th Gromov-Wittn invariants of X ar known, quation 0.5 provids an ffctiv dtrmination of th lliptic invariants n 1,β. W considr two rprsntativ xampls. 4

5 d n 1,d d n 1,d d n 1,d d n 1,d d n 1,d d n 1,d Tabl 1: Invariants n 1,d for O 1 O 1 O 1 P 2 Th most basic local Calabi-Yau 5-fold is th total spac of th bundl O 1 O 1 O 1 P Th balancd proprty of th bundl is analogous to th fundamntal local Calabi-Yau 3-fold O 1 O 1 P 1. As in th 3-fold cas, w find vry simpl closd forms in Sction 3.1 for th gnus 0 and 1 Gromov- Wittn invariants of th local Calabi-Yau 5-fold 0.6. W hav computd th invariants n 1,d via quation 0.5 up to dgr 200. All ar intgrs. Evn th first 60, shown in Tabl 1, suggst intriguing pattrns. For xampl, n 1,d = 0 for all multipls of 8. G. Martin has proposd an xplicit formula for n 1,d which holds for all th numbrs w hav computd. W stat Martin s conjctur in Sction 3.2. Th Calabi-Yau sptic hyprsurfac X 7 P 6 is a much mor complicatd xampl. Using th closd formulas for th gnus 1 and 2-pointd gnus 0 Gromov-Wittn invariants providd by [23] and [22] rspctivly, w hav computd n 1,d for d 100. All ar intgrs. Th valus of n 1,d for d 10 ar shown in Tabl 2. Th invariants n 1,d for d 4 agr with known numrativ rsults for X 7. Th invariants n 1,1 and n 1,2 vanish by gomtric considrations. Sinc vry gnus 1 curv of dgr 3 in P 6 is planar, th numbr of lliptic cubics on a gnral X 7 can b computd classically via Schubrt calculus. Th classical calculation agrs with n 1,3. Using th xprssion of non-planar gnus 1 curvs of dgr 4 as complt intrsctions of quadrics, Ellingsrud and Strömm hav numratd lliptic quartics on X 7 in Thorm 1.3 of [3]. Th rsult agrs with n 1,4. To our knowldg, th numbrs n 1,d ar inaccssibl by classical tchniqus for d BPS stats Th intgr xpansion 0.5 can b altrnativly writtn as N 1,β q β = ñ 1,β log1 q β 1 2c2 M β c M β log1 q β. 0.7 β 0 β 0 M β β 0 Th intgrality condition for th invariants ñ 1,β is quivalnt to th conjcturd intgrality for n 1,β. W viw th invariants ñ 1,β as analogous to th BPS stat counts in dimnsions 3 and 4. 5

6 d n 1,d Tabl 2: Invariants n 1,d for a dgr 7 hyprsurfac in P Highr dimnsions Th family M β of mbddd gnus 0 dgr β curvs in X is nonsingular and compact for idal Calabi-Yau n-folds for n=3, 4, 5. Th moving multipl covr rsults for n= 3, 4, 5 can b summarizd by th following quation. Th contribution of M β to th gnus 1 dgr dβ Gromov-Wittn invariant is C β dβ = 1 24d M β 2cn 3 M β c 1 M β c n 4 M β. 0.8 For dimnsion 6 and highr, th family of mbddd gnus 0 dgr β curvs in X is not compact multipl covrs can occur as limits vn in idal cass. Nvrthlss, w xpct a contribution quation of th form of 0.8 to hold. Th rsult should yild intgrality prdictions in highr dimnsions. Sinc th complxity of th Gromov-Wittn approach incrass so much in vry dimnsion, an altrnat mthod for dimnsions 6 and highr is prfrabl. It is hopd a connction to nwr shaf numration and drivd catgory tchniqus will b mad [15, 16]. 0.6 Acknowldgmnts W thank J. Bryan, I. Coskun, A. Klmm, J. Starr, and R. Thomas for svral rlatd discussions. W ar gratful to G. Martin for finding th pattrn govrning th invariants n 1,d for th local Calabi-Yau 5-fold gomtry. Th rsarch was startd during a visit to th Cntr d Rchrchs Mathématiqu in Montréal in th summr of R.P. was partial supportd by DMS A.Z. was partially supportd by th Sloan foundation and DMS Gnus 0 invariants 1.1 Configuration spacs of gnus 0 curvs Lt X b a Calabi-Yau 5-fold. W spcify hr what conditions an idal X is to satisfy with rspct to gnus 0 curvs. W dnot by H + X H 2 X, Z 0 6

7 th con of ffctiv curv classs. If β, β H + X, w writ β <β if β β is an lmnt of H + X. If J is a finit st and β H + X, w dnot by M 0,J X, β th moduli spac of gnus 0, J-markd stabl maps to X rprsnting th class β. For j J, lt L j M 0,J X, β b th univrsal tangnt lin bundl at th jth markd point. Dnot by D j Γ M 0,J X, β, HomL j, v j T X th bundl sction inducd by th diffrntial of th stabl maps at th j th markd point. If Σ is a curv, a map u: Σ X is calld simpl if u is injctiv on th complmnt of finitly many points and of th componnts of Σ on which u is constant. W will call a tupl u 1,..., u m of maps u i : Σ X simpl if th map m Σ i X, z u i z if z Σ i, i=1 is simpl. If J is a finit st and β H + X, lt M 0,J X, β M 0,JX, β b th opn subspac of stabl maps [Σ, u] such that Σ is a P 1 and u is a simpl map. If J 1 and J 2 ar two finit sts and, β 2 H + X, w dnot by M 0,J 1,J 2 X, β1, β 2 { b 1, b 2 M 0,{0} J 1 X, M {0},0 J 2 X, β 2 : v 0 b 1 =v 0 b 2 } th subst of simpl pairs of maps. Similarly, if, β 2, β 3 H + X, lt M 0, X, β1, β 2, β 3 { b 1, b 2, b 3 M 0,,{1} X,, β 2 M 0,{0} X, β 3: v 1 b 2 =v 0 b 3 } b th subst of simpl tripls of maps. If X is an idal Calabi-Yau 5-fold satisfying Conditions 1 and 2 blow, thr ar no othr configurations of simpl gnus 0 curvs in X, s Figur 1. Dnot by M 0,JX, β M 0,J X, β and M 0,J 1,J 2 X, β1, β 2 M 0,{0} J1 X, M {0},0 J2 X, β 2, th closurs of M 0,J X, β and M 0,J 1,J 2 X, β1, β 2. Lt π 1, π 2 : M 0,J 1,J 2 X, β1, β 2 M 0,{0} J1 X, β1, M0,{0} J2 X, β2, 1.1 b th componnt projction maps. Condition 1 If u : P 1 X is a simpl holomorphic map, H 1 P 1, u T X = 0. By Condition 1, M 0,J X, β is a nonsingular varity of th xpctd dimnsion 2+ J. 7

8 β β 2 β 3 β 2 Figur 1: Th thr possibl configurations of rational curvs in an idal Calabi-Yau 5-fold. Th labl nxt to ach componnt indicats th dgr. Condition 2 For all,..., β k H + X, finit sts J 1,..., J k, and a partition of J 1... J k into nonmpty disjoint substs I 1,..., I m, th rstriction of th total valuation map 2 v: k M 0,J p X, β p p=1 k X Jp, v b p p [k] p,j = v jb p p [k], j J p, p=1 to th opn subspac of simpl tupls is transvrs to th diagonal { xp,j p [k],j Jp : x p,j =x p,j if p, j, p, j I q for som q }. By Condition 2, M 0,, X, β1, β 2 and M 0, X, β1, β 2, β 3 ar nonsingular of dimnsions 1 and 0, rspctivly. Furthrmor, all simpl gnus 0 maps with rducibl domains dform to curvs with nonsingular domains. Furthrmor, for all β H + X, th opn subspac of M 0,J X, β consisting of simpl maps is nonsingular. Condition 3 For all β H + X, th rstriction of th bundl sction D 1 to M 0,1 X, β is transvrs to th zro st. For all, β 2 H + X, th bundl sction π 1 D 0 + π 2 D 0 Γ Pπ 1 L 0 π 2 L 0 M 0,, X,,β 2, Homγ, v 0 T X, whr γ Pπ 1 L 0 π 2 L 0 is th tautological lin bundl, is transvrs to th zro st. By Condition 1 and th first part of Condition 3, vry simpl holomorphic map u : P 1 X is an immrsion. By Condition 2, u is injctiv. Thus, vry irrducibl gnus 0 curv C X is nonsingular. Th normal bundl to such a curv must split as N = Oa 1 Oa 2 Oa 3 Oa 4 P 1, with a i Z, i=4 a i = 2, a i 1, i=1 th last rstriction follows from Condition 1. By th first part of Condition 4 blow, a i {0, 1} for all i. Th scond part of Condition 3 implis that vry nod of a rducibl gnus 0 curv in X is simpl. 2 By convntion, [k] = {1, 2,..., k}. 8

9 Condition 4 For all β H + X, th bundl sction dv 1 Γ PT M 0,1 X, β, Homγ, v 1 T X, whr γ PT M 0,1 X, β is th tautological lin bundl, is transvrs to th zro st. For all, β 2 H + X, th bundl sction π 1 dv 0 + π 2 D 0 Γ Pπ 1 T M 0,{0} X, π 2 L 0 M 0,, X,,β 2, Homγ, v 0 T X, whr γ Pπ 1 T M 0,{0} X, π 2 L 0 is th tautological lin bundl, is transvrs to th zro st. By Condition 4, nithr of th two bundl sctions vanishs anywhr. In th cas of th first bundl sction, th dimnsion of th bas spac and th rank of th vctor bundl both qual 5. On th othr hand, th vanishing of th bundl sction hr implis th diffrntial of th valuation map v 1 : M 0,1 X, β X is not injctiv at som simpl, dgr β, 1-markd map [P 1, x 1, u]. Hnc, th normal bundl must split as N O1 O 1 O 1 O 1. Thrfor dv 1 is not injctiv at [P 1, x, u] for all x P 1. Th zro st of th first bundl sction in Condition 4 must b at last of dimnsion on. So by transvrsality, no vanishing is possibl. Th non-vanishing of th scond bundl sction is clar from transvrsality sinc th bas spac is of dimnsion 4 and bundl is of rank 5. Lmma 1.1 Lt X b an idal Calabi-Yau 5-fold. If β H + X and J is a finit st, th spac M 0,J X, β is nonsingular of dimnsion 2 + J and consists of simpl maps. Furthrmor, th valuation map v 1 : M 0,1X, β X is an immrsion. If, β 2 H + X and J 1, J 2 ar finit sts, M 0,J 1,J 2 X, β1, β 2 is smooth of dimnsion 1+ J 1 + J 2 and consists of simpl maps. Proof. By Condition 4, th rstriction of v 1 to th opn subst M 0,JX, β M 0,JX, β is an immrsion for vry β H + X. Thrfor, by th argumnt givn in Sction 2.4, if u : Σ X is not simpl, thn no dformation of u is simpl. Hnc, u cannot li in th closur of M 0,J X, β. W conclud M 0,J X, β consists of simpl maps and thrfor nonsingular of xpctd dimnsion. Th proof of th claim for M 0,J 1,J 2 X, β1, β 2 is th sam. Conditions 1-4 can b xtndd to dfin an idal Calabi-Yau n-fold for any n. Howvr, Lmma 1.1, which dpnds on th dimnsion counting argumnt in th prcding paragraph, dos 9

10 not apply in dimnsions 6 and highr. For xampl, if X 8 P 7 is th dgr 8 Calabi-Yau hyprsurfac, M 0,1X 8, 1 = M 0,1 X 8, 1 crtainly consists of simpl maps. Howvr, a computation on G2, 8 shows th valuation map v 1 is not an immrsion along fibrs of th forgtful morphism M 0,1 X 8, 1 M 0,0 X 8, 1. A sparat computation in a projctiv bundl ovr G3, 8 shows th spac of conics in X 8 contains doubl lins. In both cass th dgnrat loci corrspond to th lins in X 8 whos normal bundl splits as O1 O 3O 1, instad of th xpctd 3O 2O 1. Whil th Calabi-Yau 6-fold X 8 is not idal, low-dgr curvs in projctiv hyprsurfacs do bhav as xpctd. Th apparanc multipl covrs as limits of simpl maps is to b xpctd in dimnsions 6 and highr, making a full numrativ tratmnt mor complicatd and likly drastically so. 1.2 Gnus 0 counts W dfin hr intgr forms of th gnus 0 Gromov-Wittn invariants of Calabi-Yau 5-folds by considring all possibl distributions of constraints and ψ-classs btwn th markd points. Th 13 rlvant typs of invariants ar indicatd in Figur 2. W stat rlations motivatd by idal gomtry which rduc all 13 to gnus 0 Gromov-Wittn invariants. Ths rlations ar takn to b th dfinition of 13 invariants for arbitrary Calabi-Yau 5-folds. If J is a finit st, J J, and β H + X, lt f J,J : M 0,J X, β M 0,J J X, β b th forgtful map dropping th markd points indxd by th st J. If j J, lt ψ j = f J,J jψ j H 2 M 0,J X, β, whr ψ j is th first chrn class of th univrsal cotangnt lin bundl for th markd point on M 0,{j} X, β. If X is an idal Calabi-Yau 5-fold and β H + X, th dimnsion of M 0,0X, β is 2. Thr ar 7 invariants of th form which w rquir: n β ψ a µ 1, µ 2,..., µ k = ψa 1 M 0,k X,β k v jµ j, j=1 1A n β µ whr µ H 6 X counting curvs through µ, 1B n β µ 1, µ 2 whr µ 1, µ 2 H 4 X counting curvs through µ 1 and µ 2, 1C n β ψµ whr µ H 4 X, 1D n β ψµ 1, µ 2 whr µ 1 H 2 X and µ 2 H 4 X, 1E n β ψ 2 µ whr µ H 2 X, a 0, µ j H 2 X, 10

11 β β β β µ µ 2 µ 1 ψµ µ 2 ψµ 1 ψ 2 µ 1A: n β µ 1B: n β µ 1, µ 2 1C: n β ψµ 1D: n β ψµ 1, µ 2 1E: n β ψ 2 µ β µ β µ β 2 µ β 2 ψ 2 1F: n β ψ 2, µ ψ 3 1G: n β ψ 3 2A: n β1 β 2 ; µ 2B: n β1 β 2 µ ; β 2 β 2 β 2 β 2 β 3 ψ 2 ψµ ψ 2 2C: n β1 β 2 ψ 2 ; 2D: n β1 β 2 ; ψµ 2E: n β1 β 2 ; ψ 2 3: m β1 β 2 β 3 Figur 2: Counts for Calabi-Yau 5-folds 1F n β ψ 2, µ whr µ H 4 X, 1G n β ψ 3. Lt M β dnot th unpointd spac M 0,0X, β. W will nd th Chrn numbr 1H γ 1 β = M β c 2 1 M β c 2 M β. Thr ar 5 typs of rlvant counts of connctd 2-componnt curvs which w rquir, n β1 β 2 ψa 1 ψ a µ 0 ψ b 1 µ 1,1, µ 1,2,..., µ 1,k1 ; ψ b 2 µ 2,1, µ 2,2,..., µ 2,k2 = π1 ψ a 1 ψ k 1 b v 0µ 0 v jµ 1,j π2 ψ a 2 ψ k 2 b v jµ 2,j, M 0,[k 1 ],[k 2 ] X,,β 2 whr π 1, π 2 ar th componnt projction maps as in 1.1, a i, b i 0, and µ 0, µ i,j H 2 X. Th 5 typs ar rprsntd by th following counts of, β 2 -curvs: 2A n β1 β 2 ; µ whr µ H 4 X, 2B n β1 β 2 µ ; whr µ H 2 X, 2C n β1 β 2 ψ 2 ;, 2D n β1 β 2 ; ψµ whr µ H 2 X, 2E n β1 β 2 ; ψ 2. j=1 j=1 11

12 ψ 1 = β β 2 =β,β 2 H + X β ψ1 = ψ β β β Figur 3: Rlations for ψ 1 and ψ 1 on M 0,3 X, β. Each curv rprsnts th divisor in M 0,3 X, β whos gnral lmnt has th domain and th dgr distribution spcifid by th curv. Finally, w dnot th cardinality of th compact 0-dimnsional spac M 0,, β 2, β 3 for tripls, β 2, β 3 H + X by m β1 β 2 β 3 : 3 m β1 β 2 β 3 is th numbr of connctd 3-componnt curvs of tridgr, β 2, β 3. Th numbrs 1A and 1B ar dtrmind from 1- and 2-pointd Gromov-Wittn invariants via 0.2. Th tautological rcursion rlation for ψ 1 can b usd to xprss ψ 1 in trms of boundary divisors on M 0,3 X, β, s Figur 3. Th divisor rlation thn givs ris to th rlations btwn th invariants 1C-1G indicatd in Figur 4, s also Sction 3 in [13]. W now dscrib ths rlations formally. If H is a divisor on X and H β =H, β, thn H 2 β n β ψµ = n β µ, H 2 2H β n β Hµ + Hβ 2 1 n β1 β 2 ; µ, +β 2 =β Hβ 2 n β ψµ 1, µ 2 = µ 1, β n β µ 2, H 2 2H β n β Hµ 1, µ 2 + µ1, Hβ 2 2 +µ 1, β 2 Hβ 2 nβ1 1 β 2 ; µ 2, +β 2 =β H 2 β n β ψ 2 µ = n β ψµ, H 2 2H β n β ψhµ + n β ψ 2, µ = n β1 β 2 ; µ, +β 2 =β H 2 β n β ψ 3 = n β ψ 2, H 2 2H β n β ψ 2 H + Hβ 2 nβ1 1 β 2 ; ψµ+n β1 β 2 µ ;, +β 2 =β Hβ 2 nβ1 1 β 2 ; ψ 2 + n β1 β 2 ψ 2 ;, +β 2 =β th fourth idntity abov is obtaind by applying th rlation of Figur 4 twic. W can similarly rmov ψ-classs from 2-componnt curvs: Hβ 2 2 n β1 β 2 ψ 2 ; = n β1 β 2 ; H 2 2H β2 n β1 β 2 H ; + Hβ 2 m β β, β+β =β 2 Hβ 2 2 n β1 β 2 ; ψµ = µ, H n β1 β 2 ; H 2 2H β2 n β1 β 2 ; Hµ + µ, βh 2 β +µ, β Hβ 2 mβ1 β β, n β1 β 2 ; ψ 2 = β+β =β 2 β+β =β 2 m β1 β β,

13 β Hβ 2 ψ c µ = 0 β β H 2 2H β ψ c 1 µ ψ c 1 Hµ + Hβ 2 1 +β 2 =β,β 2 H + X { β2 + ψ c 1 µ β 2 ψ 2 c 2 µ } H X divisor, H β = H β, H β1 = H Figur 4: Rducing th powr of ψ at markd point in th absnc of ψ-classs at othr markd points. th last idntity abov is obtaind by applying th rlation of Figur 4 twic. On th othr hand, by 1.15 and som manipulation, γ 1 β = 1 2 n β c3 X + n β ψc2 X + n β ψ3 + n β c2 X, c 2 X + 4 n β ψ2, c 2 X +β 2 =β 2 n β1 β 2 ; ψ n β 2 ψ 2 ;. 1.4 Th mting numbrs 2A, 2B, and 3 ar computd via dgr rducing rcursions analogous to Ruls i-iv of Sction 0.3 of [10] for th 4-dimnsional cas. Lt b dual bass normalizd so that PD X 2 X {ω 1,..., ω N }, {ω # 1..., ω# N } H4 X H 6 X N ω l ω # l H 2k X H 25 k X H odd X H odd X, l=1 k=0,1,4,5 whr X X 2 is th diagonal. Thn, N n β1,β 2 ; µ + n β2, ; µ, if β 2 >, n β1 β 2 ; µ = n β1 ω l n β2 ω # l, µ + n β1 β 2,β 2 ; µ, if β 2 <, l=1 n β1 c 2 X, µ + 2n β1 ψ 2, µ, if β 2 =. 1.5 In light of th fourth idntity in 1.2, th rlation diffrs from th 4-dimnsional cas only by th xpctd adjustmnt for th constraint µ. Th corrsponding rcursions for th numbrs 2B and 3 ar mor complicatd. For classs, β 2 H + X, lt γ 2, β 2 = n β1 β 2 ; c2 X + 2 n β1 β 2 ; ψ 2 + n β1 β 2 ψ 2 ; + n β2 ψ 2 ;

14 For µ H 2 X, w dfin n β2, ; ψµ + n β2, µ ; + µ, γ 2 β 2, + 1 m ββ1 β 2, if β 2 > ; β+β =β 2 C β2 β C β1 β 2 µ = 1 µ, if β 2 < ; 1.7 n β1 c 2 Xµ + n β1 ψ 2 µ + n β1 c 2 X, ψµ + µ, γ 1 2n ββ ; ψµ n ββ µ ; if =β 2., β+β =β 2 For, β 2, β 3 H + X, lt m β3,,β 2, if β 3 > ; C 1 β 2 β 3 = m β1 β 3,β 3,β 2, if β 3 < ; γ 2 β 2,, if β 3 = ; m β1,β 2,β 3 β 2, if β 3 >β 2 ; C 2 β 2 β 3 = m β1,β 3,β 2 β 3 +m β1,β 2 β 3,β 3, if β 3 <β 2 ; n β1 β 2 ; c2 X + 2 n β1 β 2 ; ψ 2, if β 3 =β 2 ; m β3 β 2,,β 2, if β 3 > +β 2 ; C 12 m β1 +β β 2 β 3 = 2 β 3,β 3 β 2,β 2, if β 2 <β 3 < +β 2 ; γ 2 β 2,, if β 3 = +β 2 ; 0, othrwis. Thn, n β1 β 2 µ ; = m β1 β 2 β 3 = N n β1 ω l µ n β2 ω # l l=1 N l=1 1.8 β<,β 2 µ, β m β1 β,β,β 2 β C β1 β 2 µ, 1.9 n β1 β 2 ; ω l n β3 ω # l C 1 β 2 β 3 C 2 β 2 β 3 C 12 β 2 β A fw low dgr 2-componnt mting numbrs for a dgr 7 hyprsurfac in P 6 ar givn in Tabl 3. Th numbr n 1,1 H ; can b confirmd via a Schubrt computation similar to Sction 3 in [9]. Configurations of rational curvs in a Calabi-Yau n-fold for can b studid for any n. If n 6, such configurations includ curvs with non-simpl nods svral componnts sharing a nod. Whil dscribing such curvs is just notationally involvd, spcifying dgr rducing rcursions for thm following th approach of Sction 1.3 blow prsnts nw difficultis. In particular, curvs with unbalancd splittings of th normal bundl will ffct xcss contributions via th loci of non-simpl tupls of maps in th closurs of simpl tupls of maps, s th nd of Sction 1.1. Thus, sparat counts must b st up for such curvs, and thir multipl-covr contributions to th appropriat topological intrsction numbrs rprsntd by th first trms on th right-hand sid of 1.5, 1.9, and 1.10 must b dtrmind. 14

15 n d1 d 2 H ; d 2 = 1 d 2 = 2 d 1 = Tabl 3: Mting invariants n d1 d 2 H ; for a dgr 7 hyprsurfac in P 6 counting th virtual numbr of d 1, d 2 -curvs with nod on a fixd hyprplan. 1.3 Justification of dgr rducing rcursions Ovrviw Each curv C of typ 2A, 2B, and 3 dtrmins a pair C, C of curvs, whr C is th last componnt of C and C consists of th rmaining componnts of C. Th curv C has 1 componnt in th first two cass and 2 componnts in th last cas. Th curvs C and C carry marking x C and y C satisfying x =y. W dnot by M and M th corrsponding compactifid spacs of curvs/maps: M Cas 2A: M 0,{} X, β { 1 φ M 0,{} X, β 2: Im φ µ }, { Cas 2B: φ M 0,{} X, : v φ µ } M 0,{} X, β 2, Cas 3: M 0,,{}X,, β 2 M 0,{}X, β 3, whr µ abov dnots a gnric rprsntativ for th Poincar dual of µ H X. Th valuation map v, : M M X X, C, x, C, y x, y, is thn a cycl of complx dimnsion 5. Th rlvant mting numbr is th cardinality of th subst of Z = v 1, X = { C, x, C, y M M : x =y } consisting of simpl pairs of maps. Th homological intrsction numbr of th cycl v, with th class of th diagonal X X 2 in X 2 is givn by th diagonal-splitting trm on th right-hand sid of 1.5, 1.9, and Th homological intrsction is th numbr of points, countd with sign, in th primag of X undr a small dformation of th map v,. All such points must li nar Z. Th points of Z at which v, is transvrs to X contribut 1 ach to th homology intrsction. Ths points includ all tupls as abov such that th curvs C and C do not hav any componnts in common. Thus, th rlvant mting numbr is th diagonal-splitting trm in 1.5, 1.9, and 1.10 minus th contribution to th homology intrsction numbr of v, with X from th subst Z of Z consisting of tupls as abov such that C and C hav at last on componnt in common. In th rst of this subsction, w dtrmin ths tupls and thir xcss contributions. 3 If X is an idal Calabi-Yau 5-fold and β H + X, th spac M β,1 = M 0,1 X, β 3 As in th 4-dimnsional cas considrd in [10], all contributions in cas 2A ar dgnrat contributions arising from loci of dimnsions 1 and 2. Howvr, in cass 2B and 3, Z includs rgular points with rspct to th valuation condition which ar isolatd and nondgnrat. M 15

16 of simpl maps to X of dgr β with 1 marking is nonsingular of dimnsion 3, and th valuation map v: M β,1 X is an immrsion, s Lmma 1.1. W dnot by T β th tangnt bundl of M β,1 and by N β th normal bundl to th immrsion v. Lt N b th normal bundl to th diagonal in X 2. If C X is a curv, lt C dnot th numbr of irrducibl componnts of C Chrn classs Lt X b an idal Calabi-Yau 5-fold, and lt β H + X. W rlat hr th Chrn classs of th normal bundl N β to th immrsion to mting numbrs. Dnot by v 1 : M β,1 X f : M β,1 M.11 th forgtful map to th nonsingular 2-dimnsional moduli spac M β =M 0,0 X, β. Using th bundl homomorphism df : T M β,1 f T M β ovr M β,1, w obtain c 1 T β = ψ + f c 1 Mβ, c 2 T β = ψ f c 1 Mβ + f c 2 Mβ, 1.12 whr ψ is th first chrn class of th cotangnt lin bundl on M β,1 viwd as a 1-pointd moduli spac and M β,1 is th locus of singular points of f points at which df is not surjctiv. On th othr hand, sinc c 1 X=0, Combining 1.12 and 1.13, w find c 1 N β = ψ f c 1 Mβ, c 1 Nβ = c1 Tβ, c 2 Nβ = v c 2 X + c Tβ c2 Tβ. c 2 N β = v c 2 X + ψ 2 ψ f c 1 Mβ + f c 2 1 M β c 2 M β If +β 2 = β and β 2, lt D β1,β 2 M β b th closur of th locus consisting of β-curvs split into a -curv and a β 2 -curv. If 2 = β, lt D β1 M β b twic th closur of th locus of consisting of β-curvs split into two -curvs. In particular, f = 1 D β1,β β 2 =β,β 2 H + X Dnot by ψ 1 +ψ 2 D β1,β 2 H 4 M β th class obtaind by capping with th first chrn class of th cotangnt lin bundl at th chosn nod for ach of th two curvs. From a Grothndick- Rimann-Roch computation applid to th dformation charactrization of T M β, w find c 1 M β = f v c 2 X + D β1,β 2, +β 2 =β,β 2 H + X 2c 2 M β c 2 1 M β = f v c 3 X + ψ v 2 c 2X + ψ ψ 1 +ψ 2 D β1,β 2. +β 2 =β,β 2 H + X 1.15

17 Th 4-dimnsional cas of th first quation abov appars in Sction of [10] and is also an immdiat consqunc of th n=4 analogu of 2.5 blow. Th scond idntity in 1.15 is 2.5 itslf Th numbrs 2A Suppos C, x, C, x is an lmnt of Z. Sinc th curv C C passs through µ, C C has at most two componnts. W hav thr possibilitis for Z. Cas 0 C =C : Hr =β 2 and Z = { C, x, C, x : C, x M }. Th normal bundl of Z in M M is isomorphic to T β1 M and th diffrntial dv, = dv : N v,n is injctiv ovr M. Thus, th contribution of Z to th homology intrsction numbr is givn by N /N, Z = c 2 N β1, M. Using th scond quation in 1.14, th first quation in 1.15, and th fourth quation in 1.2, w obtain th =β 2 cas of 1.5. Cas 1A C C : Hr <β 2 and Z = { C, x, C C, x : C, x M, C C, x Z }, whr Z M is th locus consisting of 2-componnt curvs with th markd point on th first componnt. Thus, Z is th union of th first componnts of th finitly many, β 2 - curvs passing through th constraint µ. Th normal bundl N of Z in M M contains th subbundl π 1 T and N /π 1 T is isomorphic to th normal bundl N Z of Z in M. Sinc th diffrntial dv, : N v,n is injctiv ovr Z, th contribution of Z to th homology intrsction numbr is givn by N /N, Z = c 1 N β1 c 1 N Z, Z. Sinc th dgrs of th rstrictions of N β1 and N Z to ach curv C ar 2 and 1, rspctivly, w obtain th <β 2 cas of 1.5. Cas 1B C C : Hr >β 2 and Z = { C C, x, C, x : C C, x M, C, x Z }, whr Z M is th locus of curvs mting a β 2 -curv. Thus, Z is th union of th scond componnts of th finitly many β 2, β 2 -curvs whos scond componnt passs through th constraint µ. Th normal bundl N of Z in M M contains th subbundl π 1 T and N /π 1 T is isomorphic to th normal bundl N Z of Z in M. Th lattr is trivial. Sinc th diffrntial dv, : N v, N 17

18 is injctiv ovr Z, th contribution of Z to th homology intrsction numbr is givn by N /N, Z = c 1 N β1 c 1 N Z, Z. Th >β 2 cas of 1.5 now follows from th first quation in Th numbrs 2B Suppos C, x, C, x is an lmnt of Z. Th curv C C thn has on, two, or thr componnts and carris a markd point lying on th divisor µ. Th 6 possibilitis for th connctd componnts of Z ar indicatd in Figur 5. Cas 0 C =C : Hr =β 2 and C, x, C, x is an lmnt of S = { C, x, C, x : C, x M } Z. Th normal bundl of S in M M is isomorphic to T β2 M, and th contribution of S to th homology intrsction numbr is givn by N /N, S = c 2 N β2, M. Using th scond quation in 1.14 and th first quation in 1.15, w obtain th =β 2 cas of th last trm in 1.9. Cas 1A C =2, C C : Hr <β 2 and C, x, C, x is an lmnt of S = { C, x, C C, x : C, x Z, C C, x M } Z, whr Z M is th locus consisting of curvs mting a β 2 -curv. Th normal bundl N of S in M M contains th subbundl π 2 T β 2 and N /π 2 T β 2 is isomorphic to th normal bundl N Z of Z in M. Sinc th diffrntial dv, : N v,n is injctiv ovr S, th contribution of S to th homology intrsction numbr is givn by N /N, S = c 1 N β2 c 1 N β2 + ψ 1, Z, whr ψ 1 is th untwistd ψ-class at th nod of th β 2 -componnt of a curv in Z. Using th first quations in 1.14 and in 1.15 and th fourth quation in 1.2, w obtain th <β 2 cas of th last trm in 1.9 minus th last trm in 1.7. Th lattr ariss from Cas 2A blow. Cas 1B C =2, C C : Hr >β 2 and C, x, C, x is an lmnt of S = { C C, x, C, x : C C, x Z, C, x M } Z, whr Z M is th locus of β 2, β 2 -curvs with th markd point lying on th first componnt. Th normal bundl N of S in M M contains th subbundl π 2 T β 2, N /π 2 T β 2 is isomorphic to th normal bundl N Z of Z in M, and th contribution of Z to th homology intrsction numbr is givn by N /N, S = c 1 N β2 + ψ 1 +ψ 2, Z, 18

19 β 2 β 2 β 2 = : Contr. 0 β β 2 > : Contr. 1A β 2 β β 2 β 2 < : Contr. 1B β β β β β 2 β β+β =β 2 β 2 > : Contr. 2A β <, β 2 Contr. 2B β+β = β 2 β 2 < : Contr. 2C Figur 5: Excss contributions for th mting numbr n β1 β 2 µ ;. Th labls rfr to th cass dscribd in Sction Th markd point corrsponds to th formr nod and lis on th divisor µ. Th thickr lins indicat th multipl componnt. Th spac of curvs in th first diagram in th top row is 2-dimnsional. Th othr two spacs in th first row ar 1-dimnsional. All spacs in th bottom row ar 0-dimnsional. whr ψ 1 and ψ 2 ar th ψ-classs of th first and scond componnts at th nod of a curv in Z. W obtain th >β 2 analogu of th Cas 1A contribution in 1.9. Cas 2A C = 3, C C : Hr < β 2. If C = 2, C, x, C, x is an lmnt of th spac S in Cas 1A abov. This is also th cas if C = 1 and th curv C C is connctd. In th rmaining cas, C is th middl componnt of th 3-componnt curv C and carris th markd point, which lis on th divisor µ. Each such pair C, x, C, x is a rgular lmnt of Z and thrfor contributs 1 to th homology intrsction. Th contribution of such pairs is accountd for by th last trm in 1.7. Cas 2B C = C = 2, C C : Hr th curv C C consists of thr componnts, with th middl componnt mting th hyprplan µ at th markd point. Such pairs C, x, C, x ar rgular lmnts of Z, and thir contribution is accountd for by th middl trm on th right sid of 1.9. Cas 2C C =3, C C : Th analysis is th sam as Cas 2A with and β 2 intrchangd Th numbrs 3 If C, x, C, x is an lmnt of Z, C consists of two sts of componnts, C1 and C 2, with th scond componnt carrying th markd point. Eithr C 1 or C 2 may consist of two componnts, whil th othr curv must consist of on componnt. Th total numbr of componnts in C C is ithr two or thr. Th 12 possibilitis for th connctd componnts of Z ar indicatd in 19

20 Figur 6. Cas 0 C C =2, C2 C : If C = C 2, thn β 2 =β 3 and C, x, C, x is an lmnt of S = { C 1 C, x, C, x : C 1 C, x M } Z. Similarly to Cas 0 in Sctions and 1.3.4, th normal bundl of S in M M is isomorphic to T β2 M, and th contribution of S to th homological intrsction numbr is givn by N /N, S = c 2 N β2, M. Using th scond quation in 1.14, th first quation in 1.15, and th fourth quation in 1.2, w obtain th β 3 =β 2 cas of th trm C 2 β 2 β 3 in If C = C 1 C 2, thn + β 2 =β 3 and C, x, C, x is an lmnt of S = { C, x, C, x : C, x M } Z. Th normal bundl of S in M M is isomorphic to T β1 +β 2 M, and th contribution of S to th homological intrsction numbr is givn by N /N, S = c 2 N β1 +β 2, M. Using th scond quation in 1.14, th first quation in 1.15, and th fourth quation in 1.2, w obtain th β 3 = + β 2 cas of th trm C 12 β 2 β 3 in Cas 0 C C =2, C2 C : Hr =β 3 and C, x, C, x is an lmnt of S = { C C 2, x, C, x : C C 2, x Z } Z, whr Z M consists of th pairs of 1-markd curvs with th markd point at th nod of th two curvs. Th normal bundl N of S in M M contains T β1 as a subbundl, and N /T β1 is isomorphic to th normal bundl of Z in M. Th lattr is th univrsal tangnt lin bundl at th markd point. Sinc th homomorphism dv, : N v,n is injctiv ovr Z, th contribution of Z to th homological intrsction numbr is givn by N /N, S = c 1 N β2 + ψ 2, Z. Using th first quations in 1.14 and 1.15 and th fourth quation in 1.2, w obtain th β 3 = cas of th trm C 1 β 2 β 3 in Cas 1A C C = 3, C C, C2 C : If C1 C, thn β 2 < β 3 and C, x, C, x is an lmnt of S = { C 1 C 2, x, C 2 C, x : C 1 C 2, x Z } Z, whr Z M consists of th pairs of, β 2 -curvs such that th scond componnt mts a β 3 β 2 -curv. W s S is th union of th middl componnts of, β 2, β 3 β 2 -curvs in X, 20

21 β 3 β 2 β 2 +β 2 β 3 β 2 β 2 β 3 β 3 β 3 =β 2 : Contr. 0 β 3 >β 2 : Contr. 1A β 3 > : Contr. 1A β 3 β 2 β 2 β 3 <β 3 < +β 2 : Contr. 1B β 2 β 2 β 2 β 3 β 3 β 2 +β 2 β 3 β 3 = +β 2 : Contr. 0 β 3 > +β 2 : Contr. 1A β 3 <β 2 : Contr. 1B β 2 <β 3 < +β 2 : Contr. 1B β 2 β 3 β 2 β 2 β 3 β 2 β 2 β 3 β 3 β 3 β 3 = : Contr. 0 β 3 > +β 2 : Contr. 1A β 3 <β 2 : Contr. 1B β 3 < : Contr. 1B Figur 6: Excss contributions for th mting numbr m β1 β 2 β 3. Th labls rfr to th cass dscribd in Sction Th markd point corrsponds to th formr nod joining th β 2 and β 3 curvs. For th curvs of typs 1A and 1B, indicats th nw nod on th lftovr, β 2 - curv. Th thickr lins rprsnt th multipl componnts. Th xcss loci corrsponding to Contr. 0 ar 2-dimnsional. Th loci corrsponding to Contr. 1A and 1B ar 0-dimnsional. Th rmaining loci ar 1-dimnsional. 21

22 with ach curv contributing 1 to th homological intrsction numbr. Th contribution accounts for th β 3 >β 2 cas of th trm C 2 β 2 β 3 in If C 1 C, thn +β 2 <β 3 and C, x, C, x is an lmnt of S = { C 1 C 2, x, C 1 C 2 C, x : C 1 C 2, x Z } Z, whr Z M consists of th pairs, β 2 -curvs mting a β 3 β 2 -curv with th β 2 - componnt carrying th markd point. Hr, S is th union of th last componnts of th β 3 β 2,, β 2 -curvs and th middl componnts of, β 2, β 3 β 2 -curvs. By rasoning analogous to Cas 1A in Sction 1.3.4, ach of th formr contributs 1 to th homological intrsction numbr, whil ach of th lattr contributs 0. W obtain th β 3 > +β 2 cas of th trm C 12 β 2 β 3 in Cas 1A C C = 3, C C, C2 C : Hr β 3 > and C C, x is a β 3,, β 2 - curv with th markd point lying on th nod joining th last two componnts. Each such pair C, x, C, x is a rgular lmnt of Z, contributing 1 to th homology intrsction numbr. W obtain th β 3 > cas of th trm C 1 β 2 β 3 in Cas 1B C C =3, C C, C C 1 : If C = C 2 or C = C, C, x, C, x is an lmnt of on of th spacs S dfind in Cas 0 abov. Hnc, w can assum that C C 2, C. If C C 2, thn β 2 >β 3 and C, x, C, x is an lmnt of S = { C 1 C C, x, C, x : C 1 C C, x Z } Z, whr Z M is th locus of th pairs, β 2 -curvs with th scond componnt brokn into two. As in Cas 1B of Sction 1.3.4, S is th union of th middl componnts of, β 3, β 2 β 3 -curvs and th last componnts of, β 2 β 3, β 3, with ach curv contributing 1 to th homological intrsction numbr. Th contribution accounts for th β 3 <β 2 cas of th trm C 2 β 2 β 3 in If C C 2, thn +β 2 >β 3 and C, x, C, x is an lmnt of S = { C 1 C 2 C, x, C 1 C 2, x : C 1 C 2 C, x Z } Z, whr Z M is th locus of th pairs, β 2 -curvs with on of th componnts brokn into two. Hr, S is th union of th middl componnts of, β 3, +β 2 β 3 -curvs, if β 3 >, and th last componnts of +β 2 β 3, β 3 β 2, β 2 -curvs, if β 3 >β 2. Each of th lattr curvs contributs 1 to th homological intrsction numbr, whil ach of th formr contributs 0. W obtain th β 3 <β 2 cas of th trm C 12 β 2 β 3 in Cas 1B C C = 3, C C 1 : If C = C 1, thn C, x, C, x is an lmnt of th spac S dfind in Cas 0 abov. Hnc, w can assum that C C 1. Thn, >β 3 and C, x, C, x is an lmnt of S = { C C C 2, x, C, x : C C C 2, x Z } Z, whr Z M is th locus of th pairs of 1-markd, β 2 -curvs rprsntd by a β 3, β 3, β 2 - curv in X with th markd point on th nod of th last two componnts. Each such pair C, x, C, x is a rgular lmnt of Z, contributing 1 to th homological intrsction numbr and accounting for th β 3 < cas of th trm C 1 β 2 β 3 in

23 2 Gnus 1 counts 2.1 Ovrviw For ach β H + X, N 1,β is th numbr of automorphism-wightd stabl C -maps u: Σ X from prstabl curv of gnus 1 to X of dgr β solving a prturbd Cauchy-Rimann quation, u + νu = 0, 2.1 for a small gnric multi-valud prturbation ν, s Sction 1.3 of [21] for mor dtails. If X is an idal Calabi-Yau n-fold, M 1 X, β dcomposs into strata Z T which ach hav wll-dfind contribution to N 1,β in following sns: For vry stratum Z T, thr xist C T β Q, ɛ ν R +, and a compact subst K ν of Z T with th following proprty. For vry compact subst K of Z T and an opn nighborhood U of K in th spac of stabl C -maps, thr xist an opn nighborhood U ν K of K and ɛ ν U 0, ɛ ν, rspctivly, 4 such that ± { +tν} 1 0 U = CT β if t 0, ɛ ν U, K ν K U U ν K. Whil thr ar many diffrnt strata, it turns out that C T β 0 only for strata of th thr simplst typs. If X is an idal Calabi-Yau n-fold, thr ar finitly many gnus 1 curvs in ach homology class of X. Furthrmor, vry gnus 1 curv C in X is mbddd and supr-rigid: if N is th normal bundl of C and u: Σ C is an unramifid covr, thn H 0 Σ, u N =0. Hnc, H 1 Σ, u N =0 and for vry d Z + Z 1,β/d = [C]=β/d M 1 C, d is a finit st of isolatd rgular points of M 1 X, β. Each such point u contributs 1/Autu to N 1,β. If n 1,β is th numbr of gnus 1 curvs in th homology class β, thn C 1,β/d β = σd d n 1,β/d, 2.2 whr σd is th numbr of dgr d unbranchd covrs of a gnus 1 curv by connctd gnus 1 curvs. Th intgral numbr n 1,β/d is zro unlss d β, or quivalntly, β/d is an intgral homology class. Th rmaining lmnts u : Σ X of M 1 X, β ar maps to gnus 0 curvs in X. Thy split into strata Z T indxd by combinatorial data dscribd in Sction 2.2. W will call a stratum Z T basic if ithr of th following conditions holds: B1 th domain Σ of vry lmnt [Σ, u] of Z T is a nonsingular gnus 1 curv, or 4 U νk dpnds on K, whil ɛ νu dpnds on U. 23

24 B2 th domain Σ of vry lmnt [Σ, u] of Z T is a union of a nonsingular gnus 1 curv Σ P and a P 1 and u is constant on Σ P. In both cass, th rstriction of u to th non-contractd componnt must b a d : 1 covr of a curv in th homology class β/d, for som d Z +. W will writ T ff β/d, d and T gh β/d, d for th corrsponding typs of strata B1 and B2, with ff and gh standing for ffctiv and ghost principal componnt. Thorm 2.1 Suppos X is an idal Calabi-Yau 5-fold. i If Z T is a stratum of M 1 X, β consisting of maps to rational curvs in X and is not basic, C T β=0. ii For β H + X and d Z +, In Sction 2.3, w will prov C Tff β,ddβ = d 1 d 2 C Tgh β,1β, C Tgh β,ddβ = 1 d 2 C T gh β,1β. 2.3 C Tgh β,1β = 1 24 M β 2c2 M β c 2 1 M β. 2.4 On th othr hand, th spac M β = M 0X, β consists of rgular maps to X. Thus, th contribution to N 1,β is givn by th right sid of quation 2.15 in [23]: C Tgh β,1β = 1 cx n β ψ 2 Comparing th abov idntity with 2.4, w find that M β 2c2 M β c 2 1 M β = n β +β 2 =β,β 2 H + X cx ψ 2 n β1 β 2 ψ1 +ψ 2 ;. +β 2 =β,β 2 H + X n β1 β 2 ψ 1 +ψ 2 ;. 2.5 W calculat th lft sid in trms of th Gromov-Wittn invariants of X by xpanding th right sid via th quations of Sction 1. Our proof of Thorm 2.1 applis also in dimnsions 3 and 4. In particular, th rsult provids a dirct xplanation of th 1/d-scaling in th lattr cas discovrd by othr mans in Sction 2 of [10]. Many aspcts of th proof ar applicabl in dimnsions 6 and highr as wll. 2.2 Prliminaris Lt X b an idal Calabi-Yau 5-fold. Th strata of M 1 X, β consisting of maps to rational curvs can b dscribd by dcoratd graphs whr T = Vr, Edg, d, β, κ, i, 24

25 D1 Γ = Vr, Edg is a connctd graph containing ithr xactly on loop or a distinguishd vrtx, but not both, D2 β =β i i [m] is an m-tupl of lmnts of H + X, with m {1, 2, 3}, D3 d: Vr Z 0 is a map, κ: d 1 Z + [m] is a surjctiv map, D4 i { } [m]. Th irrducibl componnts and th nods of th domain Σ of vry lmnt [Σ, u] of Z T corrspond to th sts Vr and Edg rspctivly. If v Vr is not th distinguishd vrtx of Γ, th corrsponding componnt Σ v of Σ is a P 1. Othrwis, Σ v is nonsingular of gnus 1. If v Vr, th rstriction of u to Σ v is constant if dv = 0. If dv 0, u Σv is a dv : 1 covr of th componnt C κv of C. If d dos not vanish idntically of th loop in th graph Vr, Edg or on th distinguishd vrtx, i is st to. If d vanishs idntically on th loop or on th distinguishd vrtx, th corrsponding componnts of Σ ar mappd by u to a point on th i -componnt of C. Sinc u is continuous, Z T = unlss κ satisfis crtain combinatorial conditions. 5 Givn a gnric dformation of ν of th -oprator as in 2.1 and sufficintly small t R +, w will dtrmin th numbr of solutions [Σ, u] of u + tνu = 0, 2.6 with u clos to th stratum Z T. Th assumption that ν is gnric implis that all solutions of 2.6 ar maps from nonsingular gnus 1 curvs. Th argumnts follow [19, 20]. In particular, th gluing construction for Z T will b prformd on a family of rprsntativs Σ, u for th lmnts [Σ, u] in Z T, s Sction 2.2 of [20]. Our tratmnt hr is lss xplicit in ordr to stramlin th discussion. For th rst of Sction 2, w fix a dcoratd graph T as abov. W dfin β = m β i H + X. i=1 With notation as in Sction 1.1, lt M β = M 0, X, β and M β = M 0, X, β. W dnot by M β,1 and M β,1 th spacs of pairs C, x such that C M β and x C is a nonsingular point of C in th first cas and C M β and x C is any point of C in th scond cas. Lt S M β b a family of dformations in X of curvs in M β. In othr words, th fibr S C of S ovr C M β contains C and dim S C = dim M β,1 dim M β = m. Thr is a fibration π C : S C C m Th strata Z T as dfind abov intrsct if m 2 and d vanishs on th loop or th distinguishd vrtx of Vr, Edg. Th issu can b asily addrssd by allowing i to tak valus in { } [m] {1, 2, 2, 3}. Howvr, quation 2.6 will b shown to hav no solutions nar Z T for a good choic of ν if m 2, so furthr discussion is not ndd. 25

26 giving th univrsal family of dformations of C. If m = 1, thn S = M β,1. If m = 3, S is a small nighborhood of M β,1 in M β,1. If v: M β,1 X is th valuation map at th markd point, th bundl xtnds naturally ovr M β,1 so that thr is an xact squnc Q = v T X / T S M β, f T M β Q N β 0, 2.9 whr f : M β,1 M β is th forgtful map and N β is th normal bundl to th family of simpl curvs of class β. Similarly to Sction 3.3 in [12], w choos a family of xponntial maps xp C : T X X such that xp C xv S C if x C, v T x S C, v < δc, 2.10 for som δ C M Γ; R +. Blow w will plac additional assumptions on xp C as ndd. For an idal Calabi-Yau n-fold with n 6, th abov stratification would nd to b rfind furthr basd on th dviation of th normal bundls of curvs in M β,1 from balancd splitting. Th argumnts in Sctions blow apply to th strata with balancd splitting with minor changs. Th main chang hr is that th map v is no longr an immrsion, and on would nd to pass to a blowup of M β,1 to obtain analogus of th vctor bundl Q and th short xact squnc Th strata with unbalancd splittings nd to b tratd sparatly, with th conclusion that thy do not contribut to th gnus 1 Gromov-Wittn invariants undr crtain assumptions on X. 2.3 Strata with ghost principal componnt I Hr w dscrib th contribution to N 1, from a stratum Z T consisting of maps u : Σ X that ar constant on th principal, gnus-carrying, componnts Σ P of Σ. W show Z T dos not contribut to N 1, unlss Z T is of typ B2. For ach m Z +, lt M 1,m b th moduli spac of stabl curvs of gnus 1 with m markd points. Lt E M 1,m b th Hodg lin bundl of holomorphic diffrntials. For ach i [m], dnot by L i M 1,m th univrsal tangnt lin bundl at th i th markd point. Lt s i Γ M 1,m, HomL i, E b th homomorphism inducd by th natural pairing of tangnt and cotangnt vctors at th ith markd point. Dnot by M 1,m, M ff 1,m M 1,m th subspacs consisting of nonsingular curvs and of curvs C with no bubbl componnts C is ithr a nonsingular gnus 1 curv or is a circl of rational curvs. Lt L 1 M 0,1 X, β b th univrsal tangnt lin bundl at th markd point. Dnot by D 1 Γ M 0,1 X, β, HomL 1, v 1T X 26

27 th natural homomorphism inducd by th drivativ of th map at th markd point. For m Z +, lt M 0,m X, β = { m b i i [m] M 0,{0} X, β i : β i H + X, Thr is a wll-dfind valuation map i=1 which is indpndnt of th choic of i. Lt m β i =β, i=1 v 0 b i =v 0 b i i, i [m] }. v 0 : M 0,m X, β X, b i i [m] v 0 b i, π i : M 0,m X, β β i H + X b th projction onto th i th componnt. Dnot by M ff 0,m X, β M 0,mX, β M 0,{0} X, β i th subst consisting of th tupls u i i [m] such that for ach i [m] th rstriction of u i to th domain componnt carrying th markd point 0 is not constant. Th stratum Z T admits a dcomposition Z T = Z T,P Z T,P B Z T,B / SmB, 2.11 whr Z T,P is a stratum of M ff 1,m P for som m P Z +, Z T,B is a stratum of M ff 0,m B X, β for som m B Z +, and Z T,P B is a product of moduli spacs of irrducibl stabl gnus 0 curvs. Th stratum Z T,P consists of curvs of a fixd topological typ, whil th lmnts of Z T,B ar tupls of stabl maps from domains of fixd topological typs so that th imag of th rstriction of th map to ach componnt is of a spcifid homology class and multiplicity. Th rquirmnt that Z T,P M ff 1,m P and Z T,B M ff 0,m B X, β implis that th dcomposition 2.11 is wll-dfind. Lt π P, π B : Z T,P Z T,P B Z T,B Z T,P, Z T,B dnot th projction maps. Th quotint is by th automorphism groups S mb of th data. If X is an idal CY 5-fold, Z T,B is smooth. Th cokrnls of th linarizations D b of th -oprator along Z T form th obstruction bundl O = O P B π B O B = π P E π B v 0 T X π B O B, 2.12 whr O B Z T,B is th obstruction bundl associatd with th moduli spac M 0,m X, β. Lt ν ΓZ T, O b th sction inducd by ν: νb is th projction of νb to th cokrnl of D b. W writ ν P B, ν B Γ Z T, O P B, Γ ZT, O B for th two componnts of ν. 27

28 Thr is a natural projction map snding π B [Σ, u] to uσ, uσ P. Dnot by π : Z T,B M β,1, ν P B Γ Z T, π P E π B π Q th imag of ν P B undr th natural projction map. Lt f : M 1,mP M 1,1 b th forgtful map, dropping all but th first markd point. Th rstriction of th bundl to any boundary stratum Z Γ contains a subbundl O Γ such that π 1 E π 2 Q M 1,1 M β, rk O Γ dim Z Γ > rk π 1E π 2Q dim M 1,1 M β,1 = and { f π P π π B } OΓ is a quotint of th cokrnl bundl ovr a boundary stratum of Z T. Thus, w can choos a sction ν β of 2.13 with all zros transvrs and containd in M 1,1 M β,1 and such that thr xists ν as abov satisfying ν P B = { f π P π π B } νβ. It is shown in th nxt sction that th contribution of Z T to N 1, coms from ν 1 P B 0. Thus, if Z T,P M 1,mP, thn ν 1 P B 0 is mpty for a good choic of ν by 2.14 and th stratum Z T dos not contribut to N 1,. Othrwis, ν P B 1 0 is th primag of a finit subst in M 1,1 M β,1. It dcomposs into connctd componnts ν 1 P B 0 = Z C,x, 2.15 C,x π 2 ν 1 β 0 whr C is a β-curv and x is a nonsingular point of C. Thn, C T β is th numbr of zros of a map ϕ tν from th vctor bundl F of gluing paramtrs to O ovr ach of th componnts Z C,x. Th projction of ϕ tν in th dcomposition 2.12 onto πp E T xs C /T x C is ssntially th sam as th projction of tν, which w dnot by t ν. Sinc ν is a sction of a trivial bundl ovr Z C,x, it can b chosn not to vanish if m > 1. Thus, C T β = 0 if m > 1. On th othr hand, th scond componnt of ϕ tν with rspct to th dcomposition 2.12 is ssntially t ν B. It dos not vanish on ν 1 P B 0 for dimnsional rasons if m=1, but Vr >2. Thus, C T β=0 if T is not basic. Finally, if T is basic, th principal componnt of vry lmnt of Z C,x is a fixd nonsingular gnus 1 curv Σ P with on spcial point z 1 and Z C,x M 0,1 P 1 p, d, whr M 0,1 P 1 p, d M 0,1P 1, d is th subspac of lmnts [Σ, u] such that Σ is nonsingular and v 1 [Σ, u]=p for a fixd p P 1. Lt D Γ M 1,1 M 0,1 P 1, d, Homπ 1 L 1 π 2 L 1, π 1 E π 2 v 1 T P1 28