Orbifold Gromov--Witten theory of weighted blowups
aa r X i v : . [ m a t h . S G ] S e p ORBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS
BOHUI CHEN, CHENG-YONG DU, AND RUI WANG
Abstract.
Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifoldgroupoid ( X , ω ). Let X a be the weight- a blowup of X along S , and D a = PN a be the exceptional divisor,where N is the normal bundle of S in X . In this paper we show that the absolute orbifold Gromov–Wittentheory of X a can be effectively and uniquely reconstructed from the absolute orbifold Gromov–Wittentheories of X , S and D a , the natural restriction homomorphism H ∗ CR ( X ) → H ∗ CR ( S ) and the first Chernclass of the tautological line bundle over D a . To achieve this we first prove similar results for the relativeorbifold Gromov–Witten theories of ( X a | D a ) and ( N a | D a ). As applications of these results, we prove anorbifold version of a conjecture of Maulik–Pandharipande on the Gromov–Witten theory of blowups alongcomplete intersections, a conjecture on the Gromov–Witten theory of root constructions and a conjectureon Leray–Hirsch result for orbifold Gromov–Witten theory of Tseng–You. Contents
1. Introduction 11.1. Orbifold Gromov–Witten theories of weighted projectifications 31.2. Orbifold Gromov–Witten theory of weighted blowups 31.3. Topological view for orbifold Gromov–Witten theory 71.4. Organization of the paper 8Acknowledgements 82. Weighted blowups and Chen–Ruan cohomology 82.1. Weighted projectification and projectivization 82.2. Weighted blowups 92.3. Chen–Ruan Cohomology 93. Fiber class invariants of projectification of orbifold line bundles 123.1. Projectification of line bundles 123.2. Notations for relative orbifold Gromov–Witten invariants of ( D | Y | D ∞ ) 133.3. The moduli spaces of fiber class invariants 153.4. The invariants 214. Relative orbifold Gromov–Witten theory of weighted projectification 234.1. Localization 234.2. Rubber calculus 274.3. Distinguished type II invariants 314.4. Proof of Theorem 1.1 and Theorem 1.10 395. Relative orbifold Gromov–Witten theory of weighted blowups 39References 421. Introduction
Symplectic birational geometry, proposed by Li–Ruan [32], studies symplectic birational cobordism in-variants defined via Gromov–Witten theory. Two symplectic manifolds are called symplectic birationalcobordant (cf. [22, 23]) if one can be obtained from the other by a sequence of Hamiltonian S symplecticreductions. Guillemin–Sternberg [22] proved that every symplectic birational cobordism can be realized by Mathematics Subject Classification.
Primary 53D45; Secondary 14N35.
Key words and phrases.
Orbifold Gromov–Witten theory, Leray–Hirsch result, weighted projective bundles, weightedblowup, root stack, blowup along complete intersection. finite times symplectic blowups/blow-downs and Z -linear deformations of symplectic forms . With notic-ing Gromov–Witten invariants are preserved under smooth deformations of symplectic structures (see forexample [37, 15]), to understand the relation of Gromov–Witten invariants between two symplectic bira-tional corbordant manifolds, it is enough to take care of the change of Gromov–Witten invariants after asymplectic blowup/blow-down.The first nontrivial invariant in symplectic birational geometry is the symplectic uniruledness, which wasdiscovered by Hu–Li–Ruan [23]. Since it was proved by Koll´ar [29] and Ruan [37] that a smooth projectivevariety is uniruled if and only if it is symplectically uniruled, the symplectic uniruledness can be regarded as asymplectic generalization of the uniruledness in birational algebraic geometry. Further motivated by Hu–Li–Ruan’s work, people conjectured that the symplectic rational connectedness as a symplectic generalization(cf. [24, 32]) of rational connectedness for a projective variety is also a symplectic birational invariant. Thisconjecture is still open and reader can find some recent progress along this topic from [46, 24, 40, 41]In fact, the most natural object from symplectic reduction should be symplectic orbifold instead of sym-plectic manifold. Thus it is more natural and also more interesting to study symplectic birational geometryin the category of symplectic orbifolds. At the same time, though symplectic uniruledness and symplecticrational connectedness are defined using genus zero Gromov–Witten invariants only, it is natural to studysymplectic geometry using higher genus Gromov–Witten invariants and unravel the relation between theGromov–Witten theories of a symplectic manifold/orbifold and its blowup, as blowups are building blocksof symplectic birational cobordisms.Further, for a symplectic orbifold, besides the usual symplectic blowup along a symplectic sub-orbifold,one can also consider the weighted blowup, and as it is shown in [7] by Hu and the first two authors thatthe weighted blowup instead of the usual one is the proper version for the orbifold version of symplecticbirational geometry.To be concrete, now we assume ( X , ω ) is a compact symplectic orbifold, and S is a codimension 2 n compact symplectic sub-orbifold of X . Denote by X a the weight- a blowup of X along S (cf. § a = ( a , . . . , a n ) ∈ ( Z ≥ ) n is the blowup weight. In this paper, we focus on studying the relation betweenthe orbifold Gromov–Witten theory of X and of X a .There are already some work on orbifold Gromov–Witten theory X a for the case that S is a symplecticdivisor. When X is a Deligne–Mumford stack, the weight- a = ( r ) blowup X a = X ( r ) of X along S is alsocalled a root stack (cf. [9]). In [45] Tseng–You studied the orbifold Gromov–Witten theory of X ( r ) andconjectured that the orbifold Gromov–Witten theory of X ( r ) is determined by the orbifold Gromov–Wittentheories of X and S and the restriction map H ∗ CR ( X ) → H ∗ CR ( S ). They proved their conjecture for the casethat S is a smooth manifold. The case that S is an orbifold is still open. We will prove this conjecture inthis paper. The exceptional divisor D ( r ) of X ( r ) is a Z r -gerbe over S . It is a natural example of root gerbe.Orbifold Gromov–Witten theory of gerbes is extensively studied by people. See [3, 4, 5, 39].We now describe our approaches and results. Denote by N := N S | X the normal bundle of S in X , and by D a the exceptional divisor of the weighted blowup X a . Notice that the exceptional divisor D a can be consideredas the weight- a projectivization PN a of N , it is an orbifold fiber bundle over S with fiber being the weightedprojective space P a . There is a tautological line bundle over D a = PN a coming from the tautological linebundle O ( −
1) over P a . We denote it by O PN a ( −
1) or O D a ( − D a in X a .On the other hand, let N a be the weight- a projectification of N . Then D a is the infinite divisor of N a . Thenormal line bundle of D a in N a is O D a (1), the dual line bundle of O D a ( − X X degenerate −−−−−−−→ ( X a | D a ) ∧ D a ( N a | D a ) , (1.1)where “ ∧ D a ” means the gluing is along the exceptional divisor D a ⊆ X a and the infinite divisor D a ⊆ N a .Similarly, let O D a ( −
1) be projectification of O D a ( −
1) with trivial weight, that is O D a ( −
1) = P ( O D a ( − ⊕O D a ). Let D a , ∞ = P (0 ⊕O D a ) ∼ = D a be the infinite divisor of O D a ( − O D a ( − O D a (1). Then we get a degeneration of X a as (1.1) X a degenerate −−−−−−−→ ( X a | D a ) ∧ D a ( O D a ( − | D a , ∞ ) . (1.2) A Z -linear deformation of a symplectic form ω on a manifold X is a path of symplectic form ω + tκ , t ∈ I , where κ is aclosed 2-form representing an integral class and I is an interval. See for example [23, Definition 2.5]. RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 3
As in (1.1), here “ ∧ D a ” means the gluing is along the exceptional divisor D a ⊆ X a and the infinite divisor D a ∼ = D a , ∞ ∈ O D a ( − X a and X , we only need to compare the relative orbifold Gromov–Wittentheories of ( N a | D a ) and ( O D a ( − | D a ).On the other hand, in [7], Hu and the first two authors proved that for the degeneration (1.1), there isan invertible lower triangle system which relates relative orbifold Gromov–Witten invariants of ( X a | D a ) andabsolute orbifold Gromov–Witten invariants of X relative to S , whose entries are relative orbifold Gromov–Witten invariants of ( N a | D a ). Consequently if we could determine the relative orbifold Gromov–Witteninvariants of ( O D a ( − | D a , ∞ ) and of ( N a | D a ), we could determine the orbifold Gromov–Witten invariantsof X a from the degeneration (1.2).1.1. Orbifold Gromov–Witten theories of weighted projectifications.
We first study relative orb-ifold Gromov–Witten invariants of weighted projectifications of vector bundles, which can apply to both( N a | D a ) and ( O D a ( − | D a , ∞ ).Let π : E → S be a rank 2 n symplectic orbifold vector bundle over a compact symplectic orbifold groupoid S . Let E a be the weight- a projectification of E and PE a be the infinite divisor, which is the weight- a projectivization of E (cf. § N PE a | E a of PE a in E a is O PE a (1), the dual line bundle ofthe tautological line bundle O PE a ( − Theorem 1.1.
The relative descendent orbifold Gromov–Witten theory of the pair ( E a | PE a ) can be effec-tively and uniquely reconstructed from the absolute descendent orbifold Gromov–Witten theories of S and PE a , the Chern classes of E and O PE a ( − . By virtual localization, we could first reduce the determination of relative descendent orbifold Gromov–Witten invariants of the pair ( E a | PE a ) to orbifold Gromov–Witten invariants of S twisted by E and rubberinvariants with Ψ ∞ -integrals associated to the orbifold line bundle O PE a (1) → PE a , i.e. rubber invariantswith Ψ ∞ -integrals of the projectification of O PE a (1):( D | Y | D ∞ ) := (cid:0) P (0 ⊕ O PE a ) | P ( O PE a (1) ⊕ O PE a ) | P ( O PE a (1) ⊕ (cid:1) . In § ∞ -integrals, and reduce the resulting rubber invariantswithout Ψ ∞ -integrals into certain relative descendent orbifold Gromov–Witten invariants of ( D | Y | D ∞ ),called distinguished type II invariants . Then we use an induction algorithm to determine all these distin-guished type II invariants of ( D | Y | D ∞ ). Among them, fiber class invariants are the initial values of thisinductive algorithm. We will determine fiber class invariants for general orbifold P -bundles in §
3. Theinductive algorithm is described in § § ∞ -integrals associated to a smooth line bundle L → X are determined by the Gromov–Witten theory of X and c ( L ). It is natural to expect a formula for the double ramification cycles with orbifold targets. Sucha formula would also implies that rubber invariants without Ψ ∞ -integrals associated to the orbifold linebundle O PE a (1) → PE a are determined by the orbifold Gromov–Witten theory of PE a and c ( O PE a (1)). Wewill study this in [9].1.2. Orbifold Gromov–Witten theory of weighted blowups.
Now we can determine the orbifoldGromov–Witten theory of X a , the weight- a blowup of X along S . There is a natural orbifold morphism κ : X a → X , which induces a morphism on inertia spaces I κ = a ( h ) ∈ T X a κ ( h ) : a ( h ) ∈ T X a X a ( h ) → a ( h ) ∈ T X a X ( κ t ( h )) , where κ t is the induced map on the index sets of connected components of inertia spaces. In Definition 2.4, § K := M ( h ) ∈ T X a κ ∗ ( h ) H ∗ ( X ( κ t ( h ))) , BOHUI CHEN, CHENG-YONG DU, AND RUI WANG to be the image of the induced homomorphism on cohomologies. The map κ ∗ ( h ) is injective for each ( h ) ∈ T X a . We fix in § ⋆ of H ∗ CR ( D a ) = H ∗ CR ( PN a ) (actually for the Chen–Ruan cohomology of PE a for a general symplectic orbifold vector bundle E → S ) by choosing a basis σ ⋆ of H ∗ CR ( S ). We denote thedual basis of Σ ⋆ by Σ ⋆ . Definition 1.2.
We call a relative descendent orbifold Gromov–Witten invariant of ( X a | D a ) D Y i τ k i γ i (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β admissible if γ i ∈ K and µ is a relative insertion weighted by the chosen basis Σ ⋆ , i.e. ˇ µ = (( µ , θ ( h ) ) , . . . , ( µ ℓ ( µ ) , θ ( h ℓ ( µ ) ) )) with θ ( h i ) ∈ Σ ⋆ . Given an admissible relative orbifold Gromov–Witten invariant of ( X a | D a ) D Y i τ k i γ i (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β , (1.3)we assign it an absolute orbifold Gromvo–Witten invariant of X D Y i τ k i ¯ γ i · µ S E X g,p ∗ β (1.4)by the weighted-blowup absolute/relative correspondence in [7, § γ i is the inverse image of γ i under κ ∗ ( h ) , µ S supports over IS and is determined by µ and the restriction map H ∗ CR ( X ) → H ∗ CR ( S ) (see § µ S ). There is a partial order over all admissible relative invariants of ( X a | D a )given in [7, § § N a | D a ). With Theorem1.1 we prove in § Theorem 1.3.
The admissible relative descendent orbifold Gromov–Witten theory of ( X a | D a ) can be uniquelyand effectively reconstructed from the orbifold Gromov–Witten theories of X , S and D a , the restriction map H ∗ CR ( X ) → H ∗ CR ( S ) and the first Chern class of O D a ( − . In particular, when S is a divisor and the blowup weight a = (1), the map κ : ( X a | D a ) → ( X | S ) is identity,hence K = H ∗ CR ( X a ) = H ∗ CR ( X ). Therefore all relative invariants of ( X a | D a ) = ( X | S ) are admissible. Sincenow c ( N S | X ) is determined by the restriction map H ∗ CR ( X ) → H ∗ CR ( S ) (cf. (1.6)) and O D a ( −
1) = N S | X , asa direct consequence of Theorem 1.3 we have Corollary 1.4.
When S is a divisor, the relative descendent orbifold Gromov–Witten theory of ( X | S ) can beuniquely and effectively reconstructed from the orbifold Gromov–Witten theories of X , S , and the restrictionmap H ∗ CR ( X ) → H ∗ CR ( S ) . This result extends [34, Theorem 2] to the orbifold case.After determining all admissible relative invariants of ( X a | D a ), we could determine the absolute invariantsof X a . The inertia orbifold groupoid ID a of D a is a sub-orbifold groupoid of the inertia orbifold groupoid IX a of X a . We see in § H ∗ CR ( X a ) is generated by K and forms that support over ID a . Consider anabsolute invariant of X a D Y i τ k i γ i · Y j τ k ′ j θ j E X a g,β . (1.5)with γ i belonging to K , and θ j supporting over ID a . We use degeneration formula to calculate this invariant.We degenerate X a as (1.2). For γ i ∈ K we could take the extension over ( X a | D a ) to be γ i itself and anappropriate extension γ + i over ( O PN a ( − | D a , ∞ ). For θ j we could take the extension over ( X a | D a ) to be 0,and the extension over ( O PN a ( − | D a , ∞ ) to be θ j . Then by the degeneration formula, the absolute invariant(1.5) is determined by admissible relative invariants of ( X a | D a ) and relative invariants of ( O PN a ( − | D a , ∞ ).By the linearity of Gromov–Witten invariants, all absolute descendent orbifold Gromov–Witten invariants RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 5 of X a are determined by absolute descendent orbifold Gromov–Witten invariants of X a of the form (1.5).Since D a , ∞ ∼ = D a , as a direct consequence of Theorem 1.1 and Theorem 1.3 we have proved Theorem 1.5.
The absolute descendent orbifold Gromov–Witten theory of X a can be uniquely and effectivelyreconstructed from the absolute descendent orbifold Gromov–Witten theories of X , S and D a , the restrictionmap H ∗ CR ( X ) → H ∗ CR ( S ) and the first Chern class of O PN a ( − . Our results have several applications.1.2.1.
Gromov–Witten theory of blowups along complete intersections.
Here we consider the orbifold ver-sion of a conjecture of Maulik–Pandharipande [34] on Gromov–Witten theory of blowups along completeintersections. Let X be a compact symplectic orbifold groupoid, and W , . . . , W m be m symplectic divisorsof X which intersect transversely such that S := m \ i =1 W i is a symplectic sub-orbifold groupoid of X . Let X be the blowup of X along S with trivial weight a = (1 , . . . , D be the exceptional divisor. Denote the normal bundle of W i in X by N i . Then the normal bundle of S in X is a direct sum of m line bundles N S | X = m M i =1 N i (cid:12)(cid:12) S . Therefore D = P ( N S | X ) = P (cid:16) m M i =1 N i (cid:12)(cid:12) S (cid:17) . Each line bundle N i | S gives us a section of D → SS i = P (0 ⊕ . . . ⊕ N i (cid:12)(cid:12) S ⊕ . . . ⊕ , ≤ i ≤ m. Moreover, S i ∼ = S for 1 ≤ i ≤ m . Remark 1.6.
When X = V is a smooth nonsingular projective variety, S = Z is a smooth nonsingularcomplete intersection of m smooth nonsingular divisors W , . . . , W m ⊂ V and ˜ V is the blowup of V along Z , Maulik and Pandharipande conjectured that (cf. [34, Conjecture 2])The Gromov–Witten theory of ˜ V is actually determined by the Gromov–Witten theories of V and Z and the restriction map H ∗ ( V, Q ) → H ∗ ( Z, Q ).The following Theorem 1.7 is an orbifold version of this conjecture. This conjecture was also proved by Fan[19] and Du [17]. Theorem 1.7.
The orbifold Gromov–Witten theory of X is uniquely and effectively determined by theGromov–Witten theories of X and S , and the restriction map H ∗ CR ( X , Q ) → H ∗ CR ( S , Q ) .Proof. By Theorem 1.5, the orbifold Gromov–Witten theory of X is determined by the orbifold Gromov–Witten theories of X , D and S , the first Chern class of the normal bundle O ( − → D of D in X and therestriction map H ∗ CR ( X ) → H ∗ CR ( S ). We next show that(a) the orbifold Gromov–Witten theory of D is determined by the orbifold Gromov–Witten theory of S and,(b) c ( O ( −
1) is determined by the restriction map H ∗ CR ( X ) → H ∗ CR ( S ).There is a T = ( C ∗ ) m action on N S | X which descends to a T -action on D . The fixed loci of this T -actionon D consist of S i ∼ = S , ≤ i ≤ m . The fixed lines connecting these S i are special lines in the fiber of D = P ( N S | X ) → S .The T -action on D induces a T -action on the moduli space of orbifold stable maps to D . The fixed loci aredetermined by those graphes, of which vertices represent moduli spaces of orbifold stable maps to S i , andedges represent maps from orbifold Riemann spheres to D that are totally ramified over certain S i , S j ⊂ D with i = j . By the virtual localization (cf. [31, 20, 12, 33]), the orbifold Gromov–Witten invariants of D are determined by Hodge integrals in the orbifold Gromov–Witten theories of S i ∼ = S (see § S i ∼ = S can be removed by the orbifold quantum Riemann–Rochof Tseng [42]. This proves (a). BOHUI CHEN, CHENG-YONG DU, AND RUI WANG
On the other hand, the normal bundle of D in X is the tautological line bundle O ( −
1) over D = P ( N S | X ) = P ( L mi =1 N i | S ), whose first Chern class x is determined by the classical relation (cf. Bott and Tu [6, (20.6)])( − x ) m + c ( N S | X )( − x ) m − + . . . + c m − ( N S | X )( − x ) + c m ( N S | X ) = 0 . The total Chern class c ( N S | X ) = 1 + P mi =1 c i ( N S | X ) is determined inductively by the restriction map ι ∗ : H ∗ CR ( X ) → H ∗ CR ( S ) via c ( N S | X ) ∪ c ( T S ) = ι ∗ ( c ( T X )) . (1.6)See also §
5. This proves (b). The proof is complete. (cid:3)
Orbifold Gromov–Witten theory of root constructions.
We next study a conjecture of Tseng–You onorbifold Gromov–Witten theory of root constructions. Let X be a smooth Deligne–Mumford stack, and D be a divisor, Tseng–You [45] considered the r -th root construction X r of X along D . They conjectured [45,Conjecture 1.1] that Conjecture 1.8.
The Gromov–Witten theory of X r is determined by the Gromov–Witten theories of X , D ,and the restriction map H ∗ CR ( X ) → H ∗ CR ( D ) . See [43] for more discussions on this conjecture and related conjectures.The root stack construction corresponds to the weighted blowup along symplectic divisors in the sym-plectic category. Let D ⊆ X be a symplectic divisor with normal line bundle L := N D | X and X ( r ) be theweight- a = ( r ) blowup of X along D . We have a natural projection κ : X ( r ) → X with D ( r ) := κ − ( D ) = PL ( r ) being the exceptional divisor. D ( r ) is the r -th root gerbe r p L / D of the line bundle L . Its normal line bundle N D ( r ) | X ( r ) in X ( r ) is the r -th root r √ L of L . As a Z r -gerbe over D it is a trivial band gerbe. We denote therestriction of κ on D ( r ) also by κ : D ( r ) → D . Theorem 1.9.
Conjecture 1.8 is true, i.e. the Gromov–Witten theory of X ( r ) is determined by the Gromov–Witten theories of X , D and the restriction map H ∗ CR ( X ) → H ∗ CR ( D ) .Proof. By Theorem 1.5, the Gromov–Witten theory of X ( r ) is determined by the Gromov–Witten theoriesof X , D ( r ) and D , the restriction map H ∗ CR ( X ) → H ∗ CR ( D ) and c ( N D ( r ) | X ( r ) ) = c ( r √ L ). We have c ( r √ L ) = r κ ∗ c ( L ) as ( r √ L ) ⊗ r = κ ∗ L . Therefore since c ( L ) is determined by the restriction map H ∗ CR ( X ) → H ∗ CR ( D )by (1.6), c ( r √ L ) is also determined by the restriction map H ∗ CR ( X ) → H ∗ CR ( D ).We next consider the orbifold Gromov–Witten theory of D ( r ) . The projection κ : D ( r ) → D induces amap on inertia spaces (cf. § I κ := G ( h )=( g,e π √− R ) ∈ T D ( r ) κ ( h ) : G ( h )=( g,e π √− R ) ∈ T D ( r ) H ∗ ( D ( r ) ( h )) → G ( h )=( g,e π √− R ) ∈ T D ( r ) H ∗ ( D ( g ))where for each κ ( h ) , the corresponding map | κ ( h ) | : | D ( r ) ( h ) | → | D ( g ) | on coarse spaces is a homeomorphism.The κ induces an isomorphism of Chen–Ruan cohomology groups between D ( r ) and r copies of D .Consider an orbifold Gromov–Witten invariant of D ( r ) D τ a κ ∗ ( h ) α , . . . , τ a n κ ∗ ( h n ) α n E D ( r ) g,~h,β := Z [ M g,~h,β ( D ( r ) )] vir ev ∗ D ( r ) n Y j =1 κ ∗ ( h j ) α j ∪ n Y j =1 ¯ ψ j (1.7)where • g ≥ β ∈ H ( | D ( r ) | ; Z ) is a degree 2 homology class, • α i ∈ H ∗ ( D ( g i )) with D ( g i ) obtained from κ ( h i ) : D ( r ) ( h i ) → D ( g i ), • ~h = (( h ) , . . . , ( h n )) indicates the twisted sectors of D ( r ) that the images of the evaluation map ev D ( r ) : M g,~h,β ( D ( r ) ) → ( ID ( r ) ) n are located, • ¯ ψ j is the psi-class of the line bundle over M g,~h,β ( D ( r ) ) whose fiber over a point [ f : ( C , x , . . . , x n ) → D ( r ) ] is the cotangent space of the coarse space of C at the j -th marked point. See also § Here for an orbifold groupoid X , | X | is its coarse space. RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 7
From κ : D ( r ) → D we get a moduli space M g,~g,β ( D ) of orbifold stable maps to D with ~g = (( g ) , . . . , ( g n )),and a natural map π : M g,~h,β ( D ( r ) ) → M g,~g,β ( D ) which sits in the following commutative diagram M g,~h,β ( D ( r ) ) ev D ( r ) / / π (cid:15) (cid:15) ( ID ( r ) ) n ( I κ ) n (cid:15) (cid:15) M g,π ( ~h ) ,β ( D ) ev D / / ( ID ) n , where the horizontal maps are evaluation maps. Recall that κ : D ( r ) → D is a trivial band Z r -gerbe over D .By the computation of Tang–Tseng [39, Section 5] we have π ∗ [ M g,~h,β ( D ( r ) )] vir = r g − · [ M g,~g,β ( D )] vir . Therefore from I κ ◦ ev D ( r ) = ev D ◦ π and π ∗ ¯ ψ j = ¯ ψ j for 1 ≤ j ≤ n , we have h τ a κ ∗ ( h ) α , . . . , τ a n κ ∗ ( h n ) α n i D ( r ) g,~h,β = ev ∗ D ( r ) n Y j =1 κ ∗ ( h ) α j ∪ n Y j =1 ¯ ψ a j j ∩ [ M g,~h,β ( D ( r ) )] vir = π ∗ ev ∗ D n Y j =1 ( α j ) ∪ n Y j =1 ¯ ψ a j j ∩ [ M g,~h,β ( D ( r ) )] vir = ev ∗ D n Y j =1 ( α j ) ∪ n Y j =1 ¯ ψ a j j ∩ π ∗ ([ M g,~h,β ( D ( r ) )] vir )= r g − · ev ∗ D n Y j =1 ( α j ) ∪ n Y j =1 ¯ ψ a j j ∩ [ M g,~g,β ( D )] vir = r g − · h τ a α , . . . , τ a n α n i D g,~g,β . Therefore, the orbifold Gromov–Witten invariants of D ( r ) are determined by the orbifold Gromov–Witteninvariants of D . This finishes the proof of this theorem, hence Conjecture 1.8. (cid:3) Topological view for orbifold Gromov–Witten theory.
Theorem 1.1 is proved by using virtuallocalization of relative invariants and an analogue of the rubber calculus of Maulik–Pandharipande [34].As a byproduct, we could generalize the Leray–Hirsch result for line bundles and Mayer–Vietoris result forsymplectic cutting in Gromov–Witten theory to orbifold Gromov–Witten theory.1.3.1.
Orbifold Leray–Hirsch.
Let D be a compact symplectic orbifold groupoid. Let L be a symplecticorbifold line bundle over D . Consider the projectification Y = P ( L ⊕ O D ). Y has the zero section D = P (0 ⊕ O D ) and the infinity section D ∞ = P ( L ⊕ . We have D ∼ = D ∞ ∼ = D . The normal bundles of D and D ∞ in Y are L and L ∗ respectively.Consider four orbifold Gromov–Witten theories: the absolute descendent orbifold Gromov–Witten theoryof Y and the relative descendent orbifold Gromov–Witten theories of the three pairs( Y | D ) , ( Y | D ∞ ) , and ( D | Y | D ∞ ) . Theorem 1.10.
All four theories can be uniquely and effectively reconstructed from the absolute descendentorbifold Gromov–Witten theory of D and the first Chern class of the line bundle L . We call this the orbifold Leray–Hirsch result . The relative theories of ( Y | D ) and ( Y | D ∞ ) are special caseof Theorem 1.1. On the other hand(1) when D is a compact symplectic manifold and L is a symplectic line bundle over D , this theorem isjust the Leray–Hirsch result of Maulik–Pandharipande [34, Theorem 1].(2) when D = BG is the classifying groupoid for a finite group G , this theorem was proved by Tseng–You[44] by an explicit computation of the double ramification cycles on the moduli spaces of admissiblecovers, following the approach of Janda, Pandharipande, Pixton and Zvonkine [26]. BOHUI CHEN, CHENG-YONG DU, AND RUI WANG
This orbifold Leray–Hirsch result for orbifold Gromov–Witten theory was also conjectured by Tseng andYou [45, Conjecture 2.2]. See also Tseng [43].1.3.2.
Orbifold Mayer–Vietoris.
Consider a general symplectic cutting on a compact symplectic orbifoldgroupoid X . We get a family ǫ : D → D of symplectic orbifold groupoids (see Chen–Li–Sun–Zhao [13, § D is the unit ball in C ,(ii) for all t = 0, ǫ − ( t ) ∼ = X ,(iii) X := ǫ − (0) = X + ∧ D X − , with X ± intersect with each other at the common divisor D normalcrossingly.This is a degeneration of X . There is an induced homomorphism H ( | X | ; Z ) → H ( | X | ; Z ) on homologies,and classes in the kernel of the this induced homomorphism are called vanishing cycles . There is also aninduced homomorphism on Chen–Ruan cohomologies H ∗ CR ( X + ∧ D X − ) → H ∗ CR ( X )with image called the non-vanishing cohomology of X . Theorem 1.11.
If there is no vanishing cycles, then the absolute descendent orbifold Gromov–Wittentheory of the non-vanishing cohomology of X can be uniquely and effectively reconstructed from the absolutedescendent orbifold Gromov–Witten theories of X ± and D , and the restriction maps H ∗ CR ( X + ) → H ∗ CR ( D ) , H ∗ CR ( X − ) → H ∗ CR ( D ) . Proof.
This is a direct consequence of the degeneration formula [13, 1] and Corollary 1.4. (cid:3)
It is well-known that when the symplectic cutting is proceeded by (weighted) blowup along a symplecticsub-orbifold, there is no vanishing cycle. Therefore for these cases, this theorem applies.1.4.
Organization of the paper.
This paper is organized as follows. In §
2, we review the weighted pro-jectification and weighted projectivization of orbifold bundles, and weighted blowup of symplectic orbifoldgroupoids. Then we describe the Chen–Ruan cohomologies of the projectifications of orbifold bundles, andthe Chen–Ruan cohomologies of the weighted blowups and the exceptional divisors. In § § § § Acknowledgements.
This work was supported by the National Natural Science Foundation of China(No. 11890663, No. 11821001, No. 11826102, No. 11501393, No. 12071322), by the Sichuan Science andTechnology Program (No. 2019YJ0509), and by a joint research project of Laurent Mathematics ResearchCenter of Sichuan Normal University and V. C. & V. R. Key Lab of Sichuan Province.2.
Weighted blowups and Chen–Ruan cohomology
In this paper, we study orbifolds ([38]) via proper ´etale Lie groupoids, which are called orbifold groupoids.There are some nice references on orbifold groupoids. See for example Adem–Leida–Ruan [2] and Moerdijk–Pronk [36]. One can see also [7, §
2] for a brief introduction of orbifold groupoids and Chen–Ruan cohomologyetc.We use a = ( a , . . . , a n ) to denote the blowup weight, where a i ∈ Z ≥ , for 1 ≤ i ≤ n .2.1. Weighted projectification and projectivization.
Let S act on C n by t · ( z , . . . , z n ) = ( t a z , . . . , t a n z n ) . Denote this action by S ( a ). The weight- a projective space is P a := P a ( C n ) = S n − /S ( a ) . The weight- a blowup of C n along the origin is[ C n ] a = S n − × S ( a , − C , it is the total space of the tautological line bundle O ( −
1) over P a . RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 9
Now let S = ( S ⇒ S ) be an orbifold groupoid. Consider a rank 2 n symplectic orbifold vector bundle π = ( π , π ) : E = ( E ⇒ E ) → S = ( S ⇒ S ) . So both π i : E i → S i are symplectic vector bundles for i = 0 , s ∗ E ∼ = E ∼ = t ∗ E for the source andtarget maps s, t : S → S of S . This is equivalent to an action of S = ( S ⇒ S ) on the symplectic vectorbundle π : E → S , i.e. there is an action map µ : S × s,S ,π E → E that is compatible with π : E → S , which is called the anchor map for this action, and satisfies the ruleof group actions, for instance µ ( gh, v ) = µ ( h, µ ( g, v ))for two arrows g, h ∈ S and v ∈ E . Then one has E ∼ = S × s,S ,π E . We also write E = S ⋉ E . Take a compatible complex structure. Let π : P → S be the corresponding principle bundle with structuregroup K < U ( n ). Then E = P × K C n . The weight- a projectivization of E is PE a := P × K P a , The weight- a projectification of E is E a := P × K P a , , where P a , is the weight-( a ,
1) projective space. The weight- a blowup of E is E a := P × K [ C n ] a . When the weight a = (1 , . . . ,
1) is trivial, we omit the subscript a . PE a is the exceptional divisor of the blowup E a . E a is the tautological line bundle over PE a , whoserestriction on fiber of π : PE a → S is the tautological line bundle O ( −
1) over P a . So as in the introductionwe also write E a as O PE a ( − PE a is also the infinite divisor of the weight- a projectification E a , its normalline bundle in E a is O PE a (1), the dual line bundle of O PE a ( − Weighted blowups.
Let X be a compact symplectic orbifold groupoid with S being a codimensioncodim R S = 2 n compact symplectic sub-orbifold groupoids. Denote the normal bundle of S in X by N . Thesymplectic neighborhood theorem holds for ( X , S ) (cf. [18, 7]). Locally, there is an open neighborhood U of S in X , which is symplectomorphic to the ǫ -disk bundle φ : U → D ǫ ( N )of the normal bundle N of S in X . We call ( X , S ) a symplectic pair.The weight- a blowup X a of X along S is obtained by gluing X \ U with the ǫ -disk bundle of N a , theweight- a blowup of N along the zero section S . The exceptional divisor in X a is D a := PN a , the weight- a projectivization of N . So as (1.1), the weight- a blowup of X along S gives rise to a degeneration of XX degenerates −−−−−−−→ ( X a | D a ) ∧ D a ( N a | D a ) . There is a natural projection κ : X a → X which restricts to κ : D a = PN a → S . Chen–Ruan Cohomology.
We next describe the Chen–Ruan cohomologies of E a , E a , PE a , X a and D a . For an orbifold groupoid W we use IW to denote its inertia space (see for example [7, Definition 2.10])and T W to denote the index set of its twisted sectors. For a δ ∈ T W , we denote the corresponding twistedsector by W ( δ ). When we use local chart, we also treat the conjugate class of isotropy groups as the indexof twisted sectors. Inertia space of E a and PE a . We first focus on the local case. Since E a = P × K [ C n ] a , we get aprojection π : E a → S which is also an orbifold groupoid morphism and restricts to π : PE a → S . Hence there are induced morphisms between their inertial spaces I π : IE a → IS , and I π : IPE a → IS . In particular, there are induced maps on the index sets of twisted sectors π t : T E a → T S , and π t : T PE a → T S . When restricting on twisted sectors, we write the restriction of I π as π ( h ) : E a ( h ) → S ( π t ( h )) , and π ( h ) : PE a ( h ) → S ( π t ( h )) . Remark 2.1.
Note that π : E a → PE a is an orbifold line bundle, hence T E a = T PE a . Moreover, either π ( h ) : E a ( h ) → PE a ( h ) is an orbifold line bundle or E a ( h ) = PE a ( h ) depending on the action of ( h ) on thefiber of E a is trivial or non-trivial.For a point x ∈ S , locally near x , S is modeled by G x ⋉ U x with G x being the local (or isotropy) groupof x in S . Then locally, E a and PE a are of the forms U x × ( S n − × C ) G x × S ( a , − , and U x × S n − G x × S ( a )respectively. Now consider a ( g ) ∈ T S with representative g ∈ G x , i.e. g · x = x . Suppose the g -action onthe fiber E | x is given by g · ( z , . . . , z n ) = ( e π √− b ( g )1 E o ( g ) z , . . . , e π √− b ( g ) n E o ( g ) z n )with o ( g ) = ord( g ) being the order of g , and 1 ≤ b ( g ) i E ≤ o ( g ) being the action weights. The order o ( g ) andaction weights b ( g ) i E do not depend on the choices of representative g of ( g ).If an ( h ) ∈ T E a = T PE a satisfies π t ( h ) = ( g ), then ( h ) has a representative of the form h = ( g, e π √− R ) ∈ G x × S for some R ∈ Q ∩ [0 , h -action on S n − × C is given by h · ( z , . . . , z n , w ) = ( e π √− b ( g )1 E o ( g ) + Ra ) z , . . . , e π √− b ( g ) n E o ( g ) + Ra n ) z n , e − π √− R w ) , (2.1)which restricts to the h -action on S n − = S n − × { } .Then the fiber of PE a ( h ) → S ( g ) is given by P a I ( h ) /C G x ( g )with I ( h ) := (cid:26) i (cid:12)(cid:12)(cid:12)(cid:12) b ( g ) i E o ( g ) + Ra i ∈ Z , ≤ i ≤ n (cid:27) , and a I ( h ) being a sub-weight of a obtained from the inclusion I ( h ) ⊆ { , . . . , n } .It is direct to see from (2.1) that(a) π ( h ) : E a ( h ) → PE a ( h ) is a line bundle when R = 0,(b) E a ( h ) = PE a ( h ) when R = 0.Set r ( h ) := I ( h ). Summarizing above discussions we get the following result. Lemma 2.2. π ( h ) : PE a ( h ) → S ( g ) is a weight- a I ( h ) projectivization of a rank r ( h ) sub-bundle of the pull-back bundle of E over S ( g ) via the natural evaluation morphism e ( g ) : S ( g ) ⊆ IS → S .On the other hand, since π : E → S is a vector bundle, we also have T E = T S , and (a) when ( h ) = ( g, , i.e. R = 0 , π : E a ( h ) → E ( g ) is the weight- a I ( h ) blowup of E ( g ) along S ( g ) , (b) when ( h ) = ( g, , E a ( h ) = PE a ( h ) . RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 11
Basis of Chen–Ruan cohomology of PE a . For each ( g ) ∈ T S let σ ( g ) := { δ g ) , . . . , δ k ( g )( g ) } be a basis of H ∗ ( S ( g )), then σ ⋆ := G ( g ) ∈ T S σ ( g ) = G ( g ) ∈ T S { δ g ) , . . . , δ k ( g )( g ) } (2.2)is a basis of H ∗ CR ( S ) = M ( g ) ∈ T S H ∗− ι ( g ) ( S ( g )) . Here we assume that each δ i ( g ) is of homogenous degree and δ is the identity elements in H ∗ ( S ), denotedby S , or simply . Denote the dual basis with respect to the orbifold Poincar´e duality by σ ⋆ := G ( g ) ∈ T S ˇ σ ( g ) := G ( g ) ∈ T S { ˇ δ g ) , . . . , ˇ δ k ( g )( g ) } , (2.3)i.e. ˇ δ i ( g ) is the dual of δ i ( g ) , hence ˇ δ i ( g ) ∈ H ∗ ( S ( g − )).Set Σ ( h ) := { δ j ( g ) ∪ H m ( h ) | δ j ( g ) ∈ σ ( g ) , ≤ m ≤ r ( h ) − } (2.4)with ( g ) = π t ( h ). Here H ( h ) is the hyperplane class of the projective bundle π ( h ) : PE a ( h ) → S ( g ). ThenΣ ( h ) is a basis of H ∗ ( PE a ( h )). Set Σ ⋆ to be the union of Σ ( h ) over all ( h ) ∈ T PE a . Denote elements in Σ ( h ) by θ l ( h ) , 1 ≤ l ≤ k ( g ) r ( h ). Let Σ ⋆ denote the dual basis of Σ ⋆ with respect to the orbifold Poincar´e duality.2.3.3. Basis of Chen–Ruan cohomology of E a . Similarly as PE a , E a is the weight-( a ,
1) projectivization of E ⊕ O S . Therefore there is a basis of H ∗ CR ( E a ) as (2.4), which consists ofΞ (¯ h ) := { δ j ( g ) ∪ H m (¯ h ) | δ j ( g ) ∈ σ ( g ) , ≤ m ≤ r (¯ h ) − } , (2.5)where (¯ h ) is the index of twisted sectors of E and π t (¯ h ) = ( g ). The definition of r (¯ h ) is similar as r ( h ), justnote that the g -action on O S is trivial. Denote elements in Ξ (¯ h ) by γ l (¯ h ) , 1 ≤ l ≤ k ( g ) r (¯ h ). Theorem 1.1concerns relative invariants of ( E a | PE a ) whose absolute insertions come from the basis (2.5) and relativeinsertions come from the basis (2.4). We will prove Theorem 1.1 in § Chen–Ruan cohomology of X a . Now consider a symplectic pair ( X , S ). Let N be the normal bundleof S , and ( X a , D a = PN a ) be its the weight- a blowup along S .The projection κ : X a → X also induces morphisms on twisted sectors κ ( h ) : X a ( h ) → X ( κ t ( h )) . The main difference between IX a and IX are those twisted sectors of X a intersecting with the inertia space ID a of the exceptional divisor D a = PN a . By Lemma 2.2 there are two kinds of such twisted sectors of X a . Lemma 2.3.
We have the following description of twisted sectors of X a . Let κ t ( h ) = ( g ) . (a) When X a ( h ) ∩ ID a = ∅ , we have X ( g ) ∩ IS = ∅ and κ ( h ) : X a ( h ) → X ( g ) is identity. (b) When X a ( h ) ∩ ID a = ∅ , we have X ( g ) ∩ IS = ∅ , and there are two cases: (b.i) X a ( h ) ⊇ N a ( h ) is the weighted blowup of N ( g ) along S ( g ) , i.e. ( h ) = ( g, , then κ ( h ) : X a ( h ) → X ( g ) , the sequence → H ∗ ( X ( g )) κ ∗ ( h ) −−→ H ∗ ( X a ( h )) → A ( h ) → is exact, where A ( h ) = H ∗ ( S ( g )) { H ( h ) , . . . , H r ( h ) − h ) } , is a free module over H ∗ ( S ( g )) with basis { H ( h ) , . . . , H r ( h ) − h ) } , H ( h ) is the hyperplane class of PN a ( h ) = D a ( h ) . Moreover, H ∗ ( X a ( h )) is generated by κ ∗ ( h ) H ∗ ( X ( g )) and (cid:16) H ∗ ( S ( g )) { , H ( h ) , . . . , H r ( h ) − h ) } (cid:17) ∪ [ PN a ( h )] , where [ PN a ( h )] is the Poincar´e dual of PN a ( h ) in X a ( h ) . (b.ii) X a ( h ) = N a ( h ) = PN a ( h ) , i.e. ( h ) = ( g, . Then H ∗ ( X a ( h )) = H ∗ ( S ( g )) { , H ( h ) , . . . , H r ( h ) − h ) } . Definition 2.4.
We set K := M ( h ) ∈ T X a κ ∗ ( h ) (cid:0) H ∗ ( X ( κ t ( h ))) (cid:1) . (2.6)In this paper we will deal with relative invariants of ( X a | D a ) with absolute insertions coming from K , i.e.admissible relative invariants in Definition 1.2.3. Fiber class invariants of projectification of orbifold line bundles
In the study of relative orbifold Gromov–Witten theories of weighted blowups or weighted projectifica-tions, we need to study fiber class orbifold Gromov–Witten invariants of projectifications of orbifold linebundles. In this section we study these fiber class invariants.3.1.
Projectification of line bundles.
Let π : L → D be a symplectic line bundle. That is if we write L = D ⋉ L with D = ( D ⇒ D ), then L → D is a symplectic line bundle and the linear D -action on L → D preserve the symplectic structure over L . The projectification of L is Y := P ( L ⊕ O D ) = P × U (1) P (1 , → D , where P is the principle U (1)-bundle of L , and P (1 , = P is the one dimensional projective space. In termsof notations in § Y = L (1) , i.e. Y is the projectification of L with trivial weight a = (1). Y has the zerosection D := P (0 ⊕ O D ) and the infinity section D ∞ := P ( L ⊕ D . The normalbundle of D and D ∞ in Y are L and L ∗ respectively.As the description of inertia space and Chen–Ruan cohomology of E a in § Y .The bundle π : L → D induces a bundle I π = G ( h ) ∈ T D π ( h ) : IL = G ( h ) ∈ T D L ( h ) → ID = G ( h ) ∈ T D D ( h )(3.1)of inertia spaces, which may have different ranks over different components of ID . Over each twisted sector D ( h ), for every representative h of ( h ) in local groups, there is an action of h on the fiber of L . As in § o ( h ) = ord( h ) the order of h . Then the action is given by h · z = e π √− b ( h ) L o ( h ) z with 1 ≤ b ( h ) L ≤ o ( h ) being the action weight of h on L . The order o ( h ) and action weight b ( h ) L areindependent of the choice of the representative h of ( h ) in local groups, and b ( h ) L is called the weight of( h ) on L . Definition 3.1.
For each ( h ) ∈ T D . Define l ( h ) := (cid:26) if ≤ b ( h ) L < o ( h ) , if b ( h ) L = o ( h ) . Since Y = P ( L ⊕ O D ) is the projectification of L with trivial weight a = (1), the analysis in § Lemma 3.2.
A component Y ( h ) = π − h ) ( D ( h )) of the fiber bundle I π : IY → ID is determined as follows: (a) if l ( h ) = 0 , then L ( h ) = D ( h ) and Y ( h ) is a disjoint union of two zero bundles over D ( h ) , i.e. Y ( h ) = D ( h ) ⊔ D ∞ ( h ) , and D ( h ) ∼ = D ∞ ( h ) ∼ = D ( h ) ; (b) if l ( h ) = 1 , then L ( h ) is a line bundle over D ( h ) and Y ( h ) = P ( L ( h ) ⊕ O D ( h ) ) .For the latter case, we have L ( h ) = e ∗ ( h ) L with e ( h ) : D ( h ) → D being the evaluation morphism. Moreover, Y ( h ) also contains the zero and infinitysections D ( h ) and D ∞ ( h ) , both are isomorphic to D ( h ) . Since a = (1) we identify ( h, e − π √− b ( h ) L o ( h ) ) with ( h ). RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 13
Take a basis of H ∗ CR ( D ) Σ ⋆ := G ( h ) ∈ T D Σ ( h ) := G ( h ) ∈ T D { θ h ) , . . . , θ k ( h )( h ) } . (3.2)(Since in this paper in most cases D is PE a , so for simplicity, we use the same notation as the basis (2.4) of H ∗ CR ( PE a ) to denote the basis of H ∗ CR ( D )). Then we get a basis of H ∗ CR ( Y ) as follows. When l ( h ) = 0, each θ i ( h ) contributes two elements: θ ,i ( h ) for D ( h ) , and θ ∞ ,i ( h ) for D ∞ ( h ) . When l ( h ) = 1, denote the Poincar´e dual of D ( h ) and D ∞ ( h ) in Y ( h ) = P ( L ( h ) ⊕ O D ( h ) ) by[ D ( h )] and [ D ∞ ( h )]respectively. We have [ D ( h )] = [ D ∞ ( h )] + c ( L ( h )) = [ D ∞ ( h )] + e ∗ ( h ) ( c ( L )) . (3.3)Then each θ i ( h ) contributes two elements for Y ( h ): θ i ( h ) and θ i ( h ) · [ D ( h )] . Combining these together we get a basis of H ∗ CR ( Y ): G ( h ) ∈ T D ( θ , h ) , . . . , θ , k ( h )( h ) ; θ ∞ , h ) , . . . , θ ∞ , k ( h )( h ) if l ( h ) = 0; θ h ) , . . . , θ k ( h )( h ) ; θ h ) · [ D ( h )] , . . . , θ k ( h )( h ) · [ D ( h )] if l ( h ) = 1.(3.4)3.2. Notations for relative orbifold Gromov–Witten invariants of ( D | Y | D ∞ ) . Now we fix thenotation for relative orbifold Gromov–Witten invariants of ( D | Y | D ∞ ). A relative invariant of ( D | Y | D ∞ )is of the form D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i γ l i (¯ h i ) (cid:12)(cid:12)(cid:12) ν E ( D | Y | D ∞ )Γ := 1 | Aut( µ ) | · | Aut( ν ) | (3.5) Z [ M Γ ( D | Y | D ∞ )] vir m Y i =1 ψ k i i ev ∗ i ( γ l i (¯ h i ) ) ∪ ℓ ( µ ) Y j =1 rev D , ∗ j ( θ s j ( h j ) ) ∪ ℓ ( ν ) Y k =1 rev D ∞ , ∗ k ( θ s ′ k ( h ′ k ) )with topological data (or type) Γ = ( g, β, (¯ h ) , ~µ, ~ν ), where • g is the genus, and β ∈ H ( | Y | ; Z ) is the homology class, • ¯ h = (cid:0) (¯ h ) , . . . , (¯ h m ) (cid:1) ∈ ( T Y ) m , and γ l i (¯ h i ) belongs to the basis (3.4); denote by ̟ = ( τ k γ l (¯ h ) , . . . , τ k m γ l m (¯ h m ) )the absolute insertions; denote the number of insertions in ̟ by k ̟ k = m ; • µ = (cid:16) ( µ , θ s ( h ) ) , . . . , ( µ ℓ ( µ ) , θ s ℓ ( µ ) ( h ℓ ( µ ) ) ) (cid:17) is a relative weighted partition and is weighted by the chosenbasis (3.2) of the Chen–Ruan cohomology of D with ~µ = (cid:0) ( µ , ( h )) , . . . , ( µ ℓ ( µ ) , ( h ℓ ( µ ) )) (cid:1) , and X j µ j = Z orb β [ D ] ≥ D of the relative maps; • ν = (cid:18) ( ν , θ s ′ ( h ′ ) ) , . . . , ( ν ℓ ( ν ) , θ s ′ ℓ ( ν ) ( h ′ ℓ ( ν ) ) ) (cid:19) is a relative weighted partition and is weighted by the chosenbasis (3.2) of the Chen–Ruan cohomology of D with ~ν = (cid:16) ( ν , ( h ′ )) , . . . , ( ν ℓ ( ν ) , ( h ′ ℓ ( ν ) )) (cid:17) , and X j ν j = Z orb β [ D ∞ ] ≥ D ∞ of the relative maps; • ev i , rev D j and rev D ∞ j are the evaluation maps from M Γ ( D | Y | D ∞ ) to IY , ID and ID ∞ at absolutemarked points and relative marked points respectively; • ψ i is the first Chern class of the i -th cotangent line bundle L i over the moduli space M Γ ( D | Y | D ∞ )of relative stable orbifold maps, whose fiber over a relative stable map is the cotangent line of thecoarse moduli space of the domain curve at the i -th absolute marked point. There is another i -thcotangent line bundle L i over M Γ ( D | Y | D ∞ ), whose fiber over a relative stable map is the cotangentline of the i -th absolute marked point on the domain curve. The first Chern class of L i is denotedby ψ i usually; we have L i = L ⊗ r i i , hence ψ i = r i ψ i where r i is the order of the local group of the i -th absolute marked point.For the relative weighted partition µ = (cid:16) ( µ , θ s ( h ) ) , . . . , ( µ ℓ ( µ ) , θ s ℓ ( µ ) ( h ℓ ( µ ) ) ) (cid:17) above we set(i) ˇ µ to be the dual relative weighted partition, which isˇ µ := (cid:16) ( µ , ˇ θ s ( h ) ) , . . . , ( µ ℓ ( µ ) , ˇ θ s ℓ ( µ ) ( h ℓ ( µ ) ) ) (cid:17) , therefore ~ ˇ µ = (cid:16) ( µ , ( h − )) , . . . , ( µ ℓ ( µ ) , ( h − ℓ ( µ ) )) (cid:17) , we will also denote ~ ˇ µ by ˇ ~µ ;(ii) z ( µ ) := | Aut( µ ) | · Q i µ i ;(iii) deg CR µ := ℓ ( µ ) P i =1 (deg θ ( h i ) + 2 ι D ( h i )) with ι D ( h i ) being the degree shifting number of the twistedsector D ( h i ) in D .Similar notations apply to ν .For the invariant (3.5) we also set( h ) = (cid:0) ( h ) , . . . , ( h ℓ ( µ ) ) (cid:1) and ( h ′ ) = (cid:16) ( h ′ ) , . . . , ( h ′ ℓ ( µ ) ) (cid:17) . Suppose I π : Y (¯ h i ) → D ( h i )i.e. π t (¯ h i ) = ( h i ) (cf. Lemma 3.2). Then we set ( h ) = (cid:0) ( h ) , . . . , ( h m ) (cid:1) andΓ = (cid:0) ( h ) , ( h ) , ( h ′ ) (cid:1) . Denote by n = h ) + h ) + h ′ ) = m + ℓ ( µ ) + ℓ ( ν )the number of absolute marked points and relative marked points. Finally we set ~r = ( r , . . . , r n ) := (cid:0) ord( h ) , . . . , ord( h m ) , ord( h ) , . . . , ord( h ℓ ( µ ) ) , ord( h ′ ) , . . . , ord( h ′ ℓ ( ν ) ) (cid:1) where for example the order ord( h ) is the order of a representative of ( h ) in local groups and does notdepend on the choices of representatives.Similar notations also apply to relative invariants of ( Y | D ) and ( Y | D ∞ ). We will also deal with rubberinvariants of ( D | Y | D ∞ ), where we use a superscript “ ∼ ” to indicate rubber invariants.The orbifold fibration π : Y = P ( L ⊕ O D ) → D induces a fibration of coarse space | π | : | Y | → | D | , whichis a topological P -fiber bundle. A class β ∈ H ( | Y | ; Z ) is called a fiber class if | π | ∗ ( β ) = 0 . For a relative invariant of ( D | Y | D ∞ ) as (3.5), we call it a fiber class invariant if the homology class β isa fiber class in H ( | Y | ; Z ). We next study fiber class invariants of ( D | Y | D ∞ ). RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 15
The moduli spaces of fiber class invariants.
Now consider a fiber class invariant of the form as(3.5) D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i γ l i (¯ h i ) (cid:12)(cid:12)(cid:12) ν E ( D | Y | D ∞ )Γ . So the homology class β in the topological data Γ = ( g, β, (¯ h ) , ~µ, ~ν ) is a fiber class.We next analyze the structure of the corresponding moduli space M Γ ( D | Y | D ∞ ), which we denotesimply by M Γ . It consists of equivalence classes of stable representable pseudo-holomorphic morphismsfrom (nodal) orbifold Riemann surfaces to Y of topological type indicated by Γ.We next describe a typical element in M Γ . We first recall the construction of expanded (or degenerate)targets. Given two nonnegative integers l , l ∞ , let Y [ l , l ∞ ] be the degenerate orbifold groupoid (cf. [13]) Y [ l , l ∞ ] := Y ∧ D ∞ . . . ∧ D ∞ Y | {z } l ∧ D ∞ Y ∧ D Y ∧ D . . . ∧ D Y | {z } l ∞ , (3.6)i.e. we glue l copies of Y to the original Y along the infinite section and l ∞ copies of Y to the original Y along the zero section. To distinguish them we number them as follows Y [ l , l ∞ ] = Y − l ∧ D ∞ . . . ∧ D ∞ Y − ∧ D ∞ Y ∧ D Y ∧ D . . . ∧ D Y l ∞ . We call Y [ l , l ∞ ] an expanded target when l + l ∞ >
0. The Y in Y [ l , l ∞ ] is called the root , and the restpart is called the rubber . When l = l ∞ = 0, Y [0 ,
0] = Y = Y is called the unexpanded target . Denote thezero section of Y i by D ,i and the infinite section of Y i by D ∞ ,i for − l ≤ i ≤ l ∞ . So from (3.6) we see that Y [ l , l ∞ ] is obtained by gluing D ,i in Y i with D ∞ ,i − in Y i − for − l + 1 ≤ i ≤ l ∞ . So the singular set of Y [ l , l ∞ ] is Sing Y [ l , l ∞ ] = ⊔ l ∞ − i = − l D ∞ ,i = ⊔ l ∞ i = − l +1 D ,i . Let Aut rel l ,l ∞ := Aut( Y [ l , l ∞ ] , Sing Y [ l , l ∞ ] ⊔ Y ) be the group of automorphisms of Y [ l , l ∞ ] preservingthe singular set Sing Y [ l , l ∞ ] and the root Y . Then Aut rel l ,l ∞ ∼ = ( C ∗ ) l + l ∞ , where each factor of ( C ∗ ) l + l ∞ dilates the fibers of the i -th orbifold P -bundle for i = 0. We have a natural map from Y [ l , l ∞ ] to the root, Y [ l , l ∞ ] → Y = Y , which contracts all Y i to D , for i < Y i to D ∞ , for i > D | Y | D ∞ ) of topological type Γ is a triple( ǫ, C ′ , f ) : C ǫ ←− C ′ f −→ Y [ l , l ∞ ]consists of the following ingredients:(1) C is a genus g (nodal) orbifold Riemann surface with (possible orbifold) absolute marked points x =( x , . . . , x m ) and (possible orbifold) relative marked points y = ( y , . . . , y ℓ ( µ ) ) and z = ( z , . . . , z ℓ ( ν ) ),(2) ǫ is a refinement of orbifold groupoid by an open cover of the object space,(3) f is pseudo-holomorphic and the induced map | f | : | C | → | Y [ l , l ∞ ] | → | Y | = | Y | on coarse spacessatisfies | f | ∗ [ | C | ] = β ∈ H ( | Y | ; Z ),(4) the marked points (resp. nodal points) in the coarse space | C | are divided into absolute markedpoints (resp. nodal points) and relative marked points (resp. nodal points) as follows,(4.1) the absolute marked points | x | and absolute nodal points are mapped into the nonsingular partof | Y [ l , l ∞ ] | , i.e. | Y [ l , l ∞ ] | − | Sing( Y [ l , l ∞ ]) | ,(4.2) the relative marked points | y | are mapped into | D , − l | and | f | − ( | D , − l | ) consists of onlyrelative marked points | y | , and the intersection multiplicities, i.e. contact orders, are given by µ such that the sum of all contact orders equals to D · β ,(4.3) the relative marked points | z | are mapped into | D ∞ ,l ∞ | and | f | − ( | D ∞ ,l ∞ | ) consists of onlyrelative marked points | z | , and the intersection multiplicities, i.e. contact orders, are given by ν such that the sum of all contact orders equals to D ∞ · β ,(4.4) the relative nodal points are mapped into | Sing( Y [ l , l ∞ ]) | and | f | − ( | Sing( Y [ l , l ∞ ]) | ) consistsof only relative nodal points,(4.5) the relative nodal points in | f | − ( | Sing( Y [ l , l ∞ ]) | ) satisfy the balanced condition that for eachnode q ∈ | f | − ( | D ,i | = | D ∞ ,i − | ), i = − l + 1 , . . . , l ∞ , the two branches of the domain curve | C | at the nodal point q are mapped to different irreducible components of | Y [ l , l ∞ ] | and thecontact orders to D ,i = D ∞ ,i − are equal,(5) f is representable, i.e. the induced maps on local groups are injective. The equivalence relation between such maps are generated by the following relations:(i) We say C ǫ ←− C ′ f −→ Y [ l , l ∞ ] is equivalent to C ˜ ǫ ←− ˜ C ′ ˜ f −→ Y [ l , l ∞ ] if there is a natural transformation(see for example [8]) α : f ◦ π ⇒ ˜ f ◦ π : C ′ × ǫ, C , ˜ ǫ ˜ C ′ → Y [ l , l ∞ ] . (ii) For an automorphism φ ∈ Aut rel l ,l ∞ , we say C ǫ ←− C ′ f −→ Y [ l , l ∞ ] is equivalent to C ǫ ←− C ′ φ ◦ f −−→ Y [ l , l ∞ ].A map C ǫ ←− C ′ f −→ Y [ l , l ∞ ] is called a stable map if its self-equivalences are finite. For simplicity, wewill also denote such a map by ( ǫ, C ′ , f ) : ( C , x , y , z ) → ( D | Y | D ∞ ) or ( ǫ, C ′ , f ). M Γ ( D | Y | D ∞ ) is the space(in fact groupoid) of stable maps of topological type Γ. The arrow space of M Γ ( D ∞ | Y | D ∞ ) consists ofequivalences between stable maps. Theorem 3.3.
The moduli space M Γ ( D | Y | D ∞ ) is a fibration over a certain multi-sector D ′ Γ of D thatdetermined by Γ , whose fiber is the relative moduli space of stable maps into ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]) for certain P -orbifold [ P ⋊ K Γ ] = ( K Γ × P ⇒ P ) . The finite group K Γ , the P -orbifold [ P ⋊ K Γ ] andthe topological data of the relative moduli space of ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]) are all determined by Γ . The rest of this subsection is devoted to explain and prove this theorem.3.3.1.
Universal curve of orbifold Riemann surfaces.
Let T g,n , g − n ≥ g , n marked Riemann surfaces (without nodal points). Let π : C g,n → T g,n be the correspondinguniversal curve, equipped with n canonical sections σ i , ≤ i ≤ n corresponding to those n marked points.Take a point b ∈ T g,n , then we have a stable curve C b := ( π − ( b ) , σ ( b ) , . . . , σ n ( b )) . Consider the punctured surface C ◦ b := C b \ { σ ( b ) , . . . , σ n ( b ) } . It has a canonical hyperbolic metric ofconstant curvature −
1. Then around every puncture we have a series of horocycles. The mapping classgroup
M P g,n acts on T g,n and also on C g,n . We could take n M P g,n -invariant positive functions δ i over T g,n , and use them as the radius of the horocycles at the n punctures. Then for each C ◦ b , at the i -thpuncture we remove a horodisc whose horocycle is of length δ i .In other words, for each C b we pick out n discs around the n marked points such that these discsare M P g,n -invariant. These discs together give rise to a tubular neighborhood U i of σ i ( T g,n ) in C g,n for1 ≤ i ≤ n . See for example [14].Now we construct a family of orbifold Riemann surfaces. We first cut out the sections of marked points n G i =1 σ i ( T g,n )from C g,n to get C ◦ g,n , and then patch the Z r i ⋉ U i back to C ◦ g,n . Here we identify Z r i = h ζ i i where ζ i = e π √− ri , (3.7)and Z r i acts on U i by rotating the horodiscs. Then we get an effective orbifold C g,n . Moreover, from C ◦ g,n and Z r i ⋉ U i we get an orbifold groupoid that represents C g,n , which we still denote by C g,n . The objectspace of this groupoid is C g,n := C ◦ g,n ⊔ n G i =1 U i . The arrow space C g,n is obtained accordingly, since C g,n is effective. So this orbifold groupoid is C g,n =( C g,n ⇒ C g,n ).We have a natural projection π : C g,n → T g,n . RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 17 which on C g,n is obtained from the natural projections C ◦ g,n ֒ → C g,n → T g,n and U i → T g,n . The fiber ofthis projection π : C g,n → T g,n is an orbifold Riemann surface with n orbifold points whose local groupsare Z r i , ≤ i ≤ n . This is a family of orbifold Riemann surfaces over T g,n .Moreover by the choices of U i , M P g,n acts on C g,n . First of all, it acts on C g,n = C ◦ g,n ⊔ F i U i . Secondly,since the horodiscs are M P g,n -equivariant, and the Z r i -action also commutes with the M P g,n -action onhorodiscs,
M P g,n acts on C g,n . Again, by the choices of horodiscs, the projection C g,n → T g,n is M P g,n -equivariant. Note that
M P g,n -action on C g,n and C g,n are both free. So we get a family of orbifold Riemannsurface C g,n /M P g,n → T g,n /M P g,n . (3.8)Note that the quotient T g,n /M P g,n is the top strata M g,n of M g,n , the Deligne–Mumford moduli space ofRiemann surfaces. The groupoid corresponding to this top strata is M g,n = T g,n ⋊ M P g,n = (
M P g,n × T g,n ⇒ T g,n ) . In terms of groupoid, we could write (3.8) as( C g,n × M P g,n ⇒ C g,n ) → ( M P g,n × T g,n ⇒ T g,n ) = M g,n . That is C M := ( C g,n × M P g,n ⇒ C g,n ) corresponds to C g,n /M P g,n . So C M is a family of orbifold Riemannsurfaces over M g,n . We can view it as a universal curve of orbifold Riemann surfaces over M g,n . In thesame way we could construct the universal curve of orbifold Riemann surfaces over lower strata of M g,n .3.3.2. The top strata M Γ . We next first study the top strata M Γ of M Γ ( D | Y | D ∞ ). By top strata we meanthat for each stable maps in M Γ its domain curve has no nodal points. Hence the target is the unexpandedtarget, i.e. Y itself. Take a fiber C b = π − ( b ) of C g,n over a point b ∈ T g,n . So C b is a genus g orbifoldRiemann surface with n (possible orbifold) marked points. As above we denote these n = m + ℓ ( µ ) + ℓ ( ν )marked points orderly by ( x , y , z ) with x = ( x , . . . , x m ), y = ( y , . . . , y ℓ ( µ ) ) and z = ( z , . . . , z ℓ ( ν ) ).Now consider the groupoid of stable morphisms from ( C b , x , y , z ) to ( D | Y | D ∞ ) of topological type Γ Hol Γ ( C b , Y ) = (Hol ( C b , Y ) ⇒ Hol ( C b , Y )) , where(i) Hol ( C b , Y ) is the space of stable morphisms from C b to Y of topological type Γ,(ii) Hol ( C b , Y ) is the space of natural transformations of stable morphisms in Hol ( C b , Y ).As a groupoid, the object space of the top strata M Γ is [ b ∈ T g,n Hol ( C b , Y ) . The arrow space consists of two parts. The first part is [ b ∈ T g,n Hol ( C b , Y ) . The second part comes from the action of mapping class group. The action of mapping class group commuteswith the action of S b ∈ T g,n Hol ( C b , Y ). This is similar to the arrow space of C M .We next study the structure of holomorphic morphisms in S b ∈ T g,n Hol ( C b , Y ).In the next, we omit the subscript b of C b to simplify notations. Since the curve class β in Γ is a fiberclass of the fiber bundle π : Y = ( Y ⇒ Y ) = D ⋉ Y → D = ( D ⇒ D ), by [10, Theorem 7.4] we couldassume that every morphism C ǫ ←− C ′ = ( C ′ ⇒ C ′ ) f =( f ,f ) −−−−−−→ Y in Hol ( C , Y ) satisfies that π ( f ( C ′ )) is a point in D , i.e. the image f ( C ′ ) lies in a single fiber of Y .Now take a morphism C ǫ ←− C ′ = ( C ′ ⇒ C ′ ) f =( f ,f ) −−−−−−→ Y ∈ Hol ( C , Y ). Suppose π ◦ f ( C ′ ) = p f ∈ D .Let G p f be the local group of p f in D . The fiber of Y → D over p f is P ( L p f ⊕ C ), and the fiber of Y → D over p f is modeled by [ P ( L p f ⊕ C ) ⋊ G p f ] where G p f acts on P ( L p f ⊕ C ) via acting on L p f . So C ǫ ←− C ′ = ( C ′ ⇒ C ′ ) f =( f ,f ) −−−−−−→ Y factors through C C ′ ǫ o o f / / [ P ( L p f ⊕ C ) ⋊ G p f ] (cid:31) (cid:127) / / Y . By a result of Chen–Ruan (cf. [2, Theorem 2.54]), the representable pseudo-holomorphic map f : C ′ → [ P ( L p f ⊕ C ) ⋊ G p f ] ֒ → Y is determined by the homomorphism ρ f : π orb1 ( C ′ ) → G p f , where π orb1 ( C ′ ) = π orb1 ( C ) is the orbifold fundamental group of C ′ and has a representation π orb1 ( C ′ ) = * λ , . . . , λ n , α , . . . , α g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y i =1 λ i g Y j =1 ( α j − α j α − j − α − j ) = 1 , λ r i i = 1 + , with λ i corresponding to those marked points, i.e. to the generators ζ i of local groups of those markedpoints (cf. (3.7)), and α j corresponding to the generators of the fundamental group of the coarse space | C ′ | = | C | (a smooth genus g Riemann surface). Suppose the images of the generators of π orb1 ( C ′ ) w.r.t ρ f are ρ f ( λ i ) = h i ∈ G p f , ρ f ( α j ) = ˜ h j ∈ G p f . Then π ◦ f is determined by the ( n + 2 g )-tuple h f := ( h , . . . , h n , ˜ h , . . . , ˜ h g ) . This ( n + 2 g )-tuple gives rise to a point in the object space of the ( n + 2 g )-th multiple-sector D [ n +2 g ] =(( D [ n +2 g ] ) ⇒ ( D [ n +2 g ] ) ) = D ⋉ ( D [ n +2 g ] ) of D . So p f ∈ e (( D [ n +2 g ] ) ) ⊆ D .So we have a map ρ : Hol ( C , Y ) → ( D [ n +2 g ] ) , ( ǫ, C ′ , f ) h f . Moreover, an C ǫ ←− C ′ = ( C ′ ⇒ C ′ ) f =( f ,f ) −−−−−−→ Y in Hol ( C , Y ) further factors through C C ′ ǫ o o ¯ f / / [ P ( L p f ⊕ C ) ⋊ h h f i ] (cid:31) (cid:127) / / [ P ( L p f ⊕ C ) ⋊ G p f ] (cid:31) (cid:127) / / Y . (3.9)Meanwhile, Γ determines a topological data Γ such that the induced morphism C ǫ ←− C ′ ¯ f −→ [ P ( L p f ⊕ C ) ⋊ h h f i ]to the P -orbifold [ P ( L p f ⊕ C ) ⋊ h h f i ] is of topological type Γ. Explicitly, the genus, homology class andmarked points of Γ are the same as the Γ; the only difference is that the twisted sectors of Γ are inducedfrom those of Γ via viewing those h i as elements in h h f i instead of G p f . By this further factorization (3.9)we see Lemma 3.4. ρ : Hol ( C , Y ) → ( D [ n +2 g ] ) is a fibration over its image whose fiber over a point h =( h , . . . , h n , ˜ h , . . . , ˜ h g ) is g Hol ( C , [ P ( L e ( h ) ⊕ C ) ⋊ h h i ]):= n C ǫ ←− C ′ f −→ [ P ( L e ( h ) ⊕ C ) ⋊ h h i ] (cid:12)(cid:12)(cid:12) ( ǫ, C ′ , f ) is of type Γ and ρ f ( λ i ) = h i , ρ f ( α j ) = ˜ h j o . Denote the image of ρ by D ⊆ ( D [ n +2 g ] ) . Then we get a sub-groupoid D [ n +2 g ] | D , which we denotedby D Γ = ( D ⇒ D ). So D ⊆ ( D [ n +2 g ] ) × e,s D . As the morphism is representable, the orders of h i is the same as the orders of λ i , i.e. r i for 1 ≤ i ≤ n . The multiple-sector D [ n +2 g ] is defined as follows (cf. [2]). The object space is( D [ n +2 g ] ) := { ( g , . . . , g n +2 g ) ∈ ( D ) n +2 g | s ( g i ) = t ( g j ) , ≤ i, j ≤ n + 2 g } . The D -action on ( D [ n +2 g ] ) is given by the anchor map e : ( D [ n +2 g ] ) → D , e ( g , . . . , g n +2 g ) = s ( g ) and the action map( D [ n +2 g ] ) × e,s D → ( D [ n +2 g ] ) , ( g , . . . , g n +2 g ; h ) ( h − g h, . . . , h − g n +2 g h ) . So ( D [ n +2 g ] ) = ( D [ n +2 g ] ) × e,s D . RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 19
Lemma 3.5.
With ρ : Hol ( C , Y ) → D as the anchor map, there is a D Γ -action on Hol ( C , Y ) given asfollows. For an arrow ( h , h ) ∈ D and a morphism C ǫ ←− C ′ f −→ Y with h f = h (i.e. C ǫ ←− C ′ f −→ Y belongs tothe fiber of Hol ( C , Y ) over h ), h acts on C ǫ ←− C ′ f −→ Y by transforming the image of f from the fiber of Y over s ( h ) to the fiber of Y over t ( h ) and conjugate f by h , in particular, it transfer h f = h into h − · h · h . Denote the resulting action groupoid by D Γ ⋉ Hol ( C , Y ) . Then again by the factorization (3.9) we have
Lemma 3.6.
Hol Γ ( C , Y ) ∼ = D Γ ⋉ Hol ( C , Y )Therefore we have a groupoid projection Hol Γ ( C , Y ) → D Γ . Note that D Γ may have several connected components (i.e. connected components of its coarse space). In asingle component, the group h h i is invariant up to conjugation. For simplicity, in the following we assumethat D Γ has only one component, hence the group h h i is invariant up to conjugation. Otherwise, we onlyneed to deal with components separately. Now we fix an h Γ ∈ D and set K Γ = h h Γ i . Then for every h ∈ D , K Γ ∼ = h h i via conjugation. Moreover,this isomorphism also identifies the K Γ -action on L e ( h Γ ) with the h h i -action on L e ( h ) . Then by Lemma 3.4and Lemma 3.5 we see that the fiber of ρ : Hol ( C , Y ) → D are all isomorphic to g Hol ( C , [ P ( L e ( h Γ ) ⊕ C ) ⋊ K Γ ])(3.10) := n C ǫ ←− C ′ f −→ [ P ( L e ( h Γ ) ⊕ C ) ⋊ K Γ ] (cid:12)(cid:12) ( ǫ, C ′ , f ) is of type Γ and ( . . . , ρ f ( λ i ) , . . . , ρ f ( α j ) , . . . ) = h Γ o . Moreover, this fibration ρ : Hol ( C , Y ) → D is locally trivial.By viewing Hol Γ ( C , Y ) → D Γ as a groupoid fibration (cf. [10]), we see that Hol Γ ( C , Y ) is a fibration over D Γ with fiber being the unitary/trivial groupoid (cf. [35]) g Hol ( C , [ P ( L e ( h Γ ) ⊕ C ) ⋊ K Γ ]) ⇒ g Hol ( C , [ P ( L e ( h Γ ) ⊕ C ) ⋊ K Γ ])associated to the space g Hol ( C , [ P ( L e ( h Γ ) ⊕ C ) ⋊ K Γ ]) = g Hol ( C , [ P ⋊ K Γ ]).We next interpret Hol Γ ( C , Y ) as a groupoid fibration over another groupoid D ′ Γ with fiber being groupoidsthat correspond to stable maps to the P -orbifold [ P ⋊ K Γ ]. We first construct the groupoid D ′ Γ . It is similarto the construction of the effective orbifold groupoid for an ineffective orbifold groupoid (cf. [2, Definition2.33]).Consider the following subspace of D :ker D := { ( h , h ) ∈ D | h ∈ C h h i ( h ) ⊆ C G e ( h ) ( h ) } , where, as above, G e ( h ) is the local (or isotropy) group of e ( h ) ∈ D in D and h h i is the subgroup generatedby h , C G e ( h ) ( h ) is the centralizers of h in G e ( h ) , and C h h i ( h ) is the centralizers of h in h h i , hence the centerof h h i as h h i is generated by h . We define a relation on D by( h , h ) ∼ ( h , kh )(3.11)for all k ∈ C h h i ( h ). This is obvious an equivalence relation. Remark 3.7.
In fact, the restrictions of structure maps of D over ker D gives rise to an orbifold groupoidker D Γ := (ker D ⇒ D ). Then (3.11) gives rise to an action of ker D on D , whose anchor map is thesource map s : D → D . So (3.11) is an equivalence relation. We set D ′ , := D / ∼ . Then one see that all structure maps of D Γ descends to D ′ , and D . Moreover, D ′ Γ := ( D ′ , ⇒ D )is an orbifold groupoid. Remark 3.8.
As all h ∈ D are conjugate to each other, ker D Γ is a trivial bundle over D with fiber being C K Γ ( h Γ ). So D Γ → D ′ Γ is a C K Γ ( h Γ )-gerbe over D ′ Γ .On the other hand, for a fixed h ∈ D , C h h i ( h ) is a normal subgroup of C G e ( h ) ( h ) and C h h i ( h ) ⋉ g Hol ( C , P ( L e ( h ) ⊕ C ) ⋊ h h i ) = Hol Γ ( C , P ( L e ( h ) ⊕ C ) ⋊ h h i )is the groupoid of morphisms of type Γ from C to P ( L e ( h ) ⊕ C ) ⋊ h h i . As every h is conjugate to h Γ and K Γ = h h Γ i , we have Hol Γ ( C , P ( L e ( h ) ⊕ C ) ⋊ h h i ) ∼ = Hol Γ ( C , P ⋊ K Γ ) . Lemma 3.9.
The composition of projections
Hol Γ ( C , Y ) → D Γ → D ′ Γ makes Hol Γ ( C , Y ) a fibration over D ′ Γ with fiber Hol Γ ( C , P ⋊ K Γ ) . We denote this groupoid fibration by Hol Γ ( C , P ⋊ K Γ ) ֒ → Hol Γ ( C , Y ) → D ′ Γ . See for [10, §
3] for the definition of groupoid fibration. One can think it as a groupoid fiber bundle withfiber being groupoids.
Proof.
By Lemma 3.6 we only have to show that the composed projection
Hol Γ ( C , Y ) → D Γ → D ′ Γ has fiber being isomorphic to Hol Γ ( C , P ⋊ K Γ ).First of all, the projection on object space is ρ : Hol ( C , Y ) → D . Secondly, on arrows, the projection is the composition ρ : D × s,D ,ρ Hol ( C , Y ) → D Γ → D ′ , which is (( h , h ); C ǫ ←− C ′ f −→ Y ) [ h , h ] , where [ h , h ] is the equivalence class of ( h , h ) in D ′ , = D / ∼ .Now we consider the fiber of this projection. Take a point h ∈ D and consider the identity arrow[ h , p h ] ∈ D ′ , , where p h = e ( h ) ∈ D . Then the inverse images of h and [ h , p h ] are( ρ ) − ( h ) = g Hol ( C , P ( L p h ⊕ C ) ⋊ h h i ) , and ( ρ ) − ([ h , p h ]) = C h h i ( h ) × g Hol ( C , P ( L p h ⊕ C ) ⋊ h h i )respectively. Therefore, the fiber of ρ over ([ h , p h ] ⇒ h ) is C h h i ( h ) ⋉ g Hol ( C , P ( L p h ⊕ C ) ⋊ h h i ) ∼ = Hol Γ ( C , P ⋊ K Γ ) . This finishes the proof. (cid:3)
Now we vary C in C g,n . Note that the M P g,n -action on S b ∈ T g,n Hol ( C b , Y ) commutes with the D Γ -action,we see that the top strata M Γ ( D | Y | D ∞ ) is a fibration over D ′ Γ with fiber M Γ ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]). RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 21
The moduli space M Γ . Now for the whole moduli space M Γ we have Lemma 3.10.
The moduli space M Γ ( D | Y | D ∞ ) is a groupoid fibration over D ′ Γ with fiber being M Γ ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]) .Proof. Consider a general stable map C ǫ ←− C ′ f −→ Y [ l , l ∞ ] of topological type Γ. As above by [10, Theorem7.4] we could assume that the image of objects f ( C ′ ) is a point p f in D . Then as the analysis in § f over each irreducible component of C ′ is determined by the corresponding homomorphismbetween orbifold fundamental group of the irreducible component and the local group G p f of p f in D . Eachnodal point q in C contributes a generator, say λ q, + or λ q, − of finite order (determined by the twistedsectors of D , D ∞ or Y that this nodal point mapped into), to the orbifold fundamental groups of the twoirreducible branches of C ′ at q respectively. Then one see that the induced homomorphisms on orbifoldfundamental groups of irreducible components of C ′ must satisfy that the images of λ q, + , λ q, − are inverseto each other for each nodal point q . Conversely, if the homomorphisms on orbifold fundamental groupsof irreducible components of C ′ satisfy that the images of λ q, + , λ q, − are inverse to each other for all nodalpoints, then they determine an f . Then one see that similar results as Lemma 3.4, Lemma 3.5 and Lemma3.6 hold for general stable maps. Therefore we have the analogue of Lemma 3.9. (cid:3) This also finishes the proofs of Theorem 3.3.
Remark 3.11.
The moduli spaces for fiber class relative invariants of ( Y | D ) and ( Y | D ∞ ) have similarstructures.More generally, one can consider the weighted projectification E a of general orbifold vector bundles E → S and the fiber class relative Gromov–Witten invariants of ( E a | PE a ). The corresponding moduli spaces havesimilar structure as Theorem 3.3 and Lemma 3.10 state, that is they are fibrations over certain multi-sectorsof S with fibers being relative moduli spaces of quotients of ( P a , | P a ) by certain finite groups.3.4. The invariants.
We next compute the invariants. Consider the deformation-obstruction theory of M Γ ( D | Y | D ∞ ), which at a point ( ǫ, C ′ , f ) : ( C , x , y , z ) → ( D | Y | D ∞ ) is0 → Aut( C , x , y , z ) → Def( f ) → T C , x , y , z , f ) → (3.12) → Def( C , x , y , z ) → Obs( f ) → T C , x , y , z , f ) → , where Aut( C , x , y , z ) is the space of infinitesimal automorphism of the domain ( C , x , y , z ), Def( C , x , y , z ) is thespace of infinitesimal deformation of the domain ( C , x , y , z ), Def( f ) = H ( C , f ∗ T Y ( − D − D ∞ )) is the spaceof infinitesimal deformation of the map f , and Obs( f ) = H ( C , f ∗ T Y ( − D − D ∞ )) is the space of obstructionto deforming f .According to the splitting of T Y ( − D − D ∞ ) into fiber part and base part, the (3.12) splits into0 → Aut( C , x , y , z ) → H ( C , f ∗ T fiber Y ( − D − D ∞ )) → T C , x , y , z , f ) → (3.13) → Def( C , x , y , z ) → H ( C , f ∗ T fiber Y ( − D − D ∞ )) → T C , x , y , z , f ) → , and 0 → → H ( C , ( π ◦ f ) ∗ T D ) → T C , x , y , z , f ) → (3.14) → → H ( C , ( π ◦ f ) ∗ T D ) → T C , x , y , z , f ) → , where π : Y → D .Fiberwisely, T C , x , y , z , f ) − T C , x , y , z , f ) is the deformation-obstruction theory of M Γ ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]). On the other hand from (3.14) we have T C , x , y , z , f ) − T C , x , y , z , f ) = H ( C , ( π ◦ f ) ∗ T D ) − H ( C , ( π ◦ f ) ∗ T D ) . It gives rise to the obstruction theory of stable maps of genus g , degree zero and type Γ in D . By theindex theorem of Chen–Ruan [16, Theorem 4.2.2] and the same proof of [25, Theorem 3.2] (see also [11,Proposition 1]) we haverank ( T C , x , y , z , f ) − T C , x , y , z , f ) ) = dim D (1 − g ) − ι D ( h ) − ι D ( h ′ ) − ι D ( h )= − dim D · g + (dim D − ι D ( h ) − ι D ( h ′ ) − ι D ( h )) = − dim D · g + dim D Γ − rank E Γ where E Γ is the obstruction bundle over D Γ defined in [16, § ∂ operator.The fiberwise deformation-obstruction theory gives rise to the π -relative virtual fundamental class[ M Γ ( D | Y | D ∞ )] vir π , and we have [ M Γ ( D | Y | D ∞ )] vir = ( c top ( E ⊠ T D ) ∪ c top ( E Γ )) ∩ [ M Γ ( D | Y | D ∞ )] vir π where E is the Hodge bundle over M Γ ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]).Note that c top ( E ⊠ T D ) = X q h q ( c ( E ) , c ( E ) , . . . ) t q ( c ( T D ) , c ( T D ) , . . . ) . where h q and t q are polynomials.Therefore the invariant (3.5) is Z [ M Γ ( D | Y | D ∞ )] vir m Y i =1 ψ k i i ev ∗ i ( γ l i (¯ h i ) ) ∪ ℓ ( µ ) Y j =1 rev D , ∗ j ( θ s j ( h j ) ) ∪ ℓ ( ν ) Y k =1 rev D ∞ , ∗ k ( θ s ′ k ( h ′ k ) )= Z [ M Γ ( D | Y | D ∞ )] vir π m Y i =1 ψ k i i ev ∗ i ( γ l i (¯ h i ) ) ∪ ℓ ( µ ) Y j =1 rev D , ∗ j ( θ s j ( h j ) ) ∪ ℓ ( ν ) Y k =1 rev D ∞ , ∗ k ( θ s ′ k ( h ′ k ) ) ∪ c top ( E ⊠ T D ) ∪ c top ( E Γ )Denote the integrand m Y i =1 ev ∗ i ( γ l i (¯ h i ) ) ∪ ℓ ( µ ) Y j =1 rev D , ∗ j ( θ s j ( h j ) ) ∪ ℓ ( ν ) Y k =1 rev D ∞ , ∗ k ( θ s ′ k ( h ′ k ) )by Ξ. As γ l i ¯ h i belongs to the basis (3.4), so it is of the form θ l ( h ) or θ l ( h ) · [ D ( h )]. Let Ξ F denote those possiblefactor [ D ( h i )] coming from γ l i ¯ h i , and Ξ D denote the rest part. So all classes in Ξ D are pullback classes from H ∗ CR ( D ). Then the invariant (3.5) is Z [ M Γ ( D | Y | D ∞ )] vir m Y i =1 ψ k i i ∧ Ξ(3.15) = Z [ M Γ ( D | Y | D ∞ )] vir π m Y i =1 ψ k i i ∧ Ξ ∧ c top ( E ⊠ T D ) ∪ c top ( E Γ )= X q Z D ′ Γ n Ξ D ∪ c top ( E Γ ) ∪ t q Z [ M Γ ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ])] vir Ξ F ∪ n Y i =1 ψ k i i ∪ h q o The Hodge integrals in the relative Gromov–Witten invariants of ([0 ⋊ K Γ ] | [ P ⋊ K Γ ] | [ ∞ ⋊ K Γ ]) can becomputed via virtual localizations, and the computation of the double ramification cycles on the modulispaces of admissible covers of Tseng and You [44]. The Hodge integrals reduces to the Hodge integralsover M ( BK Γ ), which can be removed by the orbifold quantum Riemann–Roch of Tseng [42]. Finally thedescendent integrations over the moduli spaces of stable curves are determined by Witten’s conjecture [47],equivalently Kontsevich’s theorem [30]. Remark 3.12.
Similar analysis applies to fiber class relative invariants of ( Y | D ) and ( Y | D ∞ ), and ananalogue of (3.15) holds for fiber class relative invariants of ( Y | D ) and ( Y | D ∞ ). Therefore, every fiber classrelative invariant of ( Y | D ) or ( Y | D ∞ ) reduces respectively to relative invariants of ([ P ⋊ G ] | [0 ⋊ G ]) or([ P ⋊ G ] | [ ∞ ⋊ G ]), where G is a finite group determined by the invariant and acts on P = P ( C ⊕ C ) byacting on the first C linearly and on the second C trivially. All relative invariants of such ([ P ⋊ G ] | [0 ⋊ G ])and ([ P ⋊ G ] | [ ∞ ⋊ G ]) were determined by Tseng and You [44]. RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 23 Relative orbifold Gromov–Witten theory of weighted projectification
In this section we determine relative orbifold Gromov–Witten theory of ( E a | PE a ). As in § E a | PE a ) D m Y i =1 τ k i γ l i (¯ h i ) (cid:12)(cid:12)(cid:12) µ E ( E a | PE a )Γ := 1 | Aut( µ ) | Z [ M Γ ( E a | PE a )] vir m Y i =1 ψ k i i ev ∗ i ( γ l i (¯ h i ) ) ∪ ℓ ( µ ) Y j =1 rev ∗ j ( θ r j ( h j ) )(4.1)with topological data Γ = ( g, β, (¯ h ) , ~µ ), where ~µ = (cid:0) ( µ , ( h )) , . . . , ( µ ℓ ( µ ) , ( h ℓ ( µ ) )) (cid:1) , and P j µ j = R orb β [ PE a ] ≥ PE a of the relative maps. The absolute insertions γ l i (¯ h i ) belongto the basis (2.5) of H ∗ CR ( E a ), and the relative insertions θ r j ( h j ) belong to the basis (2.4) of H ∗ CR ( PE a ). Asin § ̟ = ( τ k γ l (¯ h ) , . . . , τ k m γ l m (¯ h m ) )the absolute insertions and set k ̟ k = m to be the number of insertions in ̟ .4.1. Localization.
There is a fiberwise C ∗ -action on E a coming from the fiberwise C ∗ -dilation on E . Wenext apply the relative virtual localization with respect to this C ∗ -action to compute the relative invariant(4.1). The fixed loci in E a consist of the zero section S , and the infinity section PE a . The fixed linesconnecting them are lines in the fiber of E a → S that connect a point in S and points in PE a , whichcorrespond to lines in P a , connecting [0 , . . . , ,
1] with [ z , . . . , z n , M Γ , with Γ = ( g, β, (¯ h ) , µ )denoting the topological data. As in § M Γ consists of two types, those mapped to theunexpanded target ( E a | PE a ), and those mapped to an expanded target ( E a [ l ] | PE a ). Here E a [ l ] is obtainedin a parallel way of (3.6) as follows.Denote the normal line bundle of PE a in the weight- a projectification E a of E by L (which is O PE a (1),the dual line bundle of O PE a ( − Y = P ( L ⊕ O PE a ) (cf. § D and the infinity section D ∞ . Both are isomorphic to PE a . Take l copies of Y . Denotethe zero section and infinity section of the i -th copy of Y by D ,i and D ∞ ,i , ≤ i ≤ l . We first glue the l copies of Y together to get Y [ l ] via identifying D ,i with D ∞ ,i +1 for 1 ≤ i ≤ l −
1. Then E a [ l ] is obtained bygluing E a with Y [ l ] by identifying PE a with D ∞ , ∈ Y [ l ]. We also denote the PE a in E a by D , . ThenSing( E a [ l ]) = l − G i =0 D ,i = l G i =1 D ∞ ,i . As for Y [ l , l ∞ ], the E a in E a [ l ] is called the root , and the rest Y [ l ] is called the rubber .Therefore there are two types of fixed loci of the induced C ∗ -action on M Γ . A component of the fixed lociconsisting of stable maps with target E a is call a simple fixed locus. Otherwise, it is called a composite fixed locus. We denote the simple fixed locus by M simpleΓ . Denote the virtual normal bundle of M simpleΓ in M Γ by N Γ .Every element of a composite fixed locus is of the form f : C ′ ∪ C ′′ → E a [ l ] ( l ≥ , such that therestrictions f ′ : C ′ → E a and f ′′ : C ′′ → Y [ l ] agree over the nodal points { n , · · · , n k } = C ′ ∩ C ′′ . Suppose n i is mapped into PE ( h ′ i ), and the contact order of f ′ at n i , i.e. at D ∞ , (= E a ∩ Y [ l ]) in Y [ l ], is η i for 1 ≤ i ≤ k .Let Γ ′ be the topological data corresponding to f ′ and Γ ′′ the topological data corresponding to f ′′ . (HereΓ ′′ denote the genus, absolute marked points, degree and contact orders of relative marked points relativeto both D ,l in Y [ l ] and contact orders at those nodes n i , ≤ i ≤ k .) Any two of { Γ , Γ ′ , Γ ′′ } determine thethird, and Γ ′ gives us a simple fixed locus M simpleΓ ′ .Denote by M ∼ Γ ′′ the moduli space of relative stable maps to the rubber (see for example [21, 34]). Thenthe composite fixed locus F Γ ′ , Γ ′′ corresponding to a given Γ ′ and Γ ′′ is canonically isomorphic to thequotient of the moduli space M Γ ′ , Γ ′′ := M simpleΓ ′ × ( IPE ) ℓ M ∼ Γ ′′ Here and in the following for simplicity we omit the refinement of domain curve by open covers of object spaces. by the finite group Aut( ~η ), which consists of permutations of { , . . . , k } preserving ~η = (( η , ( h ′ )) , . . . , ( η k , ( h ′ k ))) . Denote the quotient map by gl : gl : M Γ ′ , Γ ′′ → F Γ ′ , Γ ′′ . Set [ M Γ ′ , Γ ′′ ] vir := ∆ ! ([ M simpleΓ ′ ] vir × [ M ∼ Γ ′ , Γ ′′ ] vir )where ∆ : ( IPE a ) k → ( IPE a ) k × ( IPE a ) k is the diagonal map. Then we have[ F Γ ′ , Γ ′′ ] vir = 1 | Aut( ~η ) | gl ∗ [ M Γ ′ , Γ ′′ ] vir . The virtual normal bundle of composite locus F Γ ′ , Γ ′′ consists of two parts. The first part is the virtualnormal bundle N Γ ′ of M simpleΓ ′ in M Γ ′ . The second part is a line bundle L corresponding to the deformation(i.e. smoothing) of the singularity D ∞ , (= E a ∩ Y [ l ]) in Y [ l ]. The fiber of this line bundle over a point in thefixed locus is canonically isomorphic to H ( PE a , N PE a | E ⊗ N D ∞ , | Y [ l ] ). The line bundle N PE a | E a ⊗ N D ∞ , | Y [ l ] = L ⊗ L ∗ = O PE a is trivial over PE a , so its space of global sections is one-dimensional, and we can canonicallyidentify this space of sections with the fiber of the line bundle at a generic point pt of PE a . Thus we canwrite the bundle L as a tensor product of bundles pulled back from the two factors separately. The onecoming from M Γ ′ is trivial, since it is globally identified with H ( pt, N PE a | E a (cid:12)(cid:12) pt ), but it has a nontrivialtorus action; we denote this weight by t . The line bundle coming from M ∼ Γ ′′ is a nontrivial line bundle,which has fiber H ( pt, N D ∞ , | Y [ l ] (cid:12)(cid:12) pt ), but has trivial torus action. We denote its first Chern class by Ψ ∞ .The relative virtual localization for relative orbifold Gromov–Witten invariants reads[ M Γ ] vir = [ M simpleΓ ] vir e ( N Γ ) + X M Γ ′ , Γ ′′ composite ( Q i η i ) gl ∗ [ M Γ ′ , Γ ′′ ] vir | Aut( ~η ) | e ( N Γ ′ )( t + Ψ ∞ ) . (4.2)Here Q i η i is the “mapping degree” of the gluing maps (cf. [13, § t is the equivariant weight ofthe C ∗ –action.We next describe explicitly the fixed loci of M Γ . We first consider the simple fixed locus M simpleΓ . Amap f : ( C , x , y ) → ( E a | PE a ) in the simple fixed locus must have the following form.(1) ( C , x , y ) with absolute marked points x = ( x , . . . , x m ) and relative marked points y = ( y , . . . , y ℓ ( µ ) )is of the form C = C ∪ C ∪ . . . ∪ C ℓ ( µ ) , where C ∩ C i = { n i } is a nodal point, called a distinguished nodal point , and C i ∩ C j = ∅ for1 ≤ i < j ≤ ℓ ( µ ). Moreover x i ∈ C , ≤ i ≤ m , and y j ∈ C j , ≤ j ≤ ℓ ( µ ).(2) C is a genus g pre-stable curves with m marked points x and ℓ ( µ ) marked points n = ( n , · · · , n ℓ ( µ ) )corresponding to the ℓ ( µ ) distinguished nodes.(3) For 1 ≤ i ≤ ℓ ( µ ), each C i is an (orbifold) Riemann sphere with a marked point y i and a markedpoint n i corresponding to the i -th distinguished nodal point.(4) f : ( C , x ⊔ n ) → S is a genus g degree π ∗ ( β ) stable maps to S , and belongs to M g,π ∗ ( β ) ,π t (¯ h ⊔ h ) ( S ).(5) For 1 ≤ i ≤ ℓ ( µ ), f : ( C i , y i , n i ) → E a is a total ramified covering of a line in the fiber of E a thatconnects a point in PE a ( h i ) and a point in S ( π t ( h − i )), the degree is determined by the contactorder at PE a . Hence it is in the simple fixed loci of the moduli space M ,µ i [ F ] ,π t ( h − i ) , ( µ i , ( h i )) ( E a | PE a ) , the moduli space of fiber class µ i [ F ] stable maps from orbifold Riemann spheres with exactlyone absolute marked point mapped to E a ( π t ( h − i )) ⊇ S ( π t ( h − i )) and one relative marked pointmapped to PE a ( h i ) with contact order µ i . For simplicity, we denote this simple fixed locus by In some literatures, − Ψ ∞ is referred as “target psi class”. − Ψ ∞ correspond to the ψ in [21, § § in[34, § out of N D ,l | Y [ l ] | pt , corresponding to the Ψ ∞ in [34, § RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 25 M simple µ i , ( h i ) . Denote the disconnected union of these M ,µ i [ F ] ,π t ( h − i ) , ( µ i , ( h i )) ( E a | PE a ) by M • ~µ , and thedisconnected union of simple fixed locus of them by M • , simple ~µ .Therefore, the simple fixed locus M simpleΓ is obtained by gluing stable maps in M g,π ∗ ( β ) ,π t (¯ h ⊔ h ) ( S ), andstable maps in M • , simple ~µ along the absolute marked points of C and C i corresponding to those distinguishednodal points. Moreover gl : M • , simple ~µ × ( IS ) ℓ ( µ ) M g,π ∗ β,π t (¯ h ⊔ h ) ( S ) → M simpleΓ is a degree | Aut( ~µ ) | cover. The fiber product is taken with respect to the evaluation maps at the markedpoints out of the ℓ ( µ ) distinguished nodal points of C .The tangent space T C , x , y , f ) and the obstruction space T C , x , y , f ) at a point f : ( C , x , y ) → ( E a | PE a ) in M simpleΓ fit in the following long exact sequence of C ∗ –representations:0 → Aut( C , x , y ) → Def( f ) → T C , x , y , f ) →→ Def( C , x , y ) → Obs( f ) → T C , x , y , f ) → , where(a) Aut( C , x , y ) = Ext (Ω C ( P i x i + P j y j ) , O C ) is the space of infinitesimal automorphism of the domain( C , x , y ). We have Aut( C , x , y ) = Aut( C , x , n ) ⊕ ℓ ( µ ) M i =1 Aut( C i , y i , n i ) , (b) Def( C , x , y ) = Ext (Ω C ( P i x i + P j y j ) , O C ) is the space of infinitesimal deformation of the domain( C , x , y ). We have a short exact sequence of C ∗ -representations:0 → Def( C , x , n ) → Def( C , x , y ) → ℓ ( µ ) M i =1 T n i C ⊗ T n i C i → , (c) Def( f ) = H ( C , f ∗ ( T E a ( − PE a ))) is the space of infinitesimal deformation of the map f , and(d) Obs( f ) = H ( C , f ∗ ( T E a ( − PE a ))) is the space of obstruction to deforming f .For i = 1 , , let T i,f and T i,m be the fixed and moving parts of T i | M simpleΓ . Then T = T ,f + T ,m , T = T ,f + T ,m . The virtual normal bundle of M simpleΓ in M Γ is N Γ = T ,m − T ,m . Let B = Aut( C , x , y ) , B = Def( f ) , B = Def( C , x , y ) , B = Obs( f )and let B fi and B mi be the fixed and moving parts of B i . Then1 e C ∗ ( N Γ ) = e C ∗ ( B m ) e C ∗ ( B m ) e C ∗ ( B m ) e C ∗ ( B m ) . On the other hand we have the following exact sequence0 → O C → M ≤ i ≤ ℓ ( µ ) O C i → M ≤ i ≤ ℓ ( µ ) O C n i → . (4.3)Denote by V = f ∗ ( T E a ( − PE a )). Then the exact sequence (4.3) gives us the following exact sequence0 → H ( C , V ) → M ≤ i ≤ ℓ ( µ ) H ( C i , V ) → M ≤ i ≤ ℓ ( µ ) V n i → H ( C , V ) → M ≤ i ≤ ℓ ( µ ) H ( C i , V ) → . Then the virtual normal bundle N Γ of M simpleΓ in M Γ consists of (i) the normal bundle N ~µ of M • , simple ~µ in M • ~µ ;(ii) the contribution from deforming maps into S , i.e. H ( C , f ∗ E ) − H ( C , f ∗ E );(iii) the contribution from deforming the distinguished nodal points: T n i C ⊗ T n i C i − E | f ( n i ) , ≤ i ≤ ℓ ( µ ),(note that only when n i is a smooth nodal point, i.e. is not an orbifold nodal point, we have theterm − E | f ( n i ) ).Denote the last two contribution by Θ Γ , it contains psi-class out of T n i C and Chern class of E . We couldwrite it in the form Θ Γ = X d ≥ X j + k = d Θ j Γ , S ℓ ( µ ) Y i =1 Θ k Γ ,i with Θ j Γ , S living over M g,π ∗ ( β ) ,π t (¯ h ⊔ h ) ( S ) and Θ k Γ ,i living over each M simple µ i , ( h i ) , where the contribution of H ( C , f ∗ E ) − H ( C , f ∗ E ) is contained in Θ j Γ , S , j ≥
0, the contribution of − E | f ( n i ) is contained in Θ k Γ ,i , k ≥ T n i C ⊗ T n i C i splits into both Θ j Γ , S , j ≥ k Γ ,i , k ≥ Z [ M simpleΓ ] vir ev ∗ x ̟ ∪ ev ∗ y µe C ∗ ( N Γ ) = 1 | Aut( ~µ ) | · X δ =( δ πt ( h ,...,δ πt ( hℓ ( µ )) )in the chosen basis(2.2) of H ∗ CR ( S ) d ≥ , j + k = d Z [ M • , simple ~µ ] vir ev ∗ y µ ∪ ev ∗ n ˇ δ ∪ Q ℓ ( µ ) i =1 Θ k Γ ,i e C ∗ ( N ~µ ) · Z [ M g,π ∗ ( β ) ,πt (¯ h ⊔ h ) ( S )] vir ev ∗ x ̟ ∪ ev ∗ n δ ∪ Θ j Γ , S ! where the sum is taken over all possible ℓ ( µ )-tuple of the chosen basis σ ⋆ of H ∗ CR ( S ) in (2.2). The integration Z [ M • , simple ~µ ] vir ev ∗ y µ ∪ ev ∗ n ˇ δ ∪ Q ℓ ( µ ) i =1 Θ k Γ ,i e C ∗ ( N ~µ ) = ℓ ( µ ) Y i =1 Z [ M simple µi, ( hi ) ] vir ev ∗ y i ( θ s i ( h i ) ) ∪ ev ∗ n i (ˇ δ π t ( h i ) ) ∪ Θ k Γ ,i e C ∗ ( N µ i )is a product of fiber class (1+1)-point relative invariants of ( E a | PE a ) and was computed by Hu and thefirst two authors [7, §
5] (see also Remark 3.11). See (5.8) in § Z [ M g,π ∗ β,πt (¯ h ⊔ h ) ( S )] vir ev ∗ x ̟ ∪ ev ∗ n δ ∪ Θ j Γ , S is a Hodge integral in the twisted Gromov–Witten invariant of S with twisting coming from the bundle E ,which by the orbifold quantum Riemann–Roch of Tseng [42] is determined by the orbifold Gromov–Wittentheory of S and the total Chern class of E → S .We next consider the composite fixed loci. Recall that a composite fixed locus is of the form F Γ ′ , Γ ′′ = gl ( M simpleΓ ′ × ( IPE a ) ℓ ( ~η ) M ∼ Γ ′′ ) . with contribution being Q i η i | Aut( ~η ) | · gl ∗ ∆ ! ([ M simpleΓ ′ × M ∼ Γ ′′ ] vir ) e ( N Γ ′ )( t + Ψ ∞ ) . By the analysis for simple fixed locus, the contribution from M simpleΓ ′ reduces to Gromov–Witten theory of S . Therefore, the contribution of F Γ ′ , Γ ′′ reduces to rubber invariants corresponding to M ∼ Γ ′′ , which arerubber invariants with Ψ k ∞ -integrals of( D | Y | D ∞ ) = ( PE a , | P ( L ⊕ O PE a ) | PE a , ∞ ) . We denote these rubber invariants by D µ (cid:12)(cid:12)(cid:12) ̟ · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E ( D | Y | D ∞ ) , ∼ g,β (4.4) RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 27 := 1 | Aut( µ ) || Aut( ν ) | Z [ M ∼ g, (¯ h ) ,β,~µ,~ν ( D | Y | D ∞ )] vir ev ∗ ̟ ∪ rev D , ∗ µ ∪ rev D ∞ , ∗ ν ∪ Ψ k ∞ . Note that here the ̟ is different from the absolute insertions in (4.1), as here ̟ consists of cohomologyclasses in H ∗ CR ( Y ), not H ∗ CR ( E a ). We will determine these rubber invariants in the following subsections.4.2. Rubber calculus.
As above denote PE a by D . Then in Y = P ( L ⊕ O PE a ) = P ( L ⊕ O D ) we have D ∼ = D ∼ = D ∞ . The projection π : Y → D induces a topological fiber bundle | π | : | Y | → | D | over the coarsespaces, whose fiber is P .In this subsection we relate rubber invariants (4.4) of ( D | Y | D ∞ ) to two kinds of relative invariants of( D | Y | D ∞ ), which are(a) fiber class invariants of the form D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E ( D | Y | D ∞ ) g,β (4.5) with β being a fiber class, θ ∈ H ∗ ( D ), and(b) general class invariants of the form D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E ( D | Y | D ∞ ) g,β (4.6) with θ ∈ H > ( D ).Here cohomology classes in ̟ come from the chosen basis (3.4), [ D ] is the Poincar´e dual of D in Y , and µ and ν denote the relative weights corresponding to D and D ∞ with cohomological weights coming from thechosen basis (3.2). The invariants in (4.6) are orbifold case analogues of Distinguished Type II invariants in [34]. Here we also call them Distinguished Type II invariants. In the following we omit the superscript( D | Y | D ∞ ) to simplify the notations.The fiber class invariants in (a) have been determined in §
3. We will determine Distinguished Type IIinvariants in (b) in § • ” to decorate disconnected invariants. However, as noted in [34],there is no product rule for rubber invariants.4.2.1. Rigidification.
Given a (possibly disconnected) rubber invariant of ( D | Y | D ∞ ) D µ (cid:12)(cid:12)(cid:12) θ · ̟ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β with θ ∈ H ∗ ( Y ) being an insertion form the non-twisted sector Y itself, we denote by M • , ∼ Γ := M • , ∼ g, (¯ h ) ,β,~µ,~ν ( D | Y | D ∞ )the moduli space of stable maps to the rubber target with Γ denote the topological data. We also denoteby M • Γ := M • g, (¯ h ) ,β,~µ,~ν ( D | Y | D ∞ )the moduli space of stable maps to Y relative to both D and D ∞ with the same topological data. Here g isthe arithmetic genus. Without loss of generality we assume ( h ) = (1) in (¯ h ) is the index of the non-twistedsector Y itself, which corresponds to θ .We have a canonical forgetful map ǫ : M • Γ → M • , ∼ Γ , (4.7)which is C ∗ -equivariant with respect to the canonical C ∗ -action on M • Γ induced from the fiber-wise C ∗ -actionon Y and the trivial C ∗ -action on M • , ∼ Γ . Lemma 4.1.
Let q be an absolute marked point corresponding to ( h ) , and the corresponding evaluationmap be ev q : M • Γ → Y . Then [ M • , ∼ Γ ] vir = ǫ ∗ ( ev ∗ q [ D ] ∩ [ M • Γ ] vir ) = ǫ ∗ ( ev ∗ q [ D ∞ ] ∩ [ M • Γ ] vir ) . (4.8) Therefore D µ (cid:12)(cid:12)(cid:12) θ · ̟ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β = D µ (cid:12)(cid:12)(cid:12) ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E • g,β = D µ (cid:12)(cid:12)(cid:12) ([ D ∞ ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E • g,β Proof.
The proof is similar to the proof of [34, Lemma 2]. We use the localization formula (4.2) for relativeorbifold Gromov–Witten theory. We only prove the first equality in (4.8). The proof of the second one issimilarly.A C ∗ –fixed stable relative map in M • Γ is a union of:(i) a nonrigid stable map to the degeneration of Y over D ,(ii) a nonrigid stable map to the degeneration of Y over D ∞ ,(iii) a collection of C ∗ -invariant, fiber class, rational Galois covers joining (i), (ii), i.e. they are mapsfrom orbifold Riemann spheres with two orbifold points and are total ramified over D and D ∞ .On the fixed locus, the forgetful map ǫ simply contracts the intermediate rational curves (iii).Obviously, simple fixed loci contributes nothing since there is at least one absolute marked point. If wehave a proper degeneration on both sides of Y , the (complex) virtual dimension of the C ∗ -fixed locus is 2less than the virtual dimension of M • Γ . However, the dimension of ǫ ∗ ( ev ∗ p ([ D ]) ∩ [ M • Γ ] vir )is only 1 less than the virtual dimension of M • Γ . Therefore we only have to consider fixed loci whose targetdegenerates on only one side of Y .Since the absolute marked point q is constrained by an insertion of [ D ], we need only consider degener-ations along D . Then we see that there is a unique composite C ∗ -fixed locus F Γ ′ , Γ = gl ( M • , ∼ Γ × ( ID ) ℓ ( ν ) M simpleΓ ′ ) . where M simpleΓ ′ consists of curves (iii), hence has no absolute marked points, and the relative marked pointsrelative to D ∞ are constrained by ν , the relative marked point relative to D are constrained by ~ ˇ ν .Then by (4.2) we get ǫ ∗ ( ev ∗ p ([ D ]) ∩ [ M • Γ ] vir )= Q i ν i Aut( ~ν ) (cid:16) ( ev ∗ p ( c ( L )) + t ) ∩ gl ∗ [ M • , ∼ Γ × ( ID ) ℓ ( ν ) M simpleΓ ′ ] vir e ( N Γ ′ )( t + Ψ ∞ ) (cid:17) = Y i ν i · X ρ =( ρ ,...,ρ ℓ ( ν ) ) in (3.2)constrained by ~ν (cid:26)(cid:16) ( ev ∗ p ( c ( L )) + t ) ev ∗ D ∞ ρ ( t + Ψ ∞ ) ∩ [ M • , ∼ Γ ] vir (cid:17) · (cid:16) ev ∗ D ˇ ρe ( N Γ ′ ) ∩ [ M simpleΓ ′ ] vir (cid:17)(cid:27) where ev ∗ p ( c ( L )) + t is the restriction of the equivariant lifting of the class [ D ] to the zero section D . Thelast term in previous equation corresponds to the following relative invariant of ( D | Y | D ∞ ) | Aut( ρ ) | · | Aut( ν ) | · D ˇ ρ (cid:12)(cid:12)(cid:12) ∅ (cid:12)(cid:12)(cid:12) ν E , | ν | , [ F ] = Z [ M Γ ′ ] vir ev ∗ D ˇ ρ ∧ ev ∗ D ∞ ν = (cid:16) ev ∗ D ˇ ρ ∪ ev ∗ D νe ( N Γ ′ ) ∩ [ M simpleΓ ′ ] vir (cid:17) = (cid:26) Q i ν i if ρ = ν ;0 if ρ = ν .Therefore by dimension counting ǫ ∗ ( ev ∗ p ([ D ]) ∩ [ M • Γ ] vir ) = (cid:16) ( ev ∗ p ( c ( L )) + t )( t + Ψ ∞ ) ∩ [ M • , ∼ Γ ] vir (cid:17) = [ M • , ∼ Γ ] vir . This finishes the proof. (cid:3)
RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 29
Dilaton and divisor equations.
As the smooth case, the rubber invariants for orbifolds also satisfy thedilaton equation and divisor equation as in [34] . The dilaton equation is D µ (cid:12)(cid:12)(cid:12) τ ( ) · m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β = (2 g − m + ℓ ( µ ) + ℓ ( ν )) D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β . The divisor equation for H ∈ H ( D ) is D µ (cid:12)(cid:12)(cid:12) τ ( H ) · m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β = ( Z orb π ∗ β H ) D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β + m X i =1 D µ (cid:12)(cid:12)(cid:12) . . . τ k i − ( γ l i ∪ CR H ) . . . Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β − ℓ ( ν ) X j =1 D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i ( γ l i ) · Ψ k − ∞ (cid:12)(cid:12)(cid:12) { . . . ( ν j , δ s j ∪ CR H ) . . . } E • , ∼ g,β · ν j . Since here the marked points corresponding to the two insertions τ ( ) and τ ( H ) are both smooth markedpoints, i.e. without orbifold structure, the standard cotangent line comparison method proves the dilatonand divisor equations above.4.2.3. Rubber calculus I: fiber class invariants.
We first consider a fiber class rubber invariant in (4.4) withdescendant insertion ̟ D µ (cid:12)(cid:12)(cid:12) ̟ · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β , (4.9)i.e. β is a multiple of the fiber class of Y → D .Note that by stability, a contracted genus zero irreducible component of the domain curve must carry atleast three absolute marked points, a contracted genus one irreducible component of the domain curve mustcarry at least one absolute marked point. A non-contracted component of the domain curve must carry atleast two relative marked points, i.e. the intersection points with D and D ∞ . Finally, by target stability,not all components of domain curve can be genus 0 and fully ramified over D and D ∞ . Hence we conclude2 g − m + ℓ ( µ ) + ℓ ( ν ) > , where m = k ̟ k is the length of ̟ . Therefore the fiber class rubber invariants (4.9) is determined by D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β (4.10)and the dilaton equation in § q denote the smooth absolute marked point that carries the insertion τ ( ) in the rubber invariants (4.10). Denote the twisted sectors of the 1 + || ̟ || marked points by (¯ h ) with( h ) = (1) corresponding to q .Denote the topological data of the moduli space of the invariant (4.10) by Γ and the corresponding rubbermoduli space by M • , ∼ Γ ( D | Y | D ∞ ). Consider a splitting Γ ∧ D Γ of Γ withΓ = ( g , β , (¯ h ) , ~µ, ~η ) , and Γ = ( g , β , (¯ h ) , ˇ ~η, ~ν ) , (4.11)satisfying ( h ) = (1) ∈ (¯ h ) , i.e. the marked point corresponding to τ ( ) is distributed to the Γ side. Then we could glue two stablemaps in the rubber moduli spaces M • , ∼ Γ ( D | Y | D ∞ ) , and M • , ∼ Γ ( D | Y | D ∞ )together to get a stable map in the rubber moduli space M • , ∼ Γ ( D | Y | D ∞ ). In this way we get a boundarycomponent of M • , ∼ Γ ( D | Y | D ∞ ), that is we have the following map gl : M • , ∼ Γ ( D | Y | D ∞ ) × ( ID ) ℓ ( ~η ) M • , ∼ Γ ( D | Y | D ∞ ) → M • , ∼ Γ ( D | Y | D ∞ ) , As noticed by Tseng–You [45], there is a typo in [34, § τ ( ) should not appear on the right side of the dilatonequation. which is a | Aut( ~η ) | cover to its image, i.e. a boundary component of M • , ∼ Γ ( D | Y | D ∞ ) corresponding tothe splitting (4.11). Denote the image by B Γ , Γ , and the normal bundle of this boundary component by N Γ , Γ . There is a line bundle N (see for example [28]) over M • , ∼ Γ whose zero set consists of disjoint unionof all these boundary components B Γ , Γ , and its restriction on B Γ , Γ is N ⊗ Q i η i Γ , Γ . The first Chern classof N is c ( N ) = − Ψ ∞ + ev ∗ q ( c ( L )) . Therefore for the invariant (4.10) we have D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β = − D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ · Ψ k − ∞ · c ( N ) (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β + D µ (cid:12)(cid:12)(cid:12) τ ( c ( L )) · ̟ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β = − X D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ (cid:12)(cid:12)(cid:12) η E • , ∼ g ,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) ̟ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g ,β + D µ (cid:12)(cid:12)(cid:12) τ ( c ( L )) · ̟ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β , (4.12)where the sum is taken over all splittings Γ ∧ D Γ of Γ of the form in (4.11), and all intermediate cohomologyweighted partitions η , and ( ̟ , ̟ ) is a distribution of ̟ according the splittings of absolute marked pointsin Γ and Γ .The firt term D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ (cid:12)(cid:12)(cid:12) η E • , ∼ g ,β in the summation in (4.12) can be expressed as a fiber class invariantsof ( D | Y | D ∞ ) by Lemma 4.1: D µ (cid:12)(cid:12)(cid:12) τ ( ) · ̟ (cid:12)(cid:12)(cid:12) η E • , ∼ g ,β = D µ (cid:12)(cid:12)(cid:12) τ ([ D ]) · ̟ (cid:12)(cid:12)(cid:12) η E • g ,β . Therefore we have reduced the original fiber class rubber invariant (4.9) to rubber invariants of the sametype with strictly fewer Ψ ∞ insertions and fiber class invariants of ( D | Y | D ∞ ). Repeating this cycle wereduce the original fiber class rubber invariant (4.9) to fiber class rubber invariants without Ψ ∞ insertions D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) k ) · ̟ ′ (cid:12)(cid:12)(cid:12) η E • , ∼ g,β , k ≥ , (4.13)and fiber class invariants of ( D | Y | D ∞ ). By Lemma 4.1, the rubber invariants in (4.13) are determined byfiber class invariants of ( D | Y | D ∞ ). Therefore the fiber class rubber invariant (4.9) is determined by fiberclass invariants of ( D | Y | D ∞ ) of the forms D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · c ( L ) k ) · ̟ ′ (cid:12)(cid:12)(cid:12) η E • g,β , k ≥ , i.e. of the form of (4.5). We have computed all fiber class invariants of ( D | Y | D ∞ ) in §
3, so all fiber classrubber invariants are determined.4.2.4.
Rubber calculus II: non-fiber class invariants.
Now suppose that the homology class β in (4.4) is nota fiber class, i.e. π ∗ ( β ) = 0. Consider a non-fiber class rubber invariants: D µ (cid:12)(cid:12)(cid:12) m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β . (4.14)There exists an H ∈ H ( D ) such that Z orb π ∗ β H > . Now consider the rubber invariant D µ (cid:12)(cid:12)(cid:12) τ ( H ) · m Y i =1 τ k i ( γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β . (4.15)By the divisor equation in § ∞ insertions, the rubberinvariant (4.14) is determined by the rubber invariant (4.15) and D µ (cid:12)(cid:12)(cid:12) . . . τ k i − ( H ∪ CR γ l i ) . . . Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β , i = 1 , · · · m. (4.16) RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 31
For invariants in (4.16) consider rubber invariants of the form D µ (cid:12)(cid:12)(cid:12) τ ( H ) . . . τ k i − ( H ∪ CR γ l i ) . . . Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β , i = 1 , · · · m, (4.17)by adding an insertion τ ( H ). Again by the divisor equation in § ∞ insertion or strictly fewer ψ i insertions, the rubber invariants in (4.16) are determined byrubber invariants in (4.17).Finally, combining (4.15) and (4.17) we see that, modulo rubber invariants with strictly fewer Ψ ∞ insertions, the rubber invariants (4.14) is determined by rubber invariants of the form D µ (cid:12)(cid:12)(cid:12) τ ( H ) · m Y i =1 τ k i − n i ( H n i ∪ CR γ l i ) · Ψ k ∞ (cid:12)(cid:12)(cid:12) ν E • , ∼ g,β , (4.18)where 1 ≤ n i ≤ k i , ≤ i ≤ m . We then apply the boundary relation in previous subsection § D µ (cid:12)(cid:12)(cid:12) τ ( H · c ( L ) k ) · ̟ ′ (cid:12)(cid:12)(cid:12) η E • , ∼ g,β , k ≥ . (4.19)Finally, we apply Lemma 4.1 to (4.19). So we can express the original rubber invariant (4.14) in terms ofdistinguished type II invariants of ( D | Y | D ∞ ) of the forms D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · H · c ( L ) k ) · ̟ ′ (cid:12)(cid:12)(cid:12) η E • g,β , k ≥ , i.e. distinguished type II invariants in (4.6). We will determines these invariants in next subsection § Distinguished type II invariants.
In this subsection we determine all distinguished Type II invari-ants via an induction algorithm with initial datum being fiber class invariants determined in §
3. We give apartial order “ ◦ ≺ ” over all distinguished Type II invariants. The partial order is different from the partialorder used in [34], since here in general we can not compare cohomology weights coming from differenttwisted sectors. The partial order we give here is a modification of the partial order given in [7]. The partialorder given in [7] is more geometric, and follows from the degeneration formula in [13, 1] directly.4.3.1. Partial order.
Consider a distinguished type II invariant D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β , (4.20)where θ ∈ H > ( D ). We also assume that µ and ν have cohomological weights from the chosen basis (3.2)and ̟ has cohomology weights from the chosen basis (3.4). We have a degeneration of ( D | Y | D ∞ ) along D ∞ ( D | Y | D ∞ ) degenerate −−−−−−−→ Y ∧ D Y = ( D | Y | D ∞ ) ∧ D ( D | Y | D ∞ )(4.21)where we glue D ∞ in Y with D in Y via D ∼ = D ∞ ∼ = D .For the invariant D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β we distribute τ ([ D ] · θ ) and insertions in ̟ with cohomologyclasses being divisible by one of { [ D ( h ) ] | ( h ) ∈ T D } to Y . Then by the degeneration formula (cf. [13, 1])we have D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β = X D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) η E • g ,β z ( η ) D ˇ η (cid:12)(cid:12)(cid:12) ̟ (cid:12)(cid:12)(cid:12) ν E • g ,β (4.22)with summation taking over all splittings of ( g, β ), all distribution of ̟ , all immediate cohomology weightsand all configurations of connected components that yield a connected total domain.There are some special summands in (4.22) with g = 0, ̟ = ∅ , β being a fiber class and ~η = ~ν . Forsuch a summand, the invariant D ˇ η (cid:12)(cid:12)(cid:12) ∅ (cid:12)(cid:12)(cid:12) ν E • ,d [ F ] = (cid:26) ν · ... · ν ℓ ( ν ) if η = ν ,0 if η = ν . Therefore there is no summands, of such type and with η = ν , on the right side of (4.22). And when η = ν we get the original invariant (4.20) on the right side of (4.22). From the summands in (4.22) and Y ∼ = Y we get a lot of distinguished type II invariants D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g ′ ,β ′ . We say that these distinguished type II invariants are all lower than (cid:10) µ (cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12) ν (cid:11) g,β and denote thisrelation by D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g ′ ,β ′ ≺ D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β . (4.23)The following theorem is a special case of [7, Theorem 6.5]. Theorem 4.2.
The relation (4.23) is a partial order among all distinguished type II invariants of ( D | Y | D ∞ ) . We will also discuss this partial order in § X a | D a ).Given a distinguished type II invariant, by the Gromov compactness there are only finite distinguishedtype II invariants lower than it according to the ordering “ ≺ ”. However, according to this partial order,two comparable distinguished type II invariants have the same relative datum over D . Next we modify thepartial order “ ≺ ” so that we could compare distinguished Type II invariants with different relative datumover D . Definition 4.3.
We say that D µ ′ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ′ ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g ′ ,β ′ ◦ ≺ D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β , if (1) β ′ < β , i.e. β − β ′ is an effective class in H ( | Y | ; Z ) , (2) equality in (1) and g ′ < g , (3) equality in (1)-(2) and k ̟ ′ k < k ̟ k , (4) equality in (1)-(3) and deg CR µ ′ > deg CR µ , (5) equality in (1)-(4) and deg CR ν ′ > deg CR ν , (6) equality in (1)-(5) and deg θ ′ > deg θ , (7) equality in (1)-(6), µ ′ = µ , θ ′ = θ and D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g ′ ,β ′ ≺ D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β . This partial order could be generalized to invariants with disconnected domains directly.
Proposition 4.4.
The relation “ ◦ ≺ ” is an partial order over all distinguished type II invariants. Moreover,given a distinguished type II invariant, there are only finite distinguished type II invariants are lower thanit under “ ◦ ≺ ”. The first assertion follows from Theorem 4.2 and the explicit inequalities in Definition 4.3. The secondassertion follows from the Gromov compactness.We next use the weighted-blowup correspondence in [7] to find three relations to determine all distin-guished type II invariants.4.3.2.
First relation.
By the weighted-blowup correspondence in [7], the relative data ν at D ∞ determinesa sequence of absolute insertions relative to D ∞ , which we denote by ν ∞ . See § µ S via µ for a weight- a blowup X a of X along S , in particular (5.5).Now consider a distinguished type II invariant D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β , (4.24)where θ ∈ H > ( D ) and π ∗ β = 0. Suppose ν is of the form ν = (cid:16) ( ν , θ s ( h ) ) , . . . , ( ν ℓ ( ν ) , θ s ℓ ( ν ) ( h ℓ ( ν ) ) ) (cid:17) . (4.25) RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 33
We could view Y as the weight-(1) blowup of Y along D ∞ , i.e. the trivial weight blowup. Then followingthe weighted-blowup correspondence in [7] (see also §
5) we set ν ∞ := (cid:16) τ [ ν ] − l ( h ([ D ( h ) ∞ ] · θ s ( h ) ) , . . . , τ [ ν ℓ ( ν ) ] − l ( hℓν ) ([ D ( h ℓ ( ν ) ) ∞ ] · θ s ℓ ( ν ) ( h ℓ ( ν ) ) ) (cid:17) , (4.26)where the integer [ ν j ] − l ( h j ) , which indicates the power of the psi-class, is obtained by applying the formula(5.5) of c j to the relative insertions in ν at D ∞ ⊆ Y , and [ ν j ] means the integral part of ν j . See for Definition3.1 for the definition of l ( h j ) .For instance when applying the formula (5.5) c j = n X i =1 " − b ( g − j ) i N o ( g j ) + a i µ j + n − − m j to ( ν j , θ s j ( h j ) ) of ν in (4.25) we have N = L ∗ , ( g j ) = ( h − j ), µ j = ν j , n = codim C D ∞ = 1, a = ( a , . . . , a n ) =(1), m j = 0. As ν j is the contact order of the j -th marked point mapped to D ∞ ( h j ) we have ν j = ( [ ν j ] + o ( h j ) − b ( h j ) L o ( h j ) if 1 ≤ b ( h j ) L < o ( h j ), i.e. l ( h j ) = 0,[ ν j ] if b ( h j ) L = o ( h j ), i.e. l ( h j ) = 1,where o ( h j ) is the order of ( h j ) and b ( h j ) L is the action weight of h j on L . On the other hand, note that b ( g − j ) u N is the action weight of ( g − j ) on N . So for the N = L ∗ and ( g − j ) = ( h j ) we have b ( h j ) L ∗ = (cid:26) o ( h j ) − b ( h j ) L if 1 ≤ b ( h j ) L < o ( h j ), i.e. l ( h j ) = 0, o ( h j ) if b ( h j ) L = o ( h j ), i.e. l ( h j ) = 1,.Therefore by noticing that o ( h j ) = o ( h − j ) and b ( g − j ) N = b ( h j ) L ∗ we have X i =1 " − b ( g − j ) i N o ( g j ) + a i ν j = " − b ( h j ) L ∗ o ( h − j ) + ν j = [ ν j ] − l ( h j ) . Now via replacing the relative insertions in ν by the absolute insertions in ν ∞ in (4.26) we get a relativeinvariant of ( Y | D ) D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β . (4.27)We denote the distinguished type II invariant (4.24) by R . In the following we view relative invariantsof ( Y | D ) , ( Y | D ∞ ) and ( D | Y | D ∞ ) with genus g and class β as principle terms , and relative invariants of( Y | D ) , ( Y | D ∞ ) and ( D | Y | D ∞ ) with β ′ < β, or β ′ = β and g ′ < g as non-principle terms . Relation 4.5.
We have C · z ( ν ) · D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β = D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β − X R ′ g,β distinguished type II R ′ ◦ ≺ R C R , R ′ · R ′ − X k ̟ ′ k≤k ̟ k deg CR µ ′ ≥ deg CR µ +1 C µ ′ ,̟ ′ · D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′ · ν ∞ E g,β − · · · , where C = D ˇ ν (cid:12)(cid:12)(cid:12) ν ∞ E • ,d [ F ] = ℓ ( ν ) Y j =1 ord ( h j ) · ν j ! = 0 , with ν j ! = (cid:26) ν j · . . . · ( ν j − [ ν j ]) if [ ν j ] < ν j , ν j · . . . · if [ ν j ] = ν j , C ∗ , ∗ are fiber class invariants of ( Y | D ) or ( D | Y | D ∞ ) and “ · · · ” stands for combinations of non-principlerelative invariants of ( Y | D ) and non-principle distinguished Type II invariants.Proof. We degenerate Y as (4.21). Then the degeneration formula express the invariant (4.27) in terms ofrelative invariants of ( D | Y | D ∞ ) and relative invariants of ( Y | D ): D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β = X D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) η E • g ,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) ̟ · ν ∞ E • g ,β , where the summation is over all splittings of g and β , all distributions of the insertions of ̟ , all intermediatecohomology weighted partitions η , and all configurations of connected components that yield a connectedtotal domain. The invariants on the right side are possible disconnected, indicated by the superscript • .The subscript g i denotes the arithmetic genus of the total map to Y i .The invariants of ( D | Y | D ∞ ) are all distinguished type II invariants. Since we only concern principleterms. We could assume that β = β or β = β . Case I: β = β . Let f i : C i → Y i be the elements of the relative moduli spaces for a fixed splitting. β = β forces β to be a fiber class. For the arithmetic genus of the glued stable map we have g = g + g + ℓ ( η ) − . Since β is a fiber class, every connected component of C contains at least one relative marked point.Therefore the arithmetic genus satisfies g ≥ − ℓ ( η ) . We conclude g ≥ g with equality if and only if C consists of rational components, each totally ramifiedover D and every component contains exactly one relative marked point. If g > g , we get non-principleterms. Hence we only consider extremal configurations.If any insertions of ̟ is distributed into Y , we get distinguished type II invariants coming from Y whichare lower than R in the order “ ◦ ≺ ”. Therefore we assume that all insertions in ̟ are distributed to Y .Hence the absolute insertions for Y all come from ν ∞ .Now every connected component of C has exactly one relative marked point. Therefore the decomposi-tion of ˇ η (hence η ) decomposes ν ∞ , hence ν , into ℓ ( η ) components ν = ℓ ( η ) a k =1 π ( k ) where empty partition is allowed as ν ∞ are absolute insertions.Write η ( k ) = (( η k , ρ k )) , π ( k ) = ( π ( k )1 , θ s ( k )1 ( h ( k )1 ) ) , . . . , ( π ( k ) ℓ ( π ( k ) ) , θ s ( k ) ℓ ( π ( k )) ( h ( k ) ℓ ( π ( k )) ) ) ! , where for a pair ( ∗ , ⋆ ), “ ∗ ” stands for the contact order and “ ⋆ ” stands for the relative insertion, i.e.cohomological weight.For 1 ≤ k ≤ ℓ ( η ), each component of C gives us a fiber class relative invariant of ( Y | D ) D ˇ η ( k ) (cid:12)(cid:12)(cid:12) π ( k ) ∞ E ,d k [ F ] (4.28)where the homology class is determined by contact order η k , the relative data is ˇ η ( k ) := (( η k , ˇ ρ k )), andthe absolute insertions π ( k ) ∞ are determined by π ( k ) via (4.26). Denote the topological data of this fiberclass invariant by Γ ( k ) . Following §
3, the fiber class invariants are obtained by integrating insertion coming The relative invariants of this form were computed by Hu and the first two authors in [7]. See for (5.8) the expression ofthese relative invariants.
RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 35 from fiber, descendent classes and Hodge class over the relative moduli space of ([ P ⋊ K Γ ( k ) ] | [0 ⋊ K Γ ( k ) ])of topological type Γ ( k ) (determined by Γ ( k ) ), and then integrating the results together with e ( E Γ ( k ) ) andinsertions from D (in fact ID ) over D ′ Γ ( k ) .Recall that in § CR ( µ ) := ℓ ( µ ) X j =1 (deg θ ( h j ) + 2 ι D ( h i ))for a relative insertion µ = (( µ , θ ( h ) ) , . . . , ( µ ℓ ( µ ) , θ ( h ℓ ( µ ) ) )). Furthermore we set deg µ = ℓ ( µ ) P j =1 deg θ ( h j ) . Thenfor the fiber class relative invariant (4.28), to get a non-zero invariants we must havedeg ˇ η ( k ) + deg π ( k ) + rank E Γ ( k ) ≤ dim R D ′ Γ ( k ) = dim R D Γ ( k ) . By the same proof of [25, Theorem 3.2] of Hu–Wang we havedim C D − dim C D Γ ( k ) + rank C E Γ ( k ) = ι D (ˇ η ( k ) ) + ι D ( π ( k ) )where ι D (ˇ η ( k ) ) is the degree shifting number of the twisted sector to which ˇ ρ k belongs and ι D ( π ( k ) ) is thesum of degree shifting numbers of the twisted sectors D ( h ( k ) i ) , ≤ i ≤ ℓ ( π ( k ) ). As a consequence we getdeg ˇ η ( k ) + deg π ( k ) ≤ dim R D − (cid:0) ι D (ˇ η ( k ) ) + ι D ( π ( k ) ) (cid:1) , which is equivalent to deg CR ˇ η ( k ) + deg CR π ( k ) ≤ dim R D . Hence by the orbifold Poincar´e duality we getdeg CR η ( k ) ≥ deg CR π ( k ) . (4.29)Summing over all components for 1 ≤ k ≤ ℓ ( η ) we getdeg CR η ≥ deg CR ν. When there is at least one strictly inequality in (4.29), we get a strictly lower distinguished Type IIinvariant from Y . So we consider the case that equality holds in (4.29) for all 1 ≤ k ≤ ℓ ( η ). Then allequalities in (1)-(6) in Definition 4.3 of the partial order “ ◦ ≺ ” holds, and we only have to consider thefollowing summands in the degeneration formula X deg CR η =deg CR ν D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) η E • g,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) ν ∞ E • ,d [ F ] . (4.30)There is exactly one summand with η = ν , for which (cf. [7, Theorem 5.29]) C = D ˇ ν (cid:12)(cid:12)(cid:12) ν ∞ E • ,d [ F ] = ℓ ( ν ) Y j =1 h j ) · ν j ! = 0 . This gives us the left side of Relation 4.5. For the rest summands in (4.30), we must have ~η = ~ν , and thenby definition of ◦ ≺ , we have D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) η E • g,β ◦ ≺ D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ (cid:12)(cid:12)(cid:12) ν E g,β and D η (cid:12)(cid:12)(cid:12) ν ∞ E • ,d [ F ] are fiber class invariants of ( Y | D ), which are determined in §
3. This gives us the secondterm on the right side of the Relation 4.5.
Case II: β = β . The principal terms from Y will be shown to be of the form of the third term on theright side of Relation 4.5.Let f i : C i → Y i be the elements of the relative moduli spaces for a fixed splitting. The condition β = β forces β to be a multiple of the fiber class [ F ]. After ignoring lower terms, we may assume C consists of ℓ ( η ) rational components, each totally ramified over D ∞ and contains exactly one relative making. Thenthe decomposition of η induces a decomposition of µ into components µ = a k π ( k ) where empty weighted partitions are not allowed, since insertions in µ are relative insertions.For 1 ≤ k ≤ ℓ ( η ), each component of f gives us a fiber class relative invariant of ( D | Y | D ∞ ) D π ( k ) (cid:12)(cid:12)(cid:12) · · · ̟ ( k )1 (cid:12)(cid:12)(cid:12) η ( k ) E ,d k [ F ] . Here “ · · · ” is empty or τ ([ D ] · θ ). We also denote the topological data of this fiber class invariant by Γ ( k ) .By the analysis in § Case I , to get a nonzero invariant we must havedeg η ( k ) + deg π ( k ) + rank E Γ ( k ) ≤ dim R D ′ Γ ( k ) = dim R D Γ ( k ) for those components which do not contain the insertion τ ([ D ] · θ ), anddeg η ( k ) + deg π ( k ) + deg θ + rank E Γ ( k ) ≤ dim R D ′ Γ ( k ) = dim R D Γ ( k ) for the component which contains the insertion τ ([ D ] · θ ).Then since by the assumption deg CR θ = deg θ ≥ CR ˇ η ≥ deg CR µ + 1 . This gives us the third term on the right side of Relation 4.5. (cid:3)
Second relation.
Next we consider the following non-fiber relative invariant of ( Y | D ∞ ) D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β , which is the first term on the right side of Relation 4.5. Relation 4.6.
We have D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β = ± X k ̟ ′ k≤k ̟ k deg CR µ ′ ≥ deg CR µ deg CR ν ′ ≥ deg CR ν +1 m ≥ C µ ′ ,̟ ′ ,ν ′ · D µ ′ (cid:12)(cid:12)(cid:12) τ ([ D ] · H · c ( L ) m ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g,β ± X k ̟ ′ k≤k ̟ k deg CR µ ′ ≥ deg CR µ +1deg CR ν ′ ≥ deg CR νm ≥ C µ ′ ,̟ ′ ,ν ′ · D µ ′ (cid:12)(cid:12)(cid:12) τ ([ D ] · H · c ( L ) m ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g,β ± X k ̟ ′ k≤k ̟ k deg CR µ ′ ≥ deg CR µ deg CR ν ′ ≥ deg CR νm ≥ C µ ′ ,̟ ′ ,ν ′ · D µ ′ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ · c ( L ) m +1 ) · ̟ ′ (cid:12)(cid:12)(cid:12) ν ′ E g,β + · · · where H ∈ H ( D ) is a class such that R orb π ∗ ( β ) H > , C ∗ , ∗ , ∗ are fiber class invariants of ( Y | D ) , ( Y | D ∞ ) ,and ( D | Y | D ∞ ) , and “ · · · ” stands for combinations of non-principle distinguished type II invariants, non-principle relative invariants of ( Y | D ) , ( Y | D ∞ ) , and descendent orbifold Gromov–Witten invariants of D .Proof. We use [ D ] = [ D ∞ ] + c ( L )to write the invariant D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · θ ) · ̟ · ν ∞ E g,β into I + I := D µ (cid:12)(cid:12)(cid:12) τ ([ D ∞ ] · θ ) · ̟ · ν ∞ E g,β + D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ · ν ∞ E g,β . RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 37
We first compute the invariant I via virtual localization with respect to the canonical C ∗ -action on Y .The insertions of I all have canonical equivariant lifts. The moduli space have two kind of fixed loci, thesimple type M simpleΓ and the composite type F Γ ′ , Γ ′′ .The simple type fixed locus M simpleΓ consists of those stable maps f : C ∪ C → Y such that C is a disjointunion of rational components and the restriction f : C → ( D | Y | D ∞ ) are totally ramified over D and D ∞ ,and the restriction f : C → Y maps into D ∞ . The insertions τ ([ D ∞ ] · θ ) and ν ∞ are all distributed to D ∞ .Therefore by the localization analysis in § M simpleΓ is Hodge integrals in the twistedGromov–Witten theory of D ∞ ∼ = D with twisting coming from N D ∞ | Y = L . Then by the orbifold quantumRiemann–Roch [42] of Tseng, these twisted invariants reduced to descendent Gromov–Witten invariants of D . We next consider composite fixed loci. Since we only consider principle terms we only need to consider F Γ ′ , Γ ′′ = gl ( M simpleΓ ′ × ID ℓ ( ρ ) M ∼ Γ ′′ ) with maps in M ∼ Γ ′′ having genus g and degree β . Let C be the sub-curveof the domain mapped to rubber, and C ∞ be the sub-curve of the domain mapped to D ∞ . Then C and C ∞ is connected by a disjoint union of rational components C which are totally ramified over D and D ∞ . C ∞ ∪ C gives a morphism to ( Y | D ) of fiber class. Note that the insertion τ ([ D ∞ ] · θ ) and insertionsin ν ∞ must be distributed to C ∞ . There would be some insertions ̟ ′′ of ̟ are distributed to C ∞ too.The rest ̟ ′ of ̟ are distributed to C . We next insert relative insertions to the connecting nodal points { n , . . . , n m } of C and rational components C by assigning η to C and ˇ η to C . Then the argument usedfor fiber class invariants in the proof of Relation 4.5 and the analysis for fiber class invariants in § CR η ≥ deg D CR ̟ ′′ + deg CR ν + deg CR θ ≥ deg CR ν + deg θ where deg D CR ̟ ′′ means the CR-degree of non-fiber parts of ̟ ′′ , i.e. those classes pulled back from H ∗ CR ( D ).Since deg θ >
0, the principle term of the localization formula of I is of the form D µ (cid:12)(cid:12)(cid:12) ̟ ′ · Ψ k ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , where k ̟ ′ k ≤ k ̟ k , deg CR ( η ) ≥ deg CR ( ν ) + 1 , with coefficient being a fiber class relative invariant of ( Y | D ). Then by repeating the rubber calculus fornon-fiber class rubber invariants in § D .We next compute I by virtual localization too. As above, the contribution from simple fixed loci reducesto descendent orbifold Gromov–Witten invariants of D . We next consider contribution from composite fixedloci. As above since we only consider principle terms we only need to consider F Γ ′ , Γ ′′ = gl ( M simpleΓ ′ ∧ ID ℓ ( ρ ) M ∼ Γ ′′ ) with maps in M ∼ Γ ′′ having genus g and degree β . We also denote by C the sub-curve of the domainmapped to rubber, and C ∞ the sub-curve of the domain mapped to D ∞ . Then C and C ∞ is connected bya disjoint union of rational components C which are totally ramified over D and D ∞ . Then C ∪ C ∞ alsogives a morphism to ( Y | D ) with fiber class. However, although the insertions in ν ∞ must be distributedto C ∞ , the insertion τ ( c ( L ) · θ ) may be distributed to C ∞ or C . Suppose there are some insertions ̟ ′′ of ̟ are distributed to C ∞ too. The rest ̟ ′ of ̟ are distributed to C . We next insert relative insertions tothe connecting nodes { n , . . . , n m } of C and rational components C by assigning η to C and ˇ η to C .If the insertion τ ( c ( L ) · θ ) is distributed to C ∞ , by the argument used for fiber class invariants in theproof of Relation 4.5 and the analysis for fiber class invariants in § CR ( η ) ≥ deg D CR ( ̟ ′′ ) + deg CR ( ν ) + deg CR ( θ ) ≥ deg CR ( ν ) + deg( θ ) . Therefore the principle term coming from this case is of the form D µ (cid:12)(cid:12)(cid:12) ̟ ′ · Ψ k ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′ k ≤ k ̟ k , deg CR ( η ) ≥ deg CR ( ν ) + 1 , whose coefficient are fiber class relative invariants of ( Y | D ). Then by repeating the rubber calculus fornon-fiber class we obtain the first term on the right side of Relation 4.6 modulo non-principle terms anddescendent orbifold Gromov–Witten invariants of D .If the insertion τ ( c ( L ) · θ ) is distribute to D we havedeg CR ( η ) ≥ deg D CR ( ̟ ′′ ) + deg CR ( ν ) ≥ deg CR ( ν ) . Therefore the principle term coming from this case is of the form D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ ′ · Ψ k ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , (4.31)with k ̟ ′ k ≤ k ̟ k , deg CR ( η ) ≥ deg CR ( ν ) , k ≥
0, whose coefficient are fiber class relative invariants of ( Y | D ).By the boundary relation in § D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ ′′ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k ≤ k ̟ k and deg CR ( η ) ≥ deg CR ( ν ),(ii) D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k ≤ k ̟ k , deg CR ( η ) ≥ deg CR ( ν ) and deg CR ( µ ′ ) ≥ deg CR ( µ )+1.(iii) D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ ′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′ k ≤ k ̟ k and deg CR ( η ) ≥ deg CR ( ν ),These three kinds of rubber invariants reduce to the second and third summands of the right hand side ofRelation 4.6 and non-principle terms as follows:(1) By rigidification, i.e. Lemma 4.1, the invariants in (i) give rise to D µ (cid:12)(cid:12)(cid:12) τ ([ D ] · c ( L ) · θ ) · ̟ ′ (cid:12)(cid:12)(cid:12) η E g,β , with k ̟ ′ k ≤ k ̟ k , and deg CR ( η ) ≥ deg CR ( ν ). These are part of the third summand of the righthand side of Relation 4.6.(2) By repeating the rubber calculus for non-fiber class, we reduce the invariants in (ii) to rubberinvariants of the form D µ ′ (cid:12)(cid:12)(cid:12) τ ( H ) · ̟ ′′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k ≤ k ̟ k , deg CR µ ′ ≥ deg CR µ + 1, and deg CR η ≥ deg CR ν . By using the boundaryrelation, modulo non-principle terms, these rubber invariants reduce to rubber invariants of theform • D µ (cid:12)(cid:12)(cid:12) τ ( H ) · ̟ ′′′ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′′ k ≤ k ̟ ′′ k and deg CR ( η ) ≥ deg CR ( ν ), • D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′′′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′′ k ≤ k ̟ ′′ k , deg CR ( η ) ≥ deg CR ( ν ) and deg CR ( µ ′ ) ≥ deg CR ( µ )+1. • D µ (cid:12)(cid:12)(cid:12) τ ( H · c ( L )) · ̟ ′′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k and deg CR ( η ) ≥ deg CR ( ν ),Then by applying the boundary relation and rigidification to these resulting rubber invariants,modulo non-principle terms we obtain the second and third summands on the right of Relation 4.6.(3) At last, applying the boundary relation to the invariants in (iii), modulo non-principle terms we get • D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ ′′ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k and deg CR ( η ) ≥ deg CR ( ν ), • D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′′ k ≤ k ̟ ′ k , deg CR ( η ) ≥ deg CR ( ν ) and deg CR ( µ ′ ) ≥ deg CR ( µ ) + 1. • D µ (cid:12)(cid:12)(cid:12) τ ( c ( L ) · θ ) · ̟ ′ · Ψ k − ∞ (cid:12)(cid:12)(cid:12) η E ∼ g,β , with k ̟ ′ k ≤ k ̟ k and deg CR ( η ) ≥ deg CR ( ν ),Similarly, by applying the boundary relation and rigidification to these resulting rubber invariants,modulo non-principle terms we obtain the second and third summands on the right of Relation 4.6.This finishes the proof of Relation 4.6. (cid:3) Third relation.
Now we compute the relative invariants of ( Y | D ) in the third term of the right sideof Relation 4.5, i.e. relative invariants of the form D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′ · ν ∞ E g,β with deg CR µ ′ ≥ deg CR µ + 1 and k ̟ ′ k ≤ k ̟ k . By the proof of Relation 4.6 we have Relation 4.7. D µ ′ (cid:12)(cid:12)(cid:12) ̟ ′ · ν ∞ E g,β = ± X k ̟ ′′ k≤k ̟ ′ k deg CR µ ′′ ≥ deg CR µ ′ deg CR ν ′′ ≥ deg CR νm ≥ C µ ′′ ,̟ ′′ ,ν ′′ · D µ ′′ (cid:12)(cid:12)(cid:12) τ ([ D ] · H · c ( L ) m ) · ̟ ′′ (cid:12)(cid:12)(cid:12) ν ′′ E g,β + · · · RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 39 where H ∈ H ( D ) satisfying R orb π ∗ ( β ) H > , C µ ′′ ,̟ ′′ ,ν ′′ are fiber class invariants of ( Y | D ) , ( Y | D ∞ ) and ( D | Y | D ∞ ) , and “ · · · ” stands for non-principle terms and descendent orbifold Gromov–Witten invariantsof D . Proof of Theorem 1.1 and Theorem 1.10.
In this subsection we give a proof of Theorem 1.1 andTheorem 1.10. We restate them here for reader’s convenience.
Theorem 4.8 (Theorem 1.1) . The relative descendent orbifold Gromov–Witten theory of the pair ( E a | PE a ) can be effectively and uniquely reconstructed from the absolute descendent orbifold Gromov–Witten theoriesof S and PE a , the Chern classes of E and O PE a ( − .Proof. By localization calculation in § E a | PE a ) are determined by descendentabsolute invariants of S and rubber invariants of ( D | Y | D ∞ ) = ( D | P ( L ⊕ O PE a ) | D ∞ ) with L = N PE a | E a = O PE a (1) and D = PE a ∼ = D ∼ = D ∞ . By the rubber calculus in § D | Y | D ∞ ) and distinguished type II invariants of ( D | Y | D ∞ ). The fiber classinvariants of ( D | Y | D ∞ ) are determined in §
3. On the other hand, by Relation 4.5, Relation 4.6 and Relation4.7, we determined distinguished type II invariants of ( D | Y | D ∞ ) by induction with respect to the partialorder ◦ ≺ with initial values being fiber class invariants determined in §
3. This finishes the proof of Theorem1.1. (cid:3)
Theorem 4.9 (Theorem 1.10) . All four theories, i.e. the absolute descendent orbifold Gromov–Wittentheory of Y = P ( L ⊕ O D ) and the relative descendent orbifold Gromov–Witten theories of the three pairs ( Y | D ) , ( Y | D ∞ ) , and ( D | Y | D ∞ ) , can be uniquely and effectively reconstructed from the absolute descendent orbifold Gromov–Witten theoryof D and the first Chern class of the line bundle L .Proof. There is a C ∗ -action on Y induced from the C ∗ -dilation on L .First of all, for an absolute invariant of Y , apply the virtual localization with respect to the C ∗ -action.Then fixed loci of the corresponding moduli space is obtained by gluing moduli space of D and D ∞ viadisjoint union of genus zero 2-marked fiber class relative moduli space of ( D | Y | D ∞ ). Then as in § D , D ∞ , hence D . Byorbifold quantum Riemann–Roch, Hodge integrals are removed.Next consider relative invariants. By relative virtual localization(a) relative invariants of ( Y | D ) are determined by Hodge integrals in twisted invariants of D ∼ = D twisted by L and rubber invariants of ( D | Y | D ∞ ).(b) relative invariants of ( Y | D ∞ ) are determined by Hodge integrals in twisted invariants of D ∞ ∼ = D twisted by L ∗ and rubber invariants of ( D | Y | D ∞ ).(c) relative invariants of ( D | Y | D ∞ ) are determined by Hodge integrals in twisted invariants of D ∼ = D twisted by L , Hodge integrals in twisted invariants of D ∞ ∼ = D twisted by L ∗ and rubber invariantsof ( D | Y | D ∞ ).By the rubber calculus in § D | Y | D ∞ ) are determined by fiber class invariantsof ( D | Y | D ∞ ) and distinguished Type II invariants of ( D | Y | D ∞ ). It is proved in § § D | Y | D ∞ ) are all determined by Gromov–Witten theoryof D and c ( L ). Again Hodge integrals are removed by orbifold quantum Riemann–Roch. This finishes theproof of Theorem 1.10. (cid:3) Relative orbifold Gromov–Witten theory of weighted blowups
In this section we determine all admissible relative descendent orbifold Gromov–Witten invariants ofweighted blowups, hence prove Theorem 1.3.We first recall the notations. Let X be a compact symplectic orbifold groupoids with S being a symplecticsuborbifold groupoid with codimension codim S = 2 n , and normal bundle N . Recall from (1.1), the weight- a blowup of X along S gives rise to a degeneration of X into X degenerate −−−−−−−→ ( X a | D a ) ∧ D a ( N a | D a ) . Here X a is the weight- a blowup of X along S , N a is the weight- a projectification of N , D a is the exceptionaldivisor in X a and also the infinity section PN a of N a , i.e. D a = PN a . So we have N D a | X a = N ∗ D a | N a = O PN a ( − , and c ( N D a | X a ) = − c ( N D a | N a ) . We have natural maps κ : ( X a | D a ) → ( X , S ) , π : N a → S . Recall that in § ⋆ , i.e. (2.4), of H ∗ CR ( D a ) = H ∗ CR ( PN a ) via fixing a basis σ ⋆ of H ∗ CR ( S ). There is also a dual basis which we denoted by Σ ⋆ in § ι : S → X being the inclusion map. Then we have an induced restriction map ι ∗ : H ∗ CR ( X ) → H ∗ CR ( S ) . By orbifold Poincar´e dual, ι ∗ determines the push-forward ι ∗ : H ∗ CR ( S ) → H ∗ CR ( X ) . The total Chern class of N is also determined by ι ∗ via c ( N ) ∪ c ( T S ) = ι ∗ ( c ( T X )) . In particular, c ( N ) = ι ∗ ( c ( T X )) − c ( T S ) . We next prove Theorem 1.3 by using Theorem 1.1 and the weighted-blowup correspondence result in [7].The main tool is the degeneration formula for orbifold Gromov–Witten theory in [13, 1].Take a relative invariant of ( X a | D a ) D ω (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β (5.1)with µ being weighted by the basis Σ ⋆ , the dual of the basis Σ ⋆ of H ∗ CR ( D a ). The degeneration (1.2) is alsoa degeneration of ( X a | D a ) ( X a | D a ) degenerate −−−−−−−→ ( X a | D a ) ∧ D a ( D a , ∞ |O D a ( − | D a , ) . Here O D a ( −
1) := P ( O D a ( − ⊕ O D a ), D a , and D a , ∞ are the zero and infinity sections of P ( O D a ( − ⊕ O D a ),and both are isomorphic to D a . The gluing is along the D a ⊆ X a and D a , ∞ ⊆ O D a ( − D ω (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β = X D ω (cid:12)(cid:12)(cid:12) η E • , ( X a | D a ) g ,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) ω (cid:12)(cid:12)(cid:12) µ E • , ( D a , ∞ |O D a ( − | D a , ) g ,β (5.2)with summation taking over all splittings of ( g, β ), and distributions of insertions ω , and all immediatecohomology weights η coming from the basis Σ ⋆ of H ∗ CR ( D a ). Therefore the cohomology weights ˇ η comefrom the basis Σ ⋆ . As in § D ω (cid:12)(cid:12)(cid:12) η E • , ( X a | D a ) g,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) ∅ (cid:12)(cid:12)(cid:12) µ E • , ( D a , ∞ |O D a ( − | D a , )0 ,d [ F ] such that every component of D ˇ η (cid:12)(cid:12)(cid:12) ∅ (cid:12)(cid:12)(cid:12) µ E • , ( D a , ∞ |O D a ( − | D a , )0 ,d [ F ] is a fiber class invariant and ~η = ~µ but η = µ ,we have D ˇ η (cid:12)(cid:12)(cid:12) ∅ (cid:12)(cid:12)(cid:12) µ E • , ( D a , ∞ |O D a ( − | D a , )0 ,d [ F ] = 0 . Therefore, such summand would not appear on the right hand side of (5.2). For the rest summands on theright side of (5.2), we define (cf. [7, Definition 6.3]) D ω (cid:12)(cid:12)(cid:12) η E • , ( X a | D a ) g ,β ≺ D ω (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β . (5.3)The following theorem is [7, Theorem 6.5]. Theorem 5.1.
The “ ≺ ” in (5.3) is a partial order on the set of (possibly disconnected) relative invariantsof ( X a | D a ) of the form (5.1) , i.e. the relative insertions coming from the chosen basis (2.4) of H ∗ CR ( D a ) .For a fixed relative invariants, there are only finite relative invariants are lower than it with respect to “ ≺ ”. RBIFOLD GROMOV–WITTEN THEORY OF WEIGHTED BLOWUPS 41
We restrict this partial order on the set of all admissible relative invariants of ( X a | D a ).Now take an admissible relative invariant D γ (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β (5.4)of ( X a | D a ) with γ = Q i τ k i γ i , γ i ∈ K and µ weighted by the chosen basis Σ ⋆ . Then ˇ µ is weighted by thechosen basis Σ ⋆ in (2.4). Supposeˇ µ = (cid:16) ( µ , δ s ( g ) ∪ H m ( h ) ) , . . . , ( µ ℓ ( µ ) , δ s ℓ ( µ ) ( g ℓµ ) ∪ H m ℓ ( µ ) ( h ℓ ( µ ) ) ) (cid:17) . For each 1 ≤ j ≤ ℓ ( µ ), since I π ( h j ) = ( g j ) and the contact order at j -th marked points is µ j , ( h j ) has arepresentative of the form ( g j , exp π √− µ j ) in local group. We next assign to each ( µ j , δ s j ( g j ) ∪ H m j ( h j ) ) in ˇ µ afiber class relative invariant of ( N a | D a ) as follows (cf. [7, § j = (cid:0) g = 0 , A = µ j [ F ] , ( g − j ) , ( µ j , ( h j )) (cid:1) , where ( g − j ) indicates the twistedsector of the unique absolute marked point and ( µ j , ( h j )) indicates the contact order and twistedsector of the unique relative marked point.(b) The relative insertion is δ s j ( g j ) ∪ H m j ( h j ) .(c) The absolute insertion is τ c j ( ι ∗ (ˇ δ s j ( g j ) )), where ˇ δ s j ( g j ) ∈ H ∗ ( S ( g − j )) is the orbifold Poincar´e dual of δ s j ( g j ) in H ∗ CR ( S ) (cf. (2.3)) and c j ∈ Z ≥ is determined by the equation (5.5) in the following.Denote the corresponding moduli space by M Γ j ( N a | D a ). The number c j is determined by dimensionconstraint, i.e. we must havevirdim C M Γ j ( N a | D a )= deg τ c j ( ι ∗ (ˇ δ s j ( g j ) )) + deg δ s j ( g j ) ∪ H m j ( h j ) = c j + codim C ( S ( g − j ) , X ( g − j )) + 12 (deg δ s j ( g j ) + deg ˇ δ s j ( g j ) ) + m j = c j + codim C ( S ( g j ) , N ( g j )) + dim C S ( g j ) + m j = c j + dim C N ( g j ) + m j . The virtual dimension of such a moduli space M Γ j ( N a | D a ) was computed in [7, Proposition 5.11]. Suppose g − j acts on the fiber of N via g − j · ( z , . . . , z n ) = (exp π √− b ( g − j )1 N o ( gj ) z , . . . , exp π √− b ( g − j ) n N o ( gj ) z n )where o ( g j ) is the order of g j and 1 ≤ b ( g − j ) i N ≤ o ( g j ) , ≤ i ≤ n are the action weights of g − j on N . Thenwe have virdim C M Γ j ( N a | D a ) = n X i =1 " − b ( g − j ) i N o ( g j ) + a i µ j + n − C N ( g j ) , where [ · ] means the integer part of a real number. Therefore c j is determined by c j = n X i =1 " − b ( g − j ) i N o ( g j ) + a i µ j + n − − m j . (5.5)So to each ( µ j , δ s j ( g j ) ∪ H m j ( h j ) ) in ˇ µ , the assigned fiber class relative invariant is D τ c j ( ι ∗ (ˇ δ s j ( g j ) )) (cid:12)(cid:12)(cid:12) ( µ j , δ s j ( g j ) ∪ H m j ( h j ) ) E N a | D a Γ j with c j determined by (5.5). Now we glue these relative invariants to the relative invariant (5.4) to get anabsolute invariant of X : D ¯ γ · µ S E X g,β := D Y i τ k i ¯ γ i · Y j τ c j ( ι ∗ (ˇ δ s j ( g j ) )) E X g,β (5.6) where ¯ γ i is the pre-image of γ i in H ∗ CR ( X ), and µ S := ( τ c ( ι ∗ (ˇ δ s ( g ) )) , . . . , τ c ℓ ( µ ) ( ι ∗ (ˇ δ s ℓ ( µ ) ( g ℓ ( µ ) ) ))) . Hu and the first two authors proved in [7, Theorem 5.29, 6.10, 6.12] the following weighted-blowupcorrespondence results.
Theorem 5.2.
The correspondence (5.4) → (5.6) is a one-to-one map from the set of admissible relative invariants of ( X a | D a ) to the set of absolute invariantsof X of the form (5.6) . This means that c j is uniquely determined by ( µ j , δ s j ( g j ) ∪ H m j ( h j ) ) , i.e. when we fix ( g j ) ,hence o ( g j ) and b ( g − j ) u N , ≤ u ≤ n are fixed, then c j is determined by these datum via (5.5) . Conversely,given c j and ( g j ) we could get ( h j ) , µ j and m j uniquely.Moreover D ¯ γ · µ S E X g,β = D γ (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β · z ( µ ) · C +(5.7) XD γ − (cid:12)(cid:12)(cid:12) η E • , X a , D a g ,β ≺ D γ (cid:12)(cid:12)(cid:12) µ E ( X a | D a ) g,β D γ − (cid:12)(cid:12)(cid:12) η E • , ( X a | D a ) g ,β · z ( η ) · D ˇ η (cid:12)(cid:12)(cid:12) γ + · Y j τ c j ( ι ∗ (ˇ δ s j ( g j ) )) E • , ( N a | D a ) g ,β with C = D ˇ µ (cid:12)(cid:12)(cid:12) Y j τ c j ( ι ∗ (ˇ δ s j ( g j ) )) E • , ( N a | D a )0 ,d [ F ] = Y j n o ( g j ) µ m j j n Y i =1 b ( g − j ) i N o ( g j ) + [ − b ( g − j ) i N o ( g j ) + a i µ j ])! o = 0 . (5.8) Here the definition of a ! for a rational number a is the same as the one in Relation 4.5, and γ − and γ + areextensions of parts of γ over X a and N a respectively. For ¯ γ i we have γ − i = γ i . Therefore (cid:10) γ − (cid:12)(cid:12) η (cid:11) • , ( X a | D a ) g ,β are admissible relative invariants of ( X a | D a ) . Now we can prove Theorem 1.3. For reader’s convenience we restate it here.
Theorem 5.3 (Theorem 1.3) . The admissible relative descendent orbifold Gromov–Witten theory of ( X a | D a ) can be uniquely and effectively reconstructed from the orbifold Gromov–Witten theories of X , S and D a , therestriction map H ∗ CR ( X ) → H ∗ CR ( S ) and the first Chern class of O D a ( − .Proof. The equation (5.7) gives us a lower triangular system determining the admissible relative invariantsof ( X a | D a ) in terms of the Gromov–Witten invariants of X and the relative Gromov–Witten invariants of( N a | D a ). By Theorem 1.1, the relative Gromov–Witten invariants of ( N a | D a ) are determined by the Gromov–Witten invariants of S and D a , c ( N ) and c ( N D a | N a ) = − c ( N D a | X a ) = − c ( O D a ( − X a | D a ) can be uniquely and effectively reconstructed from the Gromov–Wittentheories of X , S and D a , the first chern class c ( N D a | X a ) and the restriction map H ∗ CR ( X ) → H ∗ CR ( S ). Thisfinishes the proof. (cid:3) References [1] D. Abramovich and B. Fantechi. Orbifold thechniques in degeneration formulas.
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E-mail address : [email protected] School of mathematics and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, China
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