Ruan's Conjecture on Singular symplectic flops
aa r X i v : . [ m a t h . AG ] A p r RUAN’S CONJECTURE ON SINGULAR SYMPLECTICFLOPS
BOHUI CHEN, AN-MIN LI, AND GUOSONG ZHAO
Abstract.
We prove that the orbifold quantum ring is preserved undersingular symplectic flops. Hence we verify Ruan’s conjecture for thiscase.
Contents
1. Introduction 22. Relative orbifold Gromov-Witten theory and the degenerationformula 32.1. The Chen-Ruan Orbifold Cohomologies 32.2. Orbifold Gromov-Witten invariants 32.3. Ring structures on H ∗ CR ( X ). 42.4. Moduli spaces of relative stable maps for orbifold pairs 42.5. Relative orbifold Gromov-Witten invariants. 72.6. The degeneration formula 83. Singular symplectic flops 93.1. Local models and local flops 93.2. Torus action. 103.3. Symplectic orbi-conifolds and singular symplectic flops 113.4. Ruan cohomology rings 134. Relative Gromov-Witten theory on M sr and M sfr M sr and M sfr M sr , Z ) 164.3. Admissible data (Γ , T , a ) 175. Vanishing results on relative invariants 185.1. Localization via the torus action 185.2. Vanishing results on I Ω F T ( λ, u ), (I) 205.3. Vanishing results on I Ω F T ( λ, u ), (II) 226. Proof of the Main theorem 236.1. Reducing the comparison to local models 246.2. Proof of Proposition 6.3 25References 25 B.C. and A.L. are supported by NSFC, G.Z. is supported by a grant of NSFC andQiushi Funding. Introduction
One of deep discovery in Gromov-Witten theory is its intimate relationwith the birational geometry. A famous conjecture of Ruan asserts thatany two K -equivalent manifolds have isomorphic quantum cohomology rings([R1]) (see also [Wang]). Ruan’s conjecture was proved by Li-Ruan forsmooth algebraic 3-folds ([LR]) almost ten years ago. Only recently, it wasgeneralized to simple flops and Mukai flops in arbitrary dimensions by Lee-Lin-Wang ([LLW]). In a slightly different context, there has been a lot ofactivities regarding Ruan’s conjecture in the case of McKay correspondence.On the other hand, it is well known that the appropriate category tostudy the birational geometry is not smooth manifolds. Instead, one shouldconsider the singular manifolds with terminal singularities. In the complexdimension three, the terminal singularities are the finite quotients of hyper-surface singularities and hence the deformation of them are orbifolds. Ittherefore raises the important questions if Ruan’s conjecture still holds forthe orbifolds where there are several very interesting classes of flops. Thisis the main topic of the current article.Li-Ruan’s proof of the case of smooth 3-folds consists of two steps. Thefirst step is to interpret flops in the symplectic category, then, they usealmost complex deformation to reduce the problem to the simple flop; thesecond step is to calculate the change of quantum cohomology under thesimple flop. The description of a smooth simple flop is closely related to theconifold singularity W = { ( x, y, z, t ) | xy − z + t = 0 } . In [CLZZ], we initiate a program to understand the flop associated withthe singularities W r = { ( x, y, z, t ) | xy − z r + t = 0 } /µ r ( a, − a, , .W r appears in the list of terminal singularities in [K]. The singularitieswithout quotient are also studied in [La] and [BKL].The program is along the same framework of that in [LR]. The first stepis to describe the flops with respect to W r symplectically. This is done inthe previous paper([CLZZ]). Our main theorem in this paper is Theorem 1.1.
Suppose that Y s is a symplectic 3-fold with orbifold singu-larities of type W r , . . . , W r n and Y sf is its singular flop, then QH CR ( Y s ) = QH CR ( Y sf ) . Theorem 1.1 verifies Ruan’s conjecture in this particular case. We shouldmention that Ruan also proposed a simplified version of the above conjecturein terms of Ruan cohomology RH CR which has been established in [CLZZ]as well. Furthermore, our previous results enters the proof of this generalconjecture in a crucial way. UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 3
The technique of the proof is a combination of the degeneration formula oforbifold Gromov-Witten invariants, the localization techniques and dimen-sion counting arguments. The theory of relative orbifold Gromov-Witten in-variants and its degeneration formula involves heavy duty analysis on modulispaces and will appear elsewhere ([CLS]).The paper is organized as following. We first describe the relative orb-ifold GW-invariants and state the degeneration formula (without proof)( § § § § § Acknowledge.
We would like to thank Yongbin Ruan for suggesting theproblem and for many valuable discussions. We also wish to thank Qi Zhangfor many discussions.2.
Relative orbifold Gromov-Witten theory and thedegeneration formula
The Chen-Ruan Orbifold Cohomologies.
Let X be an orbifold.For x ∈ X , if its small neighborhood U x is given by a uniformization system( ˜ U , G, π ), we say G is the isotropy group of x and denoted by G x . Let T = [ x ∈ X G x ! / ∼ . Here ∼ is certain equivalence relation. For each ( g ) ∈ T , it defines a twistedsector X ( g ) . At the mean while, the twisted sector is associated with adegree-shifting number ι ( g ). The Chen-Ruan orbifold cohomology is definedto be H ∗ CR ( X ) = H ∗ ( X ) ⊕ M ( g ) ∈T H ∗− ι ( g ) ( X ( g ) ) . For details, readers are referred to [CR1].2.2.
Orbifold Gromov-Witten invariants.
Let M g,n,A ( X ) be the mod-uli space of representable orbifold morphism of genus g, n -marked points and A ∈ H ( X, Z ) (cf. [CR2],[CR3]). By specifying the monodromy h = (( h ) , . . . , ( h n ))at each marked points, we can decompose M g,n,A ( X ) = G h M g,n,A ( X, h ) . Let ev i : M g,n,A ( X, h ) → X ( h i ) , ≤ i ≤ n BOHUI CHEN, AN-MIN LI, AND GUOSONG ZHAO be the evaluation maps. The primary orbifold Gromov-Witten invariantsare defined as h α , . . . , α n i Xg,A = Z virt M g,n,A ( X, h ) m Y i =1 ev ∗ i ( α i ) , where α i ∈ H ∗ ( X ( h i ) ).In particular, set h α , α , α i CR = h α , α , α i , , h α , α , α i = h α , α , α i CR + X A =0 h α , α , α i ,A . Ring structures on H ∗ CR ( X ) . Let V be a vector space over R . Let h : V ⊗ V → R be a non-degenerate pairing and A : V ⊗ V ⊗ V → R be a triple form. Then it is well known that one can define a product ∗ on V by h ( u ∗ v, w ) = A ( u, v, w ) . Different A ’s give different products. Remark 2.1.
Suppose we have ( V, h, A ) and ( V ′ , h ′ , A ′ ) . A map φ : V → V ′ induces an isomorphism (with respect to the product) if φ is a groupisomorphism and φ ∗ h ′ = h, and φ ∗ A ′ = A. Now let V = H ∗ CR ( X ) and h be the Poincare pairing on V . If A ( α , α , α ) = h α , α , α i CR , it defines the Chen-Ruan product. If A ( α , α , α ) = h α , α , α i , it defines the Chen-Ruan quantum product. We denote the ring to be QH ∗ CR ( X ).2.4. Moduli spaces of relative stable maps for orbifold pairs.
Forthe relative stable maps for the smooth case , there are two equivariantversions. One is on the symplectic manifolds with respect to cylinder ends,each of which admits a Hamiltonian S action ([LR]), the other is on theclosed symplectic manifolds with respect to divisors([LR],[Li]). This is alsotrue for orbifolds. We adapt the second version here.Let X be a symplectic orbifold with disjoint divisors { Z , . . . , Z k } . For simplicity, we assume k = 1 and Z = Z . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 5
By a relative stable map in (
X, Z ), we mean a stable map f ∈ M g,n,A ( X )with additional data that record how it intersects with Z . Be precisely,suppose f : (Σ g , z ) → X. The additional data are collected in order: • Set x = f − ( Z ) = { x , . . . , x k } . We call x i the relative marked points . The rest of marked points aredenoted by p = { p , . . . , p m } , i.e, z = x ∪ p ; • Let g = (( g ) , . . . , ( g k ))denote the monodromy of f (with respect to Z ) at each point in x .The rest are denoted by h = (( h ) , . . . , ( h m ))which are the monodromy of f (with respect to X ) at each point in p . • the multiplicity of the tangency ℓ j of f with Z at z j = f ( x j ) isdefined by the following: Locally, the neighborhood of z j is given by( ˜ V × C → ˜ V ) /G z j , where ˜ V /G z j ⊂ Z . Suppose the lift of f is˜ f : ˜ D → ˜ V × C ˜ f ( t ) = ( v ( t ) , u ( t ))Suppose the multiplicity of u is α and g j ∈ G z j acts on the fiber over z j with multiplicity c . Then the multiplicity is set to be ℓ j = α · c | g j | . We say f maps x j to Z g j at ℓ j z j . Set l = ( ℓ , . . . , ℓ k ) . We may write f − ( Z ) = l · x = k X j =1 ℓ j x j . As a relative object, we say f is in the moduli space of relative map M g,n,A ( X, Z, h , g , l ) . We denote the map by f : (Σ , p , l · x ) → ( X, Z ) . BOHUI CHEN, AN-MIN LI, AND GUOSONG ZHAO
We now describe the compactification of this moduli space. The construc-tion is similar to the smooth case([LR]).The target space of a stable relative map is no longer X . Instead, it isextended in the following sense: let L → Z be the normal bundle of Z in X and P Z = P ( L ⊕ C )be its projectification, then given an integer b ≥
0, we have an extendedtarget space X ♯b := X ∪ [ ≤ α ≤ b P Z α . Here
P Z α denotes the α -th copy of P Z . Let Z α be the 0-section and Z α ∞ bethe ∞ -section of P Z α . X is called the root component of X ♯b . Z b is calledthe divisor of X ♯b and is (again) denoted by Z . Definition 2.1.
A relative map in X ♯b consists of following data: on eachcomponent, there is a relative map: on the root component, the map isdenoted by f : (Σ , p , l · x ) → ( X, Z ); and on each component P Z α , the map is denoted by f α : (Σ α , p α , l α · x α ∪ ¯ l α · ¯ x α ) → ( P Z α , Z α ∪ Z α ∞ ) . Here l α · x α = f − ( Z α ) and ¯ l α · ¯ x α = f − ( Z α ∞ ) . Moreover, we require f α at Z α matches f α +1 at Z α +1 ∞ . (see Remark 2.2.)We denote such a map by f = ( f , f , . . . , f b ) . Set x = x b and g j = g x bj , g = ( g , . . . , g | x | ) ℓ j = ℓ bj , l = ( ℓ , . . . , ℓ | x | ) . We say that f maps x j to the divisor Z of X ♯b at ℓ j z j ∈ Z ( g j ) . Similarly, h records the twisted sector for p = p ∪ [ α p α . The homology class A in X represented by f can be defined properly. Collectthe data Γ = ( g, A, h , g , l ) , T = ( g , l ) . We say that f is a relative orbifold map in X ♯b of type (Γ , T ) . Denote themoduli space by ˜ M Γ , T ( X ♯b , Z ) . Remark 2.2.
Let f α and f α +1 be as in the definition. Suppose that • f α maps x αj ∈ x α to ( Z α ) ( g αj ) at ℓ αk z αj ; • f α +1 maps ¯ x α +1 i ∈ ¯ x α +1 to ( Z α +1 ∞ ) (¯ g α +1 i ) at ¯ ℓ α +1 i ¯ z α +1 i , UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 7 then by saying that f α at Z α matches f α +1 at Z α +1 ∞ we mean that ℓ αi = ¯ ℓ α +1 i , z αi = ¯ z α +1 i , and g αi = ¯ g α +1 i . Note that there is a C ∗ action on P Z α . Let T be the product of these b copies of C ∗ . Then T acts on ˜ M Γ , T ( X ♯b , Z ). Define M Γ , T ( X ♯b , Z ) = ˜ M Γ , T ( X ♯b , Z ) /T. It is standard to show that
Proposition 2.3.
There exists a large integer B which depends on topolog-ical data (Γ , T ) such that M Γ , T ( X ♯b , Z ) is empty when b ≥ B . Hence,
Definition 2.2.
The compactified moduli space is M Γ , T ( X, Z ) = [ b ∈ Z ≥ M Γ , T ( X ♯b , Z ) . The following technique theorem is proved in [CLS]
Theorem 2.4. M Γ , T ( X, Z ) is a smooth compact virtual orbifold withoutboundary with virtual dimension c ( A ) + 2(dim C X − − g ) + 2 m X i =1 (1 − ι ( h i )) + 2 n X j =1 (1 − ι ( g j ) − [ ℓ j ]) , where [ ℓ j ] is the largest integer that is less or equal to ℓ j . Relative orbifold Gromov-Witten invariants.
Let M Γ , T ( X, Z )be the moduli space given above. There are evaluation maps ev Xi : M Γ , T ( X, Z ) → X ( h i ) , ev Xi ( f ) = f ( p i ) , ≤ i ≤ m ;and ev Zj : M Γ , T ( X, Z ) → Z ( g j ) , ev Zj ( f ) = f ( x j ) , ≤ j ≤ k. Then for α i ∈ H ∗ ( X ( h i ) ) , ≤ i ≤ m, β j ∈ H ∗ ( Z ( g j ) ) , ≤ j ≤ k the relative invariant is defined as h α , . . . , α m | β , . . . , β k , T i ( X,Z )Γ = 1 | Aut ( T ) | Z vir M Γ , T ( X,Z ) m Y i =1 ( ev Xi ) ∗ α i k Y j =1 ( ev Zj ) ∗ β j . In this paper, we usually set a = ( α , . . . , α m ) , b = ( β , . . . , β k ) , then the invariant is denoted by h a | b , T i ( X,Z )Γ . BOHUI CHEN, AN-MIN LI, AND GUOSONG ZHAO
Moreover, if Γ = ` γ Γ γ , the relative invariants (with disconnected domaincurves) is defined to be the product of each connected component h a | b , T i • ( X,Z )Γ = Y γ h a | b , T i ( X,Z )Γ γ . The degeneration formula.
The symplectic cutting also holds fororbifolds. Let X be a symplectic orbifold. Suppose that there is a local S Hamiltonian action on U ⊂ X . We assume that U ∼ = Y × ( − , π : U → ( − , Y × { } splits X into two orbifolds withboundary Y , denoted by X ± . Then the routine symplectic cutting gives thedegeneration π : X → ¯ X + ∪ Z ¯ X − . Topologically, ¯ X ± is obtained by collapsing the S -orbits of the boundariesof X ± .There are maps π ∗ : H ( X ) → H ( X + ∪ Z X − ) , π ∗ : H ∗ ( X + ∪ Z X − ) → H ∗ ( X ) . For A ∈ H ( M ) we set [ A ] ⊂ H ( X ) to be π − ∗ ( π ∗ ( A )) and denote π ∗ ( A )by ( A + , A − ). On the other hand, for α ± ∈ H ∗ ( X ± ) with α + | Z = α − | Z ,it defines a class on H ∗ ( X + ∪ Z X − ) which is denoted by ( α + , α − ). Let α = π ∗ ( α + , α − ). Theorem 2.5.
Suppose π : X → X + ∪ Z X − is the degeneration. Then (2.1) h a i X Γ = X I X η =(Γ + , Γ − ,I ρ ) C η h a + | b I , T i • ( X + ,Z )Γ + h a − | b I , T i • ( X − ,Z )Γ − . Notations in the formula are explained in order. Γ is a data for Gromov-Witten invariants, it includes ( g, [ A ]); (Γ + , Γ − , I ρ ) is an admissible triplewhich consists of (possible disconnected) topological types Γ ± with the samerelative data T under the identification I ρ and they glue back to Γ. (Forinstance, one may refer to [HLR] who interpret Γ’s as graphs and then thegluing has an obvious geometric meaning); the relative classes β i ∈ b I runsover a basis of Z ( g i ) and at the mean while β i runs over the dual basis; finally C η = | Aut ( T ) | k Y i =1 ℓ i for T = ( g , l ). UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 9 Singular symplectic flops
Local models and local flops.
Locally, we are concern those resolu-tions of ˜ W r = { ( x, y, z, t ) | xy − z r + t = 0 } and their quotients. ˜ W r − { } inherits a symplectic form ˜ ω ◦ r from C .By blow-ups, we have two small resolutions of ˜ W r :˜ W sr = { (( x, y, z, t ) , [ p, q ]) ∈ C × P | xy − z r + t = 0 , pq = xz r − t = z r + ty } ˜ W sfr = { (( x, y, z, t ) , [ p, q ]) ∈ C × P | xy − z r + t = 0 , pq = xz r + t = z r − ty } . Let ˜ π sr : ˜ W sr → ˜ W r , ˜ π sfr : ˜ W sfr → ˜ W r be the projections. The exceptional curves (˜ π sr ) − (0) and (˜ π sfr ) − (0) aredenoted by ˜Γ sr and ˜Γ sfr respectively. Both of them are isomorphic to P .Let µ r = h ξ i , ξ = e πir be the cyclic group of r -th roots of 1. We denote its action on C by µ r ( a, b, c, d ) if the action is given by ξ · ( x, y, z, t ) = ( ξ a x, ξ b y, ξ c z, ξ d t ) . Then µ r ( a, − a, ,
0) acts on ˜ W r , and naturally extending to its small reso-lutions. Set W r = ˜ W r /µ r , W sr = ˜ W sr /µ r , W sfr = ˜ W sfr /µ r . Similarly, Γ sr = ˜Γ sr /µ r Γ sfr = ˜Γ sfr /µ r . We call that W s and W sf are the small resolutions of W r . We say that W sf is the flop of W s and vice versa. They are both orbifolds with singular pointson Γ s and Γ sf . Note that the symplectic form ˜ ω ◦ r reduces to a symplecticform ω ◦ r on W r .It is known that Proposition 3.1.
For r ≥ , the normal bundle of ˜Γ sr (˜Γ sfr ) in ˜ W sr ( ˜ W sfr ) is O ⊕ O ( − . Proof.
We take ˜ W sr as an example. For the setΛ p = { q = 0 } , set u = p/q . Then ( u, z, y ) gives a coordinate chart for Λ p . Similarly, forthe set Λ q = { p = 0 } , set v = q/p . Then ( v, z, x ) gives a coordinate chart for Λ q . The transitionmap is given by(3.1) v = u − ; z = z ; x = − u y + 2 uz r . By linearize this equation, it is easy to get the conclusion. q.e.d.
Corollary 3.2.
For r ≥ , the normal bundle of Γ sr (Γ sfr ) in W sr ( W sfr ) is ( O ⊕ O ( − /µ r . On ˜Γ sr (˜Γ sfr ), there are two special points. In term of [ p, q ] coordinates,they are 0 = [0 , ∞ = [1 , . We denote them by p s and q s ( p sf and q sf ) respectively. After taking quo-tients, they become singular points. By the proof of Proposition 3.1, theuniformization system of p s is { ( p, x, y, z, t ) | x = t = 0 } with µ r action given by ξ ( p, y, z ) = ( ξ a p, ξ − a y, ξz ) . At p s , for each given ξ k = exp(2 πik/r ) , ≤ k ≤ r , there is a correspondingtwisted sector([CR1]). As a set, it is same as p s . We denote this twistedsector by [ p s ] k . For each twisted sector, a degree shifting number is assigned.We conclude that Lemma 3.3.
For ξ k = exp(2 πik/r ) , ≤ k ≤ r , the degree shifting ι ([ p s ] k ) = 1 + kr . Proof.
This follows directly from the definition of degree shifting. q.e.d.Similar results hold for the singular point q s . Hence we also have twistedsector [ q s ] k and ι ([ q s ] k ) = 1 + kr . Similarly, on W sf , there are twisted sectors [ p sf ] k , [ q sf ] k and ι ([ p sf ] k ) = ι ([ q sf ] k ) = 1 + kr . Torus action.
We introduce a T -action on ˜ W r :( t , t )( x, y, z, t ) = ( t t r x, t − t r y, t z, t r t ) . For an action t a t b · , we write the weight of action by aλ + bu . This actionnaturally extends to the actions on all models generated from ˜ W r , such as W r , W sr and W sfr . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 11
It then induces an action on ˜Γ sr (˜Γ sfr ):( t , t )[ p, q ] = [ t p, q ] . Recall that the normal bundle of ˜Γ sr in ˜ W sr is O ⊕ O ( − Lemma 3.4.
The action weights at O p and O q are u . The action weightsat O p ( − and O q ( − are − λ + ru and λ + ru . Proof.
This follows directly from the model given by § p s , q s ( p sf , q sf ) are fixed points of the action.On the other hand, there are four special lines connecting to these pointsthat are invariant with respect to the action. Let us look at ˜ W sr . For thepoint p s , two lines are in Λ p and are given by˜ L sp,y = { x = z = t = 0 , u = 0 } , ˜ L sp,z = { x = y = 0 , z r + t = 0 , u = 0 } . To the point q s , two lines are in Λ q and are given by˜ L sq,x = { y = z = t = 0 , v = 0 } , ˜ L sq,z = { x = y = 0 , z r − t = 0 , v = 0 } . Similarly, for ˜ W sfr we have˜ L sfp,y = { x = z = t = 0 , u = 0 } , ˜ L sfp,z = { x = y = 0 , z r − t = 0 , u = 0 } , ˜ L sfq,x = { y = z = t = 0 , v = 0 } , ˜ L sfq,z = { x = y = 0 , z r + t = 0 , v = 0 } . Correspondingly, these lines in W sr and W sfr are denoted by the same nota-tions without tildes. Remark 3.5.
Note that the defining equations for the pairs L sq,x and L sfq,x , L sp,y and L sfp,y are same. Symplectic orbi-conifolds and singular symplectic flops.
Anorbi-conifold ([CLZZ]) is a topological space Z with a set of (singular) points P = { p , . . . , p k } such that Z − P is an orbifold and for each p i ∈ P there exists a neighborhood U i that is isomorphic to W r i for some integer r i ≥
1. By a symplecticstructure on Z we mean a symplectic form ω on Z − P and it is ω ◦ r i in U i .We call Z a symplectic orbi-conifold. There exists 2 k resolutions of Z . Let Y s be such a resolution, its flop is defined to be the one that is obtained byflops each local model of Y s . We denote it by Y sf . In [CLZZ] we prove that Theorem 3.6. Y s is a symplectic orbifold if and only if Y sf is. So Y sf is called the (singular) symplectic flop of Y s and vice versa.Now for simplicity, we assume that Z contains only one singular point p and is smooth away from p . Suppose Y s and Y sf are two resolutions thatare flops of each other and, locally, Y s contains W sr and Y sf contains W sfr .Then H ∗ CR ( Y s ) = H ∗ ( Y s ) ⊕ r M k =1 C [ p s ] k ⊕ r M k =1 C [ q s ] k ; H ∗ CR ( Y sf ) = H ∗ ( Y sf ) ⊕ r M k =1 C [ p sf ] k ⊕ r M k =1 C [ q sf ] k . Lemma 3.7.
There are natural isomorphisms ψ k : H k ( Y s ) → H k ( Y sf ) . Proof.
We know that Y s − Γ s = Y sf − Γ sf . We also have the exact sequence · · · → H k ( Y, Y \ Γ) → H k ( Y ) → H k ( Y \ Γ) → H k +1 ( Y, Y \ Γ) → · · · and H k ( Y, Y \ Γ) ∼ = H kc ( O ⊕ O ( − ∼ = H k − ( P ) .Y is either Y s or Y sf and Γ is the exceptional curve in Y .Suppose we have ω s ∈ H k ( Y s ). Suppose X sω s is its Poincare dual. If k >
2, we may require that X sω s ∩ Γ s = ∅ . Hence X sω s is in Y s \ Γ s = Y sf \ Γ sf . Using this, we get a class ω sf ∈ H k ( Y sf ). Set ψ k ( ω s ) = ω sf .If k ≤
2, since H mcomp ( O ⊕ O ( − , m ≤ , we have H k ( Y s ) ∼ = H k ( Y s \ Γ s ) ∼ = H k ( Y sf \ Γ sf ) ∼ = H k ( Y sf ) . The isomorphism gives ψ k . q.e.d.On the other hand, we set ψ o ([ p s ] k ) = [ p sf ] k , ψ o ([ q s ] k ) = [ q sf ] k , Totally, we combine ψ k and ψ o to get a map(3.2) Ψ ∗ : H ∗ CR ( Y s ) → H ∗ CR ( Y sf ) . It can be shown that
Proposition 3.8. Ψ ∗ preserves the Poincare pairing. UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 13
Without considering the extra classes from twisted sectors, the proof isstandard. When the cohomology classes from twisted sectors are involved,it is proved in [CLZZ].On the other hand, there is a natural isomorphismΨ ∗ : H ( Y s ) → H ( Y sf )with Ψ ∗ ([Γ sr ]) = − [Γ sfr ].Now suppose that we do the symplectic cutting on Y s and Y sf at W sr and W sfr respectively. Then π s : Y s degenerate −−−−−−−→ Y − ∪ Z M sr ; π sf : Y sf degenerate −−−−−−−→ Y − ∪ Z M sfr . It is clear that M sr and M sfr are flops of each other. Then similarly, we havea map Ψ ∗ r : H ∗ orb ( M sr ) → H ∗ orb ( M sfr ) . It is easy to see that the diagram(3.3) H ∗ CR ( Y − ∪ Z M sr ) ( id,σ ∗ ) −−−−→ H ∗ CR ( Y − ∪ Z M sfr ) π ∗ s y y π ∗ sf H ∗ CR ( Y s ) Σ ∗ −−−−→ H ∗ CR ( Y sf )commutes.3.4. Ruan cohomology rings.
As explained in § A . In the current situation, we candefine a ring structure on Y s (and Y sf ) that plays a role between Chen-Ruan (classical) ring structure and Chen-Ruan quantum ring structure. Thetriple forms on Y s and Y sf are given by(3.4) h α , α , α i R = h α , α , α i CR + X A = d [Γ sr ] ,d> h α , α , α i ,A q ds , (3.5) h α , α , α i R = h α , α , α i CR + X A = d [Γ sfr ] ,d> h α , α , α i ,A q dsf respectively. Here q s and q sf are formal variables that represent classes [Γ sr ]and [Γ sfr ]. They define Ruan rings RH ( Y s ) and RH ( Y sf ). In [CLZZ], wealready proved that Theorem 3.9. Ψ ∗ gives the isomorphism RH ( Y s ) ∼ = RH ( Y sf ) . Relative Gromov-Witten theory on M sr and M sfr Local models M sr and M sfr . M sr and M sfr are obtained from W sr and W sfr by cutting at infinity. We explain this precisely.We introduce an S action on C γ ( x, y, z, t ) = ( γ r x, γ r y, γz, γ r t ) . Using this action, we collapse ˜ W r at ∞ . The infinity divisor is identified as˜ Z = ˜ W sr ∩ S S . By this way, we get an orbifold with singularity at 0, denoted by ˜ M r . Byblowing-up ˜ M r at 0, we have ˜ M sr and ˜ M sfr . µ r -action can then naturallyextend to ˜ M r , ˜ M sr and ˜ M sfr . By taking quotients, we have M r , M sr and M sfr . M sr and M sfr are the collapsing of W sr and W sfr at infinity. Note thatthe T -action given in § P := P ( r, r, , r,
1) be the weighted projective space. Then ˜ M r can beembedded in ˜ P and is given by the equation xy − z r + t = 0 . The original ˜ W r is embedded in { w = 0 } and ˜ Z is in { w = 0 } . µ r -actionextends to ˜ P by ξ ( x, y, z, t, w ) = ( ξ a x, ξ − a y, ξz, t, w ) . Then M r is embedded in P := ˜ P /µ r . Set Z = ˜ Z/µ r . To understand the localbehavior of M sr and M sfr at Z , it is sufficient to use this model at { w = 0 } .We now study the singular points at Z . Combining the S and µ r actions,we have( γ, ξ )( x, y, z, t, w ) = ( γ r ξ a x, γ r ξ − a y, γξz, γ r t, γw ) , ( γ, ξ ) ∈ S × Z r . Lemma 4.1.
There are four singular points x = [1 , , , , y = [0 , , , , z + = [0 , , , , z − = [0 , , , − , and a singular set S = { xy + t = 0 , z = 0 } on Z . Their stabilizers are Z r , Z r , Z r , Z r and Z r respectively. Proof.
Take a point ( x, y, z, t, w ). We find those points with nontrivialstabilizers.Case 1, assume that z = 0. Then γξ = 1. Therefore,( γ, ξ )( x, y, z, t, w ) = ( γ r ξ a x, γ r ξ − a y, γξz, γ r t,
0) = ( ξ a x, ξ − a y, z, t ) . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 15
In order to have nontrivial stabilizers, we must have x = y = 0. Therefore,only z + and z − survive. Their stabilizer are both Z r .Case 2, assume that z = 0. If t = 0, γ should be a r -root. Then( γ, ξ )( x, y, z, t, w ) = ( ξ a x, ξ − a y, , t,
0) = ( ξ a x, ξ − a y, z, t ) . Hence, when xy are not 0, the set { xy + t = 0 } has the stabilizer Z r .Case 3, assume that z = t = 0. Then xy = 0 by the equation. Hence we canonly have x and y . Clearly, their stabilizers are both Z r . q.e.d.We now look at the local models at z + and z − . The coordinate chart at z + is given by ( x, y, w ) and the action is(4.1) ξ ( x, y, w ) = ( ξ − a x, ξ a y, ξw ) , ξ ∈ Z r . The model at z − is same.Now look at the local models at x and y . At x , the local coordinate chartis given by ( z, t, w ). The action is given by(4.2) ξ ( z, t, w ) = ( ξηz, ξ r t, ξw ) , where η a ξ r = 1 , ξ ∈ Z r . Suppose(4.3) η = exp 2 πiµ, ≤ µ < . Similarly, at y , the local coordinate chart is given by ( z, t, w ). The action isgiven by(4.4) ξ ( z, t, w ) = ( ξηz, ξ r t, ξw ) , where η − a ξ r = 1 , ξ ∈ Z r . For points on S , the action on the normal direction is given by(4.5) ξ ( z, w ) = ( ξz, ξw ) , ξ ∈ Z r . Recall that we have four lines described in § M sr ( M sfr ). Take M sr as an example. We have L sp,y : p s ↔ y ; L sq,x : q s ↔ x ; L sp,z : p s ↔ z − ; L sq,z : q s ↔ z + ; . In the table, for each line we give the name of the curve and the ends itconnects.For M sfr , we have L sfp,y : p sf ↔ y ; L sfq,x : q sf ↔ x ; L sfp,z : p sf ↔ z + ; L sfq,z : q sf ↔ z − ; . Regarding the T -action, we have following two lemmas. The proof isstraightforward, we leave it to readers. Lemma 4.2.
The fixed points on M sr ( M sfr ) are p s , q s ( p sf , q sf ) on Γ sr ( Γ sfr )and x , y , z + , z − on Z . Lemma 4.3. In M sr , the invariant curves with respect to the torus actionare L sp,y , L sq,x , L sp,z and L sq,z . Relative Moduli spaces for the pair ( M sr , Z ) . We explain the rela-tive moduli spaces for the pair ( M sr , Z ). Similar explanations can be appliedto ( M sfr , Z ).Let Γ = ( g, A, h , g , l ) , T = ( g , l )be as before. Let M Γ , T ( M sr , Z ) be the moduli space. Recall that the virtualdimension of the moduli space isdim = c ( A ) + m X i =1 (1 − ι ( h i )) + k X j =1 (1 − [ ℓ j ] − ι ( g j )) . Here we use the complex dimension.First, we note that[ c ( M sr ) · Z ] = ( r + 2)[ c ( L Z ) · Z ] = X x ∈ x ( r + 2) ℓ x . Therefore dim = m X i =1 (1 − ι ( h i )) + k X j =1 (( r + 2) ℓ j + 1 − [ ℓ j ] − ι ( g j ))Set u i = 1 − ι ( h i ) , ≤ i ≤ m, v j = ( r + 2) ℓ j + 1 − [ ℓ j ] − ι ( g j ) , ≤ j ≤ k. Then dim = m X i =1 u i + k X j =1 v j . For u i and v j we have following facts:(1) if x j is mapped to x with g j = exp(2 πi αr ). Then ℓ j = [ ℓ j ] + αr . So v j = ( r + 2)( αr + [ ℓ j ]) + 1 − ([ ℓ j ] + αr + { αr + µ } + { αr } ) . Here { z } := z − [ z ]. Note that this can be( r + 1)[ ℓ j ] + n − µ, n ≥ . Here µ is defined in (4.3). The degree shifting numbers are given by(4.2).(2) If x j is mapped to y with g j = exp(2 πi αr ), it is same as the previouscase. UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 17 (3) If x j is mapped to S with g j = exp(2 πi αr ), then v j = ( r + 1)[ ℓ j ] + α + 1 . The degree shifting numbers are given by (4.5).(4) If x j is mapped to z + with g j = exp(2 πi αr ). Then ℓ j = [ ℓ j ] + αr . So v j = ( r + 1)[ l x ] + α + αr . Here the degree shifting numbers are given by (4.1).(5) If x j is mapped to z − with g j = exp(2 πi αr ), then it is same as theprevious case.(6) whenever g j = 1 v j = ( r + 1) ℓ j + 1 . (7) when h i = exp(2 πi αr ), u i = − αr . (8) when h i = 1, u i = 1.4.3. Admissible data (Γ , T , a ) . Suppose we are computing the relativeGromov-Witten invariant(4.6) h a | b , T i ( M s ,Z )Γ . Let | α | denote the degree of a form α . Set N = dim − m X i =1 | α i | . On the other hand, set N ′ = k X j =1 dim( Z g j ) . Definition 4.1.
The data (Γ , T , a ) is called admissible if N ≤ N ′ . By the definition, we have
Lemma 4.4. If (Γ , T , a ) is not admissible, the invariant (4.6) is 0. In this paper, we may assume that:
Assumption 4.5. (i) | α i | = 0 for all p i , (ii) | a | ≤ . Since N − N ′ = m X i =1 u i + k X j =1 ( v j − dim( Z g j )) , we make the following definition. Definition 4.2. we say that u i is the contribution of marked point p i to N − N ′ and v j − dim( Z g j ) is that of x j . Proposition 4.6.
Suppose that Assumption 4.5 holds. If (Γ , T , a ) is ad-missible, then one of the following cases holds: (1) x consists of only one smooth point, then p consists of three singularpoints ( p , p , p ) such that ι ( h ) + ι ( h ) + ι ( h ) = 5 . For this case, N = N ′ . (2) if x contains a point x maps to z − or z + , then one of the followingshould hold: | a | = 2 , ι ( h ) + ι ( h ) = 3 + 1 r ; | a | = 3 , ι ( h ) + ι ( h ) + ι ( h ) = 4 + 1 r . (3) x consists of only singular points, the multiplicities at x ∈ x are allless than 1. Furthermore, x can not be mapped to either z + or z − .Moreover, ℓ x ≤ . Proof.
First, we suppose that x is smooth point. It contributes ( r +1) l x + 1 to N and contributes 2 to N ′ . Hence its contribution to N − N ′ is( r + 1) l x − ≥ r. If x a singular point, its contribution to N − N ′ is given by the followinglist • x → x but not in S , the contribution is ( r + 1)[ l x ] + n − µ, n ≥ • x → S , the contribution is ( r + 1)[ l x ] + α + 1 , • x → z ± , the contribution is ( r + 1)[ l x ] + α + α/r .Note that they are all positive. Hence we conclude that, if x contains asmooth point x , then only the following situation survives: | x | = 1, r =2 , | a | = 3, and ι ( h ) + ι ( h ) + ι ( h ) = 5 . Furthermore, N = N ′ .Now suppose that x contains a point x mapping to z − (or z + ), only thefollowing situation survives: α = 1 and one of the following holds: | a | = 2 , ι ( h ) + ι ( h ) = 3 + 1 r ; | a | = 3 , ι ( h ) + ι ( h ) + ι ( h ) = 4 + 1 r . The rest admissible data belong to the third case. q.e.d.5.
Vanishing results on relative invariants
Localization via the torus action.
The torus action T on M sr in-duces an action on the moduli space. We now study the fix loci of the modulispace with respect to the action. We use the notations in § X = M sr . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 19
A relative stable map f = ( f , f , . . . , f b , . . . , f k , . . . , f b k k )is invariant if and only if each f is invariant. Since we only consider theinvariants for admissable data, the invariant maps in such moduli spacesonly have f . In fact, the fact ℓ x ≤ f is a stable map in X whose components are invariant maps (maybeconstant map) and nodal points are mapped to fix points, which are p s , q s and x , y , z + , z − . The constant map should also map to these points, whilethe nontrivial invariant curves should cover one of those four lines or Γ sr .Set FT = { p s , q s , x , y , z + , z − } , IC = { Γ sr , L sp,y , L sq,x , L sp,z , L sq,z } As in the Gromov-Witten theory, we introduce graphs to describe thecomponents of fix loci. We now describe the graph T for f . Let V T and E T be the set of vertices and edges of T . • each vertex is assigned to a connected component of the pre-imageof FT; on each vertex, the image point is recorded; • each edge is assigned to the component that is non-constant map;the image with multiplicity is recorded; • on each flag, a twisted sector (or the group element of the sector) isrecorded.Let F T be the fix loci that correspond to the graph T . Since T only describes f , F T may contain several components. Let T be the collection of graphsand F be the collection of F T .We recall the virtual localization formula. Suppose that Ω is a form on M ∆ ( X, Z ) and Ω T is its equivariant extension if exists. Then I (Ω) = Z vir M ∆ ( X,Z ) Ω = X T ∈ T Z F T Ω T e T ( N virF T ) . Here, N virF T is the virtual normal bundle of F T in the virtual moduli space, e T is the T -equivariant Euler class of the bundle. The right hand side isa function in ( λ, u ), which is rational in λ and polynomial in u . We denoteeach term in the summation as I Ω F T ( λ, u ) and the sum by I Ω ( λ, u ). Lemma 5.1.
Let (Γ , T , a ) be an admissible data in Proposition 4.6. Thenthe nontrivial relative invariants h a | b i ( M s ,Z )Γ , T can be computed via localization. Proof.
It is sufficient to show that the forms in a and b have equivariantextensions. By our assumption, we always take α ∈ a to be 1. It is alreadyequivariant.Now suppose that β j is assigned to x j . If Z ( g j ) is a single point, β j aretaken to be 1, which is equivariant. If Z g j = S , β j is either a 0 or 2-form, both have equivariant representatives. The last case is that x is smooth.For this case, since N = N ′ (cf. case (1) in Proposition 4.6), we must havedeg( β ) = 4 to get nontrivial invariants. Since β is of top degree, it has anequivariant representative as well. q.e.d.5.2. Vanishing results on I Ω F T ( λ, u ) , (I). By localization, we have I (Ω) = I Ω ( λ, u ) = X T ∈ T I Ω F T ( λ, u ) . Since the left hand side is independent of u , we have I (Ω) = lim u → I Ω ( λ, u ) = X T ∈ T lim u → I Ω F T ( λ, u ) . Theorem 5.2.
Suppose that | x | > . If T contains an edge e that recordsa map cover Γ sr , then lim u → I F T ( λ, u ) = 0 . Edge e records a map: f : S → Γ sr .f can be either a smooth or an orbifold map. Hence, we restate the theoremas, Proposition 5.3.
If the map f for e is smooth, then lim u → I F T ( λ, u ) = 0 . and Proposition 5.4.
If the map f for e is singular, then lim u → I F T ( λ, u ) = 0 . Clearly, Proposition 5.3 and 5.4 imply Theorem 5.2.Suppose that T is a graph. Let C be a curve in F T .First assume that all components in C are smooth. Then copied from[GP], we have0 → O C → M vertices O C v ⊕ M edges O Ce → M flags O F → , then (write E = f ∗ T M sr )0 → H ( C, E ) → M vertices H ( C v , E ) ⊕ M edges H ( C e , E ) → M flags E p ( F ) → H ( C, E ) → M edges H ( C v , E ) ⊕ M edges H ( C e , E ) → . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 21
Please refer to [GP] for flags. Here, by p ( F ) we mean the fixed point assignedto the flag. Hence the contribution of H /H is(5.1) H ( C, E ) H ( C, E ) = L vertices E val ( v ) − p ( v ) ⊕ L vertices H ( C e , E ) L edges H ( C e , E )We translate each term to the equivariant Euler class, i.e, a polynomial in λ and u .If C is not smooth, each space in the long complex should be replacedby the invariant subspace with respect to the proper finite group actions.Hence, each term in the right hand side of (5.1) should be replaced accord-ingly.Recall that(5.2) I Ω F T ( λ, u ) = Z F T Ω T e T ( N virF T ) . It is known that the equivariant form of H /H gives(5.3) 1 e T ( N virF T ) . It is easy to show that for each e , whose map f e does not Γ sr , e T (cid:18) H ( C e , f ∗ e E ) H ( C e , f ∗ e E ) (cid:19) contains no either u or u − . We now focus other terms in H /H . Claim 1: if f is smooth, the equivariant Euler form of the above term H /H contains a positive power of u . We count the possible contributions for u . Case 1, there is v = p s or q s with val ( v ) >
1, we then have u val ( v ) − ; Case 2, there is a component C e that is a multiple of Γ sr , we then actually have u in the denorminator for H ( C e , f ∗ O ) and ru in the numerator for H ( C e , f ∗ O ( − ✷ Claim 2: if f is an orbifold map, the equivariant Euler of H /H containsa factor of positive power of u . Suppose f is given by f : [ S ] → Γ sr ⊂ W sr . Such a map can be realized by a map˜ f : S → ˜Γ sr ⊂ ˜ W sr with a quotient by Z r . Here [ S ] = S / Z r .Unlike the smooth case, H ([ S ] , f ∗ E ) contains no u . By the computa-tions given below, in Corollary 5.7 we conclude that H ([ S ] , f ∗ E ) containsa factor u . ✷ We compute H ([ S ] , f ∗ E ) and its weight.Suppose that ˜ f is a d -cover. Then on the sphere ˜ S , the torus actionweight at 0 is λ/d and at ∞ is − λ/d , and suppose that Z r action is µ at0 and µ − at ∞ ; for the pull-back bundle O ( − d ) the torus action weightat fiber over 0 is − λ + ru and at fiber over ∞ is λ + ru , the Z r action are µ − d and µ d for the fibers over at 0 and ∞ . These data are ready for us tocompute the action on H ([ S ] , f ∗ E ).By Serre-duality, we have H ( S , O ( − d )) = ( H ( S , O (2 d − ∗ . The induced torus action weights on O (2 d −
2) at the fibers over 0 and ∞ are d − d λ − ru, − dd λ − ru respectively. The induced Z r action are µ d − and µ − d +1 for fibers on O (2 d −
2) over 0 and ∞ . Lemma 5.5.
The sections of H ( S , O (2 d − are given by { x a y b | a + b = 2 d − , a, b ≥ } . The torus action weight for section x a y b is d − − ad λ − ru . The action of µ is µ d − − a . Hence
Lemma 5.6.
The section x a y b that is Z r -invariant if and only if r | d − − a ,and the torus action weight is d − − ad λ − ru . Corollary 5.7. H ([ S ] , f ∗ O ( − contains a factor ru . Proof.
By the above lemma, we know that x d − y d − is Z r -invariantand it is action weight is − ru . By taking the dual, the corresponding factoris ru . q.e.d. Proof of Proposition 5.3 : Claim 1 implies the proposition.
Proof of Proposition 5.4 : Claim 2 implies the proposition.5.3.
Vanishing results on I Ω F T ( λ, u ) , (II). Now suppose that (Γ , T , a ) isan admissible data given in Proposition 4.6. Theorem 5.8. If (Γ , T , a ) is of case (1) and (2) in Proposition 4.6, h a | b i ( M s ,Z )Γ , T = 0 . Proof.
Suppose that h a | b i ( M s ,Z )Γ , T = I (Ω)for some Ω which has equivariant extension Ω T . (cf. Lemma 5.1). UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 23
Let F T be a fix component of the torus action. It contributes 0 to theinvariant unless the fixed curves C ∈ F T contains no component covering Γ s (cf. Theorem 5.2). It is easy to conclude that C must contains a ghost map f : ( S , q , q , · · · , q l ) → M s such that the image of f is either p s or q s and sum of the degree shiftingnumbers for all twisted sectors defined by q i is 2 + l . Suppose p s = f ( S ).Again, we claim that I Ω F T ( λ, u ) contains factor u . In fact, e T ( H ( S , f ∗ O s p )) = u. This is proved in [CH]. Hence, it is easy to see the claim follows. q.e.d.
Definition 5.1. If lim u → I Ω F T ( λ, u ) = 0 , we say the component F T contributes trivial to the invariant I (Ω) . Corollary 5.9.
Let (Γ , T , a ) be admissible, and I (Ω) = h a | b i ( M s ,Z )Γ , T . If F T contributes nontrivial to the invariant, then for any curve C ∈ F T ,all its connected components in the root component M s must cover L sp,y or L sq,x . Proof.
By the previous theorem, the admissible data must be of the3rd case in Proposition 4.6. Hence, points in C ∩ Z must be either x or y .Therefore, the invariant curves that C lives on must be L sp,y and L sq,x . q.e.d.6. Proof of the Main theorem
Combining § Theorem 6.1. X A Z Γ sr h α s , α s , α s i A = X A Z Γ sfr h Ψ ∗ α s , Ψ ∗ α s , Ψ ∗ α s i A . The rest of the section is devoted to the proof of this theorem.
Remark 6.2.
We should point out that ” = ” is rather strong from the pointof view of Ruan’s conjecture. Usually, it is conjectured that h α s , α s , α s i ∼ = h Ψ ∗ α s , Ψ ∗ α s , Ψ ∗ α s i . By ” ∼ = ”, we mean that both sides equal up to analytic continuations. Thisis necessary when classes [Γ s ] and [Γ sf ] involved. For example, in [CLZZ] we proved h α s , α s , α s i R ∼ = h Ψ ∗ α s , Ψ ∗ α s , Ψ ∗ α s i R for Theorem 3.9.But for the invariants that correspond to A = d [Γ s ] , it turns out thatwe do not need the analytic continuation argument, the reason is because ofTheorem 5.2: as long as [Γ s ] (so is [Γ sf ] ) appears, the invariant vanishes. Reducing the comparison to local models.
We now apply thedegeneration formula to reduce the comparing three point functions only onlocal models. We explain this: consider a three point function h α s , α s , α s i ,A s . First, we observe that [ A s ] = A s since π ∗ has no kernel (cf [LR]). Denotethe topology data by Γ s = (0 , A s )and forms by a s = ( α s , α s , α s ). Correspondingly, on Y sf we introduce a sf = (Ψ ∗ ) − a s ,A sf = Ψ ∗ A s , Γ sf = (0 , A sf ) . We write Γ sf = Ψ(Γ s ).Consider the degenerations π s : Y s degenerate −−−−−−−→ M sr ∪ Z Y − ; π sf : Y sf degenerate −−−−−−−→ M sfr ∪ Z Y − . Let η s = (Γ + ,s , Γ − , I ρ ) be a possible splitting of Γ s . Correspondingly,Ψ( η s ) := (Ψ r (Γ + ,s ) , Γ − , I ρ )gives a splitting of Ψ(Γ s ). On the other hand, suppose that a s = π ∗ s ( a + ,s , a − ) . Then by the diagram (3.3), a sf := Ψ ∗ ( a s ) = π ∗ sf (Ψ ∗ r ( a + ,s ) , a − ) . Proposition 6.3.
Suppose that a s and b are given on M sr , then (6.1) h a + ,s | b , T i ∗ ( M s ,Z )Γ + ,s = h Ψ ∗ r a + ,s | b , T i ∗ ( M s ,Z )Ψ r (Γ + ,s ) . Unlike hi • ( M s ,Z ) , here hi ∗ ( M s ,Z ) only sums over all admissible data. Proposition 6.4.
Proposition 6.3 ⇒ Theorem 6.1.
Proof.
Applying the degeneration formula to h a s i Γ s , we have h a s i Y s Γ s = X I X η =(Γ + ,s , Γ − ,I ρ ) C η h a + ,s | b I , T i • ( M s ,Z )Γ + ,s h a − | b I , T i • ( Y − ,Z )Γ − . Similarly, h a sf i Y sf Γ sf = X I X η sf =(Γ + ,sf , Γ − ,I ρ ) C η h a + ,sf | b I , T i • ( M s ,Z )Γ + ,sf h a − | b I , T i • ( Y − ,Z )Γ − . Here ∗ sf is always the correspondence of ∗ s via Ψ or Ψ r . UAN’S CONJECTURE ON SINGULAR SYMPLECTIC FLOPS 25
Since only admissible data contributes on the right hand sides of twoequations, (6.1) implies h a s i Y s Γ s = h a sf i Y sf Γ sf . which is exactly what Theorem 6.1 asserts.6.2. Proof of Proposition 6.3.
We now proceed to proof Proposition 6.3.Since the moduli spaces in local models admit torus actions. By localiza-tions, we know the contributions only come from those fix loci. So it issufficient to compare fix loci and the invariants they contribute.For ( M s , Z ), let T be a graph, and F sT be the component of fix loci. Thenby Corollary 5.9, F sT makes a nontrivial contribution only when each curvesin F T consists of only components on L sp,y and L sq,x .Similarly, for ( M sf , Z ), F sfT makes a nontrivial contribution only wheneach curves in F sfT consists of only components on L sfp,y and L sfq,x .Since, the flop identifies L sp,y ↔ L sfp,y , L sq,x ↔ L sfq,x and their normal bundles, therefore the flop identifies F sT and F sfT and theirvirtual normal bundles in their moduli spaces. Hence, I F sT ( λ, u ) = I F sfT ( λ, u ) . Proposition 6.3 then follows.
References [BKL] J. Bryant, S. Katz, N. Leung, Multiple covers and the integrality conjecture forrational curves in CY threefolds , J. ALgebraic Geometry 10(2001),no.3.,549-568.[CH] B. Chen, S. Hu, A de Rham model of Chen-Ruan cohomology ring of abelian orb-ifolds, Math. Ann. 2006 (336) 1, 51-71.[CL] B. Chen, A-M. Li, Symplectic Virtual Localization of Gromov-Witten invariants,arXiv:math.DG/0610370.[CLS] B. Chen, A-M. Li, S. Sun, Relative Gromov-Witten invariants and glue formula, inpreparation.[CLZZ] B. Chen, A-M. Li, Q. Zhang, G. Zhao, Singular symplectic flops and Ruan coho-mology, accepted by Topology.[CT] B. Chen, G. Tian, Virtual orbifolds and Localization, arXiv:math.DG/0610369 .[CR1] W. Chen, Y. Ruan, A new cohomology theory for orbifold, AG/0004129, Commun.Math. Phys., 248(2004), 1-31.[CR2] W. Chen, Y. Ruan, orbifold Gromov-Witten theory, AG/0011149. Cont. Math.,310, 25-86.[CR3] W. Chen, Y. Ruan, orbifold quantum cohomology, Preprint AG/0005198.[FP] Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent.Math., 139 (2000), 173C199, math.AG/9810173[F] R. Friedman, Simultaneous resolutions of threefold double points, Math. Ann.274(1986) 671-689.[GP] T. Graber, R. Pandharipande,Localization of virtual classes. Invent. Math. 135(1999), no. 2, 487-518.[Gr] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. math., 82(1985), 307-347. [HLR] J. Hu, T.-J. Li, Y. Ruan, Birational cobordism invariance of uniruled symplecticmanifolds, to appear on Invent. Math.[HZ] J. Hu, W. Zhang, Mukai flop and Ruan cohomology, Math. Ann. 330, No.3, 577-599(2004).[K] J. Koll´ar, Flips, Flops, Minimal Models, Etc., Surveys in Differential Geometry,1(1991),113-199.[La] Henry B. Laufer, On CP as an xceptional set, In recent developments in severalcomplex variables ,261-275, Ann. of Math. Studies 100, Princeton, 1981.[LLW] Y.-P. Lee, H.-W. Lin C.-L. Wang, Flops, Motives and Invariance of QuantumRings, To appear in Ann. of Math.[L] E. Lerman, Symplectic cuts, Math Research Let 2(1985) 247-258[LR] A-M. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau3-folds, Invent. Math. 145, 151-218(2001)[LZZ] A-M. Li, G. Zhao, Q. Zheng, The number of ramified covering of a Riemann surfaceby Riemann surface, Commu. Math. Phys, 213(2000), 3, 685–696.[Li] J. Li, Stable morphisms to singular schemes and relative stable morphisms, JDG 57(2001), 509-578.[Reid] M. Reid, Young Person’s Guide to Canonical Singularities, Proceedings of Sym-posia in Pure Mathematics, V.46 (1987).[R1] Y. Ruan, Surgery, quantum cohomology and birational geometry, math.AG/9810039.[R2] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, alg-geom/9611021.[S] I. Satake, The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9(1957),464-492.[STY] I. Smith, R.P. Thomas, S.-T. Yau, Symplectic conifold transitions. SG/0209319. J.Diff. Geom., 62(2002), 209-232.[Wang] C.-L. Wang, K-equivalence in birational geometry, in ”Proceeding of Second In-ternational Congress of Chinese Mathematicians (Grand Hotel, Taipei 2001)”, Inter-national Press 2003. Department of Mathematics, Sichuan University, Chengdu,610064, China
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