Singular symplectic flops and Ruan cohomology
aa r X i v : . [ m a t h . S G ] A p r SINGULAR SYMPLECTIC FLOPS AND RUANCOHOMOLOGY
BOHUI CHEN, AN-MIN LI, QI ZHANG, AND GUOSONG ZHAO
Abstract.
In this paper, we study the symplectic geometry ofsingular conifolds of the finite group quotient W r = { ( x, y, z, t ) | xy − z r + t = 0 } /µ r ( a, − a, , , r ≥ , which we call orbi-conifolds. The related orbifold symplectic coni-fold transition and orbifold symplectic flops are constructed. Let X and Y be two symplectic orbifolds connected by such a flop. Westudy orbifold Gromov-Witten invariants of exceptional classes on X and Y and show that they have isomorphic Ruan cohomologies.Hence, we verify a conjecture of Ruan. Introduction
In [LR], the authors proved an elegant result that any two smoothminimal models in dimension three have the same quantum cohomol-ogy. Besides the key role of the relative invariants introduced in thepaper, one of the main building blocks towards this result is the under-standing of how the Gromov-Witten invariants change under flops. Thedescription of a smooth flop is closely related to the conifold singularity W = { ( x, y, z, t ) | xy − z + t = 0 } . A crucial step in their proof is a symplectic description of a flop andhence symplectic techniques can be applied. However, it is well-knownthat the appropriate category for birational geometry is singular mani-folds with terminal singularities. In complex dimension three, terminalsingularities are deformations of orbifolds. In this paper and its se-quel, we initiate a program to study the quantum cohomology underbirational transformation of orbifolds.In the singular category, W r = { ( x, y, z, t ) | xy − z r + t = 0 } /µ r ( a, − a, , . is a natural replacement for the smooth conifold. The orbifold sym-plectic flops coming from this model are defined in the first part of the B.C. and A.L. are supported by NSFC, G.Z. is supported by a grant of NSFCand Qiushi Funding. paper (cf. § X and Y connected via orbifold symplectic flops,their Ruan cohomology rings are isomorphic.1.1. Orbifold symplectic flops.
The singularity given by W hasbeen studied intensively. Let ω o be the symplectic form on W \ { } induced from that of C . It has two small resolutions, denoted by W s and W sf , and a smoothing via deformation which is denoted by Q .The transformations W s ↔ Q , W sf ↔ Q are called conifold transitions . And the transformation W s ↔ W sf is called a flop .A symplectic conifold([STY]) ( Z, ω ) is a space with conifold singu-larities P = { p , . . . } such that ( Z \ P, ω ) is a symplectic manifold and ω coincides with ω o locally at p i ∈ P . Now suppose that Z is compact and | P | = κ < ∞ .Such Z admits a smoothing, denoted by X , and 2 κ resolutions Y = { Y , . . . , Y κ } . In X each p i is replaced by an exceptional sphere L i ∼ = S , while foreach Y j , p i is replaced by an extremal ray P .In [STY], they studied a necessary and sufficient condition for theexistence of a symplectic structure on one of the Y in Y in terms ofcertain topological condition on X . They showed that one of the κ small resolutions admits a symplectic stucture if and only if on X wehave the following homology relation (1.1) [ κ X i =1 λ i L i ] = 0 ∈ H ( X, Z ) with λ i = 0 f or all i. Here the L i are exceptional spheres on X . One can rephrase their theorem using cohomological language. Then,equation (1.1) reads as(1.2) [ κ X i =1 λ i Θ i ] = 0 ∈ H ( X, Z ) with λ i = 0 f or all i. Here Θ i is the Thom form of the normal bundle of L i . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 3
The cohomological version will be generalized to the general modelwith finite group quotient. Our model is(1.3) W r = { ( x, y, z, t ) | xy − z r + t = 0 } /µ r ( a, − a, , , r ≥ . (see [K] and [Reid] for references). Such a local model is called r -conifold or an orbi-conifold in our paper. Such (terminal) singularitiesappear naturally in the Minimal Model Program. They are the sim-plest examples in the list of singularities in [K]. W r without the finitequotient has been considered in [La]. It also has two resolutions ˜ W sr and ˜ W sfr . We can take quotients W sr = ˜ W sr /µ r , W sfr = ˜ W sfr /µ r . Both of them are orbifolds. In this paper, we propose a smoothing Q r as well. The transformations W sr ↔ Q r , W sfr ↔ Q are called (orbi)-conifold transitions . And the transformation W s ↔ W sf is called a (orbi)-flop .We are interested in symplectic geometry of the orbi-conifold ( Z, ω Z ).It has a smoothing X and 2 κ small resolutions Y = { Y i , ≤ i ≤ κ . } A theorem generalizing that of Smith-Thomas-Yau is
Theorem 1.1.
One of the κ small resolutions admits a symplecticstucture if and only if on X we have the following cohomology relation (1.4) [ κ X i =1 λ i Θ r i ] = 0 ∈ H ( X, R ) with λ i = 0 f or all i. As a corollary of this theorem, we show that if one of Y i ∈ Y issymplectic then so is its flop Y fi ∈ Y (refer to § The ring structures and Ruan’s conjecture.
Let X be anorbifold. It is well known that H ∗ ( X ) does not suffice for quantumcohomology. One should consider the so-called twisted sectors X ( g ) on X and study a bigger space H ∗ CR := H ∗ ( X ) ⊕ M ( g ) | g =1 H ∗ ( X ( g ) ) . Using the orbifold Gromov-Witten invariants [CR2], one can define theorbifold quantum ring QH ∗ CR ( X ). The analogue of classical cohomol-ogy is known as the Chen-Ruan orbifold cohomology ring. BOHUI CHEN, AN-MIN LI, QI ZHANG, AND GUOSONG ZHAO
Motivated by the work of Li-Ruan ([LR]) on the transformation ofthe quantum cohomology rings with respect to a smooth flop, we mayask how the orbifold quantum cohomology ring transforms (or evenhow the orbifold Gromov-Witten invariants change) via orbi-conifoldtransitions or orbifold flops. It can be formulated as the followingconjecture
Conjecture 1.2.
Let Y be the orbifold symplectic flop of X , then QH ∗ CR ( X ) ∼ = QH ∗ CR ( Y ) . To completely answer the question, one needs a full package of tech-nique, such as relative orbifold Gromov-Witten invariants and degen-eration formulae. These techniques are out of reach at this momentand will be studied in future papers([CLZZ]).On the other hand, it is easy to show that H ∗ CR ( X ) ∼ = H ∗ CR ( Y )additively. In general, they will have different ring structures. In thispaper, we study a new ring structure that it is in a sense between H ∗ CR and QH ∗ CR . It was first introduced by Ruan [ ? ] in the smooth case andcan be naturally extended to orbifolds. Let’s review the construction.Let Γ si , Γ sfi , ≤ i ≤ κ be extremal rays in X and Y respectively. On X , (and on Y ), we use only moduli spaces of J-curves representingmultiples of Γ i ’s and define 3-point functions on H ∗ CR ( X ) by(1.5) Ψ Xqc ( β , β , β ) = Ψ Xd =0 ( β , β , β ) + κ X i =1 ∞ X d =1 Ψ X ( d [Γ s ] , , ( β , β , β ) . Such functions also yield a product on H ∗ CR ( X ). This ring is calledthe Ruan cohomology ring [HZ] and denoted by RH ∗ CR ( X ). Ruanconjectures that if X , Y are K-equivalent, RH ∗ CR ( X ) is isomorphicto RH ∗ CR ( Y ) . Our second theorem is
Theorem 1.3.
Suppose that X and Y are connected by a sequenceof symplectic flops constructed out of r -conifolds. Then RH ∗ CR ( X ) isisomorphic to RH ∗ CR ( Y ) . Hence, Ruan’s conjecture holds in this case.Acknowledge. We would like to thank Yongbin Ruan for telling usabout the program and for many valuable discussions. We also wish tothank Qi Zhang, Shengda Hu and Quan Zheng for many discussions.The second and third authors also would like to thank University ofWisconsin- Madison and MSRI for their hospitality.
INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 5 Local Models
Local r -orbi-conifolds. 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 ) . Let ˜ W r ⊂ C be the complex hypersurface given by˜ W r = { ( x, y, z, t ) | xy − z r + t = 0 } , r ≥ . It has an isolated singularity at the origin. We call ˜ W r the local r -conifold. Set ˜ W ◦ r = ˜ W r \ { } . It is clear that, for any integer a that is prime to r , the action µ r ( a, − a, ,
0) preserves ˜ W r . Set W r = ˜ W r /µ r , W ◦ r = ˜ W ◦ r /µ r . We call W r the local r -orbi-conifold. Let ˜ ω ◦ r,w be the symplectic struc-ture on ˜ W ◦ r induced from C . It yields a symplectic structure ω ◦ r,w on W ◦ r .2.2. The small resolutions of W r and flops. By blow-ups, we havetwo small resolutions of ˜ W r . They are˜ 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 sr , ˜ π sfr : ˜ W sfr → W sfr be the projections. The extremal rays (˜ π sr ) − (0) and (˜ π sfr ) − (0) aredenoted by ˜Γ sr and ˜Γ sfr respectively. Both of them are isomorphic to P . The action of µ r extends naturally to both resolutions by setting ξ · [ p, q ] = [ ξ a p, q ]for the first model and ξ · [ p, q ] = [ ξ − a p, q ]for the second one. BOHUI CHEN, AN-MIN LI, QI ZHANG, AND GUOSONG ZHAO
Set W sr = ˜ W sr /µ r , W sfr = ˜ W sfr /µ r Γ sr = ˜Γ sr /µ r Γ sfr = ˜Γ sfr /µ r . We call W s and W sf small resolutions of W r . We say that W sf is theflop of W s and vice versa. They are both orbifolds with singular pointson Γ s and Γ sf .Another important fact we use in this paper is Proposition 2.1.
For r ≥ , the normal bundle of ˜Γ sr (˜Γ sfr ) in ˜ W sr ( ˜ W sfr ) is O ⊕ O ( − . Proof.
The proof is given in [La].2.3.
Orbifold structures on W s and W sf . Let us take W s . Thesingular points are points 0 and ∞ on Γ s . In term of [ p, q ] coordinates,they are 0 = [0 , ∞ = [1 , . We denote them by p s and q s respectively. Since ˜ W s ⊂ C near p s , the(tangent) of a uniformizing system of p s is given by { ( p, x, y, z, t ) | x = t = 0 } .µ r acts on this space 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 corre-sponding twisted sector([CR1]). As a set, it is same as p s . We denotethis twisted sector by [ p s ] k . For each twisted sector, a degree shiftingnumber is assigned. We conclude that Lemma 2.2.
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 havetwisted sector [ q s ] k and ι ([ q s ] k ) = 1 + kr . A similar structure applies to W sf . There are two singular points,denoted by p sf , q sf . The corresponding twisted sectors are [ p sf ] k , [ q sf ] k .Then ι ([ p sf ] k ) = ι ([ q sf ] k ) = 1 + kr . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 7
The deformation of W r . For convenience, we change coordi-nates: x = z + √− z , y = z − √− z , z = r √− z , t = z . Thus in terms of the new coordinates ˜ W r is given by a new equation(2.1) z + z + z r + z = 0 . It is also convenient to use real coordinates( x , y , x , y , x , y , x , y ) = ( z , z , z , z ) . In terms of real coordinates, µ r ( a, − a, ,
0) action is given by e πir · x y x y = cos πar − sin πar
00 cos πar − sin πar sin πar πar
00 sin πar πar x y x y , and e πir · x y x y = cos πr − sin πr πr cos πr x y x y . The equation for ˜ W r is (cid:26) x + x + f ( x , y ) + x = y + y + g ( x , y ) + y x y + x y + f ( x , y ) g ( x , y ) + x y = 0 . Here f and g are defined by f ( x, y ) + √− g ( x, y ) = ( x + √− y ) r . We propose
Definition 2.1.
The deformation of ˜ W r is the set ˜ Q r defined by (cid:26) x + x + f ( x , y ) + x = 1 ,x y + x y + f ( x , y ) g ( x , y ) + x y = 0 . The action µ r ( a, − a, , preserves ˜ Q r . Hence we set Q r = ˜ Q r /µ r and called it the deformation of W r . Lemma 2.3. ˜ Q r is a 6-dimensional symplectic submanifold of R × R . BOHUI CHEN, AN-MIN LI, QI ZHANG, AND GUOSONG ZHAO
Proof.
Consider the map F : R × R → R ( x, y ) → ( F ( x, y ) , F ( x, y )) . given by F ( x, y ) = x + x + f ( x , y ) + x − ,F ( x, y ) = x y + x y + f ( x , y ) g ( x y ) + x y . Then F − (0) = ˜ Q r . The Jacobian of F is x x f ∂f∂x x f ∂f∂y y y g ∂f∂x + f ∂g∂x y x x g ∂f∂y + f ∂g∂y x ! . We claim that this is a rank 2 matrix: if one of x , x , x , say x i , isnonzero, the above matrix has a rank 2 submatrix (cid:18) x i y i x i (cid:19) . Otherwise, say ( x , x , x ) = (0 , , Q r wehave f ( x , y ) = 0 , and g ( x , y ) = 0 . Then since f + √− g is a holo-morphic function of x + √− y , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ∂f∂x f ∂f∂y g ∂f∂x + f ∂g∂x g ∂f∂y + f ∂g∂y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( ∂f∂x ) + ( ∂f∂y ) = 0 . Hence F has rank 2 everywhere on ˜ Q r and 0 is its regular value. Thisimplies that ˜ Q r is a smooth 6-dimensional submanifold of R × R . Next we prove that ˜ Q r has a canonical symplectic structure ω ˜ Q r induced from ( R × R , ω o = − Σ dx i ∧ dy i ) . It is sufficient to prove that ω o ( ∇ F , ∇ F ) = 0 . By direct computations, ∇ F = (2 x , x , f ∂f∂x , x , , , f ∂f∂y , y ) , ∇ F = ( y , y , f ∂g∂x + g ∂f∂x , y , x , x , f ∂g∂y + g ∂f∂y , x ) , INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 9
Therefore − ω o ( ∇ F , ∇ F ) = X dx i ( ∇ F ) dy i ( ∇ F ) − dx i ( ∇ F ) dy i ( ∇ F )= 2 x + 2 x + 2 f (( ∂f∂x ) + ( ∂g∂x ) ) + 2 x = 0 . Hence ˜ Q r is a symplectic submanifold with a canonical symplecticstructure induced from R × R . q.e.d.We denote the symplectic structure by ˜ ω ◦ r,q .Put ˜ L r := { ( x, y ) ∈ ˜ Q r | y = y = g ( x , y ) = y = 0 } . and set ˜ Q ◦ r = ˜ Q r \ ˜ L r . The µ r -action preserves ˜ L r ; we set L r = ˜ L r /µ r , Q ◦ r = ˜ Q ◦ r /µ r .L r is the exceptional set in Q r with respect to the deformation in thefollowing sense: Lemma 2.4.
There is a natural diffeomorphism between W ◦ r and Q ◦ r . Proof.
We denote by [ x, y ] ∈ W ◦ r the equivalence class of ( x, y ) ∈ ˜ W r with respect to the quotient by µ r .For any λ > W r,λ ⊂ ˜ W r be the set of ( x, y ) satisfying x + x + f ( x , y ) + x = y + y + g ( x , y ) + y = λ and x y + x y + f ( x , y ) g ( x , x ) + x y = 0 . It is not hard to see that • ˜ W r,λ is preserved by the µ r action; set W r,λ = ˜ W r,λ /µ r ; • ˜ W ◦ r is foliated by ˜ W r,λ , λ ∈ R + .On the other hand, ˜ Q ◦ r has a similar foliation: for λ >
0, let ˜ Q r,λ ⊂ ˜ Q r be the set of ( x, y ) satisfying x + x + f ( x , y ) + x = 1 ,y + y + g ( x , y ) + y = λ ,x y + x y + f ( x , y ) g ( x , x ) + x y = 0 . Then • ˜ Q r,λ is preserved by the µ r action; set Q r,λ = ˜ Q r,λ /µ r ; • ˜ Q ◦ r is foliated by ˜ Q r,λ , λ ∈ R + .We next introduce the identification between W r,λ and Q r,λ . Let u λ ( x , y )and v λ ( x , y ) be functions that solve( u + iv ) r = λ − f ( x , y ) + √− λg ( x , y ) . Such a pair u + iv exists up to a factor ξ k . Then[ x , x , x , x , y , y , y , y ] ←→ [ λ − x , λ − x , u ( x , y ) , λ − x , λy , λy , v ( x , y ) , λy ]induces an identification between W r,λ and Q r,λ , and therefore between W ◦ r and Q ◦ r . q.e.d.We denote the identification map constructed in the proof byΦ r : W ◦ r → Q ◦ r . In particular, we note that the restriction of Φ r to W r, is the identity.2.5. The comparison between local r -orbi-conifolds and localconifolds. When r = 1, the local model is the well-known conifold.Since µ r = µ = { } is trivial, there is no orbifold structure. It is wellknown that • W s and W sf are O ( − ⊕ O ( − → P , where Γ s and Γ sf are the zero section P ; They are flops of eachother; • Q is diffeomorphic to the cotangent bundle of S . The inducedsymplectic structure from R × R coincides with the canonicalsymplectic structure on T ∗ S . • the map Φ : ( W , ω ◦ ,w ) → ( Q , ω ◦ ,q )is a symplectomorphism.There are natural (projection) maps π r,w : ˜ W r → W , π r,q : ˜ Q r → Q given by x i → x i , y i → y i , i = 3 , and ( x , y ) → ( f ( x , y ) , g ( x , y )) . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 11
Similarly, there are maps π sr,w : ˜ W sr → W s , π sfr,w : ˜ W sfr → W sf . We note that all these projection maps are almost r -coverings. They arecoverings except on x = y = 0, where the maps are only r -branchedcoverings. Note that ˜ L r = π − r,q L . It is the union of r copies of S intersecting at (cid:26) x + x + x = 1 x y + x y + x y = 0 (cid:27) ∩ { x = y = 0 } . Cohomologies
Definitions.
Let (Ω ∗ ( ˜ W ◦ r ) , d ) be the de Rham complex of ˜ W ◦ r . µ r has a natural representation on this complex. letΩ ∗ µ r ( ˜ W ◦ r ) ⊂ Ω ∗ ( ˜ W ◦ r )be the subcomplex of µ r -invariant forms. We have H ∗ ( W ◦ r ) = H ∗ (Ω ∗ µ r ( ˜ W ◦ r ) , d ) . Similar definitions apply to W sr , W sfr , Q ◦ r , Q r , W r, = Q r, etc.Then Lemma 3.1. H ∗ ( W ◦ r ) = H ∗ ( W r, ) . Proof.
We note that there is a µ r -isomorphism˜ W ◦ r ∼ = ˜ W r, × R + . In fact, it is induced by a natural identification˜ W r,λ ↔ ˜ W r, × { λ } ; x i ↔ λ − x i , i = 3; x ↔ λ − r x ,y i ↔ λ − y i , i = 3; y ↔ λ − r y . Hence ˜ W ◦ r is µ r -homotopy equivalent to ˜ W r, . Hence the claim follows.q.e.d.The result also follows from W ◦ r ∼ = W r, × R + directly. Similarly, we have Q ◦ r ∼ = Q r, × R + . Hence H ∗ ( Q ◦ r ) = H ∗ ( Q r, ) . Note that Q r, = W r, . We have H ∗ ( W ◦ r ) = H ∗ ( W r, ) = H ∗ ( Q r, ) = H ∗ ( Q ◦ r ) . Computation of cohomologies.
We first study H ∗ ( W r, ).Recall that we have a map π r,w : ˜ W r, → W , given by π r,w ( x, y ) = ( x , x , f ( x , y ) , x , y , y , g ( x , y ) , y ) . We now introduce a µ r action on W , . For convenience, we use coor-dinates ( u, v ) for the R × R in which W , is embedded. Then e πir · u v u v = cos πar − sin πar
00 cos πar − sin πar sin πar πar
00 sin πar πar u v u v , and acts trivially on u , v , u and v . Then it is clear that π r,w is µ r -equivariant. It induces a morphism between complexes(3.1) π ∗ r,w : (Ω ∗ µ r ( W , ) , d ) → (Ω ∗ µ r ( ˜ W r, ) , d ) . Here Ω G always represents the subspace that is G -invariant if Ω is a G -representation. Proposition 3.2. π ∗ r,w in (3.1) is an isomorphism between the coho-mologies of the two complexes. Proof.
The idea of the proof is to consider a larger connected Liegroup action on spaces: Let S = { e πiθ } . Suppose its action on ( x, y )is given by e πiθ · x y x y = cos θ − sin θ
00 cos θ − sin θ sin θ θ
00 sin θ θ x y x y , and the trivial action on x , y , x and y . The same action is definedon ( u, v ). Again, π r,w is S -equivariant.Since S is a connected Lie group and its actions commutes with µ r -actions on both spaces, the subcomplex((Ω ∗ µ r ( ˜ W r, )) S , d ) ⊂ (Ω ∗ µ r ( ˜ W r, ) , d )of S -invariant forms yields same cohomology as the original one, i.e, H ∗ ((Ω ∗ µ r ( ˜ W r, )) S , d ) = H ∗ (Ω ∗ µ r ( ˜ W r, ) , d ) INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 13
Similarly, H ∗ ((Ω ∗ µ r ( W , )) S , d ) = H ∗ (Ω ∗ µ r ( W , ) , d )It is then sufficient to show that(3.2) π ∗ r,w : H ∗ ((Ω ∗ µ r ( W , )) S , d ) → H ∗ ((Ω ∗ µ r ( ˜ W r, )) S , d )is an isomorphism. By the definition of the actions, we note that(3.3) (Ω ∗ µ r ( W , )) S = Ω ∗ S ( W , ) . We now show (3.2). Recall that π r,w is an r -branched covering ram-ified over R = (cid:26) u + u + u = v + v + v = 1 u v + u v + u v = 0 (cid:27) ∩ { u = v = 0 } Set ˜ R r = π − r,w ( R ) and˜ U r = ˜ W r, \ ˜ R r , U = W , \ R . Then π r,w : ˜ R r → R is 1-1 and π r,w : ˜ U r → U is an r -covering.Let V be an S -invariant tubular neighborhood of R in W , . Bythe implicit function theorem, we know that V ∼ = R × D , where D is the unit disk in the complex plane C = { u + √− v } . Let˜ V r = π − r,w ( V ). Then ˜ V r ∼ = ˜ R r × D , where D is the unit disk in the complex plane C = { x + iy } . Interms of these identifications, π r,w can be rewritten as π r,w : ˜ R r × D → R × D π r,w ( γ, z ) = ( γ, z r ) , where γ ∈ ˜ R r = R , z = x + iy . Consider the short exact sequences0 → (Ω ∗ µ r ( W , )) S → (Ω ∗ µ r ( U )) S ⊕ (Ω ∗ µ r ( V )) S → (Ω ∗ µ r ( U ∩ V )) S → → (Ω ∗ µ r ( ˜ W r, )) S → (Ω ∗ µ r ( ˜ U r )) S ⊕ (Ω ∗ µ r ( ˜ V r )) S → (Ω ∗ µ r ( ˜ U r ∩ ˜ V r )) S → .π ∗ r,w is a morphism between two complexes. We assert that π ∗ r,w : H ∗ ((Ω ∗ µ r ( U )) S , d ) ∼ = −→ H ∗ ((Ω ∗ µ r ( ˜ U r )) S , d ) , (3.4) π ∗ r,w : H ∗ ((Ω ∗ µ r ( V )) S , d ) ∼ = −→ H ∗ ((Ω ∗ µ r ( ˜ V r )) S , d ) , (3.5) π ∗ r,w : H ∗ ((Ω ∗ µ r ( U ∩ V )) S , d ) ∼ = −→ H ∗ ((Ω ∗ µ r ( ˜ U r ∩ ˜ V )) S , d ) . (3.6) Once these are proved, by the five-lemma, we know that π ∗ r,w : H ∗ ((Ω ∗ µ r ( W , )) S , d ) ∼ = −→ H ∗ ((Ω ∗ µ r ( ˜ W r, )) S , d )which is (3.2).We now explain (3.4), (3.5) and (3.6). The proof of (3.4) . We observe that π ∗ r,w : (Ω ∗ µ r ( U )) S ∼ = −→ (Ω ∗ µ r ( ˜ U r )) S . Hence it induces an isomorphism on cohomology level.
The proof of (3.5) . Since ˜ V r is µ r × S -homotopy equivalent to ˜ R r , wehave H ∗ ((Ω ∗ µ r ( ˜ V r )) S , d ) ∼ = H ∗ ((Ω ∗ µ r ( ˜ R r )) S , d ) . Similarly, H ∗ ((Ω ∗ µ r ( V )) S , d ) ∼ = H ∗ ((Ω ∗ µ r ( R )) S , d ) . Because H ∗ ((Ω ∗ µ r ( ˜ R r )) S , d ) = H ∗ ((Ω ∗ µ r ( R )) S , d ) , we have (3.5). The proof of (3.6) . The proof is the same as that of (3.4).This completes the proof of the theorem. q.e.d.So far, we have shown that H ∗ ( W r, ) = H ∗ (Ω ∗ µ r ( ˜ W r, ) , d ) ∼ = H ∗ (Ω ∗ µ r ( W , ) , d ) = H ∗ ((Ω ∗ µ r ( W , )) S , d ) . Furthermore, by (3.3) we have H ∗ ((Ω ∗ µ r ( W , )) S , d ) = H ∗ (Ω ∗ S ( W , ) , d ) = H ∗ ( W , ) . Since W , ∼ = S × S we have Corollary 3.3. H ∗ ( W r, ) ∼ = H ∗ ( S × S ) . Let H be a generator of H ( S × S ) such that Z S H = 1 . Here S is any fiber { x } × S in S × S . Set˜ H r = π ∗ r,w H and let H r be its induced form on W r, . This is a generator of H ( W r, ).Without loss of generality, we also assume that it is a generator of H ( W ◦ r ). INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 15
Let ω r,w and ω r,q be symplectic forms on W ◦ r and Q ◦ r respectively.Suppose that [ ω r,w | W r, ] = [ ω r,q | Q r, ] . Here [ ω ] denotes the cohomology class of ω . Then there exists a sym-plectomorphism Φ ′ r : ( W ◦ r , ω r,w ) → ( Q ◦ r , ω r,q ) . In fact, by the assumption, we have[ ω r,w ] = [Φ ∗ r ω r,q ] . Then, by the standard Moser argument, there exists a diffeomorphism f : W ◦ r → W ◦ r such that f ∗ ω r,w = Φ ∗ r ω r,q . Now we can set Φ ′ r = Φ r ◦ f − . In particular,by applying it to ω ◦ r,w and ω ◦ r,q we have Corollary 3.4.
There exists a symplectomorphism Φ ′ r : ( W ◦ r , ω ◦ r,w ) → ( Q ◦ r , ω ◦ r,q ) . Proof.
We observe that both symplectic forms are exact. Hencethey represent the same cohomology class, namely 0. q.e.d.Next we consider H ∗ ( W sr ). The argument is same as above: we alsohave a map π r,w : ˜ W sr → W s . This map will induce an isomorphism
Proposition 3.5. H ∗ ( W sr ) = H ∗ ( W s ) . Proof.
Since the proof is parallel to that of proposition 3.2, we onlysketch the proof.We use complex coordinates ( x, y, z, t, [ p, q ]) for ˜ W sr and ( u, v, w, s, [ m, n ])for W s . Then π r,w is induced by the map u = x, v = y, w = z r , s = t, mn = pq . We can introduce a µ r -action on W s by ξ ( u, v, w, s, [ m, n ]) = ( ξ a u, ξ − a v, w, s, [ ξ a m, n ]) , ξ = e πir . Then π r,w is µ r -equivariant.Moreover, both spaces admit an S -action such that π r,w is S -equivariant:for ξ ∈ S : ξ ( x, y, z, t, [ p, q ]) = ( ξ a x, ξ − a y, z, t, [ ξ a p, q ]) ξ ( u, v, w, s, [ m, n ]) = ( ξ a u, ξ − a v, w, s, [ ξ a m, n ]) . π r,w is an r -branched covering ramified over W s ∩ { w = 0 } . Then the rest of the proof is simply a copy of the argument in Propo-sition 3.2. q.e.d.Since W s ∼ = O ( − ⊕ O ( − ,H ( W s ) = H ( P ) is 1-dimensional. So is H ( W sr ). Let H sr be thegenerator of H ( W sr ) such that Z Γ sr H sr = 1 . Since the normal bundle of ˜Γ sr is O ⊕ O ( − ω ′ . We normalize it by Z Γ sr ω ′ = 1 . It induces a symplectic structure, denoted by ω sr on the neighborhood U of Γ sr . It is easy to see that this symplectic structure is tamed by thecomplex structure on U . Hence we conclude that Corollary 3.6.
There is a symplectic form on W sr that represents theclass H sr and is tamed by its complex structure. This form is denotedby ω sr . Orbifold symplectic flops
The global orbi-conifolds.
Following [STY] we give the defini-tion of orbi-conifolds.
Definition 4.1. A real 6-dimensional orbi-conifold is a topologicalspace Z covered by an atlas of charts { ( U i , φ i ) } of the following twotypes: either ( U i , φ i ) is an orbifold chart or φ j : U j → W r j is a homeomorphism onto W r j defined in § φ − j (0) a singularity of Z .Moreover, the transition maps φ ij = φ i ◦ φ − j must be smooth in theorbifold sense away from singularities and if p ∈ U i ∩ U j is a singularitythen we have r i = r j (denote it by r ), and there must be an open subset N ⊂ C containing 0 such that the lifting of φ ij , ˜ φ ij : ˜ W r ∩ N −→ ˜ W r ∩ N INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 17 in the uniformizing system is the restriction of an analytic isomorphism ˜ φ : C → C which is smooth away from the origin, C at the originwith d ˜ φ ∈ Sp (8 , R ) , and set-wise fixes ˜ W r . We call such charts smooth admissible coordinates.
Note that in thecase r = 1 the singularity is the ordinary double point discussed in[STY].¿From now on, we label the set of singularities P = { p , p , . . . } , and for each point p i its local model is given by a standard model W r i . Definition 4.2. A symplectic structure on an orbi-conifold Z is asmooth orbifold symplectic form ω Z on the orbifold Z \ P which, aroundeach singularity p i , coincides with ω ◦ w,r i . ( Z, ω Z ) is called a symplecticorbi-conifold . ¿From now on, we assume that Z is compact and | P | = κ . One canperform a smoothing for each singularity of Z as in § p i by a neighborhood of L r i in Q r i -to get an orbifold. We denote this orbifold by X .For each singularity p i of Z we can perform two small resolutions,i.e., we replace the neighborhood of the singularity by W sr i or W sfr i asin § κ choices of small resolutions, and so we get 2 κ orbifolds Y , · · · , Y κ . Definition 4.3.
Two small resolutions Y and Y ′ are said to be flopsof each other if at each p i , one is obtained by replacing W sr i and theother by W sfr i . We write Y ′ = Y f and vice versa. Symplectic structures on Y i ’s and flops. Not every small res-olution Y α , ≤ α ≤ κ admits a symplectic structure. Our first maintheorem of the paper gives a necessary and sufficient condition for Y to have a symplectic structure in terms of the topology of X .Let L r i ⊂ X . For simplicity, we assume its neighborhood to be Q r i .Recall that there is a projection map π r i ,q : ˜ Q r i → Q . Let Θ be the Thom form of the normal bundle of L = S in Q . Weassume it is supported in a small neighborhood of L . Set˜Θ r i = π ∗ r i ,q Θ . We can choose Θ properly such that ˜Θ r i is µ r i invariant. Hence itinduces a local form Θ r i on Q r i and hence on X . Then we restate Theorem 1.1:
One of the κ small resolutions ad-mits a symplectic stucture if and only if on X we have the followingcohomology relation (4.1) [ κ X i =1 λ i Θ r i ] = 0 ∈ H ( X, R ) with λ i = 0 f or all i. As a corollary,
Corollary 4.1.
Suppose we have a pair of resolution Y and Y f thatare flops of each other. Then Y admits a symplectic structure if andonly if Y f does. Y f is then called the symplectic flop of Y .5. Proof of theorem 1.1
Necessity.
We first prove that (4.1) is necessary.Suppose that we have a Y that admits a symplectic structure ω . Forsimplicity, we assume that at each singular point p i ∈ Z , it is replacedby W sr i to get Y . The extremal ray is Γ si . Set λ i = Z Γ sri ω = 1 r i Z ˜Γ sri ˜ ω. Now we consider the pair of spaces (
X, X \∪ L r i ). The exact sequenceof the (orbifold) de Rham complex of the pair is0 → Ω ∗− ( X \ ∪ L r i ) γ −→ Ω ∗ ( X, X \ ∪ L r i ) δ −→ Ω ∗ ( X ) → . given by γ ( f ) = (0 , f ) , δ ( α, f ) = α. It induces a long exact sequence on (orbifold) cohomology · · · → H ( X \ ∪ L r i ) → H ( X, X \ ∪ L r i ) → H ( X ) → · · · And applying this to ω on Z \ P ∼ = X \ ∪ i L r i , we have ω (0 , ω ) . This says that [ δ ◦ γ ( ω )] = 0 . We compute the left hand side of the equation. First, by applying theexcision principle we get H ( X, X \ ∪ i L r i ) ∼ = M i H ( Q r i , Q ◦ r i ) . This reduces the computation to the local case.
INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 19
Let ω r i ,w be the restriction of ω in the neighborhood, simply denotedby W sr i , of Γ sr i . It induces a form ω r i ,q on Q ◦ r i . Suppose that ω r i ,q = c i H r i , where H r i is the generator on Q r i , , hence on Q ◦ r i . Let β be a cut-offfunction such that β ( t ) = (cid:26) , if t > . , if t < . . By direct computation, we have δ ◦ γ ( H r i ) = d ( β ( λ ) H r i ) = Θ r i . Therefore, we conclude that κ X i =1 c i Θ r i = 0In order to show (4.1), it remains to prove that Proposition 5.1. c i = − λ i . Proof.
The computation is done on ˜ W sr i .Take an S in Q , as B = { (1 , , , , , v , v , v ) ∈ ˜ Q r i | v + v + v = 1 } Let ˜ B r = π − r,q ( B ). It is˜ B r i = { (1 , , x , , , y , y , y ) ∈ ˜ Q r i | y + g ( x , y )+ v = 1 , f ( x , y ) = 0 } Then Z ˜ B ri ˜ H r i = r i Z B H = r i . Hence Z ˜ B ri ω r i ,q = c i r i . Next we explain that(5.1) Z ˜ B ri ω r i ,q = − λ i r i . Then the claim follows from these two identities.
Proof of (5.1) : We treat B and ˜ B r i as subsets of W s and ˜ W sr i .By Proposition 3.2, we assume ω r i ,w is homologous to π ∗ r i ,w ω for some ω ∈ H ( W s ). Then Z ˜ B ri ω r i ,q = r i Z B ω. On the other hand, B is homotopic to − Γ s : via complex coordinates W is given by uv − ( w − s )( w + s ) = 0 . The equation of the small resolution W s in the chart { q = 0 } is ζ v − ( w − s ) = 0 , , where ζ = mn = uw + s is the coordinate of the exceptional curve Γ s . Recall that on B the complex coordinates are x = 1 + y , y = 1 − y , z = √− y , t = y . We have a projection map B −→ Γ s given by η = xz + t = 1 + p − y − y √− y + y . Here we take y , y as coordinates on B . It is easy to see that this isa one to one map and the point with √− y + y = 0 corresponds tothe point ” ∞ ” of − Γ s . The sign is due to the orientation.Let ( ζ , y, z, t ) = ( 1 + p − y − y √− y + y , − y , iy , y )be any point in B ; then( ζ , , ,
0) = ( 1 + p − y − y √− y + y , , , s . We construct a subset Λ of W s ρ ( y , y , s ) = { ( 1 + p − y − y √− y + y , s (1 − y ) , s √− y , sy ) } where 0 ≤ s ≤ y , y are the coordinates of N . This is a smooth3-dimensional submanifold with boundary { ρ ( y , y ,
0) = − Γ s } ∪ { ρ ( y , y ,
1) = B } . It gives us a homotopy between − Γ s and B . Then Z ˜ B ri ω r i ,w = r i Z B ω = − r i Z Γ s ω = − Z ˜Γ sri ω r i ,w = − r i λ i . This shows (5.1).We have completed the proof of the proposition. q.e.d.This completes the proof of necessity.
INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 21
Remark 5.2.
If the local resolution is W sfr i , [ δ ◦ γ ( ω r i ,w )] = λ i Θ r i . Sufficiency.
Suppose that (4.1) holds for X : i.e, there exists λ i such that X i λ i Θ r i = 0 . For the moment we assume that all λ i <
0. Let Y be a small resolutionof Z obtained by replacing the neighborhood of p i by W sr i . We assertthat Y admits a symplectic structure.¿From the exact sequence of the pair of spaces ( X, X \ ∪ i L r i ) H ( X \ ∪ i L r i ) γ −→ H ( X, X \ L r i ) → H ( X )we conclude that there exists a 2-form σ ∗ ∈ H ( X \ ∪ i L r i ) such that γ ( σ ∗ ) = X λ i Θ r i . since X \ ∪ i L r i ∼ = Y \ ∪ i Γ sr i ,σ ∗ ∈ H ( Y \ ∪ i Γ sr i ) . On the other hand, we consider the exact sequenceof the pair of spaces (
Y, Y \ ∪ i Γ sr i ) H ( Y ) → H ( Y \ ∪ i Γ sr i ) → H ( Y, Y \ ∪ Γ sr i ) ∼ = M i H ( W sr i , W ◦ r i ) . It is known that locally ˜ W sr i is diffeomorphic to its normal bundle O L O ( −
2) of ˜Γ r i , thus H ( Y, Y \ ∪ Γ sr i ) = 0 . It follows that there exist a 2-form σ ∈ H ( Y ) which extends σ ∗ .Let U i be a small neighborhood of Γ sr i in Y and ˜ U i ⊂ ˜ W sr i be itspre-image in the uniformizing system. Set σ i = σ | U i . By the proof of necessity, we know that[ σ i ] = [ − λ i ω sr i ] . Then we can deform σ i in its cohomology class near ˜Γ sr i such that σ i = − λ i ω sr i . Hence we get a new form σ on Y that gives symplectic forms near Γ si .On the other hand, we have a form ω Z on Z that is symplectic away from P . This form extends to Y , still denoted by ω Z , but is degenerateat the Γ sr i . For sufficiently large N we haveΩ = σ + N ω Z . This is a symplectic structure on Y : Ω is non-degenerate away from asmall neighborhood of the Γ sr i for large N ; both σ and ω Z are tamedby the complex structure in the U i , i.e, σ ( · , J · ) > , ω Z ( · , J · ) ≥ , therefore Ω( · , J · ) > , which says that Ω is also a symplectic structure near the Γ sr i . Hence( Y, Ω) is symplectic.We now remark that the assumption on the sign of λ i is inessential:suppose that λ >
0; then we alter Y by replacing the neighborhood of p by W sfr . Then the construction of the symplectic structure on this Y is the same.5.3. Proof of corollary 4.1.
This follows from remark 5.2. If Y and Y f are a pair of flops, then one of them satisfies some equation X i λ i Θ r i = 0and the other one satisfies − X i λ i Θ r i = 0 . Therefore, the symplectic structures exist on them simultaneously.6.
Orbifold Gromov-Witten invariants of W sr and W sfr We first introduce the cohomology group for an orbifold in the stringysense. Then we compute the orbifold Gromov-Witten invariants.¿From now on, r ≥ r from W sr and W sfr .6.1. Chen-Ruan orbifold cohomology of W s and W sf . The stringyorbifold cohomology of W s is H ∗ CR ( W s ) = H ∗ ( W s ) ⊕ M k C [ p s ] k ⊕ M k C [ q s ] k . We abuse the notation here such that [ p s ] k represents the 0-cohomologyof the sector [ p s ] k . On the other hand, the grading should be treatedcarefully: the degree of an element in H ∗ ( W s ) remains the same, how-ever the degree of [ p s ] k is 0 + ι ([ p s ] k ) and the same treatment appliesto [ q s ] k . We call these new classes twisted classes . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 23
A similar definition applies to W sf . H ∗ CR ( W sf ) = H ∗ ( W sf ) ⊕ M k C [ p sf ] k ⊕ M k C [ q sf ] k . Moduli spaces M ,l,k ( W s , d [Γ s ] , x ) , k ≥ . Here x = ( T , . . . , T k )consists of k twisted sectors in W s .By the definition in [CR2], the moduli space M ,l,k ( W s , d [Γ s ] , x )consists of orbifold stable holomorphic maps from genus 0 curves, onwhich there are l smooth marked points and k orbifold points y , . . . , y k ,to W s such that • y i are sent to Y i ; • the isotropy group at y i is Z | ξ a | if y i = [ p ] a (or [ q ] a ), where | ξ a | is the order of ξ a ; • the image of the map represents the homology class d [Γ s ].By a genus 0 curve we mean S , or a bubble tree consisting of several S ’s. The stability is the same as in the smooth case. Remark 6.1.
There is an extra feature for orbifold stable holomorphicmaps. That is, the nodal points on a bubble tree may also be orbifoldsingular points on its component: for example, say y is a nodal pointthat is the intersection of two spheres S and S − ; then y can be asingular points, denoted by y + and y − respectively, on both spheres.Moreover if y + is mapped to [ p ] a , y − has to be mapped to [ p ] r − a . When we write M ,l,k ( W s , d [Γ s ] , x ), we mean the map whose domainis S . Usually, we call M the compactified space of M and M the topstratum of M . Lemma 6.2.
For k ≥ , the virtual dimension dim M , ,k ( W s , d [Γ s ] , x ) < . Proof.
We recall that the virtual dimension is given by2 c ( d [Γ s ]) + 2( n −
3) + k − k X i =1 ι ( Y i ) = k − k X i =1 ι ( Y i ) < k − k = 0 . Here we use Lemma 2.2. q.e.d.
Lemma 6.3. M , , ( W s , d [Γ s ] , x ) = ∅ . Proof.
This also follows from the dimension formula: the virtualdimension of this moduli space is a rational number. q.e.d.
Moduli spaces M , , ( W s , d [Γ s ]) . Convention of notations: If k = 0, it is dropped and the moduli space is denoted by M ,l ( W s , d [Γ s ]);if k = l = 0, then the moduli space is denoted by M ( W s , d [Γ s ]).We have shown that M , ,k ( W s , d [Γ s ] , x ) for k ≥ k = 0. Although its top stratum M ( W s , d [Γ s ]) consists of only”smooth” maps, there may be orbifold maps in lower strata. Here,by the smoothness of a map we mean that the domain of the map iswithout orbifold singularities. The next lemma rules out this possibil-ity. Lemma 6.4. M ( W s , d [Γ s ]) only consists of smooth maps. Proof.
If not, suppose we have a map f ∈ M ( W s , d [Γ s ]) thatconsists of orbifold type nodal points in the domain. By looking atthe bubble tree, we start searching from the leaves to look for thefirst component, say S i , that containing a singular nodal point. Thiscomponent must contain only one singular point. So f | S i is an elementin some moduli space M , , ( W s , d [Γ s ] , x ). But it is claimed in Lemma6.3 that such an element does not exist. This proves the lemma. q.e.d.Notice that W s = ˜ W s /µ r and Γ s = ˜Γ s /µ r . We may like to comparethe moduli space M ( W s , d [Γ s ]) with M ( ˜ W s , d [˜Γ s ]). Note that µ r acts naturally on the latter space. We claim that Proposition 6.5. M ( W s , d [Γ s ]) = ∅ if r ∤ d . Otherwise, there is anatural isomorphism M ( W s , mr [Γ s ]) = M ( ˜ W s , m [˜Γ s ]) /µ r . if d = mr . Proof.
Since M ( W s , d [Γ s ]) = M (Γ s , d [Γ s ])and M ( ˜ W s , d [˜Γ s ]) = M (˜Γ s , d [˜Γ s ]) , it is sufficient to show that M ( W s , d [Γ s ]) = ∅ if r ∤ d and M (Γ s , mr [Γ s ]) = M (˜Γ s , m [˜Γ s ]) /µ r . We need the following lemma. Let π : ˜Γ s → Γ s be the projection givenby the quotient of µ r . We claim that Lemma 6.6. for any smooth map f : S → Γ s there is a lifting ˜ f : S → ˜Γ s such that ˜Π( ˜ f ) = f . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 25
Now suppose the lemma is proved. Then we have that M ( W s , d [Γ s ]) = ∅ for r ∤ d .To prove the second statement, we define a map:˜Π : M (˜Γ s , m [˜Γ s ]) → M (Γ s , mr [Γ s ])given by ˜Π( ˜ f ) = π ◦ ˜ f . It is clear that this induces an injective mapΠ : M (˜Γ s , m [˜Γ s ]) /µ r → M (Γ s , mr [Γ s ]) . On the other hand, since a stable smooth map on a bubble tree consistsof smooth maps on each component of the tree that match at nodalpoints, therefore, by Lemma 6.6 the map can be components wise lifted.This shows that Π is surjective. q.e.d.
Proof of Lemma 6.6: S and Γ s are P . We identify them as C ∪{∞} as usual. On Γ s , we assume p s and q s are 0 and ∞ respectively.Suppose thatΛ = f − ( p s ) = { x , . . . , x m } , , Λ ∞ = f − ( q s ) = { y , . . . , y n } . Let z be the coordinate of the first sphere; we write f ( z ) = [ p ( z ) , q ( z )] . Now since f is assumed to be smooth at the x i , the map can be liftedwith respect to the uniformizing system of p s : namely, suppose that π s p : D ǫ (0) ⊂ C → D ǫ r ( p s ) C ; π s p ( w ) = w r gives the uniformizing system of the neighborhood of p s for some ǫ ; f ,restricted to a small neighborhood U x i , can be lifted to˜ f : U x i → D ǫ such that f = π s p ◦ ˜ f . Without loss of generality, we assume that f ( U x i ) = D ǫ (0). Therefore we have a lifting˜ f : [ i U x i ∪ [ j U y j → D ǫ (0) ∪ D ǫ ( ∞ )for f . Now we look at the rest of the map f : S − [ i U x i ∪ [ j U y j → Γ s − D ǫ r ( p s ) ∪ D ǫ r ( q s ) . We ask if this map can be lifted to the covering space˜Γ s − D ǫ r (0) ∪ D ǫ r ( ∞ ) → Γ s − D ǫ r ( p s ) ∪ D ǫ r ( q s ) . The answer is affirmative by the elementary lifting theory for the cov-ering space. Therefore, the whole map f has a lifting ˜ f . The ambiguityof the lifting is up to the µ r action. q.e.d.6.4. Orbifold Gromov-Witten invariants on W s . We study theGromov-Witten invariants that are needed in this paper.Given a moduli space M ,l,k ( W s , d [Γ s ] , x ), one can define the Gromov-Witten invariants via evaluation maps: ev i : M ,l,k ( W s , d [Γ s ] , x ) → X, ≤ i ≤ l ; ev orbj : M ,l,k ( W s , d [Γ s ] , x ) → Y j , ≤ j ≤ k. The Gromov-Witten invariants are given byΨ W s ( d [Γ s ] , ,l,k, x ) ( α , . . . , α l , γ , . . . , γ k )= Z [ M ,l,k ( W s ,d [Γ s ] , x )] vir [ i ev ∗ i ( α i ) ∪ [ j ev orb, ∗ j ( β j ) . Here α i ∈ H ∗ ( X ) and β j ∈ H ∗ ( Y j ). Note that l, k and x are specifiedby the α i and β j . For the sake of simplicity and consistency, we alsore-denote the invariants byΨ W s ( d [Γ s ] , ,l + k ) ( α , . . . , α l , γ , . . . , γ k ) , when the α i and β j are given. Lemma 6.7.
For k ≥ and d ≥ W s ( d [Γ s ] , , ,k, x ) = 0 . Proof.
As explained in Lemma 6.2, this moduli space has nega-tive dimension. Therefore the Gromov-Witten invariants have to be 0.q.e.d.
Proposition 6.8.
For d ≥ , if r ∤ d , Ψ W s ( d [Γ s ] , vanishes. Otherwise, if d = mr Ψ W s ( mr [Γ s ] , = 1 m . Proof.
We have shown that M ( W s , mr [Γ s ]) = M ( ˜ W s , m [˜Γ s ]) /µ r . This would suggest that(6.1) Ψ W s ( mr [Γ s ] , = 1 r Ψ ˜ W s ( m [˜Γ s ] , . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 27
This has to be shown by virtual techniques. Following the standardconstruction of virtual neighborhoods of moduli spaces, we have asmooth virtual moduli space U ( ˜ W s , m [˜Γ s ]) ⊃ M ( ˜ W s , m [˜Γ s ]) , with an obstruction bundle ˜ O . The Gromov-Witten invariant is thengiven by Ψ ˜ W s ( m [˜Γ s ] , = Z U ( ˜ W s ,m [˜Γ s ]) Θ( ˜ O ) . Here Θ( ˜ O ) is the Thom form of the bundle. See the construction ofvirtual neighborhood in [CL] (and orginally in [R2]). The constructionof virtual neighborhoods for M ( W s , mr [Γ s ]) is parallel. We also have U ( W s , mr [Γ s ])with obstruction bundle O . The model can be suggestively expressedas ( U ( W s , mr [Γ s ]) , O ) = ( U ( ˜ W s , m [˜Γ s ]) , ˜ O ) /µ r . Therefore, we conclude thatΨ W s ( mr [Γ s ] , = 1 r Z U ( ˜ W s ,m [˜Γ s ]) Θ( ˜ O ) = 1 r Ψ ˜ W s ( m [˜Γ s ] , . On the other hand, Ψ ˜ W s ( m [˜Γ s ] , = rm . This is computed in [BKL]. Therefore the proposition is proved. q.e.d.6.5. H ∗ CR ( W s ) and H ∗ CR ( W sf ) . On W s , H ∗ CR ( W s ) = C [1] ⊕ C ( H s ) ⊕ r − M i =1 C [ p s ] i ⊕ r − M j =1 C [ q s ] j . Given β i , ≤ i ≤ , in H ∗ CR ( X ) one defines the 3-point function asfollowing:Ψ W s ( β , β , β ) = Ψ W s CR ( β , β , β ) + X d ≥ Ψ W s ( d [Γ s ] , , ( β , β , β ) q d [Γ s ] . Here the first termΨ W s CR ( β , β , β ) = Ψ W s ([0] , , ( β , β , β )is the 3-point function defining the Chen-Ruan product. In the smoothcase, this is just Z β ∧ β ∧ β . A similar expression for the orbifold case still holds. This is proved in[CH]: by introducing twisting factors, one can turn a twisted form β on twisted sector into a formal form ˜ β on the global orbifold. Then westill have Ψ W s CR ( β , β , β ) = Z orbW s ˜ β ∧ ˜ β ∧ ˜ β . Remark 6.9.
Unfortunately, for the local model, Ψ W s cr ( β , β , β ) doesnot make sense if and only if all β i are smooth classes, for the modulispace of the latter case is non-compact. Hence Ψ W s CR ( β , β , β ) is onlya notation at the moment. But we will need it when we move on tostudy compact symplectic conifolds. By the computation in § Proposition 6.10.
If at least one of the β i is a twisted class, Ψ W s ( β , β , β ) = Ψ W s CR ( β , β , β ) . Proof.
Case 1, if all β i are twisted classes,Ψ W s ( d [Γ s ] , , ( β , β , β ) = 0if d ≥ β is not twisted and the other two are.Case 2: Suppose β = 1; then it is well known thatΨ W s ( d [Γ s ] , , ( β , β ,
1) = 0if d ≥ β = nH s ; thenΨ W s ( d [Γ s ] , , ( β , β , β ) = β ( d [Γ s ])Ψ W s ( d [Γ s ] , , ( β , β ) = 0 . Similar arguments can be applied to the case in which only one ofthe β i is twisted. Hence the claim follows. q.e.d.Now suppose deg( β i ) = 2, i.e. β i = n i H s . Then X m ≥ Ψ W s ( mr [Γ s ] , , ( β , β , β ) q mr [Γ s ] = β ([ r Γ s ]) β ([ r Γ s ]) β ([ r Γ s ]) q [ r Γ s ] − q [ r Γ s ] . The last statement follows from Proposition 6.8. HenceΨ W s ( β , β , β ) = Z orbW s β ∧ β ∧ β + β ([ r Γ s ]) β ([ r Γ s ]) β ([ r Γ s ]) q [ r Γ s ] − q [ r Γ s ] . Formally, we write [˜Γ s ] = [ r Γ s ]. To summarize, INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 29
Proposition 6.11.
The three-point function Ψ W s ( β , β , β ) of W s is Ψ W s CR ( β , β , β ) if at least one of the β i is twisted or of degree 0, or Ψ W s cr ( β , β , β ) + β (˜Γ s ) β (˜Γ s ) β (˜Γ s ) q [˜Γ s ] − q [˜Γ s ] , if deg( β i ) = 2 , ≤ i ≤ . This proposition says that the quantum product β ⋆ β is the usualproduct( in the sense of the Chen-Ruan ring structure) except for thecase in which deg( β ) = deg( β ) = 2. Next, we write down the Chen-Ruan ring structure for twisted classes: Proposition 6.12.
The Chen-Ruan products for twisted classes aregiven by [ p s ] i ⋆ [ q s ] j = 0 , [ p s ] i ⋆ [ p s ] j = δ i + j,r Θ p , [ q s ] i ⋆ [ q s ] j = δ i + j,r Θ q . Here Θ p and Θ q are Thom forms of the normal bundles of p and q in W s . Also β ⋆ H s = 0 if β is a twisted class. Proof.
This follows from the theorem in [CH]. As an example, weverify [ p s ] i ⋆ [ p s ] j = δ i + j,r Θ p = 0 . For other cases, the proof is similar. The normal bundle of p is a rank3 orbi-bundle which splits as three lines C p , C y and C z (cf. S2.3). LetΘ p , Θ y and Θ z be the corresponding Thom forms. Then the twistingfactor(cf. [CH]) of [ p s ] i is t ([ p s ] i ) = Θ bp Θ r − by Θ iz . Here b ≡ ai ( mod r ) is an integer between 0 and r −
1. Similarly, wewrite t ([ p s ] j ) = Θ cp Θ r − cy Θ jz . Here c ≡ aj ( mod r ) is an integer between 0 and r −
1. Then we havea formal computation[ p s ] i ⋆ [ p s ] j = t ([ p s ] i ) ∧ t ([ p s ] j ) = δ i + j,r Θ p . q.e.d.Equivalently, this can be restated in terms of Ψ W s cr as Proposition 6.13.
Suppose at least one of the β i is twisted in thethree-point function Ψ W s cr ( β , β , β ) . Then only the following functionsare nontrivial: Ψ W s cr ([ p s ] i , [ p s ] j ,
1) = δ i + j,r r ;Ψ W s cr ([ q s ] i , [ q s ] j ,
1) = δ i + j,r r . Identification of three-point functions Ψ W s and Ψ W sf . Wefollow the argument in [LR]. Define a map φ : H ∗ CR ( W sf ) → H ∗ CR ( W s ) . On twisted classes, we define φ ([ p sf ] k ) = [ p s ] k , φ ([ q sf ] k ) = [ q s ] k . And on H ∗ CR ( W sf ), φ is defined as in the smooth case in [LR]. Since atthe moment we are working in the local model, we should avoid usingPoincare duality. We give a direct geometric construction of the map.On the other hand, a technical issue mentioned in Remark 6.9 is dealtwith: let β sfi , ≤ i ≤ , be 2-forms on W sf representing the classes[ β sfi ]; by the identification of W sf − Γ sf with W s − Γ s , we then alsohave 2-forms in W s − Γ s which as cohomology classes can be uniquely extended over W s . The cohomology classes are denoted by[ α i ] = φ ([ β i ]) . Moreover we can require that the representing forms, denoted by α i , coincide with β i away from the Γ’s.Then we can defineΨ W s CR ([ α ] , [ α ] , [ α ]) − Ψ W sf CR ([ β ] , [ β ] , [ β ]):= Z orbW s α ∧ α ∧ α − Z orbW sf β ∧ β ∧ β . The well-definedness can be easily seen due to the coincidence of the α i and β i outside a compact set. Moreover, Lemma 6.14.
Suppose that deg β i = 2 ; then Ψ W s CR ([ α ] , [ α ] , [ α ]) − Ψ W sf CR ([ β ] , [ β ] , [ β ]) = α (˜Γ s ) α (˜Γ s ) α (˜Γ s )= − β (˜Γ sf ) β (˜Γ sf ) β (˜Γ sf ) . Proof.
We lift the problem to ˜ W s and ˜ W sf . Then we can furtherdeform both models simultaneously to ˜ V s and ˜ V sf as [F]. Each of themconsists r copies of the standard model O ( − ⊕ O ( − → P . ˜ V sf isa flop of ˜ V s . Therefore, the computations are essentially r copies of INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 31 the computation on the standard model. By the argument in [LR], wehave Z orbW s α ∧ α ∧ α − Z orbW sf β ∧ β ∧ β = 1 r (cid:18)Z ˜ W s α ∧ α ∧ α − Z ˜ W sf β ∧ β ∧ β (cid:19) = 1 r · r · α (˜Γ s ) α (˜Γ s ) α (˜Γ s )= α (˜Γ s ) α (˜Γ s ) α (˜Γ s ) . Now we conclude that
Theorem 6.15.
Let β i ∈ H ∗ CR ( W sf ) , ≤ i ≤ , and α i = φ ( β i ) . Then Ψ W s ( α , α , α ) = Ψ W sf ( β , β , β ) with the identification of [Γ s ] ↔ − [Γ sf ] . Proof.
The only nontrivial verification is for all deg β i = 2. Supposethis is the case. Then the differenceΨ W s ( α , α , α ) − Ψ W sf ( β , β , β )includes two parts. Part(I) isΨ W s cr ([ α ] , [ α ] , [ α ]) − Ψ W sf cr ([ β ] , [ β ] , [ β ]) = α (˜Γ s ) α (˜Γ s ) α (˜Γ s )and part(II) is α (˜Γ s ) α (˜Γ s ) α (˜Γ s ) q [˜Γ s ] − q [˜Γ s ] − β (˜Γ sf ) β (˜Γ sf ) β (˜Γ sf ) q [˜Γ sf ] − q [˜Γ sf ] = α (˜Γ s ) α (˜Γ s ) α (˜Γ s ) q [˜Γ s ] − q [˜Γ s ] + α (˜Γ s ) α (˜Γ s ) α (˜Γ s ) q [ − ˜Γ s ] − q [ − ˜Γ s ] = − α (˜Γ s ) α (˜Γ s ) α (˜Γ s ) . Here we use [Γ s ] ↔ − [Γ sf ]. Part(I) cancels part (II), thereforeΨ W s ( α , α , α ) = Ψ W sf ( β , β , β ) . q.e.d.7. Ruan’s conjecture on orbifold symplectic flops
Ruan cohomology.
Let X and Y be compact symplectic orb-ifolds related by symplectic flops. Correspondingly, Γ si and Γ sfi , 1 ≤ i ≤ k , are extremal rays on X and Y respectively. We define three-point functions on X (similarly on Y ):Ψ Xqc ( β , β , β ) = Ψ XCR ( β , β , β ) + k X i =1 ∞ X d =1 Ψ X ( d [Γ si ] , , ( β , β , β ) . This induces a ring structure on H ∗ CR ( X ) Definition 7.1.
Define the product on H ∗ CR ( X ) by h β ⋆ r β , β i = Ψ Xqc ( β , β , β ) . We call this the Ruan product on X . This cohomology ring is denotedby RH CR ( X ) . Similarly, we can define RH ∗ CR ( Y ) by using the three-point functionsgiven by Γ sfi . Ruan conjectures that Conjecture 7.1 (Ruan) . RH ∗ CR ( X ) is isomorphic to RH ∗ CR ( Y ) . Verification of Ruan’s conjecture.
SetΦ([Γ su ]) = − [Γ sfu ] . This induces an obvious identificationΦ : H ( X ) → H ( Y ) . As explained in the local model, there is a natural isomorphism φ : H ∗ CR ( Y ) → H ∗ CR ( X ) . We explain φ . For twisted classes [ p sfs ] i and [ q sft ] j we define φ ([ p sfu ] i ) = [ p su ] i , , φ ([ q sfv ] j ) = [ q sv ] j . For degree 0 or 6-forms, φ is defined in an obvious way. For α ∈ H orb ( Y ), φ ( α ) is defined to be the unique extension of α | X −∪ Γ su = α | Y −∪ Γ sfv over X . For β ∈ H ( Y ), define φ ( β ) ∈ H ( X ) to be the extension asabove such that Z X φ ( β ) ∧ φ ( α ) = Z Y β ∧ α, for any α ∈ H ( Y ). Then Theorem 7.2.
For any classes β i ∈ H ∗ CR ( Y ) , ≤ i ≤ , Φ ∗ (Ψ Xqc,r ( φ ( β ) , φ ( β ) , φ ( β ))) = Ψ Yqc,r ( α , α , α ) . INGULAR SYMPLECTIC FLOPS AND RUAN COHOMOLOGY 33
Proof.
If one of β i , say β , has degree ≥
4, the quantum correctionterm vanishes. Therefore, we need only verifyΨ
XCR ( φ ( β ) , φ ( β ) , φ ( β )) = Ψ YCR ( α , α , α ) . We choose β to be supported away from the Γ sf . Then we have fol-lowing observations: • whenever β or β is a twisted class, both sides are equal to 0; • if β and β are in H ∗ ( Y ), thenΨ Xcr ( φ ( β ) , φ ( β ) , φ ( β )) = Z X φ ( β ) ∧ φ ( β ) ∧ φ ( β )= Z Y β ∧ β ∧ β = Ψ Ycr ( α , α , α ) . Now we assume that β i are either twisted classes or degree 2 classes.Then the verification is exactly same as that in Theorem 6.15. q.e.d.As an corollary, we have proved Theorem 7.3.
Suppose X and Y are related via an orbifold symplecticflops, Via the map φ and coordinate change Φ , RH ∗ CR ( X ) ∼ = RH ∗ CR ( Y ) . This explicitly realizes the claim of Theorem 1.3.
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