FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS Y.-P. LEE, H.-W. LIN, AND C.-L. WANG

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1 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS Y.-P. LEE, H.-W. LIN, AND C.-L. WANG ABSTRACT. For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish. 0. INTRODUCTION 0.. Statement of main results. Let X be a smooth complex projective manifold and ψ : X X a flopping contraction in the sense of minimal model theory, with ψ : Z S the restriction map on the exceptional loci. Assume that (i) ψ equips Z with a P r -bundle structure ψ : Z = P S (F) S for some rank r + vector bundle F over a smooth base S, (ii) N Z/X Zs = OP r( ) (r+) for each ψ-fiber Z s, s S. It is not hard to see that the corresponding ordinary P r flop f : X X exists. An ordinary flop is called simple if S is a point. For a P r flop f : X X, the graph closure [ Γ f ] A (X X ) identifies the Chow motives ˆX of X and ˆX of X. Indeed, let F := [ Γ f ] then the transpose F is [ Γ f ]. One has the following theorem. Theorem 0.. For an ordinary P r flop f : X X, the graph closure F := [ Γ f ] induces ˆX = ˆX via F F = X and F F = X. In particular, F preserves the Poincaré pairing on cohomology groups. While the ring structure is in general not preserved under F, the quantum cohomology ring is, when the analytic continuation on the Novikov variables is allowed. Theorem 0.2. The big quantum cohomology ring is invariant under simple ordinary flops, after an analytic continuation over the extended Kähler moduli space. A contraction (ψ, ψ) : (X, Z) ( X, S) is of Mukai type if N Z/X = T Z/S. The corresponding algebraic flop f : X X exists and its local analytic

2 2 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG model can be realized as a slice of an ordinary flop. The following result is proved based upon our understanding of local geometry of Mukai flops. Theorem 0.3. Let f : X X be a Mukai flop. Then X and X are diffeomorphic, and have isomorphic Hodge structures and full Gromov Witten theory. In fact, any local Mukai flop is a limit of isomorphisms and all quantum corrections attached to the extremal ray vanish Motivations. This paper is the first of our study of the relationship between birational geometry and Gromov Witten theory. Our motivations come from both fields. In birational geometry, two varieties X and X are K-equivalent if there exist birational morphisms φ : Y X and φ : Y X such that φ K X = φ KX. K-equivalent smooth varieties have the same Betti numbers ([] [22], see also [23] for a survey on recent development). However, the cohomology ring structures are in general different. Two natural questions arise here: () Is there a canonical correspondence between the cohomology groups of K-equivalent smooth varieties? (2) Is there a modified ring structure which is invariant under the K- equivalence relation? The following conjecture was advanced by Y. Ruan [2] and the third author [23] in response to these questions. Conjecture 0.4. K-equivalent smooth varieties have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces. The choice to start with ordinary flops is almost obvious. Ordinary flops are not only the first examples of K-equivalent maps, but also crucial to the general theory. In fact, one of the goals of this paper is to study some of their fundamental properties. In the Gromov Witten theory, one is led to consider the problem of functoriality in quantum cohomology. Quantum cohomology is not functorial with respect to the usual operations: pull-backs, push-forwards, etc.. Y. Ruan [20] has proposed to study the Quantum Naturality Problem: finding the morphisms in the category of symplectic manifolds for which the quantum cohomology is natural. Conjecture 0.4 suggests that the K- equivalent maps play a key role in this direction. In particular, Theorem 0.2 establishes some naturality for the quantum cohomology. Conjecture 0.4 can also be interpreted as a consistency check for the Crepant Resolution Conjecture [2]. In general, there are more than one possible crepant resolution, but different crepant resolutions are K-equivalent. The consistency check naturally leads to a special version of Conjecture Contents of the paper. Section studies the geometry of ordinary flops. The existence of ordinary flops is proved and explicit description of local analytic models is given.

3 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 3 Section 2 is devoted to the correspondences and Chow motives of projective smooth varieties under an ordinary flop. The main result of this section is Theorem 0. alluded above. The ring structure is, however, not preserved. For a simple P r -flop, let h be the hyperplane class of Z = P r and let α i H 2l i (X), with li r and l + l 2 + l 3 = dim X = 2r +. Proposition 0.5. (Fα.Fα 2.Fα 3 ) = (α.α 2.α 3 ) + ( ) r (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ). For Calabi-Yau threefolds under a simple P flop, it is well known in the context of string theory (see e.g. [25]) that the defect of the classical product is exactly remedied by the quantum corrections attached to the extremal rays. This picture also emerged as part of Morrison s cone conjecture on birational Calabi-Yau threefolds [8] where Conjecture 0.4 for Calabi-Yau threefolds was proposed. For threefolds Conjecture 0.4 was proved by A. Li and Y. Ruan [2]. Their proof has three ingredients: () A symplectic deformation and decomposition of K-equivalent maps into composite of ordinary P flops, (2) the multiple cover formula for P = C X with N C/X = O( ) 2, and their main contribution: (3) the theory of relative Gromov-Witten invariants and the degeneration formula. In 3 a higher dimensional version of ingredient (2) is proved: Theorem 0.6. Let Z = P r X with N Z/X = O( ) r+. Let l be the line class in Z. Then for all α i H 2l i (X) with li r and i= n l i = 2r + + (n 3), α,..., α n 0,n,d e [M 0,n (X,dl)] virt α enα n = ( ) (d )(r+) N l,...,l n d n 3 (α.h r l ) (α n.h r l n ). where N l,...,l n are recursively determined universal constants. N l,...,l n are independent of d and N l,...,l n = for n = 2 or 3. All other (primary) Gromov-Witten invariants with degree in Zl vanish. This proposition, together with some algebraic manipulations, implies that for simple P r flops the quantum corrections attached to the extremal ray exactly remedy the defect caused by the classical product for any r N and the big quantum products restricted to exceptional curve classes are invariant under simple ordinary flops. Note that there are Novikov variables q involved in these transformations (c.f. Remark 3.3), and F(q β ) = q Fβ. The proof has two ingredients: Localization and the divisor relations. Localization has been widely used in calculating Gromov Witten invariants. For genus zero one-pointed descendent invariants twisted by a direct

4 4 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG sum of negative line bundles, this was carried out in [3] and [4]. The divisor relations studied in [0] gives a reconstruction theorem, which allows us to go from one-point invariants to multiple-point ones. To achieve the invariance of big quantum product, non-extremal curve classes need to be analyzed. The main purpose of 4 is to reduce the case of general X to the local case. Briefly, the degeneration formula expresses α X in terms of relative invariants α (Y,E) and α 2 (Ẽ,E), where Y X is the blow-up of X over Z and Ẽ = P Z (N Z/X O). Similarly for X, one has Y, Ẽ, E. By definition of ordinary flops, Y = Y and E = E. It is possible to match all output on the part of (Y, E) from X and X. Thus, the problem is transformed to one for the relative cases of (Ẽ, E) and (Ẽ, E). Following ideas in the work of D. Maulik and R. Pandharipande [7], a further reduction from relative invariants to absolute invariants is made. The problem is thus reduced to X = Ẽ = P P r(o( ) (r+) O), which is a semi-fano projective bundle. Remark 0.7. For simple flops, we may and will consider only cohomology insertions of real even degrees throughout all our discussions on GW invariants. This is allowed since Ẽ has only algebraic classes and any real odd degree insertion must go to the Y side after degeneration. The proof of the local case is carried out in 5 by exploring the compatibility of functional equations of n-point functions under the reconstruction procedure of genus zero invariants. It is easy to see that the Mori cone NE(X) = Z + l Z + γ with l the line class in Z and γ the fiber line class of X = Ẽ Z. The proof is based on an induction on d 2 and n with degree β = d l + d 2 γ. The case d 2 = 0 is Proposition 0.6. For d 2 > 0, the starting case, namely the one-point invariant, is again based on localization technique on semi-fano toric manifolds [4] and [4]. Theorem 0.8 (Functional equations for local models). Consider an n-point function on X = P P r(o( ) (r+) O), α = α,..., α n β q β β NE(X) where α i lies in the span of cohomology classes in X and descendents of (pushforward of) cohomology classes in E. For β = d l + d 2 γ, the summands are non-trivial only for a fixed d 2. If d 2 = 0 then F α X = Fα X. (Here = stands for equality up to analytic continuations.) Combining all the previous results Theorem 0.2 is proved.

5 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 5 Remark 0.9. Concerning ingredient (), it is very important to understand the closure of ordinary flops. To the authors knowledge, no serious attempt was made toward a higher dimensional version of () except some much weaker topological results [24]. Even in dimension three, the only known proof of () relies on the minimal model theory and classifications of terminal singularities. It is desirable to have a direct proof in the symplectic category. Such a proof should shed important light toward the higher dimensional cases. Our main theorem applies to K-equivalent maps that are composite of simple ordinary flops and their limits. As an application of the construction of ordinary flops in, we discuss (generalized) Mukai flops in 6. Some new understanding of the local geometry of Mukai flops is presented and this leads to a proof of Theorem 0.3. Theorem 0.3 can also be interpreted as a generalization of a local version of Huybrechts results on hyper-kähler manifolds [6], with the flexibility of allowing the exceptional fiber Z s = P r to be any projective space including the odd dimensional ones and the base S to be any smooth variety. As in the hyper-kähler case, it also implies that the correspondence induced by the fiber product [X X X ] = [ Γ f ] + [Z S Z ] A (X X ) is the one which gives an isomorphism of Chow motives. Besides dimension three [2] and the hyper-kähler case [6], our results provide the first known series of examples in all high dimensions which support Conjecture Acknowledgments. We would like to thank A. Givental, C.-H. Liu, D. Maulik, Y. Ruan, S.-T. Yau and J. Zhou for useful discussions. C.-L. is grateful to C.-S. Lin and J. Yu for their encouragement. Y.-P. is partially supported by NSF and AMS Centennial Fellowship. H.- W. is partially supported by NSC. C.-L. is partially supported by NSC and the NCTS Chern Fellowship. We are grateful to NCTS (Taiwan) for providing stimulating and delightful environment which makes the cooperation possible.. ORDINARY FLOPS.. Ordinary P r flops. Let ψ : X X be a flopping contraction as defined in 0.. Our first task is to show that the corresponding algebraic ordinary flop X X exists. The construction of the desired flop is rather straightforward. First blow up X along Z to get φ : Y X. The exceptional divisor E is a P r P r -bundle over S. The key point is that one may blow down E along another fiber direction φ : Y X, with exceptional loci ψ : Z = P S (F ) S for F another rank r + vector bundle over S and also N Z /X ψ fiber = O P r( ) (r+). We start with the following elementary lemma.

6 6 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG Lemma.. Let p : Z = P S (F) S be a projective bundle over S and V Z a vector bundle such that V p (s) is trivial for every s S. Then V = p F for some vector bundle F over S. Proof. Recall that H i (P r, O) is zero for i = 0 and H 0 (P r, O) = C. By the theorem on Cohomology and Base Change we conclude immediately that p O(V) is locally free over S of the same rank as V. The natural map between locally free sheaves p p O(V) O(V) induces isomorphisms over each fiber and hence by the Nakayama Lemma it is indeed an isomorphism. The desired F is simply the vector bundle associated to p O(V). Now apply the lemma to V = O PS (F)() N Z/X, and we conclude that N Z/X = OPS (F)( ) ψ F. Therefore, on the blow-up φ : Y = Bl Z X X, N E/Y = O PZ (N Z/X )( ). From the Euler sequence which defines the universal sub-line bundle we see easily that O PZ (L F)( ) = φ L O PZ (F)( ) for any line bundle L over Z. Since the projectivization functor commutes with pull-backs, we have E = P Z (N Z/X ) = P Z ( ψ F ) = ψ P S (F ) = P S (F) S P S (F ). For future reference we denote the projection map Z := P S (F ) S by ψ and E Z by φ. The various sets and maps are summarized in the following commutative diagram. E = P S (F) S P S (F ) Y φ φ Z = P S (F) X Z ψ = P S (F ) ψ S X with normal bundle of E in Y being N E/Y = O PZ (N Z/X )( ) = O PZ (O Z ( ) ψ F )( ) = φ O PS (F)( ) O PZ ( ψ F )( ) = φ O PS (F)( ) φ O PS (F )( ). Remark.2. Notice that the bundles F and F are uniquely determined up to a twisting by a line bundle. Namely, the pair (F, F ) is equivalent to (F L, F L ) for any line bundle L on S. The next step is to show that there is a blow-down map φ : Y X which contracts the left ruling of E and restricts to the projection map φ : E Z. The existence of the contraction ψ : X X is essential here. Let us denote a line in the left ruling by C Y such that φ(c Y ) = C.

7 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 7 Proposition.3. Ordinary P r flops exist. Proof. Firstly, we will show that C Y is K Y -negative. From the exact sequence 0 T C T X C N C/X 0 and N C/X = OC () (r ) O C ( ) (r+), we find that (K X.C) = 2g(C) 2 ((r ) (r + )) = 0. Together with K Y = φ K X + re, we get (K Y.C Y ) = (K X.C) + r(e.c Y ) = r < 0. Next we will show C Y is extremal, i.e. it has supporting (big and nef) divisors. Let H be a very ample divisor on X and L a supporting divisor for C (e.g. take L = φ H for an ample divisor H on X). Let c = (H.C), then φ H + ce has type (0, c) on each P r P r fiber of E. The divisor kφ L (φ H + ce) is clearly big and nef for large k and vanishes precisely on the class [C Y ]. Thus C Y is a K Y -negative extremal ray and the contraction morphism φ : Y X fits into by the cone theorem on Y flop. φ Y X ψ φ ψ X X (c.f. [8]). X X is then the desired Remark.4. Notice that (K X.C) = 0, (K X.C ) = 0 (C is a line in the fiber of Z S) and φ K X = φ K X (K-equivalence). It is clear from the proof that for the existence of φ one needs only the (weaker) assumption that C is extremal instead of the existence of the contraction ψ : X X. However, since (K X.C) = 0 these two are equivalent by the cone theorem. In general (i) and (ii) are not enough to construct φ, a well known phenomenon already in the case of Atiyah flop (r = and S = {pt}). In the analytic category the situation is better behaved, and will be demonstrated in the next subsection..2. Analytic local models. We would like to localize the picture along the exceptional sets in the classical topology. Consider a complex manifold S and two holomorphic vector bundles F S and F S. Let ψ : Z := P S (F) S and ψ : Z := P S (F ) S be the induced morphisms and let E = P S (F) S P S (F ) with two projections φ : E Z and φ : E Z. Let Y be the total space of N := φ O Z ( ) φ O Z ( ) and we identify E as its zero section. It is clear that N E/Y = N.

8 8 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG We intend to show unconditionally a contraction diagram j E Y φ φ =π 2 φ i Z X ψ ψ Z i X ψ ψ S j X π = φ in the analytic category, with X (resp. X ) being the total space of O PS (F)( ) ψ F (resp. O PS (F )( ) ψ F). As we have mentioned before, two pairs of bundles (F, F ) and (F, F ) define the same analytic local model if and only if (F, F ) = (F L, F L ) for some line bundle L on S. Indeed, when S reduces to a point this is a standard result in complex geometry since N E/Y is then a negative line bundle. For S a small open disk this also holds since the whole picture is a trivial product with S. The general case follows from patching the local data over an open cover of S. It is accurate to say that the local analytic model of an ordinary P r flop is a locally trivial family of simple ordinary P r flops. Proposition.5. Conditions (i) and (ii) imply the analytic contractibility of ψ and the existence of analytic ordinary P r flops. 2. CORRESPONDENCES AND MOTIVES 2.. Grothendieck s category of Chow motives. General references of Chow motives can be found in [6] and [3] Let M be category of motives (over C). For each smooth variety X, one associates an object ˆX in M. The morphisms are given by correspondences Hom M ( ˆX, ˆX 2 ) = A (X X 2 ). For U A (X X 2 ), V A (X 2 X 3 ), let p ij : X X 2 X 3 X i X j be the projection maps. The composition law is given by V U = p 3 (p 2 U.p 23V). A correspondence U has associated maps on Chow groups: U : A (X ) A (X 2 ); a p 2 (U.p a) as well as induced maps on T-valued points Hom( ˆT, ˆX i ): U T : A (T X ) U A (T X 2 ). Then we have Manin s identity principle: Let U, V Hom( ˆX, ˆX ). Then U = V if and only if U T = V T for all T. (Since U X X = V X X implies U = V.) Theorem 2.. For an ordinary P r flop f : X X, the graph closure F := [ Γ f ] induces ˆX = ˆX via F F = X and F F = X.

9 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 9 Proof. For any T, id T f : T X T X is also an ordinary P r flop. Hence to prove that F F = X, by the identity principle, we only need to show that F F = id on A (X) for any ordinary P r flop. From the definition of pull-back, FW = p ( Γ f.p W) = φ φ W. We also have the formulae for pull-back from the intersection theory (c.f. [3], Theorem 6.7, Blow-up formula): φ W = W + j ( c(e). φ s(w Z, W) ) dim W where W is the proper transform of W in Y and E is the excess normal bundle defined by (2.) 0 N E/Y φ N Z/X E 0 and s(w Z, W) is the relative Segre class. The key observation is that the error term is lying over W Z. Let W A k (X). By Chow s moving lemma we may assume that W intersects Z transversally, so l := dim W Z = k + (r + s) (r + r + s + ) = k r. Since dim φ (W Z) = l + r = k < k, the error term in the pull-back formula must be zero and we get φ W = W. Hence FW = W, the proper transform of W in X. Notice that W is almost never transversal to Z. Let B be an irreducible component of W Z and B = ψ(b) S with dimension l B l. Notice that W Z has irreducible components {B := ψ ( B)} B (different B with the same B will give rise to the same B ). Let φ W = W + E B, where E B varies over irreducible components lying over B, hence E B φ ψ ( B), a P r P r bundle over B. For the generic point s ψ(φ(e B )) B, we thus have dim E B,s k l B = r + + (l l B ) > r. In particular, E B,s contains positive dimensional fibers of φ (as well as φ ). Hence φ (E B ) = 0 and F FW = W. By the same argument we have also that F F = X, thus the proof is completed. Remark 2.2. (i) For a general ground field k, if the flop diagram under consideration is defined over k then the theorem works for motives over k. (ii) Since every geometric cohomology theory (a graded ring functor H with Poincaré duality, Künneth formula and a cycle map A H etc.) factors through M, the theorem also holds on such a specialized theory. Corollary 2.3. Let f : X X be a P r flop. If dim α + dim α 2 = dim X, then (Fα.Fα 2 ) = (α.α 2 ). That is, F is an isometry with respect to (. ).

10 0 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG Proof. We may assume that α, α 2 are transversal to Z. Then (α.α 2 ) = (φ α.φ α 2 ) = ((φ Fα ξ).φ α 2 ) = ((φ Fα ).φ α 2 ) = (Fα.(φ φ α 2 )) = (Fα.Fα 2 ). Here we use the fact proved in the above theorem that ξ has positive fiber dimension in the φ direction. Thus for ordinary flops, F = F both in the sense of correspondences and Poincaré pairing. Remark 2.4. It is an easy fact that if X = K X then X and X are isomorphic in codimension one and in particular the graph closure gives canonical isomorphisms F on A (X) = A (X ) and A (X) = A (X ) respectively. In this more general setting, the above proof still implies that the Poincaré pairing on A A (and H 2 H 2 ) is preserved under F Triple product for simple flops. Let f : X X be a simple P r flop with S being a point. Let h be the hyperplane class of Z = P r and h be the hyperplane class of Z. Let also x = φ h = [h P r ], y = φ h = [P r h ] in E = P r P r. Lemma 2.5. For classes inside Z, we have φ [h l ] = j (x l y r x l+ y r + + ( ) r l x r y l ). Hence by symmetry we get F[h l ] = ( ) r l [h l ]. In particular, F[C] = [C ]. Proof. Recall that N E/Y = O Pr P r(, ) := φ O P r( ) φ O P r( ) and N Z/X = O P r( ) (r+). From (2.), c(e) = ( x) r+ ( x y). Taking degree r terms from both sides, we have c r (E) = [( x) r+ ( (x + y)) ] (r) = (x + y) r C r+ (x + y) r x + + ( ) r Cr r+ = (x + y) ((x + y) x) r+ ( ) r+ x r+ ) = (y r+ ( ) r+ x r+ )/(y + x) = y r y r x + y r 2 x 2 + ( ) r x r. The basic pull-back formula ([3], Proposition 6.7) then implies that φ [h l ] = j (c r (E). φ [h l ]) = j (c r (E).x l ) = j r t=0 ( )t y r t x t+l. x r If t + l r + then y r t x t+l = 0. The result follows.

11 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS Lemma 2.6. For a class α H 2l (X) with l r, let α = Fα in X. Then φ α = φ α + (α.h r l x ) l ( y) l j. x + y Proof. Since the difference φ α φ α has support in E, we may write φ α = φ α + j (a x l + + a k x l k y k + + a l y l ). By intersecting this equation with x r l y r in X and noticing that E (x + y) on E, we get by the projection formula 0 = φ α.x r l y r a x l (x + y)x r l y r = (α.h r l ) a. Similarly by intersecting with x r l+ y r we get 0 = a x l (x + y)x r l+ y r a 2 x l 2 (x + y)x r l+ y r = a a 2. Continuing in this way by intersecting with x p y q with p + q = 2r l we get a k = ( ) k (α.h r l ) for all k =,..., l. This proves the lemma. These formulae allow us to compare the triple products of classes in X and X : Proposition 2.7. For simple P r -flop f : X X, let α i H 2l i (X), with li r, l + l 2 + l 3 = dim X = 2r +. Then (Fα.Fα 2.Fα 3 ) = (α.α 2.α 3 ) + ( ) r (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ). Proof. The proof consists of straightforward computations. (Fα.Fα 2.Fα 3 ) = (φ Fα.φ Fα 2.φ Fα 3 ) ( = φ α + (α.h r l x l ( y) l )j x + y ( φ α 3 + (α 3.h r l 3 x l 3 ( y) l 3 )j x + y ) ( φ α 2 + (α 2.h r l 2 x l 2 ( y) l 2 )j x + y Among the resulting eight terms, the first term is clearly equal to α.α 2.α 3. For those three terms with two pull-backs like φ α.φ α 2, the intersection values are zero since the remaining part necessarily contains the φ fiber (from the formula the power in y is at most l 3 ). The term with φ α and two exceptional parts contributes φ x l 2 ( y) l 2 x l 3 ( y) l 3 α.j.j x + y x + y = φ ( α.j (x l 2 ( y) l 2 )(x l3 + x l3 2 ( y) + + ( y) l3 ) ) times (α 2.h r l 2)(α 3.h r l 3). The terms with non-trivial contribution must contain y r, hence there is only one such term, namely (notice that l + l 2 + l 3 = 2r ) ). ( y) l 2 x l 3 (r l 2 ) ( y) r l 2 = ( ) r x r l y r )

12 2 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG and the contribution is ( ) r (α.h r l )(α2.h r l 2)(α 3.h r l 3). There are three such terms. It remains to consider the term of triple product of three exceptional parts. It is (α.h r l )(α2.h r l 2)(α 3.h r l 3) times (x l ( y) l )(x l 2 ( y) l 2 )(x l 3 + x l 3 2 ( y) + + ( y) l 3 ). The terms with non-trivial values are precisely multiples of x r y r. Since l + l 2 > r, there are two such terms x l ( y) l 2 x r l ( y) l 3 (r l ) x l 2 ( y) l x r l 2 ( y) l 3 (r l 2 ) which give 2( ) r. Summing together we then finish the proof. 3. QUANTUM CORRECTIONS ATTACHED TO EXTREMAL RAYS The theorem above on triple product suggests that one needs to correct the product structure by some contributions from the extremal ray. In this section we show that for simple ordinary flops the quantum corrections attached to the extremal ray exactly remedy the defect of the ordinary product. 3.. Quantum cohomology. We use [2] as our general reference on moduli spaces of stable maps, Gromov-Witten theory and quantum cohomology. Let β NE(X), the Mori cone of numerical classes of effective one cycles. Let M g,n (X, β) be the moduli space of n-pointed stable maps f : (C; x,..., x n ) X from a nodal cure C with arithmetic genus g(c) = g and with degree [ f (C)] = β. Let e i : M g,n (X, β) X be the evaluation morphism f f (x i ). The Gromov-Witten invariant for classes α i H (X), i n, is given by α,..., α n g,n,β := e [ M g,n (X,β)] virt α enα n. The genus zero three-point functions (as formal power series) α, α 2, α 3 := β A (X) α, α 2, α 3 0,3,β q β together with the Poincaré pairing (, ) determine the small quantum product. More precisely, let T = t i T i with {T i } a cohomology basis and t i being formal variables. Let {T i } be the dual basis with (T i, T j ) = δ ij. The (genus zero) pre-potential combines all n-point functions together: Φ(T) = n=0 β NE(X) n! Tn β q β, where T n β = T,..., T 0,n,β. The big quantum product is defined by T i T j = k Φ ijk T k

13 where Φ ijk = FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 3 3 Φ = t i t j t k n=0 Ti β NE(X), T j, T n! k, T n β qβ. The n = 0 part Φ ijk (0) gives the small quantum product Analytic continuation. Let f : X X be a simple P r flop. Since X and X have the same Poincaré pairing under F, in order to compare their quantum products we only need to compare their n-point functions. For three-point functions, write α, α 2, α 3 = (α.α 2.α 3 ) + d N α, α 2, α 3 dl q dl + β Zl α, α 2, α 3 β q β. The difference (Fα.Fα 2.Fα 3 ) (α.α 2.α 3 ) is already determined in last section. The next step is to compute the middle term, namely quantum corrections coming from the extremal ray l = [C]. The third term will be discussed in later sections. The virtual dimension of M g,n (X, dl) is given by (c (X).dl) + (2r + )( g) + (3g 3) + n. Since (K X.l) = 0, for g = 0 we need only consider classes α i A l i (X) with n i= l i = 2r + + (n 3). For n = 3 this is 2r + = dim X. Theorem 3.. For all α i H 2l i (X) with li r and i= n l i = 2r + + (n 3), α,..., α n 0,n,d e [M 0,n (X,dl)] virt α enα n = ( ) (d )(r+) N l,...,l n d n 3 (α.h r l ) (α n.h r l n ). where N l,...,l n are recursively determined universal constants. N l,...,l n are independent of d and N l,...,l n = for n = 2 or 3. All other (primary) Gromov-Witten invariants with degree in Zl vanish. Corollary 3.2. Both the small and big quantum products restricted to exceptional curve classes are invariant under simple ordinary flops. In fact the three-point functions attached to the extremal ray exactly remedy the defect caused by the classical product. Proof. Since (Fα i.h (r l i) ) = ( ) l i (Fαi.Fh r l i ) = ( ) l i(αi.h r l i ), for three point functions we get Fα, Fα 2, Fα 3 α, α 2, α 3 = ( ) r (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ) ( ) + (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ( ) ) 2r+ q l q l ( ) r+ q l ( ) r+ q l. Under the correspondence F, we shall identify q l with q l. Plug in this into the last bracket we get when r is odd and get when r is even.

14 4 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG In both cases the right hand side cancels out and then Fα, Fα 2, Fα 3 = α, α 2, α 3. This proves the statement on small quantum product. For general n = 3 + k point invariants with k, we get α,..., α n = N l,...,l n (α.h r l ) (α n.h r l n ) d=0 ( = N l,...,l n (α.h r l ) (α n.h r l n ) q l ( ) (d )(r+) d k q dl d ) k ( ) r+ dq l ( ) r+ q l. Similarly, since ( ) l i = ( ) k+, Fα,..., Fα n equals ( ( ) k+ N l,...,l n (α.h r l ) (α n.h r l n ) q d ) k ( ) r+ l dq l ( ) r+. q l Taking into account of q l d = d ql dq l dq l and ( ) r+ q l = ( ) r+ q l we get Fα,..., Fα n = α,..., α n for all k (n 4). The proof for the statement on big quantum product is thus completed. To put the result into perspective, we interpret the change of variable l by l in terms of analytic continuation over the extended complexified Kähler moduli space. Without lose of generality we illustrate this by writing out the small quantum part. This is simply a word by word adoption of the treatment in the r = case (cf. [25] 5.5, [8] 4). The quantum cohomology is parameterized by the complexified Kähler class ω = B + ih with q β = exp(2πi(ω.β)), where B H, R (X) and H K X, the Kähler cone of X. For a simple P r flop X X, F identifies H,, A and the Poincaré pairing (, ) on X and X. Then α, α 2, α 3 X restricted to Zl converges in the region and equals H+, = {ω (H.l) > 0} H, R i K X e 2πi(ω.l) (α.α 2.α 3 ) + (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ) ( ) r+ e 2πi(ω.l). This is a well-defined analytic function of ω on the whole H,, which defines the analytic continuation of α, α 2, α 3 X from H, R i K X to H,. Similarly, Fα, Fα 2, Fα 3 X restricted to Zl converges in the region {ω (H.l ) > 0} = {ω (H.l) < 0} = H, H, R i K X and equals e 2πi(ω.l) (Fα.Fα 2.Fα 3 ) (α.h r l )(α 2.h r l 2 )(α 3.h r l 3 ) ( ) r+ e 2πi(ω.l)

15 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 5 which is the analytic continuation of the previous one from H, + to H,. Remark 3.3. It was conjectured that the total series Φijk X converges for B K X, at least for B large enough, hence the large radius limit goes back to the classical cubic product. The Novikov variables {q β } β NE(X) are introduced to avoid the convergence issue. Since K X K X = for non-isomorphic K-equivalent models, the collection of Kähler cones among them form a chamber structure. The conjectural canonical isomorphism F : H (X) = H (X ) assigns to each model X a coordinate system H (X) of the fixed H and F serves as the (linear) transition function. The conjecture asserts that Φ X ijk can be analytically continued from K X to K X and agrees with Φ X ijk. Equivalently, Φ ijk is well-defined on K X K X which verifies the functional equation FΦ ijk (ω, T) = Φ ijk (ω, FT). For simple ordinary flops, this is verified from 3 to 5 for each given cohomology insertions. The convergence has just been verified for extremal rays and will be verified for local models in One-point functions with descendents. In order to prove Theorem 3., we first reduce the problem to one for projective spaces. Let U d := R f t e n+ N be the obstruction bundle, where N = N Z/X and f t is the forgetting morphism in M 0,n+ (P r, d) e n+ f t M 0,n (P r, d) It is well known (see e.g. [2]) that (3.) [M 0,n (X, dl)] virt = e(u d ) [M 0,n (P r, dl)]. Since U d is functorial under f t, we use the same notation for all n. Define a generating function of one point GW invariants with descendents (3.2) In our case J X (q, z ) := β NE(X) := q β e X β NE(X) P r q β J X (β, z ) H (X)[[z ]][[q]] (. ) z(z ψ) [M 0,(X, β)] virt J X (dl, z e(u ) e Pr d ) z(z ψ)

16 6 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG has been calculated: Lemma 3.4 ([3], also [4] [2]). J X (dl, z ) = P d := ( ) (d )(r+) (h + dz) r+. Remark 3.5. This calculation can also be interpreted as quantum Lefschetz hyperplane theorem for concave bundles over P r. From this viewpoint, the mirror transformation from J X (dl, z ) to P d is not needed since the rank of the bundle O( ) r+ is greater than one. See e.g. [9]. Corollary 3.6. For l + k = 2r, l r, τ k h l = ( )d(r+)+k d d k+2 Cr k+ where Cr k = k!/r!(k r)!. The invariant is zero if l + k = 2r by dimensional constraints. Proof. We start with A := h l.p d = P r k 0 z k+2 τ k h l. d By Lemma 3.4 A = ( ) (d )(r+) P r h l (h + dz) r+ = P r h l ( d r+ z r+ + h ) (r+). dz The result follows from the Taylor expansion and the elementary fact that C (r+) r l = ( ) k+(r+) C k+ r Multiple-point functions via divisor relations. We recall the following rational equivalence in rational Chow groups A (M 0,n (X, β)) from [0], Corollary : For L Pic(X) and i = j, (3.3) e i L [M 0,n(X, β)] virt =(e j L + (β, L)ψ j) [M 0,n (X, β)] virt (3.4) ψ i + ψ j = [D i j ] virt, (β, L)[D i,β j,β 2 ] virt, β +β 2 =β where [D i,β j,β 2 ] virt A (M 0,n (X, β)) is the push-forward of the virtual classes of the corresponding boundary divisor components D i,β j,β 2 = i A,j B; A B={,...,n} D(A, B; β, β 2 ) and D i j = D i,β j,β 2. β +β 2 =β

17 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 7 Here is a simple observation which will be repeatedly used in the sequel: Lemma 3.7 (Vanishing lemma). Let P r X with N P r /X = j O( m j ), m j N. Let l be the line class in P r. Then for deg T > r and d = 0,..., T dl = 0. Proof. Since [M 0,n (X, dl)] virt equals [M 0,n (P r, d)] cut out by e(u d ), the evaluation morphisms factor through P r. But then en(t Z ) = 0. Here deg T := l if T H 2l (X). As we had mentioned in the introduction, only real even degree classes will be relevant throughout our discussions. Proposition 3.8. For k + k 2 + l + l 2 = 2r, l i r, τ k h l, τ k2 h l 2 d = ( )d(r+)+l +k 2 + C 2r (l +l 2 ) r l, d k +k 2 + and other descendent invariants vanish. In particular, the only non-trivial twopoint function without descendents in degree dl is given by h r, h r d = ( ) (d )(r+) d. Proof. We consider the invariant without descendents first. Since the virtual dimension is 2r, only h r, h r d survives. Using the above equivalence relations, we may decrease the power of e h one by one. In each step only the second term in the resulting three terms has nontrivial contribution. Indeed, for the first term any addition to the power of e2 hr leads to zero. For the third boundary splitting terms, write [ (X)] = i T i T i. For each i, since dim X = 2r + one of T i or T i must have degree strictly bigger than r. If β = d l, β 2 = d 2 l with d i = 0 then one of the integral, hence the product, must vanish by the vanishing lemma. This is what happens now. We apply the divisor relation to i = and j = 2. Since n = 2, we find n = A =, n 2 = B = and in the splitting we have sum of product of two-point invariants. The degree in each side is non-zero since there is no constant genus zero stable map with two marked points. So the splitting terms vanish. We apply the divisor relation repeatedly to compute h r, h r d = d h r, τ h r d = = d r h, τ r h r d = d r τ r h r where the last equality is by the divisor axiom. Now we plug in Corollary 3.6 with (k, l) = (r, r) and the statement follows. For descendent invariants we proceed in the same manner. For simplicity we abuse the notation by denoting, ψ s α, β =, τ s α, β. Let s, l + m + s = 2r and consider h l, ψ s h m h = l, ψ s h m+ + (h, dl) h l, ψ s+ h m d d d = h l, (h + dψ)ψ s h m = d = h, (h + dψ) l ψ s h m. d

18 8 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG Notice that the splitting terms are all zero as before. Now the divisor axiom of descendent invariants gives d (h + dψ) l ψ s h m + (h + dψ) l ψ s h m+, d d which leads to the reduction formula: h l, ψ s h m d = (h + dψ) l ψ s h m d. Notice that this equals the constant term in z in ψ k z k 0 k (h + dz)l z s h m d ( = z s+ e(ud ) e z(z ψ).e ((h + dz) l.h m)) which is = ( ) (d )(r+) z s+ (h + dz) l (r+).h m = ( ) (d )(r+) z r m ( + h ) l (r+).h m, dz ( ) d(r+)+r+ d r+ l+r m C l (r+) r m d (r+) l = ( )d(r+)+l+s+ d s+ In general from ψ = ψ 2 + [D 2 ] virt, we find τ k h l, τ k2 h l 2 d = τ k h l, τ k2 +h l 2 = = d ( )k since the splitting terms all vanishes. The result follows. C 2r (l+m) r m. h l, τ k +k 2 h l 2 d For n 3, it is known that for any three different markings i, j and k, ψ j = [D ik j ] virt. By plugging this into (3.3), we get e i L = e j L + In our special case this reads as e i h = e j h + ((β 2.L)[D ik,β j,β 2 ] virt (β.l)[d i,β jk,β 2 ] virt ). β +β 2 =β d +d 2 =d Notice that now d i is allowed to be zero. Lemma 3.9. For n 3, h l +, h l 2, h l 3,... n,d (d 2 [D ik,d j,d 2 ] virt d [D i,d jk,d 2 ] virt ). = h l, h l 2+, h l 3,... n,d + d h l +l 3, h l 2,... n,d d h l, h l 2+l 3,... n,d. Note that for l = 0 this recovers the divisor axiom.

19 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 9 Proof. As in the previous theorem, the boundary terms with non-trivial degree must vanish. For degree zero, the only non-trivial invariants are threepoint functions, hence we are left with h l +, h l 2, h l 3,... n,d = h l, h l2+, h l 3,... n,d + d h l, h l 3, T i 0 T i, h l 2,... d d T i, h l,... d h l 2, h l 3, T i 0. i i For the first boundary sum, in the diagonal decomposition [ (X)] = T i T i we may choose basis so that h l +l 3 appear in {T i }. Then the above degree zero invariants survive only in one term which is equal to. The same argument applies to the second sum too. So the above expression equals as expected. h l, h l 2+, h l 3,... d + d h l +l 3, h l 2,... d d h l, h l 2+l 3,... d In light of (3.) and results above, Theorem 3. can be reformulated as the following equation (3.5) h l, h l 2,..., h l n d = ( ) (d )(r+) N l,...,l n d n 3, which we will now prove. Proof. (of (3.5), or equivalently Theorem 3..) We will prove the theorem by induction on n N. The case n 2 are already proven before. We treat the case n = 3 first. Consider h l, h l 2, h l 3 d with l + l 2 + l 3 = 2r + and l l 2 l 3. If l = then l 2 = l 3 = r and so If l 2, then l 2 r and h, h r, h r d = d h r, h r d = ( ) (d )(r+). h l, h l 2, h l 3 d = h l, h l 2+, h l 3 d + d h l +l 3, h l 2 d d h l, h l 2+l 3 d. But then both l + l 3 and l 2 + l 3 are larger than r + and the boundary terms vanish individually. By reordering l 2, l 3 if necessary, and repeating this procedure we are reduced to the case l = and proof for n = 3 is completed. Suppose the theorem holds up to n (with n 4). The above lemma and the induction hypothesis imply that h l, h l 2, h l 3,... d = h l, h l 2+, h l 3,... d + d h l +l 3, h l 2,... d d h l, h l 2+l 3,... d = h l, h l 2+, h l 3,... d + (N l +l 3,l 2,... N l,l 2 +l 3,...)d n 3. By repeating this procedure, l is decreased to one and we get h l, h l 2,..., h l n d = ( ) (d )(r+) N l,...,l n d n 3,

20 20 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG where N l,...,l n is given by N s in one lower level. The proof is complete. Similar methods apply to descendent invariants: Theorem 3.0. The only three-point descendent invariants of extremal classes dl, up to permutations of insertions, are given by h l, h l 2, τ k3 h l 3 d = ( )d(r+)+l 3+ d k 3 C k 3+ r (l +l 2 ), where l + l 2 + l 3 + k 3 = 2r + and by convention Cn m = 0 if n < 0. More generally, an n-point descendent invariant i= n τ k i h l i d with n 3 is non-zero only if there are at least two insertions being free of descendents, say k = k 2 = 0. In such cases, there are universal constants N k,l Z such that h l, h l 2, τ k3 h l 3,..., τ kn h l n = N k,l d n 3 k i. d Proof. Let n 3 and assume that 0 k k 2 k n. If k 2 then use ψ 2 = [D 2 3 ] virt we get τ k h l,..., τ kn h l n d = i; d +d 2 =d τ k2 h l 2,, T i T d i, τ k h l, τ k3 h l 3,. d 2 We separate two cases. If the first factor is a two-point function then it is non-zero only if T i = h j for some j r. But then deg T i > r and the right factor vanishes since it contains ψ classes. For other cases, both factors contain ψ classes hence the factor with deg T i > r (or deg T i > r) must vanish. For three-point invariants, from ψ 3 = [D 3 2 ] virt we get as before that h l, h l 2, τ k3 h l 3 = d i; d +d 2 =d τ k3 h l 3, T i d T i, h l, h l 2 = τ k3 h l 3, h l +l 2 d and the formula follows from the two-point case. Similarly, for n 4, if k i = 0 then from ψ i = [D i 2 ] virt we get h l, h l 2,..., τ ki h l i,... The result follows from an induction on n. d = τ ki h l i, h l +l 2,... d. d 2 4. DEGENERATION ANALYSIS Our next task is to compare the genus zero Gromov Witten invariants of X and X for curve classes other than the flopped curve. Naively, one may wish to decompose the varieties into the neighborhoods of exceptional loci and their complements. As the latter s are obviously isomorphic, one is reduced to study the local case. The degeneration formula, [2] [] [7], provides a rigorous formulation of the above naive picture.

21 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS The degeneration formula. Recall the relative invariants of a smooth pair (Y, E) with E Y a smooth divisor: Let Γ = (g, n, β, ρ, µ) with µ = (µ,..., µ ρ ) N ρ a partition of µ := ρ i= µ i = (β.e). For A H (Y) n and ε H (E) ρ, the relative invariant of stable maps with topological type Γ (i.e. with contact order µ i in E at the i-th contact point) is A ε, µ (Y,E) Γ := [M Γ (Y,E)] virt e Y A e E ε where e Y : M Γ (Y, E) Y n, e E : M Γ (Y, E) E ρ are evaluation maps on marked points and contact points respectively. If Γ = π Γ π, the relative invariants (with disconnected domain curves) A ε, µ (Y,E) Γ := π A ε, µ (Y,E) Γ π is defined to be the product of each connected component. Except for using the numerical form of relative invariants, our presentation of degeneration formula below mostly follows that of [] and [5]. Given an ordinary flop f : X X, we apply deformations to the normal cone to both X and X. Let X = X A and Φ : W X be the blow-up along Z {0}. We use t A as the deformation parameter. Then W t = X for all t = 0 and W 0 = Y Y 2 with the blow-up along Z and φ = Φ Y : Y = Y X p = Φ Ẽ : Y 2 = Ẽ := P Z (N Z/X O) Z X the compactified normal bundle. Also Y Ẽ = E = P Z (N Z/X ) is the φ- exceptional divisor which consists of the infinity part of Ẽ. Similar construction gives Φ : W X = X A and W 0 = Y Ẽ. By definition of ordinary flops, Y = Y and E = E. Remark 4.. For simple P r flops, Y 2 = PP r(o( ) (r+) O) = Y 2. However the gluing maps of Y and Y 2 along E for X and X differ by a twist which interchanges the order of factors in E = P r P r. Thus W 0 = W 0 and it is necessary to study the details of the degenerations. In general, f induces an ordinary flop f : Y 2 Y 2 of the same type which is the local model of f. Since the family W A comes from a trivial family, all cohomology classes α H (X, Z) n have global liftings and the restriction α(t) on W t is defined for all t. Let j i : Y i W 0 be the inclusion maps for i =, 2. For {e i } a basis of H (E) with {e i } its dual basis, {e I } forms a basis of H (E ρ ) with dual basis {e I } where I = ρ, e I = e i e iρ. The degeneration formula expresses the absolute invariants of X in terms of the

22 22 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG relative invariants of the two smooth pairs (Y, E) and (Y 2, E): α X g,n,β = j α(0) (Y,E) ei, µ j 2α(0) e I, µ I Γ η Ω β C η (Y2,E). Γ 2 Here η = (Γ, Γ 2, I ρ ) is an admissible triple which consists of (possibly disconnected) topological types Γ i = Γ i π= Γπ i with the same contact order partition µ under the identification I ρ of contact points. The gluing Γ + Iρ Γ 2 has type (g, n, β) and is connected. In particular, ρ = 0 if and only if that one of the Γ i is empty. The total genus g i, total number of marked points n i and the total degree β i NE(Y i ) satisfy the splitting relations g = g + g 2 + ρ + Γ Γ 2, n + n 2 = n and φ β + p β 2 = β. The constants C η = m(µ)/ Aut η, where m(µ) = µ i and Aut η = { σ S ρ η σ = η }. We denote by Ω the equivalence class of all admissible triples, also by Ω β and Ω µ the subset with fixed degree β and fixed contact order µ respectively Liftings of cohomology insertions. Next we discuss the presentation of α(0). Denote by ι j : E Y = Y and ι 2 : E Y 2 = Ẽ the natural inclusions. The class α(0) can be represented by (j α(0), j 2 α(0)) = (α, α 2 ) with α i A (Y i ) such that ι α = ι 2α 2 and φ α + p α 2 = α. Such representatives are not unique. The flexibility on different choices is of key importance. The simplest choice is α = φ α and α 2 = p (α Z ) since they restrict to the same class in E and they push forward to α and 0 in X respectively, whose sum is α. More generally: Lemma 4.2. For e being a class in E, if α(0) = (α, α 2 ) then it can also be represented by α(0) = (α ι e, α 2 + ι 2 e). Proof. This follows from the facts that ι ι e = (e.c (N E/Y )) E = (e.c (N E/Ẽ )) E = ι 2ι 2 e and φ ι e + p ι 2 e = 0 (since φ ι = p ι 2 = φ : E Z). For an ordinary flop f : X X, we compare the degeneration expressions of X and X. For a given admissible triple η = (Γ, Γ 2, I ρ ) on X, we pick the corresponding η = (Γ, Γ 2, I ρ) on X with Γ = Γ. We modify the When a map is decomposed into two parts, an (extra) ordering to the contact points is assigned. The automorphism of the decomposed curves will also introduce an extra factor. These contribute to Aut η.

23 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 23 choices j α(0) = φ α and j Fα(0) = φ Fα by adding suitable classes in E to make them equal. This is possible since φ α φ Fα ι H (E). The relative invariants on the Y = Y part from both sides are then the same and we are left with the comparison of the part on Ẽ and Ẽ. The following is clear, e.g. by a dimension count. Lemma 4.3. Let Ẽ = P Z (N O) be a projective bundle with base i : Z Ẽ and infinity divisor ι 2 : E = P Z (N) Ẽ. Then the kernel of the the restriction map ι 2 : H (Ẽ) H (E) is i H (Z). Notice that f induces an ordinary flop f on the local model Ẽ Ẽ. Moreover it is a family of simple ordinary flops f t : Ẽ t Ẽ t over the base S, where t S and Ẽ t is the fiber of Ẽ Z S etc.. Denote again by F the cohomology correspondence induced by the graph closure. Then Lemma 4.4 (Cohomology reduction to local models). For f : X X a P r - flop over base S with dim S = s. Let α H 2l (X), l dim Z = r + s (otherwise α Z = 0) with representatives α(0) = (α, α 2 ) and Fα(0) = (α, α 2 ). 2 α 2 and the fact that f is an isomor- Proof. From ι 2 α 2 = ι α = ι phism outside Z, we get If α = α then Fα 2 = α 2. α = ι ι 2 (Fα 2 α 2) = Fι 2α 2 ι 2 α 2 = ι 2α 2 ι 2 α 2 = 0. Thus Fα 2 α 2 = i z for some z H 2(l (r+)) (Z ) since codimẽ Z = r +. For simple flops, s = 0 and then l (r + ) s < 0. So z = 0 and we are done. In general we restrict the equation to each fiber f t : Ẽ t Ẽ t. Since Γ f t = Γ ft, by the case of simple flops we get (Fα 2 α 2 ) Ẽ t = 0 for all t S. That is, z is a class supported in the fiber of p : Z S. But then codimẽ z s + r + > l, which implies that z = 0. Remark 4.5. Using this lemma we may give an alternative proof of equivalence of Chow motives under ordinary flops. Indeed the equivalence of Chow groups for simple flops is easy to establish. The deformation to normal cone then allows us to reduce the general case to the local case and then to the local simple case Reduction to relative local models. First notice that A (Ẽ) = ι 2 A (E) since both are projective bundles over Z. We then have φ β = β + β 2 by regarding β 2 as a class in E Y. Indeed φ (β + β 2 ) = φ β + p β 2 = β and ((β + β 2 ).E) Y = (β.e) Y (β 2.E)Ẽ = µ µ = 0 (where N E/Ẽ = N E/Y is used). These characterize the class φ β.

24 24 Y.-P. LEE, H.-W. LIN, AND C.-L. WANG We consider only the case g = 0. Consider the generating series A ε, µ := A ε, µ (Ẽ,E) Aut µ β 2 q β 2. β 2 NE(Ẽ) and the similar one with possibly disconnected domain curves A ε, µ (Ẽ,E) := A ε, µ (Ẽ,E) Γ q βγ Γ; µ Γ =µ Aut Γ Proposition 4.6. To prove F α X = Fα X, it is enough to show that F A ε, µ = FA ε, µ. Proof. For the n-point function α X = β NE(X) α X β qβ we have α X = β NE(X) η Ω β I = µ I η Ω µ C η C η α e I, µ (Y,E) Γ α 2 e I, µ (Y 2,E) Γ 2 q φ β ( α e I, µ (Y,E) Γ q β ) ( ) α 2 e I, µ (Y 2,E) Γ 2 q β 2. To simplify the generating series, we consider also absolute invariants α X with possibly disconnected domain curves as before. Then by comparing the order of automorphisms, α X = µ m(µ) α e I, µ (Y,E) α 2 e I, µ (Y2,E). I To compare F α X and Fα X, by Lemma 4.4 we may assume that α = α and α 2 = Fα 2. By comparing with the similar expression for Fα X, the relative terms for (Y, E) are identical. It remains to compare α 2 e I, µ (Ẽ,E) and Fα 2 e I, µ (Ẽ,E). We further split the sum into connected invariants. Denote by Γ π a connected part. Γ π has its contact order µ π induced from µ. We consider partitions P : µ = π P µ π and denote by P(µ) the set of all such partitions. Then A ε, µ (Ẽ,E) = Γ Aut µ π Aπ ε π, µ π (Ẽ,E) Γ q π βγπ. π P P(µ) π P Notice that only β Γπ can vary in the sum over Γ π and we may denote the generating series of connected relative invariants as sum over β 2 NE(Ẽ). This reduces the problem to A π ε π, µ π and we are done. Remark 4.7. Here is a brief comment on the term F α 2 e I, µ (Ẽ,E) = β 2 NE(Ẽ) Aut µ α 2 e I, µ (Ẽ,E) β 2 q Fβ 2.

25 FLOPS, MOTIVES AND INVARIANCE OF QUANTUM RINGS 25 Since Ẽ is a projective bundle, NE(Ẽ) = i NE(Z) Z + γ with γ the fiber line class of Ẽ Z. The point is that, for β 2 NE(Ẽ) it is in general not true that Fβ 2 β 2 (in E) is effective in Ẽ. Indeed, for simple ordinary flops, let γ = δ, δ = γ be the two line classes in E = P r P r. It is easily checked that l δ γ in Ẽ. Hence l = l and γ = γ + l and β 2 = d l + d 2 γ = (d 2 d )l + d 2 γ. Fβ 2 NE(Ẽ ) if and only if d 2 d. Thus it is too optimistic to expect that α 2 e I, µ (Ẽ,E) = Fα 2 e I, µ (Ẽ,E) term by term. Analytic continuations are in general needed Relative to absolute. Now we shall combine a method of Maulik and Pandharipande ([7], Lemma 4) to further reduce the relative cases to the absolute cases with at most descendent insertions along E. Following [7], we call the pair (ε, µ) = {(ε, µ ),, (ε ρ, µ ρ )} with ε i H (E), µ i N a weighted partition, a partition of contact orders weighted by cohomology classes in E. Proposition 4.8. For simple ordinary flops Ẽ Ẽ, to prove F A ε, µ = FA ε, µ for any A and (ε, µ), it is enough to show that F A, τ k ε,..., τ kρ ε ρ = FA, τ k ε,..., τ kρ ε ρ for any possible insertions A H (Ẽ) n, k j N {0} and ε j H (E). (Here we abuse the notations by denoting ι 2 ε H (Ẽ) by the same symbol ε.) Proof. We apply the deformation to the normal cone for Z Ẽ to get W A. Then W 0 = Y Y 2 with Y = PE (O E (, ) O) a P bundle and Y 2 = Ẽ. Denote E 0 = E = Y Y 2 and E = E the infinity divisor of Y. Since ε i Z = 0, in the degeneration formula we may represent ε i (0) = (ι ε i, 0) so it must contributes to the Y side. As before the relative invariants on (Y, E) can be regarded as constants under F by Lemma 4.4. We will prove the proposition first by induction on (d 2 = µ, n, ρ) with ρ in the reverse ordering, and then on weighted partition (e I, µ) in the reverse ordering of the following order: For pairs in H (E) N, we define the size relation (ε, m ) > (ε, m ) m > m or m = m, deg ε > deg ε. Then (e I, µ ) > (e I, µ ) if in their size-decreasing sequences we have for the first size unequal term the one from µ is larger. It is clear that for any (d 2, n, ρ, (e I, µ)) there are only finitely many such data of lower order.