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Flops, motives and

invariance of quantum rings

By Yuan-Pin Lee, Hui-Wen Lin and Chin-Lung Wang

Abstract

For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincar´e pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended K¨ahler 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 0.1. 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 restric- tion map on the exceptional loci. Assume that

(i) ¯ψ equips Z with a Pr-bundle structure ¯ψ : Z = PS(F ) → S for some rank r + 1 vector bundle F over a smooth base S,

(ii) NZ/X|Zs = OPr(−1)⊕(r+1) for each ¯ψ-fiber Zs, s ∈ S.

It is not hard to see that the corresponding ordinary Pr flop f : X 99K X0 exists. An ordinary flop is called simple if S is a point.

For a Pr flop f : X 99K X0, the graph closure [¯Γf] ∈ A(X × X0) identifies the Chow motives ˆX of X and ˆX0 of X0. Indeed, let F := [¯Γf] then the transpose F is [¯Γf−1]. One has the following theorem.

Theorem 0.1. For an ordinary Pr flop f : X 99K X0, the graph closure F := [¯Γf] induces ˆX ∼= ˆX0 via F◦ F = ∆X and F ◦ F = ∆X0. In particular, F preserves the Poincar´e pairing on cohomology groups.

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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 sim- ple ordinary flops, after an analytic continuation over the extended K¨ahler moduli space.

A contraction (ψ, ¯ψ) : (X, Z) → ( ¯X, S) is of Mukai type if Z = PS(F ) → S is a projective bundle under ¯ψ and NZ/X = TZ/S . The corresponding algebraic flop f : X 99K X0 exists and its local 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 99K X0 be a Mukai flop. Then X and X0 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.

0.2. Motivations

This paper is the first of our study of the relationship between birational ge- ometry and Gromov–Witten theory. Our motivations come from both fields.

K-equivalence in birational geometry

Two (Q-Gorenstein) varieties X and X0 are K-equivalent if there exist bira- tional morphisms φ : Y → X and φ0 : Y → X0 with Y smooth such that

φKX = φ0∗KX0.

K-equivalent smooth varieties have the same Betti numbers ([1] [25], see also [26] for a survey on recent development). However, the cohomology ring struc- tures are in general different. Two natural questions arise here:

(1) 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 [24] and the third author [26] in response to these questions.

Conjecture 0.4. K-equivalent smooth varieties have canonically iso- morphic quantum cohomology rings over the extended K¨ahler moduli spaces.

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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.

Functoriality in Gromov–Witten theory

In the Gromov–Witten theory, one is led to consider the problem of functo- riality in quantum cohomology. Quantum cohomology is not functorial with respect to the usual operations: pull-backs, push-forwards, etc.. Y. Ruan [23]

has proposed to study the Quantum Naturality Problem: finding the “mor- phisms” in the “category” of symplectic manifolds for which the quantum cohomology is “natural”.

The main reason for lack of functoriality comes from the dimension count of the moduli of stable maps, where Gromov–Witten invariants are defined.

(See §3.1 for the relevant definitions.) For example, given a birational mor- phism f : Y → X, there is an induced morphism from moduli of maps to Y to moduli of maps to X. However, the (virtual) dimensions of the two moduli spaces are equal only if Y and X are K-equivalent. When the virtual dimen- sions of moduli spaces are different, the non-zero integral on moduli space of maps to X will be “pulled-back” to a zero integral on moduli space of maps to Y . Therefore, K-equivalence appears to be a necessary condition for this type of functoriality. Conjecture 0.4 suggests that the K-equivalence is also sufficient. We note here that there is of course no K-equivalent morphism between smooth varieties and a “flop-type” transformation is needed.

Theorem 0.2 can therefore be considered as establishing some functoriality of the genus zero Gromov–Witten theory in this direction. The higher genus case will be discussed in a separate paper.

Crepant resolution conjecture

Conjecture 0.4 can also be interpreted as a consistency check for the Crepant Resolution Conjecture [24] [3]. 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 0.4.

0.3. Contents of the paper

§1 studies the geometry of ordinary flops. The existence of ordinary flops is proved and explicit description of local models is given.

§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 0.1 alluded above. The ring structure is, however, not preserved. For

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a simple Pr-flop, let h be the hyperplane class of Z = Pr and let αi ∈ H2li(X), with li ≤ r and l1+ l2+ l3 = dim X = 2r + 1.

Proposition 0.5.

(Fα1.Fα2.Fα3) = (α123) + (−1)r1.hr−l1)(α2.hr−l2)(α3.hr−l3).

For Calabi-Yau threefolds under a simple P1 flop, it is well known in the context of string theory (see e.g. [28]) 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 [21] where Conjecture 0.4 for Calabi-Yau threefolds was proposed. For threefolds Conjecture 0.4 was proved by A. Li and Y. Ruan [15].

Their proof has three ingredients:

(1) A symplectic deformation and decomposition of K-equivalent maps into composite of ordinary P1 flops,

(2) the multiple cover formula for P1∼= C ⊂ X with NC/X ∼= O(−1)⊕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 = Pr⊂ X with NZ/X ∼= O(−1)r+1. Let ` be the line class in Z. Then for all αi ∈ H2li(X) with 1 ≤ li ≤ r,Pn

i=1li = 2r+1+(n−3) and d ∈ N,

1, . . . , αni0,n,d Z

[M0,n(X,d`)]virt

e1α1· · · enαn

= (−1)(d−1)(r+1)Nl1,...,lndn−31.hr−l1) · · · (αn.hr−ln).

where Nl1,...,ln are recursively determined universal constants. Nl1,...,ln are in- dependent of d and Nl1,...,ln = 1 for n = 2 or 3. All other (primary) Gromov- Witten invariants with degree in Z` vanish.

This formula, together with some algebraic manipulations, implies that for simple Pr 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(qβ) = q.

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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 sum of negative line bundles, this was carried out in [16] and [7] in the context of the study of mirror symmetry. The divisor relations studied in [13] 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 hαiX in terms of relative invariants hα1i(Y,E) and hα2i( ˜E,E), where Y → X is the blow-up of X over Z and ˜E = PZ(NZ/X⊕ O). Similarly for X0, one has Y0, ˜E0, E0. By definition of ordinary flops, Y = Y0 and E = E0. It is possible to match all output on the part of (Y, E) from X and X0. Thus, the problem is transformed to one for the relative cases of ( ˜E, E) and ( ˜E0, E). Following ideas in the work of D. Maulik and R. Pandharipande [20], a further reduction from relative invariants to absolute invariants is made. The problem is thus reduced to

X = ˜E = PPr(O(−1)⊕(r+1)⊕ 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 ˜E 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 compat- ibility of functional equations of n-point functions under the reconstruction procedure of genus zero invariants. It is easy to see that the Mori cone

N E(X) = Z+` ⊕ Z+γ

with ` the line class in Z and γ the fiber line class of X = ˜E → Z. The proof is based on an induction on d2 and n with degree β = d1` + d2γ. The case d2 = 0 is handled by Theorem 0.6. For d2 > 0, the starting case, namely the one-point invariant, is again based on localization technique on semi-Fano toric manifolds [7] and [17].

Theorem 0.8 (Functional equations for local models). Consider an n- point function on X = PPr(O(−1)⊕(r+1)⊕ O),

hαi = X

β∈N E(X)

1, . . . , αniβqβ

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where αi lies in the span of cohomology classes in X and descendents of (push- forward of) cohomology classes in E. For β = d1` + d2γ, the summands are non-trivial only for a fixed d2. If d26= 0 then

FhαiX ∼= hFαiX0.

(Here ∼= stands for equality up to analytic continuations.) Combining all the previous results Theorem 0.2 is proved.

Remark 0.9. Concerning ingredient (1), it is very important to under- stand the closure of ordinary flops. To the authors’ knowledge, no serious attempt was made toward a higher dimensional version of (1) except some much weaker topological results [27]. Even in dimension three, the only known proof of (1) 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 §1, we discuss (twisted) 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-K¨ahler manifolds [9], with the flexibility of allowing the base S to be any smooth variety. As in the hyper-K¨ahler case, it also implies that the correspondence induced by the fiber product

[X ×X¯ X0] = [¯Γf] + [Z ×SZ0] ∈ A(X × X0) is the one which gives an isomorphism of Chow motives.

Besides dimension three [15] and the hyper-K¨ahler case [9], our results provide the first known series of examples in all high dimensions which support Conjecture 0.4.

0.4. Acknowledgements

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. W. is grateful to C.-S. Lin and J. Yu for their encouragement.

We are grateful to the National Center for Theoretic Sciences (NCTS, Taiwan) for providing stimulating and delightful environment which makes the collaboration possible.

Y.-P. L. is partially supported by NSF and AMS Centennial Fellowship.

H.-W. L. is partially supported by NSC. C.-L. W. is partially supported by NSC and the NCTS Chern Fellowship.

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1. Ordinary flops 1.1. Ordinary Pr flops.

Let ψ : X → ¯X be a flopping contraction as defined in §0.1. Our first task is to show that the corresponding algebraic ordinary flop X 99K X0 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 Pr× Pr-bundle over S. The key point is that one may blow down E along another fiber direction φ0 : Y → X0, with exceptional loci ¯ψ0 : Z0 = PS(F0) → S for F0 another rank r + 1 vector bundle over S and also NZ0/X0|ψ¯0−fiber= OPr(−1)⊕(r+1). We start with the following elementary lemma.

Lemma 1.1. Let p : Z = PS(F ) → S be a projective bundle over S and V → Z a vector bundle such that V |p−1(s) is trivial for every s ∈ S. Then V ∼= pF0 for some vector bundle F0 over S.

Proof. Recall that Hi(Pr, O) is zero for i 6= 0 and H0(Pr, O) ∼= C. By the theorem on Cohomology and Base Change we conclude immediately that pO(V ) is locally free over S of the same rank as V . The natural map between locally free sheaves ppO(V ) → O(V ) induces isomorphisms over each fiber and hence by the Nakayama Lemma it is indeed an isomorphism. The desired F0 is simply the vector bundle associated to pO(V ).

Now apply the lemma to V = OPS(F )(1) ⊗ NZ/X, and we conclude that NZ/X = OPS(F )(−1) ⊗ ¯ψF0.

Therefore, on the blow-up φ : Y = BlZX → X, NE/Y = OPZ(NZ/X)(−1).

¿From the Euler sequence which defines the universal sub-line bundle we see easily that OPZ(L⊗F )(−1) = ¯φL ⊗ OPZ(F )(−1) for any line bundle L over Z.

Since the projectivization functor commutes with pull-backs, we have E = PZ(NZ/X) ∼= PZ( ¯ψF0) = ¯ψPS(F0) = PS(F ) ×SPS(F0).

For future reference we denote the projection map Z0 := PS(F0) → S by ¯ψ0 and E → Z0 by ¯φ0. The various sets and maps are summarized in the following commutative diagram.

E = PS(F ) ×SPS(F0) ⊂ Y

φ¯

ttiiiiiiiiiiiiiiii φ¯0

))TT TT TT TT TT TT TT T

Z = PS(F ) ⊂ X

ψ¯

**UU UU UU UU UU UU UU UU

UU Z0 = PS(F0)

ψ¯0

ttjjjjjjjjjjjjjjjj S ⊂ ¯X

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with normal bundle of E in Y being

NE/Y = OPZ(NZ/X)(−1) = OP

Z(OZ(−1)⊗ ¯ψF0)(−1)

= ¯φOPS(F )(−1) ⊗ OP

Z( ¯ψF0)(−1)

= ¯φOPS(F )(−1) ⊗ ¯φ0∗OPS(F0)(−1).

Remark 1.2. Notice that the bundles F and F0 are uniquely determined up to a twisting by a line bundle. Namely, the pair (F, F0) is equivalent to (F ⊗ L, F0⊗ L) for any line bundle L on S.

The next step is to show that there is a blow-down map φ0 : Y → X0 which contracts the left ruling of E and restricts to the projection map ¯φ0: E → Z0. The existence of the contraction ψ : X → ¯X is essential here. Let us denote a line in the left ruling by CY such that φ(CY) = C.

Proposition 1.3. Ordinary Pr flops exist.

Proof. Firstly, we will show that CY is KY-negative. From the ex- act sequence 0 → TC → TX|C → NC/X → 0 and NC/X = OC(1)⊕(r−1) OC(−1)⊕(r+1)⊕ Odim SC , we find that

(KX.C) = 2g(C) − 2 − ((r − 1) − (r + 1)) = 0.

Together with KY = φKX+ rE, we get

(KY.CY) = (KX.C) + r(E.CY) = −r < 0.

Next we will show CY 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 = φH for an ample divisor ¯¯ H on ¯X). Let c = (H.C), then φH + cE has type (0, −c) on each Pr× Pr fiber of E. The divisor

L − (φH + cE)

is clearly big and nef for large k and vanishes precisely on the class [CY]. Thus CY is a KY-negative extremal ray and the contraction morphism φ0: Y → X0 fits into

Y φ

0 //

ψ◦φ@@@@@@ÂÂ@ X0

ψ0

~~}}}}}}}}

X¯

by the cone theorem on Y → ¯X (c.f. [11]). X 99K X0 is then the desired flop.

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Remark 1.4. Notice that (KX.C) = 0, (KX0.C0) = 0 (C0 is a line in the fiber of Z0 → S) and φKX = φ0∗KX0 (K-equivalence).

It is clear from the proof that for the existence of φ0 one needs only the (weaker) assumption that C is extremal instead of the existence of the contrac- tion ψ : X → ¯X. However, since (KX.C) = 0 these two are indeed equivalent by the cone theorem.

1.2. Local models

In general, without assuming the existence of ψ, (i) and (ii) are not sufficient to construct φ0 in the projective category. This is well known already in the case of Atiyah flop (r = 1 and S = {pt}). In the analytic category results of Cornalba [4] do imply the contractibility of ψ, φ0 and ψ0 hence lead to the existence of analytic ordinary Pr flops under (i) and (ii). The situation is particularly simple in the case of local models which we now describe.

Consider a complex manifold S and two holomorphic vector bundles F → S and F0 → S. Let ¯ψ : Z := PS(F ) → S and ¯ψ0 : Z0 := PS(F0) → S be the induced morphisms and let E = PS(F )×SPS(F0) with two projections ¯φ : E → Z and ¯φ0 : E → Z0. Let Y be the total space of N := ¯φOZ(−1) ⊗ ¯φ0∗OZ0(−1) with E the zero section. It is clear that NE/Y = N . There is a contraction diagram

E

π1= ¯φ

}}|||||||| CCC

φ¯02

C!!C C Â Ä j // Y

φ

}}{{{{{{{{

φ0

D!!D DD DD DD

Z

ψ¯

BÃÃB BB BB BB Â Ä i // X

ψ

B!!B BB BB BB Z0

|||

ψ¯0

}}|||

Â Ä i0 // X0

ψ0

}}{{{{{{{{

S Â Ä j0 // ¯X

in the analytic category, with X (resp. X0) being the total space of OPS(F )(−1)⊗

ψ¯F0 (resp. OPS(F0)(−1) ⊗ ¯ψ0∗F ).

First of all, the discussion in §1.1 implies that φ and φ0 are simply the blow-up maps along Z and Z0 respectively. For ψ and ψ0, when S reduces to a point the existence of contraction morphism g : (Y, E) → ( ¯X, pt) is a classical result of Grauert since NE/Y is a negative line bundle. From the universal property the induced maps ψ and ψ0 are then analytic. For S a small Stein open set, g : (Y, E) → ( ¯X, S), as well as ψ and ψ0, also exists 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. In summary the local analytic model of an ordinary Pr flop is a locally trivial family (over S) of simple ordinary Pr flops.

It is convenient to consider compactified local models ˜X, ˜Y etc. by adding the common infinity divisor E∼= E to X, Y etc. respectively. Denote by

p : ˜X = PZ(NZ/X⊕ OZ) → Z.

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Proposition 1.5. If S is projective, for any bundles F , F0 of rank r + 1 the compactified local models of Pr flops exist in the projective category.

Proof. ˜X is clearly projective and E is p-ample. By Remark 1.4 and Proposition 1.3 we only need to construct a supporting divisor L for the fiber line of ¯ψ : Z → S. Let H be ample in S then ¯ψH is a supporting divisor for the fiber line in Z. Hence we may take L := pψ¯H + E.

The projective local models will be used extensively in §4–§6.

2. Correspondences and motives 2.1. Grothendieck’s category of Chow motives

General references of Chow motives can be found in [19] and [6]. Let M be the category of Chow motives (over C). For each smooth variety X, one associates an object ˆX in M. The morphisms are given by correspondences

HomM( ˆX1, ˆX2) = A(X1× X2).

For U ∈ A(X1× X2), V ∈ A(X2× X3), let pij : X1× X2× X3 → Xi× Xj be the projection maps. The composition law is given by

V ◦ U = p13∗(p12U.p23V ).

A correspondence U has associated maps on Chow groups:

U : A(X1) → A(X2); a 7→ p2∗(U.p1a) as well as induced maps on T -valued points Hom( ˆT , ˆXi):

UT : A(T × X1)−→ AU ◦ (T × X2).

Then we have Manin’s identity principle: Let U, V ∈ Hom( ˆX, ˆX0). Then U = V if and only if UT = VT for all T . (Since U = UX(∆X) = VX(∆X) = V .) Theorem 2.1. For an ordinary Pr flop f : X 99K X0, the graph closure F := [¯Γf] induces ˆX ∼= ˆX0 via F◦ F = ∆X and F ◦ F = ∆X0.

Proof. For any T , idT × f : T × X 99K T × X0 is also an ordinary Pr flop.

Hence to prove that F◦ F = ∆X, by the identity principle, we only need to show that FF = id on A(X) for any ordinary Pr flop. From the definition of pull-back,

FW = p0(¯Γf.pW ) = φ0φW.

We also have the formulae for pull-back from the intersection theory (c.f. [6], Theorem 6.7, Blow-up formula):

φW = ˜W + j¡

c(E). ¯φs(W ∩ Z, W )¢

dim W

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where ˜W is the proper transform of W in Y and E is the excess normal bundle defined by

(2.1.1) 0 → NE/Y → φNZ/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 ∈ Ak(X). By Chow’s moving lemma we may assume that W intersects Z transversally, so

` := dim W ∩ Z = k + (r + s) − (r + r + s + 1) = k − r − 1.

Since dim φ−1(W ∩ Z) = ` + r = k − 1 < k, the error term in the pull-back formula must be zero and we get φW = ˜W . Hence FW = W0, the proper transform of W in X0. Notice that W0 is almost never transversal to Z0.

Let B be an irreducible component of W ∩ Z and ¯B = ¯ψ(B) ⊂ S with dimension `B ≤ `. Notice that W0 ∩ Z0 has irreducible components {B0 :=

ψ¯0−1( ¯B)}B0 (different B with the same ¯B will give rise to the same B0).

Let φ0∗W0 = ˜W +P

EB0, where EB0 varies over irreducible components lying over B0, hence EB0 ⊂ ¯φ0−1ψ¯0−1( ¯B), a Pr× Pr bundle over ¯B. For the generic point s ∈ ψ(φ(EB0)) ⊂ ¯B, we thus have

dim EB0,s≥ k − `B= r + 1 + (` − `B) > r.

In particular, EB0,s contains positive dimensional fibers of φ (as well as φ0).

Hence φ(EB0) = 0 and FFW = W .

By the same argument we have also that F ◦ F = ∆X0, thus the proof is completed.

Remark 2.2. For a general ground field k, if the flop diagram under con- sideration is defined over k then the theorem works for motives over k.

Corollary 2.3. Let f : X 99K X0 be a Pr flop. If dim α1+ dim α2 = dim X, then

(Fα1.Fα2) = (α12).

That is, F is an isometry with respect to (−.−).

Proof. We may assume that α1, α2 are transversal to Z. Then 12) = (φα1α2) = ((φ0∗1− ξ).φα2)

= ((φ0∗1).φα2) = (Fα1.(φ0φα2)) = (Fα1.Fα2).

Here we use the fact proved in the above theorem that ξ has positive fiber dimension in the φ direction.

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Thus for ordinary flops, F−1 = F both in the sense of correspondences and Poincar´e pairing.

Remark 2.4. It is an easy fact that if X =KX0 then X and X0 are iso- morphic in codimension one and in particular the graph closure gives canonical isomorphisms F on A1(X) ∼= A1(X0) and A1(X) ∼= A1(X0) respectively. In this more general setting, the above proof still implies that the Poincar´e pairing on A1× A1 (and H2× H2) is preserved under F.

2.2. Triple product for simple flops

Let f : X 99K X0 be a simple Pr flop with S being a point. Let h be the hyperplane class of Z = Pr and h0 be the hyperplane class of Z0. Let also x = ¯φh = [h × Pr], y = ¯φ0∗h0 = [Pr× h0] in E = Pr× Pr.

Lemma 2.5. For classes inside Z, we have

φ[hl] = j(xlyr− xl+1yr−1+ · · · + (−1)r−lxryl).

Hence by symmetry we get F[hl] = (−1)r−l[h0l]. In particular, F[C] = −[C0].

Proof. Recall that

NE/Y = OPr×Pr(−1, −1) := ¯φOPr(−1) ⊗ ¯φ0∗OPr(−1) and NZ/X = OPr(−1)⊕(r+1). ¿From (2.1.1),

c(E) = (1 − x)r+1(1 − x − y)−1. Taking degree r terms from both sides, we have

cr(E) = [(1 − x)r+1(1 − (x + y))−1](r)

= (x + y)r− C1r+1(x + y)r−1x + · · · + (−1)rCrr+1xr

= (x + y)−1((x + y) − x)r+1− (−1)r+1xr+1)

= (yr+1− (−1)r+1xr+1)/(y + x)

= yr− yr−1x + yr−2x2− · · · + (−1)rxr.

The basic pull-back formula ([6], Proposition 6.7) then implies that φ[hl] = j(cr(E). ¯φ[hl]) = j(cr(E).xl) = jXr

t=0(−1)tyr−txt+l. If t + l ≥ r + 1 then yr−txt+l= 0. The result follows.

Lemma 2.6. For a class α ∈ H2l(X) with l ≤ r, let α0= Fα in X0. Then φ0∗α0 = φα + (α.hr−l) jxl− (−y)l

x + y .

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Proof. Since the difference φ0∗α0− φα has support in E, we may write φ0∗α0 = φα + j(a1xl−1+ · · · + akxl−kyk−1+ · · · + alyl−1).

By intersecting this equation with xr−lyr in X and noticing that E ∼ −(x + y) on E, we get by the projection formula

0 = φα.xr−lyr− a1xl−1(x + y)xr−lyr= (α.hr−l) − a1. Similarly by intersecting with xr−l+1yr−1 we get

0 = −a1xl−1(x + y)xr−l+1yr−1− a2xl−2(x + y)xr−l+1yr−1= −a1− a2. Continuing in this way by intersecting with xpyq with p + q = 2r − l we get ak= (−1)k−1(α.hr−l) for all k = 1, . . . , l. This proves the lemma.

These formulae allow us to compare the triple products of classes in X and X0:

Proposition 2.7. For a simple Pr-flop f : X 99K X0, let αi ∈ H2li(X), with li≤ r, l1+ l2+ l3 = dim X = 2r + 1. Then

(Fα1.Fα2.Fα3) = (α123) + (−1)r1.hr−l1)(α2.hr−l2)(α3.hr−l3).

Proof. The proof consists of straightforward computations.

(Fα1.Fα2.Fα3) = (φ0∗10∗20∗3)

= µ

φα1+ (α1.hr−l1)jxl1− (−y)l1 x + y

¶ µ

φα2+ (α2.hr−l2)jxl2− (−y)l2 x + y

× µ

φα3+ (α3.hr−l3)jxl3− (−y)l3 x + y

.

Among the resulting eight terms, the first term is clearly equal to α123. For those three terms with two pull-backs like φα1α2, the intersection values are zero since the remaining part necessarily contains the φ fiber (from the formula the power in y is at most l3− 1).

The term with φα1 and two exceptional parts contributes φα1.jxl2− (−y)l2

x + y .jxl3− (−y)l3 x + y

= −φα1.j¡

(xl2− (−y)l2)(xl3−1+ xl3−2(−y) + · · · + (−y)l3−1times (α2.hr−l2)(α3.hr−l3). The terms with non-trivial contribution must con- tain yr, hence there is only one such term, namely (notice that l1+ l2+ l3 = 2r + 1)

−(−y)l2× xl3−1−(r−l2)(−y)r−l2 = −(−1)rxr−l1yr

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and the contribution is (−1)r1.hr−l1)(α2.hr−l2)(α3.hr−l3). There are three such terms.

It remains to consider the term of triple product of three exceptional parts.

It is (α1.hr−l1)(α2.hr−l2)(α3.hr−l3) times

(xl1− (−y)l1)(xl2− (−y)l2)(xl3−1+ xl3−2(−y) + · · · + (−y)l3−1).

The terms with non-trivial values are precisely multiples of xryr. Since l1+l2>

r, there are two such terms

−xl1(−y)l2× xr−l1(−y)l3−1−(r−l1)− xl2(−y)l1× xr−l2(−y)l3−1−(r−l2) which give −2(−1)r. Summing together we then finish the proof.

2.3. Motives and ordinary flips

Results in §1 and §2 extend straightforwardly to the case of ordinary flips.

Before we move to quantum corrections for ordinary flops, we shall summarized here the classical aspects, especially the motivic aspects, of ordinary flips. The proofs are identical with the flop case and are thus omitted.

Consider (ψ, ¯ψ) : (X, Z) → ( ¯X, S) a log-extremal contraction as before.

ψ is an ordinary (r, r0) flipping contraction if

(i) Z = PS(F ) for some rank r + 1 vector bundle F over S, (ii) NZ/X|Zs = OPr(−1)⊕(r0+1) for each ¯ψ-fiber Zs, s ∈ S.

Then the (r, r0) flip f : X 99K X0 exists with explicit local model as in §1.2.

In terms of the K-partial order within a birational class, X ≤K X0 if and only if r ≤ r0. For f a (r, r0) flip with r ≤ r0, the graph closure F = [¯Γf] ∈ A(X × X0) identifies the Chow motive ˆX of X as a sub-motive of ˆX0 which preserves also the Poincar´e pairing on cohomology groups.

More precisely, a self correspondence p ∈ A(X × X) is a projector if p2 = p. There is a natural pseduo-abelian extension ˜M of M to include all pairs (X, p) as its objects. (X, p) is regarded as the image of p. Moreover, X = (X, p) ⊕ (X, 1 − p) in ˜ˆ M. With this notion, for an ordinary (r, r0) flip f : X 99K X0 with r ≤ r0, the graph closure F := [¯Γf] induces ˆX ∼= (X0, p0) via F◦ F = ∆X, where p0 = F ◦ F is a projector.

Since every geometric cohomology theory (a graded ring functor H with Poincar´e duality, K¨unneth formula and a cycle map A → H etc.) factors through ˜M, the result also holds on such a specialized theory.

For simple (r, r0) flips (i.e. S = pt) with l ≤ min{r, r0},

φ[hr−l] = j(xr−lyr0 − xr−l+1yr0−1+ · · · + (−1)lxryr0−l).

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In particular F[hr−l] = (−1)l[h0r0−l]. For α ∈ Al(X) with l ≤ min{r, r0}, φ0∗Fα = φα + (α.hr−l) jxl− (−y)l

x + y .

Let αi ∈ H2li(X), 1 ≤ i ≤ 3 with li ≤ min{r, r0}, l1+ l2+ l3 = dim X = r + r0+ 1. The defect of the triple product is again given by

(Fα1.Fα2.Fα3) = (α123) + (−1)r01.hr−l1)(α2.hr−l2)(α3.hr−l3).

3. Quantum corrections attached to extremal rays

Proposition 2.7 on triple products 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.

3.1. Quantum cohomology

We use [5] as our general reference on moduli spaces of stable maps, Gromov- Witten theory and quantum cohomology.

Let β ∈ N E(X), the Mori cone of numerical classes of effective one cycles. Let Mg,n(X, β) be the moduli space of n-pointed stable maps f : (C; x1, . . . , xn) → X from a nodal cure C with arithmetic genus g(C) = g and with degree [f (C)] = β. Let ei: Mg,n(X, β) → X be the evaluation morphism f 7→ f (xi). The Gromov-Witten invariant for classes αi ∈ H(X), 1 ≤ i ≤ n, is given by

1, . . . , αnig,n,β :=

Z

[ ¯Mg,n(X,β)]virt

e1α1· · · enαn. The genus zero three-point functions (as formal power series)

1, α2, α3i :=X

β∈A1(X)1, α2, α3i0,3,βqβ

together with the Poincar´e pairing (−, −) determine the small quantum prod- uct.

More precisely, let T =P

tiTi with {Ti} a cohomology basis and ti being formal variables. Let {Ti} be the dual basis with (Ti, Tj) = δij. The (genus zero) pre-potential combines all n-point functions together:

Φ(T ) =X

n=0

X

β∈N E(X)

1

n!hTniβ qβ, where ­

Tn®

β = hT, . . . , T i0,n,β. The big quantum product is defined by TitTj =X

kΦijkTk

參考文獻

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