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 Poincar´e pairing. In the case of simple ordinary flops, this correspon-dence preserves the big quantum cohomology ring after an analytic con-tinuation over the extended K¨ahler moduli space.
For Mukai flops, it is shown that the birational map for the local mod-els is deformation equivalent to isomorphisms. This implies that the bi-rational 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 man-ifold 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) ¯ψequips Z with aPr-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 ordinaryPr flop f : X 99K X0
exists. An ordinary flop is called simple if S is a point.
For aPrflop f : X 99KX0, the graph closure[Γ¯f] ∈ A∗(X×X0)identifies the Chow motives ˆX of X and ˆX0 of X0. Indeed, let F := [Γ¯
f] then the
transposeF∗ is[Γ¯f−1]. One has the following theorem.
Theorem 0.1. For an ordinaryPrflop f : X 99KX0, the graph closureF := [Γ¯f] induces ˆX ∼= Xˆ0 viaF∗◦F = ∆X andF◦F∗ = ∆X0. In particular,F preserves the Poincar´e pairing on cohomology groups.
While the ring structure is in general not preserved underF, the quan-tum 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
ordi-nary 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 NZ/X = TZ/S∗ .
The corresponding algebraic flop f : X 99KX0 exists and its local analytic
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 99KX0 be a Mukai flop. Then X and X0 are diffeomor-phic, 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 geometry and Gromov–Witten theory. Our motivations come from both fields.
In birational geometry, two varieties X and X0 are K-equivalent if there exist birational morphisms φ : Y →X and φ0 : Y→X0such that
φ∗KX= φ0∗KX0.
K-equivalent smooth varieties have the same Betti numbers ([1] [22], see also [23] for a survey on recent development). However, the cohomology ring structures 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 [21] and the third author [23] in response to these questions.
Conjecture 0.4. K-equivalent smooth varieties have canonically isomorphic
quan-tum cohomology rings over the extended K¨ahler 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 func-toriality in quantum cohomology. Quantum cohomology is not functor-ial with respect to the usual operations: pull-backs, push-forwards, etc.. Y. Ruan [20] has proposed to study the Quantum Naturality Problem: find-ing 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 [21]. In general, there are more than one pos-sible 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. Section 1 studies the geometry of ordinary flops. The existence of ordinary flops is proved and explicit description of local analytic models is given.
Section 2 is devoted to the correspondences and Chow motives of pro-jective 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 a simplePr-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) = (α1.α2.α3) + (−1)r(α1.hr−l1)(α2.hr−l2)(α3.hr−l3).
For Calabi-Yau threefolds under a simpleP1flop, 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 [18] where Conjecture 0.4 for Calabi-Yau threefolds was proposed. For threefolds Conjecture 0.4 was proved by A. Li and Y. Ruan [12]. Their proof has three ingredients:
(1) A symplectic deformation and decomposition of K-equivalent maps into composite of ordinaryP1flops,
(2) the multiple cover formula forP1∼= C⊂ X with NC/X ∼=O(−1)⊕2,
and their main contribution:
(3) the theory of relative Gromov-Witten invariants and the degenera-tion 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 and∑ni=1li =2r+1+ (n−3),
hα1, . . . , αni0,n,d ≡ Z [M0,n(X,d`)]virt e∗1α1· · ·e∗nαn = (−1)(d−1)(r+1)Nl1,...,lnd n−3( α1.hr−l1) · · · (αn.hr−ln).
where Nl1,...,ln are recursively determined universal constants. Nl1,...,ln are
inde-pendent of d and Nl1,...,ln = 1 for n=2 or 3. All other (primary) Gromov-Witten
invariants with degree inZ`vanish.
This proposition, together with some algebraic manipulations, implies that for simplePr 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) =qFβ.
The proof has two ingredients: Localization and the divisor relations. Localization has been widely used in calculating Gromov–Witten invari-ants. For genus zero one-pointed descendent invariants twisted by a direct
sum of negative line bundles, this was carried out in [13] and [4]. The divi-sor relations studied in [10] 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 invariantshα1i(Y,E) andhα2i(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(E˜0, E). Following ideas in the work of D. Maulik and R. Pandharipande [17], 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 invari-ants. 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 compati-bility 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+` ⊕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 d2and n with degree β=d1` +d2γ. The
case d2 = 0 is Proposition 0.6. For d2 > 0, the starting case, namely the
one-point invariant, is again based on localization technique on semi-Fano toric manifolds [4] and [14].
Theorem 0.8(Functional equations for local models). Consider an n-point function on X=PPr(O(−1)⊕(r+1)⊕O),
hαi =
∑
β∈NE(X)
hα1, . . . , αniβq β
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 d2 6=0 then
FhαiX ∼= hFαiX
0 .
(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 understand 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 [24]. Even in dimension three, the only known proof of (1) relies on the minimal model theory and classifications of ter-minal singularities. It is desirable to have a direct proof in the symplectic category. Such a proof should shed important light toward the higher di-mensional 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 dis-cuss (generalized) Mukai flops in §6. Some new understanding of the local geometry of Mukai flops is presented and this leads to a proof of Theo-rem 0.3. TheoTheo-rem 0.3 can also be interpreted as a generalization of a local version of Huybrechts’ results on hyper-K¨ahler manifolds [6], with the flex-ibility of allowing the exceptional fiber Zs ∼= Pr to be any projective space
including the odd dimensional ones and the base S to be any smooth va-riety. As in the hyper-K¨ahler case, it also implies that the correspondence induced by the fiber product
[X×X¯ X0] = [Γ¯f] + [Z×SZ0] ∈A∗(X×X0)
is the one which gives an isomorphism of Chow motives.
Besides dimension three [12] and the hyper-K¨ahler case [6], our results provide the first known series of examples in all high dimensions which support Conjecture 0.4.
0.4. 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 provid-ing stimulatprovid-ing and delightful environment which makes the cooperation possible.
1. ORDINARY FLOPS
1.1. OrdinaryPr flops. Let ψ : X → X be a flopping contraction as de-¯
fined 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 aPr×Pr-bundle over S. The key point is that one
may blow down E along another fiber direction φ0 : Y → X0, with excep-tional loci ¯ψ0 : Z0 = PS(F0) → S for F0 another rank r+1 vector bundle
over S and also NZ0/X0|ψ0−fiber ∼= OPr(−1)⊕(r+1). We start with the
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 ∼= p∗F0 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 p∗O(V)is locally free over S of the same rank as V. The natural map
be-tween 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 F0 is simply the vector bundle associated to p∗O(V).
Now apply the lemma to V=OP
S(F)(1) ⊗NZ/X, and we conclude that
NZ/X ∼=OPS(F)(−1) ⊗ψ¯∗F0.
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) = φ¯∗L⊗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(ψ¯∗F0) =ψ¯∗PS(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 ttiiiiiiiiii iiiiii φ¯0 ))T T T T T T T T T T T T T T T Z=PS(F) ⊂X ¯ ψ UUUUUUU **U U U U U U U U U U U Z0=PS(F0) ¯ ψ0 uujjjjjjjjjj jjjjjj S⊂X¯
with normal bundle of E in Y being NE/Y =OP
Z(NZ/X)(−1) =OPZ(OZ(−1)⊗ψ¯∗F0)(−1)
=φ¯∗OPS(F)(−1) ⊗OPZ(ψ¯∗F0)(−1)
=φ¯∗OPS(F)(−1) ⊗φ¯0∗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 CYsuch that φ(CY) =C.
Proposition 1.3. OrdinaryPrflops exist.
Proof. Firstly, we will show that CYis KY-negative. From the exact sequence
0 → TC → TX|C → NC/X → 0 and NC/X ∼= OC(1)⊕(r−1)⊕OC(−1)⊕(r+1),
we find that
(KX.C) =2g(C) −2− ((r−1) − (r+1)) =0.
Together with KY =φ∗KX+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=φ∗H for an ample divisor ¯¯ H on ¯X). Let c= (H.C), then φ∗H+cE has type(0,−c)on eachPr×Prfiber of E. The divisor
kφ∗L− (φ∗H+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→X0fits into Y φ 0 // ψ◦φ >> > > > > > X0 ψ0 ~~~~ ~~~ ¯ X
by the cone theorem on Y → X (c.f. [8]). X¯ 99K X0 is then the desired
flop.
Remark 1.4. Notice that(KX.C) =0,(KX0.C0) =0 (C0 is a line in the fiber of Z0 →S) and φ∗KX=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 con-traction ψ : X → X. However, since¯ (KX.C) = 0 these two are equivalent
by the cone theorem.
In general (i) and (ii) are not enough to construct φ0, a well known phe-nomenon already in the case of Atiyah flop (r = 1 and S = {pt}). In the analytic category the situation is better behaved, and will be demonstrated in the next subsection.
1.2. 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 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 := φ¯∗OZ(−1) ⊗φ¯0∗OZ0(−1)and we identify E as its zero section. It is clear that NE/Y = N.
We intend to show unconditionally a contraction diagram E π1= ¯φ ~~}}}}}} }} BBB ¯ φ0=π2 !!B B B j // Y φ }}|||||| || φ0 !!C C C C C C C C Z ¯ ψ @@@ @ @ @ @ @ i // X ψ A A A A A A A A Z0 }}}¯ ψ0 ~~}}} i0 // X0 ψ0 }}|||||| || S j 0 //X
in the analytic category, with X (resp. X0) being the total space ofOPS(F)(−1) ⊗
¯
ψ∗F0 (resp. OPS(F0)(−1) ⊗ψ¯0∗F). As we have mentioned before, two pairs
of bundles (F, F0)and(F1, F10)define the same analytic local model if and
only if(F1, F10) = (F⊗L, F0⊗L∗)for some line bundle L on S.
Indeed, when S reduces to a point this is a standard result in complex geometry since NE/Yis 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 ordinaryPrflop is a locally trivial family of simple ordinaryPrflops.
Proposition 1.5. Conditions (i) and (ii) imply the analytic contractibility of ψ
and the existence of analytic ordinaryPrflops.
2. CORRESPONDENCES AND MOTIVES
2.1. Grothendieck’s category of Chow motives. General references of Chow motives can be found in [16] and [3] 16.1.12.
LetMbe category of motives (overC). For each smooth variety X, one associates an object ˆX inM. The morphisms are given by correspondences
HomM(Xˆ1, 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∗(p∗12U.p∗23V).
A correspondence U has associated maps on Chow groups: U : A∗(X1) →A∗(X2); a7→ p2∗(U.p∗1a)
as well as induced maps on T-valued points Hom(T, ˆˆ Xi):
UT : A∗(T×X1) U
◦
−→A∗(T×X2).
Then we have Manin’s identity principle: Let U, V ∈ Hom(X, ˆˆ X0). Then U =V if and only if UT = VT for all T. (Since UX◦∆X =VX◦∆X0 implies U=V.)
Theorem 2.1. For an ordinaryPrflop f : X 99KX0, the graph closureF := [Γ¯f] induces ˆX∼=Xˆ0viaF∗◦F=∆XandF◦F∗ =∆
Proof. For any T, idT × f : T×X 99K T×X0 is also an ordinary Pr flop.
Hence to prove thatF∗◦F= ∆X, by the identity principle, we only need to
show thatF∗F=id on A∗(X)for any ordinaryPrflop. From the definition of pull-back,
FW = p0∗(Γ¯f.p∗W) =φ∗0φ∗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
(2.1) 0→NE/Y →φ∗NZ/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 φ∗W =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 :=
¯
ψ0−1(B¯)}B0(different B with the same ¯B will give rise to the same B0). Let φ0∗W0 = W˜ +∑ EB0, where EB0 varies over irreducible components lying over B0, hence EB0 ⊂ φ¯0−1ψ¯−1(B¯), aPr×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,scontains positive dimensional fibers of φ (as well as φ0). Hence φ∗(EB) =0 andF∗FW =W.
By the same argument we have also thatF◦F∗ = ∆X0, thus the proof is
completed.
Remark 2.2. (i) 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.
(ii) 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 theorem also holds on such a specialized theory.
Corollary 2.3. Let f : X99KX0be aPrflop. If dim α1+dim α2=dim X, then
(Fα1.Fα2) = (α1.α2).
Proof. We may assume that α1, α2are transversal to Z. Then
(α1.α2) = (φ∗α1.φ∗α2) = ((φ0∗Fα1−ξ).φ∗α2)
= ((φ0∗Fα1).φ∗α2) = (Fα1.(φ∗0φ∗α2)) = (Fα1.Fα2).
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−1 = F∗ both in the sense of correspondences
and Poincar´e pairing.
Remark 2.4. It is an easy fact that if X =K X0then X and X0 are isomorphic in codimension one and in particular the graph closure gives canonical iso-morphisms 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 underF.
2.2. Triple product for simple flops. Let f : X 99K X0 be a simplePrflop with S being a point. Let h be the hyperplane class of Z =Prand h0be the hyperplane class of Z0. Let also x = φ¯∗h = [h×Pr], y = φ¯0∗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 getF[hl] = (−1)r−l[h0l]. In particular,F[C] = −[C0]. Proof. Recall that
NE/Y =OPr×Pr(−1,−1):=φ¯∗OPr(−1) ⊗φ¯0∗OPr(−1)
and NZ/X =OPr(−1)⊕(r+1). From (2.1),
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 ([3], Proposition 6.7) then implies that
φ∗[hl] =j∗(cr(E). ¯φ∗[hl]) =j∗(cr(E).xl) =j∗
∑
rt=0(−1)tyr−txt+l.Lemma 2.6. For a class α∈ H2l(X)with l≤r, let α0 =Fα in X0. Then
φ0∗α0 =φ∗α+ (α.hr−l)j∗x
l− (−y)l
x+y .
Proof. Since the difference φ0∗α0−φ∗αhas support in E, we may write φ0∗α0 =φ∗α+j∗(a1xl−1+ · · · +akxl−kyk−1+ · · · +alyl−1).
By intersecting this equation with xr−lyrin X and noticing that E∼ −(x+ y)on E, we get by the projection formula
0=φ∗α.xr−lyr−a1xl−1(x+y)xr−lyr= (α.hr−l) −a1.
Similarly by intersecting with xr−l+1yr−1we 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 xpyqwith 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 simplePr-flop f : X99KX0, let αi ∈ H2li(X), with li ≤r,
l1+l2+l3 =dim X=2r+1. Then
(Fα1.Fα2.Fα3) = (α1.α2.α3) + (−1)r(α1.hr−l1)(α2.hr−l2)(α3.hr−l3).
Proof. The proof consists of straightforward computations. (Fα1.Fα2.Fα3) = (φ0∗Fα1.φ0∗Fα2.φ∗Fα3) = φ∗α1+ (α1.hr−l1)j∗x l1− (−y)l1 x+y φ∗α2+ (α2.hr−l2)j∗x l2 − (−y)l2 x+y × φ∗α3+ (α3.hr−l3)j∗x l3 − (−y)l3 x+y .
Among the resulting eight terms, the first term is clearly equal to α1.α2.α3.
For those three terms with two pull-backs like φ∗α1.φ∗α2, 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 φ∗α1and two exceptional parts contributes
φ∗α1.j∗x l2− (−y)l2 x+y .j∗ xl3 − (−y)l3 x+y = −φ∗α1.j∗ (xl2 − (−y)l2)(xl3−1+xl3−2(−y) + · · · + (−y)l3−1)
times (α2.hr−l2)(α3.hr−l3). The terms with non-trivial contribution must
contain yr, hence there is only one such term, namely (notice that l
1+l2+
l3=2r−1)
and the contribution is (−1)r(α1.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.
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 at-tached to the extremal ray exactly remedy the defect of the ordinary prod-uct.
3.1. 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 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
mor-phism f 7→ f(xi). The Gromov-Witten invariant for classes αi ∈ H∗(X),
1≤i≤n, is given by
hα1, . . . , αnig,n,β:=
Z
[M¯g,n(X,β)]virt
e∗1α1· · ·e∗nαn.
The genus zero three-point functions (as formal power series) hα1, α2, α3i:=
∑
β∈A1(X)hα1, α2, α3i0,3,βq
β
together with the Poincar´e pairing (−,−) determine the small quantum product.
More precisely, let T =∑ 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) =
∑
∞ n=0∑
β∈NE(X) 1 n!hT ni β q β, whereTnβ =hT, . . . , Ti0,n,β. The big quantum product is defined by
where Φijk = ∂3Φ ∂ti∂tj∂tk =
∑
∞ n=0∑
β∈NE(X) 1 n!Ti, Tj, Tk, T n βq β.The n=0 partΦijk(0)gives the small quantum product.
3.2. Analytic continuation. Let f : X 99KX0 be a simplePrflop. Since X and X0 have the same Poincar´e pairing underF, in order to compare their quantum products we only need to compare their n-point functions. For three-point functions, write
hα1, α2, α3i = (α1.α2.α3) +
∑
d∈Nhα1, α2, α3id`qd `+∑
β6∈Z`hα1, α2, α3iβq β.The difference(Fα1.Fα2.Fα3) − (α1.α2.α3)is already determined in last
sec-tion. The next step is to compute the middle term, namely quantum cor-rections coming from the extremal ray ` = [C]. The third term will be discussed in later sections.
The virtual dimension of Mg,n(X, d`)is given by
(c1(X).d`) + (2r+1)(1−g) + (3g−3) +n.
Since(KX.`) =0, for g= 0 we need only consider classes αi ∈ Ali(X)with
∑n
i=1li =2r+1+ (n−3). For n=3 this is 2r+1=dim X.
Theorem 3.1. For all αi ∈ H2li(X)with 1 ≤ li ≤ r and∑ni=1li = 2r+1+
(n−3), hα1, . . . , αni0,n,d ≡ Z [M0,n(X,d`)]virt e∗1α1· · ·e∗nαn = (−1)(d−1)(r+1)Nl1,...,lnd n−3( α1.hr−l1) · · · (αn.hr−ln).
where Nl1,...,ln are recursively determined universal constants. Nl1,...,ln are inde-pendent of d and Nl1,...,ln = 1 for n=2 or 3. All other (primary) Gromov-Witten
invariants with degree inZ`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(Fαi.h0(r−li)) = (−1)li(Fα
i.Fhr−li) = (−1)li(αi.hr−li), for three
point functions we get
hFα1,Fα2,Fα3i − hα1, α2, α3i = (−1)r(α1.hr−l1)(α2.hr−l2)(α3.hr−l3) + (α1.hr−l1)(α2.hr−l2)(α3.hr−l3) (−1)2r+1q`0 1− (−1)r+1q`0 − q` 1− (−1)r+1q` ! .
Under the correspondenceF, we shall identify q`0 with q−`. Plug in this into the last bracket we get 1 when r is odd and get −1 when r is even.
In both cases the right hand side cancels out and then hFα1,Fα2,Fα3i =
hα1, α2, α3i. This proves the statement on small quantum product.
For general n=3+k point invariants with k≥1, we get
hα1, . . . , αni = Nl1,...,ln(α1.h r−l1) · · · (α n.hr−ln) ∞
∑
d=0 (−1)(d−1)(r+1)dkqd` = Nl1,...,ln(α1.h r−l1) · · · (α n.hr−ln) q` d dq` k (−1)r+1 1− (−1)r+1q`. Similarly, since(−1)∑ li = (−1)k+1,hFα 1, . . . ,Fαniequals (−1)k+1Nl1,...,ln(α1.h r−l1) · · · (α n.hr−ln) q`0 d dq`0 k (− 1)r+1 1− (−1)r+1q`0. Taking into account ofq−` d dq−` = −q ` d dq` and 1 1− (−1)r+1q−` =1− 1 1− (−1)r+1q`
we gethFα1, . . . ,Fαni = hα1, . . . , αnifor all k≥1 (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 `0 by−`in terms of analytic continuation over the extended complexified K¨ahler 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=1 case (cf. [25] §5.5, [18] §4).
The quantum cohomology is parameterized by the complexified K¨ahler class ω = B+iH with qβ = exp(2πi(
ω.β)), where B ∈ HR1,1(X)and H ∈
KX, the K¨ahler cone of X. For a simple Pr flop X 99K X0, F identifies H1,1, A1 and the Poincar´e pairing(−,−)on X and X0. Thenhα1, α2, α3iX
restricted toZ`converges in the region
H+1,1= {ω| (H.`) >0} ⊃H1,1R ×iKX
and equals
(α1.α2.α3) + (α1.hr−l1)(α2.hr−l2)(α3.hr−l3) e 2πi(ω.`)
1− (−1)r+1e2πi(ω.`).
This is a well-defined analytic function of ω on the whole H1,1, which
de-fines the analytic continuation ofhα1, α2, α3iXfrom HR1,1×iKXto H1,1.
Similarly,hFα1,Fα2,Fα3iX
0
restricted toZ`0converges in the region {ω| (H.`0) >0} = {ω| (H.`) <0} =H−1,1⊃ HR1,1×iKX0 and equals
(Fα1.Fα2.Fα3) − (α1.hr−l1)(α2.hr−l2)(α3.hr−l3)
e−2πi(ω.`)
which is the analytic continuation of the previous one from H1,1+ to H−1,1.
Remark 3.3. It was conjectured that the total series ΦXijk converges for B ∈ KX, at least for B large enough, hence the large radius limit goes back to the
classical cubic product. The Novikov variables{qβ}
β∈NE(X)are introduced
to avoid the convergence issue.
SinceKX∩ KX0 = ∅ for non-isomorphic K-equivalent models, the collec-tion of K¨ahler cones among them form a chamber structure. The conjectural canonical isomorphism
F : H∗(
X) ∼= H∗(X0)
assigns to each model X a coordinate system H∗(X)of the fixed H∗ andF serves as the (linear) transition function. The conjecture asserts thatΦXijkcan be analytically continued fromKXtoKX0and agrees withΦX
0
ijk. Equivalently,
Φijkis well-defined onKX∪ KX0which verifies the functional equation FΦijk(ω, T) ∼= Φijk(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 §5.
3.3. One-point functions with descendents. In order to prove Theorem 3.1, we first reduce the problem to one for projective spaces. Let
Ud:=R1f t∗e∗n+1N
be the obstruction bundle, where N = NZ/X and f t is the forgetting mor-phism in M0,n+1(Pr, d) en+1 // f t Pr M0,n(Pr, d) .
It is well known (see e.g. [2]) that
(3.1) [M0,n(X, d`)]virt =e(Ud) ∩ [M0,n(Pr, d`)].
Since Udis functorial under f t∗, we use the same notation for all n.
Define a generating function of one point GW invariants with descen-dents JX(q, z−1):=
∑
β∈NE(X) qβJ X(β, z−1) ∈ H∗(X)[[z−1]][[q]] :=∑
β∈NE(X) qβeX 1∗ 1 z(z−ψ)∩ [M0,1(X, β)] virt (3.2) In our case JX(d`, z−1) ≡eP r 1∗ e(Ud) z(z−ψ)has been calculated:
Lemma 3.4([13], also [4] [2]).
JX(d`, z−1) =Pd:= (−1)(d−1)(r+1)
1 (h+dz)r+1.
Remark 3.5. This calculation can also be interpreted as quantum Lefschetz hyperplane theorem for concave bundles overPr. From this viewpoint, the “mirror transformation” from JX(d`, z−1)to Pdis not needed since the rank
of the bundleO(−1)r+1is greater than one. See e.g. [9].
Corollary 3.6. For l+k=2r−1, 1≤l≤r, D τkhl E d = (−1)d(r+1)+k dk+2 C k+1 r
where Ckr = k!/r!(k−r)!. The invariant is zero if l+k 6=2r−1 by dimensional constraints.
Proof. We start with A := Z Prh l.P d=
∑
k≥0 1 zk+2 D τkhl E d. By Lemma 3.4 A= (−1)(d−1)(r+1) Z Pr hl (h+dz)r+1 = Z Pr hl dr+1zr+1 1+ h dz −(r+1) . The result follows from the Taylor expansion and the elementary fact thatCr−(−lr+1)= (−1)k+(r+1)Ckr+1.
3.4. Multiple-point functions via divisor relations. We recall the follow-ing rational equivalence in rational Chow groups A∗(M0,n(X, β))from [10],
Corollary 1: For L∈Pic(X)and i6=j, e∗iL∩ [M0,n(X, β)]virt =(e∗jL+ (β, L)ψj) ∩ [M0,n(X, β)]virt−
∑
β1+β2=β (β1, L)[Di,β1|j,β2] virt, (3.3) (3.4) ψi+ψj = [Di|j]virt, where [Di,β1|j,β2] virt ∈ A∗(M0,n(X, β)) is the push-forward of the virtual
classes of the corresponding boundary divisor components Di,β1|j,β2 =
∑
i∈A,j∈B; Aä B={1,...,n} D(A, B; β1, β2) and Di|j =∑
β1+β2=β Di,β1|j,β2.Here is a simple observation which will be repeatedly used in the sequel:
Lemma 3.7(Vanishing lemma). LetPr ⊂ X with NPr/X = ⊕jO(−mj), mj ∈
N. Let`be the line class inPr. Then for deg T >r and d6=0,h. . . , Tid`=0.
Proof. Since[M0,n(X, d`)]virtequals[M0,n(Pr, d)]cut out by e(Ud), the
eval-uation morphisms factor throughPr. But then e∗n(T|Z) =0. Here deg T :=l if T∈ H2l(X). As we had mentioned in the introduction, only real even degree classes will be relevant throughout our discussions.
Proposition 3.8. For k1+k2+l1+l2 =2r, 1≤li ≤r, D τk1hl1, τk2hl2 E d = (−1)d(r+1)+l1+k2+1 dk1+k2+1 C 2r−(l1+l2) r−l1 ,
and other descendent invariants vanish. In particular, the only non-trivial two-point function without descendents in degree d`is given by
hhr, hrid= (−1)(d−1)(r+1)1 d.
Proof. We consider the invariant without descendents first. Since the vir-tual dimension is 2r, onlyhhr, hrid survives. Using the above equivalence relations, we may decrease the power of e∗1h 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 e∗2hrleads to zero.
For the third boundary splitting terms, write [∆(X)] = ∑iTi⊗Ti. For
each i, since dim X=2r+1 one of Tior Timust have degree strictly bigger
than r. If β1 =d1`, β2 =d2`with di 6=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 = 1 and j=2. Since n= 2, we find n1 = |A| = 1, n2 = |B| =1 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
hhr, hrid=dhhr−1, τ1hrid= · · · =dr−1hh, τr−1hrid=drhτr−1hri
where the last equality is by the divisor axiom. Now we plug in Corol-lary 3.6 with(k, l) = (r−1, r)and the statement follows.
For descendent invariants we proceed in the same manner. For simplic-ity we abuse the notation by denotingh· · · , ψsα,· · · iβ = h· · · , τsα,· · · iβ.
Let s≥1, l+m+s=2r and consider D hl, ψshmE d= D hl−1, ψshm+1E d+ (h, d`) D hl−1, ψs+1hmE d = Dhl−1,(h+dψ)ψshm E d= · · · = Dh,(h+dψ)l−1ψshm E d.
Notice that the splitting terms are all zero as before. Now the divisor axiom of descendent invariants gives
dD(h+dψ)l−1ψshm E d+ D (h+dψ)l−1ψs−1hm+1 E d,
which leads to the reduction formula: D hl, ψshmE d = D (h+dψ)lψs−1hm E d.
Notice that this equals the constant term in z in *
∑
k≥0 ψk zk(h+dz) lzs−1hm + d = zs+1e1∗ e(Ud) z(z−ψ).e ∗ 1 (h+dz)l.hm = (−1)(d−1)(r+1)zs+1(h+dz)l−(r+1).hm = (−1)(d−1)(r+1) z r−m d(r+1)−l 1+ h dz l−(r+1) .hm, which is (−1)d(r+1)+r+1 dr+1−l+r−m C l−(r+1) r−m = (−1)d(r+1)+l+s+1 ds+1 C 2r−(l+m) r−m .In general from ψ1= −ψ2+ [D1|2]virt, we find
D τk1h l1, τ k2h l2E d = − D τk1−1h l1, τ k2+1h l2E d = · · · = (−1) k1 D hl1, τ k1+k2h l2E d
since the splitting terms all vanishes. The result follows. For n ≥ 3, it is known that for any three different markings i, j and k,
ψj = [Dik|j]virt. By plugging this into (3.3), we get
e∗iL=e∗jL+
∑
β1+β2=β ((β2.L)[Dik,β1|j,β2] virt− ( β1.L)[Di,β1|jk,β2] virt).In our special case this reads as ei∗h=e∗jh+
∑
d1+d2=d (d2[Dik,d1|j,d2] virt−d 1[Di,d1|jk,d2] virt).Notice that now di is allowed to be zero. Lemma 3.9. For n≥3,
hhl1+1, hl2, hl3, . . .i
n,d
= hhl1, hl2+1, hl3, . . .i
n,d+dhhl1+l3, hl2, . . .in−1,d−dhhl1, hl2+l3, . . .in−1,d.
Proof. As in the previous theorem, the boundary terms with non-trivial de-gree must vanish. For dede-gree zero, the only non-trivial invariants are three-point functions, hence we are left with
hhl1+1, hl2, hl3, . . .i n,d = hhl1, hl2+1, hl3, . . .i n,d +
∑
i dhhl1, hl3, T ii0hTi, hl2, . . .id−∑
i dhTi, hl1, . . .i dhhl2, hl3, Tii0.For the first boundary sum, in the diagonal decomposition[∆(X)] =∑ Ti⊗ Tiwe may choose basis so that hl1+l3appear in{Ti}. Then the above degree
zero invariants survive only in one term which is equal to 1. The same argument applies to the second sum too. So the above expression equals
hhl1, hl2+1, hl3, . . .i
d+dhhl1+l3, hl2, . . .id−dhhl1, hl2+l3, . . .id
as expected.
In light of (3.1) and results above, Theorem 3.1 can be reformulated as the following equation
(3.5) hhl1, hl2, . . . , hlni
d= (−1)(d−1)(r+1)Nl1,...,lnd
n−3,
which we will now prove.
Proof. (of (3.5), or equivalently Theorem 3.1.)
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.
Considerhhl1, hl2, hl3i dwith l1+l2+l3 =2r+1 and l1 ≤l2≤l3. If l1 =1 then l2 =l3=r and so hh, hr, hrid=dhhr, hrid= (−1)(d−1)(r+1). If l1≥2, then l2≤r−1 and hhl1, hl2, hl3i d = hhl1−1, hl2+1, hl3id+dhhl1+l3−1, hl2id−dhhl1−1, hl2+l3id.
But then both l1+l3−1 and l2+l3are larger than r+1 and the boundary
terms vanish individually. By reordering l2, l3 if necessary, and repeating
this procedure we are reduced to the case l1 = 1 and proof for n = 3 is
completed.
Suppose the theorem holds up to n−1 (with n ≥ 4). The above lemma and the induction hypothesis imply that
hhl1, hl2, hl3, . . .i d = hhl1−1, hl2+1, hl3, . . .i d+dhhl1+l3−1, hl2, . . .id−dhhl1−1, hl2+l3, . . .id = hhl1−1, hl2+1, hl3, . . .i d+ (Nl1+l3−1,l2,...−Nl1−1,l2+l3,...)d n−3.
By repeating this procedure, l1is decreased to one and we get
hhl1, hl2, . . . , hlni
d= (−1)(d−1)(r+1)Nl1,...,lnd
where Nl1,...,lnis given by N∗’s in one lower level. The proof is complete.
Similar methods apply to descendent invariants:
Theorem 3.10. The only three-point descendent invariants of extremal classes d`, up to permutations of insertions, are given by
D hl1, hl2, τ k3h l3E d= (−1)d(r+1)+l3+1 dk3 C k3+1 r−(l1+l2),
where l1+l2+l3+k3=2r+1 and by convention Cnm =0 if n<0.
More generally, an n-point descendent invariant h∏n i=1τkih
lii
d with n ≥ 3
is non-zero only if there are at least two insertions being free of descendents, say k1= k2 =0. In such cases, there are universal constants Nk,l ∈Z such that
D hl1, hl2, τ k3h l3, . . . , τ knh ln E d= Nk,ld n−3−∑ ki.
Proof. Let n ≥ 3 and assume that 0 ≤ k1 ≤ k2 ≤ · · · ≤ kn. If k2 ≥ 1 then
use ψ2 = [D2|13]virtwe get
D τk1h l1, . . . , τ knh lnE d =
∑
i; d1+d2=d D τk2−1h l2,· · · , T i E d1 D Ti, τk1h l1, τ k3h l3,· · ·E d2 .We separate two cases. If the first factor is a two-point function then it is non-zero only if Ti = hj for some j ≤ r. But then deg Ti > r and the right
factor vanishes since it contains ψ classes. For other cases, both factors contain ψ classes hence the factor with deg Ti > r (or deg Ti > r) must
vanish.
For three-point invariants, from ψ3= [D3|12]virtwe get as before that
D hl1, hl2, τ k3h l3 E d =
∑
i; d1+d2=d D τk3−1h l3, T i E d1 D Ti, hl1, hl2 E d2 =Dτk3−1hl3, hl1+l2 E dand the formula follows from the two-point case.
Similarly, for n≥4, if ki 6=0 then from ψi = [Di|12]virtwe get
D hl1, hl2, . . . , τ kih li, . . . E d= D τki−1h li, hl1+l2, . . .E d.
The result follows from an induction on n.
4. DEGENERATION ANALYSIS
Our next task is to compare the genus zero Gromov–Witten invariants of X and X0 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, [12] [11] [7], provides a rigorous formulation of the above naive picture.
4.1. The degeneration formula. Recall the relative invariants of a smooth pair(Y, E)with E ,→ Y a smooth divisor: LetΓ = (g, n, β, ρ, µ)with µ = (µ1, . . . , µρ) ∈ Nρ a partition of|µ| := ∑
ρ
i=1µi = (β.E). For A ∈ H∗(Y)⊗n
and ε ∈ H∗(E)⊗ρ, the relative invariant of stable maps with topological
typeΓ (i.e. with contact order µiin E at the i-th contact point) is
hA|ε, µi(ΓY,E):=
Z
[MΓ(Y,E)]virte ∗
YA∪e∗Eε
where eY : MΓ(Y, E) → Yn, eE : MΓ(Y, E) → Eρ are evaluation maps on
marked points and contact points respectively.
IfΓ=äπΓπ, the relative invariants (with disconnected domain curves)
hA|ε, µi•(ΓY,E):=
∏
πhA|ε, µi
(Y,E)
Γπ is defined to be the product of each connected component.
Except for using the numerical form of relative invariants, our presenta-tion of degenerapresenta-tion formula below mostly follows that of [11] and [15].
Given an ordinary flop f : X 99KX0, we apply deformations to the normal cone to both X and X0. LetX = X×A1 andΦ : W → X be the blow-up
along Z× {0}. We use t∈A1as the deformation parameter. Then W t∼= X
for all t6=0 and W0=Y1∪Y2with
φ=Φ|Y : Y1=Y→X
the blow-up along Z and
p= Φ|E˜ : Y2 =E :˜ =PZ(NZ/X⊕O) →Z⊂ X
the compactified normal bundle. Also Y∩E˜ = E = PZ(NZ/X)is the φ-exceptional divisor which consists of the infinity part of ˜E. Similar con-struction givesΦ0 : W0 →X0 =X0×A1and W0
0 =Y0∪E˜0. By definition of
ordinary flops, Y=Y0and E= E0. Remark 4.1. For simplePr flops, Y
2 ∼= PPr(O(−1)⊕(r+1)⊕O) ∼= Y0
2.
How-ever the gluing maps of Y1 and Y2 along E for X and X0 differ by a twist
which interchanges the order of factors in E = Pr×Pr. Thus W
0 6∼= W00
and it is necessary to study the details of the degenerations. In general, f induces an ordinary flop ˜f : Y2 99K Y20 of the same type which is the local
model of f .
Since the family W → A1 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 ji : Yi ,→W0be the inclusion maps for i=1, 2.
For {ei}a basis of H∗(E)with {ei}its dual basis, {eI}forms a basis of
H∗(Eρ) with dual basis {eI}where |I| = ρ, e
I = ei1 ⊗ · · · ⊗eiρ. The de-generation formula expresses the absolute invariants of X in terms of the
relative invariants of the two smooth pairs(Y1, E)and(Y2, E): hαiXg,n,β =
∑
I η∑
∈Ωβ Cη D j∗1α(0) eI, µ E•(Y1,E) Γ1 D j2∗α(0) e I, µE•(Y2,E) Γ2 .Here η = (Γ1,Γ2, Iρ) is an admissible triple which consists of (possibly
disconnected) topological types Γi =
ä
|Γi|
π=1Γ
π
i
with the same contact order partition µ under the identification Iρof contact
points. The gluingΓ1+IρΓ2has type(g, n, β)and is connected. In particu-lar, ρ =0 if and only if that one of theΓi is empty. The total genus gi, total
number of marked points ni and the total degree βi ∈ NE(Yi)satisfy the
splitting relations g=g1+g2+ρ+1− |Γ1| − |Γ2|, n1+n2=n and φ∗β1+p∗β2= β.
The constants Cη = m(µ)/|Aut η|, where m(µ) = ∏ µi and Aut η =
{σ∈Sρ|ησ =η}1. We denote byΩ the equivalence class of all admissible
triples, also byΩβandΩµthe subset with fixed degree β and fixed contact
order µ respectively.
4.2. Liftings of cohomology insertions. Next we discuss the presentation of α(0). Denote by ι1 ≡ j : E ,→ Y1 = Y and ι2 : E ,→ Y2 = E the natural˜
inclusions. The class α(0)can be represented by(j∗1α(0), j∗2α(0)) = (α1, α2)
with αi ∈ A∗(Yi)such that
ι∗1α1 =ι∗2α2 and φ∗α1+p∗α2 =α.
Such representatives are not unique. The flexibility on different choices is of key importance. The simplest choice is α1 =φ∗αand α2 = 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) = (α1, α2) then it can also be
represented by
α(0) = (α1−ι1∗e, α2+ι2∗e).
Proof. This follows from the facts that
ι∗1ι1∗e = (e.c1(NE/Y))E = −(e.c1(NE/ ˜E))E = −ι∗2ι2∗e
and−φ∗ι1∗e+p∗ι2∗e =0 (since φ◦ι1 = p◦ι2 =φ¯ : E→Z).
For an ordinary flop f : X99KX0, we compare the degeneration expres-sions of X and X0. For a given admissible triple η = (Γ1,Γ2, Iρ)on X, we
pick the corresponding η0 = (Γ01,Γ02, Iρ0)on X0 withΓ1=Γ01. We modify the 1When 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.
choices j1∗α(0) = φ∗αand j0∗1Fα(0) = φ0∗Fα by adding suitable classes in E
to make them equal. This is possible since
φ∗α−φ0∗Fα∈ι1∗H∗(E).
The relative invariants on the Y1=Y part from both sides are then the same
and we are left with the comparison of the part on ˜E and ˜E0. The following is clear, e.g. by a dimension count.
Lemma 4.3. Let ˜E = PZ(N⊕O)be a projective bundle with base i : Z ,→ E˜ and infinity divisor ι2 : E =PZ(N) ,→ E. Then the kernel of the the restriction˜
map ι∗2 : H∗(E˜) →H∗(E)is i∗H∗(Z).
Notice that f induces an ordinary flop ˜f on the local model ˜E 99K E˜0. Moreover it is a family of simple ordinary flops ˜ft : ˜Et 99KE˜t0 over the base
S, where t∈ S and ˜Etis the fiber of ˜E→Z→S etc.. Denote again byF the
cohomology correspondence induced by the graph closure. Then
Lemma 4.4(Cohomology reduction to local models). For f : X99KX0 aPr -flop over base S with dim S=s. Let α∈ H2l(X), l ≤dim Z =r+s (otherwise
α|Z=0) with representatives α(0) = (α1, α2)andFα(0) = (α01, α02).
If α1 =α01 then Fα2=α02.
Proof. From ι∗2α2 = ι∗1α1 = ι0∗1α01 = ι0∗2 α20 and the fact that ˜f is an
isomor-phism outside Z, we get
ι0∗2(Fα2−α20) =Fι∗2α2−ι20∗α20 =ι∗2α2−ι0∗2α02=0.
ThusFα2−α20 =i∗z0for some z0 ∈ H2(l−(r+1))(Z0)since codimE˜0Z0 = r+1. For simple flops, s= 0 and then l− (r+1) ≤ s−1 < 0. So z0 = 0 and we are done. In general we restrict the equation to each fiber ˜ft : ˜Et → E˜0t.
Since ¯Γ˜f|t =Γ¯ ˜f
t, by the case of simple flops we get(Fα2−α 0
2)|E˜0t =0 for all
t ∈ S. That is, z0 is a class supported in the fiber of p0 : Z0 → S. But then codimE˜0z0 ≥ s+r+1>l, which implies that z0 =0. Remark 4.5. Using this lemma we may give an alternative proof of equiv-alence of Chow motives under ordinary flops. Indeed the equivequiv-alence of Chow groups for simple flops is easy to establish. The deformation to nor-mal cone then allows us to reduce the general case to the local case and then to the local simple case.
4.3. Reduction to relative local models. First notice that A1(E˜) =ι2∗A1(E)
since both are projective bundles over Z. We then have
φ∗β= β1+β2
by regarding β2as a class in E⊂Y. Indeed φ∗(β1+β2) =φ∗β1+p∗β2= β
and
((β1+β2).E)Y = (β1.E)Y− (β2.E)E˜ = |µ| − |µ| =0
We consider only the case g=0. Consider the generating series hA|ε, µi:=
∑
β2∈NE(E˜) 1 |Aut µ|hA|ε, µi (E,E˜ ) β2 q β2.and the similar one with possibly disconnected domain curves
hA|ε, µi•(E,E˜ ):=
∑
Γ; µΓ=µ 1 |AutΓ|hA|ε, µi •(E,E˜ ) Γ qβ ΓProposition 4.6. To proveFhαiX∼= hFαiX0, it is enough to show that
FhA|ε, µi ∼= hFA|ε, µi.
Proof. For the n-point functionhαiX=∑β∈NE(X)hαiXβ q
βwe have hαiX=
∑
β∈NE(X)∑
η∈Ωβ∑
I Cηhα1|eI, µi •(Y1,E) Γ1 hα2 |e I, µi•(Y2,E) Γ2 q φ∗β =∑
µ∑
I η∑
∈Ωµ Cη hα1|eI, µiΓ•(1Y1,E)qβ1 hα2|eI, µiΓ•(2Y2,E)qβ2 .To simplify the generating series, we consider also absolute invariants hαi•X with possibly disconnected domain curves as before. Then by
com-paring the order of automorphisms, hαi•X =
∑
µ m(µ)∑
I hα1 |eI, µi•(Y1,E)hα2|eI, µi•(Y2,E). To compareFhαi•XandhFαi•X 0, by Lemma 4.4 we may assume that α1 = α01and α02=Fα2. By comparing with the similar expression forhFαi•X
0 , the relative terms for(Y, E)are identical. It remains to compare
hα2|eI, µi•(E,E˜ ) and hFα2|eI, µi•(E˜
0,E) .
We further split the sum into connected invariants. Denote byΓπa
con-nected part. Γπ has its contact order µπ induced from µ. We consider
par-titions P : µ = ∑π∈Pµπ and denote by P(µ)the set of all such partitions.
Then hA|ε, µi•(E,E˜ )=
∑
P∈P(µ)π∏
∈P∑
Γπ 1 |Aut µπ|hA π | επ, µπi(ΓE,Eπ˜ )q βΓπ.Notice that only βΓπ can vary in the sum overΓπand we may denote the
generating series of connected relative invariants as sum over β2∈ NE(E˜).
This reduces the problem tohAπ |επ, µπiand we are done.
Remark 4.7. Here is a brief comment on the term Fhα2 |eI, µi(E,E˜ )=
∑
β2∈NE(E˜) 1 |Aut µ|hα2|e I, µi(E,E˜ ) β2 q Fβ2.Since ˜E is a projective bundle, NE(E˜) =i∗NE(Z) ⊕Z+γwith γ the fiber
line class of ˜E → Z. The point is that, for β2 ∈ NE(E˜)it is in general not
true thatFβ2≡ β2(in E) is effective in ˜E0.
Indeed, for simple ordinary flops, let γ = δ0, δ = γ0 be the two line
classes in E ∼= Pr×Pr. It is easily checked that ` ∼
δ−γ in ˜E. Hence
` = −`0and γ=γ0+ `0 and
β2 =d1` +d2γ= (d2−d1)`0+d2γ0.
Fβ2 ∈NE(E˜0)if and only if d2 ≥d1. Thus it is too optimistic to expect that
hα2 |eI, µi(E,E˜ )= hFα2 |eI, µi(E˜
0,E)
term by term. Analytic continuations are in general needed.
4.4. Relative to absolute. Now we shall combine a method of Maulik and Pandharipande ([17], Lemma 4) to further reduce the relative cases to the absolute cases with at most descendent insertions along E. Following [17], we call the pair
(ε, µ) = {(ε1, µ1),· · · ,(ερ, µρ)}
with εi ∈ H∗(E), µi ∈ N a weighted partition, a partition of contact orders
weighted by cohomology classes in E.
Proposition 4.8. For simple ordinary flops ˜E99KE˜0, to prove FhA|ε, µi ∼= hFA|ε, µi
for any A and(ε, µ), it is enough to show that
FhA, τk1ε1, . . . , τkρερi ∼= hFA, τk1ε1, . . . , τkρερi
for any possible insertions A ∈ H∗(E˜)⊕n, kj ∈ N∪ {0}and εj ∈ H∗(E). (Here
we abuse the notations by denoting ι2∗ε ∈ H∗(E˜)by the same symbol ε.)
Proof. We apply the deformation to the normal cone for Z,→ E to get W˜ →
A1. Then W
0 = Y1∪Y2 with Y1 ∼= PE(OE(−1,−1) ⊕O)aP1bundle and
Y2∼=E. Denote E˜ 0 =E=Y1∩Y2and E∞ ∼=E the infinity divisor of Y1.
Since εi|Z = 0, in the degeneration formula we may represent εi(0) =
(ι1∗εi, 0)so it must contributes to the Y1side. As before the relative
invari-ants on(Y1, E)can be regarded as constants underF by Lemma 4.4.
We will prove the proposition first by induction on (d2= |µ|, n, ρ) with ρ
in the reverse ordering, and then on weighted partition(eI, µ)in the reverse
ordering of the following order:
For pairs in H∗(E) ×N, we define the size relation
(ε0, m0) > (ε00, m00) ⇔m0 >m00 or m0 =m00, deg ε0 >deg ε00.
Then(eI0, µ0) > (eI00, µ00)if in their size-decreasing sequences we have for the first size unequal term the one from µ0 is larger.
It is clear that for any(d2, n, ρ,(eI, µ))there are only finitely many such
The first observation is that if a curve class β = d1` +d2γ ∈ NE(E˜)is
split into β1 ∈ NE(Y1)and β2 ∈NE(Y2), then since
NE(Y1) =Z+δ+Z+γ¯+Z+γ and NE(Y2) =Z+` +Z+γ
( ¯γis the fiber class of Y1), we have
(β2, β1) = (a` +bγ, cδ+dγ+eγ¯)
subject to
a, b, c, d, e≥0, a+c=d1, b+c+d =d2
and the total contact order condition
b= (β2, E)E˜ = (β1, E)Y1 = −c−d+e.
In particular, e=d2and b≤d2with b=d2if and only if that c=d =0. In
this case β1= d2γ¯ and the invariants on(Y1, E)are fiber class integrals.
We follow the procedure used in the proof of Proposition 4.6 to split the generating series of invariants with possibly disconnected domain curves, according to the contact order. We consider only the case g = 0 and it is enough to consider that(ε1, . . . , ερ) =eI = (ei1, . . . , eiρ). Then
hα1, . . . , αn, τµ1−1ei1, . . . , τµρ−1eiρi •E˜ =
∑
µ0 m(µ0)×∑
I0 hτµ1−1ei1, . . . , τµρ−1eiρ |e I0, µ0i•(Y1,E)hα 1, . . . , αn |eI0, µ0i(E,E˜ )+R, where R denotes the remaining terms which either have total contact order smaller than d2or have smaller n on the ˜E. By induction R isF-invariant.For the main terms, the invariants on (E, E˜ ) are connected invariants. The total contact order d2 = |µ0|equals|µ| =∑iρ=1µi. This follows from the
dimension counting on ˜E and(E, E˜ ). For this we need a simple lemma:
Lemma 4.9. For any projective normal bundle Y2=PZ(N⊕O)of a pair Z⊂ X we have that
c1(Y2) = (rk N+1)E+p∗c1(X)|Z.
Proof. Indeed, from 0 → O → O(1) ⊗p∗(N⊕O) → TY2/Z → 0 we get
c1(TY2/Z) = (rk N+1)E+p ∗c
1(N), so the formula follows from c1(Y2) =
c1(TY2/Z) +p ∗c
1(Z).
Now let D = c1(E˜).β2+dim ˜E−3 = (r+2)d2+ (2r−2). For the
ab-solute invariant on ˜E,
∑
n j=1deg αj+ |µ| −ρ+∑
ρ j=1deg(eij+1) = D+n+ρ. While on(E, E˜ ),∑
n j=1deg αj+∑
ρ0 j=1deg ei0j = D+n+ρ 0− | µ0|.Hence(eI, µ)appears as one of the(eI0, µ0)’s. From|µ0| = |µ|, we get also deg eI−deg eI0 =ρ−ρ0.
We will show that the highest order term in the sum is given by C(µ)hα1, . . . , αn |eI, µi(E,E˜ )
where C(µ) 6=0. The following argument follows [17] closely.
For any(eI0, µ0), consider the splitting of weighted partitions (eI, µ) =
ä
ρ0
k=1(eIk, µ k)
according to the connected components of the relative moduli of(Y1, E)
in-dexed by the contact points of µ0. Since fiber class invariants onP1bundles can be computed by pairing cohomology classes in E with GW invariants in the fiberP1(c.f. [17], §1.2), we must have deg eIk+deg ei
0
k ≤dim E to get
non-trivial invariants. That is deg eIk =
∑
jdeg eik
j ≤dim E−deg e
ik0 ≡deg e i0k
for each k. In particular, deg eI ≤deg eI0, hence also ρ≤ρ0.
The case ρ < ρ0 is handled by the induction hypothesis, so we assume
that ρ=ρ0and then deg eIk =deg ei0
k for each k=1, . . . , ρ.
Since c1(Y1) = (r+2)E∞−rE0, the dimension count for each k
(con-nected component of the relative virtual moduli) on Y1reads as
2µ0k+2r−2+#{µk} +1−µ0k =
∑
j (µkj −1) + (deg eIk j +1) + (2r−deg ei0 k), which simplifies to µ0k−1=∑
j (µkj −1).Let (ei1, µ1)be the largest pair in (eI, µ)and let (eIk, µ
k)contain it. The
above equality shows that either(ei0
k, µ 0
k) > (ei1, µ1)or they are equal in size
with all other pairs in(eIk, µk)are of the form([E], 1). Here deg eIk =deg ei0
k
is used. In the later case we repeat the argument for the second largest pair in(eI, µ)and continue. In the process, either(eI0, µ0) > (eI, µ)is achieved and then the invariants is of lower order and handled by the induction hypothesis, or(eI0, µ0)and(eI, µ)are equal in size.
In the later case it remains to show that (eI0, µ0) = (eI, µ). Notice that from the fiber class invariants consideration we now have that
deg eik+deg e
i0k =dim E
for each k. Hence they must be Poincar´e dual classes to get non-trivial integral over E. That is, eI = eI0 is proved. This gives the term we expect for with C(µ)a nontrivial fiber class invariants. The proof is complete.
The functional equations for these special absolute invariants with de-scendents will be handled in §5.
