Quantum Invariance of Simple Flops
Hui-Wen Lin
∗Abstract
This note is a supplementary reading for the joint paper [8]
with Yuan-Pin Lee and Chin-Lung Wang, entitled ”Flops, motives and invariance of quantum rings”. In this note, I report our main result by a conceptual description instead of giving logically strict proofs. About the degeneration part which consists of complicated induction procedures, I provide some examples to illustrate it. I hope that readers can catch the key idea of our paper quickly through this note.
2000 Mathematics Subject Classification: 14N35, 14E30.
Keywords and Phrases: Ordinary flops, Chow motives, quan- tum cohomology, analytic continuations, K¨ahler moduli space.
1. Motivation
If not specifically stated, the ground field is assumed to be the complex numbers C. Two algebraic varieties are called birational if they have an isomorphic Zariski open subset. The problem of classifying varieties up to birational equivalence is usually the main interest of algebraic geometers. One of the main goals in birational geometry is to find a good geometric model that is convenient for the study of the given algebraic variety or its function field.
For 1-dimensional case, there is a unique nonsingular projective curve in a fixed birational equivalence class. For 2-dimensional case, there are possibly many smooth surfaces in a fixed birational equivalence class. At the beginning of the 20th century (c.f. [2]), Italian algebraic geometers applied the Castelnuovo’s contraction theorem to a smooth surface X repeatedly to obtain a minimal surface which contains no (-1) rational curve. When κ(X) = −∞, that is Γ(X, KXm) = 0 for all m ∈ N,
∗Department of Mathematics, National Central University, Chung-Li, Taiwan.
Email: linhw@math.ncu.edu.tw
Enrique’s theorem says that X is birational to a ruled surface C × P1. When κ(X) ≥ 0, that is Γ(X, KXm) 6= 0 for some m ∈ N, the minimal model is unique.
In 1982, Mori proved the three dimensional generalization of Castel- nuovo’s contraction theorem to continue the minimal model program.
The existence of minimal models has later been achieved in dimensions three and four (c.f. [6] and the references therein). However, since min- imal models in a fixed birational equivalence class are in general not unique, one of the most important problem remaining is to find invari- ants among birational minimal models.
For this purpose, C.-L. Wang raised the notion of K-equivalent va- rieties to generalize the one of minimal models [19]. Two (Q-Gorenstein) varieties X and X0 are K-equivalent if there exist birational morphisms φ : Y → X and φ0: Y → X0 with Y smooth
Y
φ
~~~~~~~~~~ φ0
A A AA AA AA
X X0
such that
φ∗KX = φ0∗KX0.
Two birational minimal models are automatically K-equivalent, so we turn our attention to study K-equivalent varieties.
V. Batyrev [1] and C.-L. Wang [19] showed that K-equivalent smooth varieties have the same Betti numbers. 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 [18] and C.- L. Wang [20] in response to these questions.
Conjecture 1 K-equivalent smooth varieties have canonically isomor- phic quantum cohomology rings over the extended K¨ahler moduli spaces.
For threefolds this Conjecture was proved by A. Li and Y. Ruan [10]. Our work is to study this conjecture in higher dimensional case.
In dimension three, by generalizing Koll´ar’s result [7], any K-equivalent map can be connected by a finite sequence of algebraic surgeries called flops. Roughly, each flop is obtained by removing one chain of rational curves C in X with KX|C = 0 then gluing back C into the open space
X\C in a different manner. Among them, the Atiyah P1 flop is the simplest one.
For higher dimensions, the natural generalizations of the Atiyah flop are called ordinary Pr flops. Ordinary flops are not only of the simplest type, but also crucial to the general theory of minimal models and K equivalence, so we make the choice to start with them.
Our main results in [8] answer the first question for general ordinary Prflops and the second question for simple Pr flops.
Theorem 2 For ordinary flops, the correspondence defined by the graph closure gives equivalence of Chow motives and preserves the Poincar´e pairing.
While the ring structure is in general not preserved under this cor- respondence, the quantum cohomology ring is, when the analytic con- tinuation on the Novikov variables is allowed.
Theorem 3 The big quantum cohomology ring is invariant under simple ordinary flops, after an analytic continuation over the extended K¨ahler moduli space.
2. Cohomology correspondence
2.1. Ordinary P
rflops.
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 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.
To construct the corresponding flops, we blow up X along Z to get φ : Y → X and the exceptional divisor E is a Pr× Pr-bundle over S.
Then we can blow down E along another fiber direction to get φ0: Y → X0, with exceptional loci ¯ψ0 : Z0 = PS(F0) → S for F0being another rank r + 1 vector bundle over S and also NZ0/X0|ψ0−fiber∼=OPr(−1)⊕(r+1).
We call f : X 99K X0 constructed as above an ordinary Pr flop.
The various sets and maps are summarized in the following commutative diagram.
E = Z ×SZ0
φ¯
vvnnnnnnnnnnnn QQQQQQ
φ¯0
Q((Q QQ QQ
j // Y vvmmmmφ mmmmmmmmmmmm
φ0
L %%L LL LL LL LL L
Z = PS(F )
ψ¯PPPP ((P PP PP PP PP
P i // X
ψQQQQQ ((Q QQ QQ QQ QQ
QQ Z0= PS(F0) mmmmmm
ψ¯0
vvmmmmmmmm
i0 // X0
ψ0
yyrrrrrrrrrrrr S
j0
//X
When S consists of a point, we call f a simple Prflop.
2.2. Equivalence of Chow motives
Instead of comparing special cohomology groups, we are devoted to the universal cohomology theory, namely Grothendieck’s category of Chow motives. General references of Chow motives can be found in [16].
Let M be the category of motives (over C). For each smooth variety X, one associates an object ˆX in M. A morphism from ˆX1 to ˆX2 is a correspondence U ∈ A∗(X1× X2) which has induced maps on T -valued points Hom( ˆT , ˆXi):
UT : A∗(T × X1)−→ AU ◦ ∗(T × X2).
and the composition law is given by
V ◦ U = p13∗(p∗12U.p∗23V )
where U ∈ A∗(X1× X2), V ∈ A∗(X2× X3) and pij : X1× X2× X3 → Xi× Xj are the projection maps.
The basic tool in motives is Manin’s identity principle: Let U, V ∈ Hom( ˆX, ˆX0). Then U = V if and only if UT = VT for all T .
For a Pr flop f : X 99K X0, to see that the graph closure [¯Γf] ∈ A∗(X × X0) identifies the Chow motives ˆX of X and ˆX0 of X0, our strategy is to apply the identity principle to show thatF∗◦F = ∆Xand F ◦ F∗= ∆X0.
For any T , idT× f : T × X 99K T × X0 is also an ordinary Prflop.
Hence to prove thatF∗◦F = ∆X, we only need to show thatF∗F = id on A∗(X) for any ordinary Prflop. The associated map on Chow groups of the correspondenceF is
F : A∗(X) → A∗(X0); W 7→ p0∗(¯Γf.p∗W ) = φ0∗φ∗W.
Let ˜W be the proper transform of W in Y and W0 be the proper trans- form of W in X0. By the precise formulae for pull-back from the inter- section theory ([4], Theorem 6.7) and dimensional consideration, we get
φ∗W = ˜W and thusFW = W0. By more delicate dimensional computa- tion, we find that the error term of φ∗W and φ0∗W0contains both fibers of φ as well as φ0 and thusF∗FW = W . By symmetry, FF∗W0 = W0. Hence we have our first result:
Theorem 4 For an ordinary Pr flop f : X 99K X0, the graph closure F := [¯Γf] induces ˆX ∼= ˆX0 via F∗◦F = ∆X andF ◦ F∗= ∆X0.
Using again the fact that the difference of φ∗αi and φ0∗Fαi has positive fiber dimension in both the φ direction and the φ0 direction, it follows thatF preserves the Poincar´e pairing.
Corollary 5 Let f : X 99K X0be a Prflop. If dim α1+dim α2= dim X, then
(Fα1.Fα2) = (α1.α2).
Remark 6 (i) 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.
(ii) However the ring structure, i.e. the cohomology product struc- ture, is not preserved under the correspondence F (c.f. next section).
To investigate a general product structure ∗ on H∗(X), let {Ti} be a cohomology basis and {Ti} be the dual basis with (Ti.Tj) = δij. Write
Ti∗ Tj =X
kcijkTk. We usually require thatR
XTi∗Tj= (Ti.Tj). Then the structure constants cijk= (Ti∗ Tj.Tk) = (Ti∗ Tj∗ Tk).
Hence under the Poincar´e pairing, ∗ is determined by its triple product.
2.3. The defect of triple products
In this section, I am going to determine the defect of triple products under a simple ordinary flop.
Let f : X 99K X0be a simple Prflop. Let h be the hyperplane class of Z = Pr and h0 be the hyperplane class of Z0. Let also x = ¯φ∗h = [h × Pr], y = ¯φ0∗h0 = [Pr× h0] in E = Pr× Pr. First of all, we seek out the correspondence of classes in Z and Z0:
Lemma 7 For classes inside Z, we have F[hk] = (−1)r−k[h0k].
In particular, F[C] = −[C0] with C, C0 being the line classes in Z, Z0 respectively.
Next we determine the difference of two pull-backs of α and Fα with classes α in X. The proof given below is slightly more concise than the original one in [8] from the structural viewpoint.
Lemma 8 For a class α ∈ H2k(X) with k ≤ r, let α0 = Fα in X0. Then
φ0∗α0= φ∗α + (α.hr−k) j∗xk− (−y)k x + y .
Proof. Since the difference φ0∗α0− φ∗α has support in E, we may write φ0∗α0− φ∗α = j∗λ for some λ ∈ H2(k−1)(E). Then
(φ0∗α0)|E− (φ∗α)|E= j∗j∗λ = c1(NE/Y)λ = −(x + y)λ.
By the Lefschetz hyperplane theorem, we have λ = − 1
x + y((φ0∗α0)|E− (φ∗α)|E) = − 1
x + y( ¯φ0∗(α0|Z0) − ¯φ∗(α|Z))
= − 1 x + y
φ¯0∗((α0.h0r−k)h0k)) − ¯φ∗((α.hr−k)hk).
SinceF preserves the Poincar´e pairing,
(α0.h0r−k) = (Fα.F((−1)khr−k)) = (−1)k(α.hr−k).
Hence we have λ = −(α.hr−k)
φ¯0∗(−1)kh0k− ¯φ∗hk
x + y = (α.hr−k)xk− (−y)k x + y .
These formulae allow us to compare the triple products of classes in X and X0. Besides I would like to simplify the proof in [8] a bit.
Proposition 9 For a simple Pr-flop f : X 99K X0, let αi ∈ H2ki(X), with ki≤ r, k1+ k2+ k3= dim X = 2r + 1. Then
(Fα1.Fα2.Fα3) = (α1.α2.α3) + (−1)r(α1.hr−k1)(α2.hr−k2)(α3.hr−k3).
Proof. Since for all i = 1, 2, 3, φ0∗Fαi= φ∗αi+ j∗λi with λi= (αi.hr−ki) j∗
xki− (−y)ki x + y
which contains both fiber directions of ¯φ and ¯φ0, we have
(Fα1.Fα2.Fα3) = (φ0∗Fα1.φ0∗Fα2.(φ∗α3+ j∗λ3)) = (φ0∗Fα1.φ0∗Fα2.φ∗α3)
= ((φ∗α1+ j∗λ1).(φ∗α2+ j∗λ2).φ∗α3).
Among the resulting terms, the first term is clearly equal to (α1.α2.α3).
For those terms with two pull-backs like φ∗α1.φ∗α3, the intersection val- ues are zero since the remaining part contains the φ fiber. The remaining term contributes
φ∗α3.j∗
xk1− (−y)k1 x + y .j∗
xk2− (−y)k2 x + y
= −φ∗α3.j∗ (xk1− (−y)k1)(xk2−1+ xk2−2(−y) + · · · + (−y)k2−1) times (α1.hr−k1)(α2.hr−k2). The terms with non-trivial contribution must contain yr, hence there is only one such term, namely (notice that k1+ k2+ k3= 2r + 1)
−(−y)k1× xk2−1−(r−k1)(−y)r−k1 = −(−1)rxr−k3yr
and the contribution is (−1)r(α1.hr−k1)(α2.hr−k2)(α3.hr−k3).
3. Quantum corrections
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 illustrate the reason why the quantum corrections at- tached to the extremal ray exactly remedy the defect of the ordinary product for simple ordinary flops.
3.1. Gromov-Witten invariants
We use [3] 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 curve 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
hα1, . . . , αnig,n,β :=
Z
[ ¯Mg,n(X,β)]virt
e∗1α1· · · e∗nαn.
The idea of Gromov-Witten invariants is that if we want to compute the relation of classes in X via stable maps, then we may use these evalu- ation morphisms to pull back the classes to the moduli space ¯Mg,n(X, β) and take integration. There are something subtle here. Because the moduli space usually does not have correct dimension, Li and Tian con- structed the virtual moduli cycle [ ¯Mg,n(X, β)]virt to have the expected
dimension [12]. The virtual (expected) dimension of ¯Mg,n(X, β) is given by
(c1(X).β) + dim X(1 − g) + (3g − 3) + n.
In our case, ψ : X → ¯X is a simple Pr flopping contraction with Z = Pr ⊂ X and NZ/X ∼= OPr(−1)⊕(r+1). If we deal with the case of β = d` with ` = [C], the extremal ray contracted by ψ, then since (KX.`) = 0, for g = 0, the virtual dimension of ¯M0,n(X, d`) equals 2r + 1 + (n − 3).
In practice, we may represent [ ¯M0,n(X, d`)]virt by the Euler class of the obstruction bundle
Ud= R1ρ∗e∗n+1NZ/X,
where ρ : ¯M0,n+1(Pr, d) → ¯M0,n(Pr, d) is the forgetting morphism. Then Z
[ ¯M0,n(X,d`)]virt
e∗1α1· · · e∗nαn = Z
M¯0,n(Pr,d)
e∗1(α1|
Pr) · · · e∗n(αn|
Pr).e(Ud).
3.2. Quantum product
Let T =P tiTi with {Ti} a cohomology basis and ti being formal vari- ables. 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, whereTn
β= hT, . . . , T i0,n,β. The big quantum product is defined by Ti∗ Tj=X
kΦijkTk where
Φijk = ∂3Φ
∂ti∂tj∂tk =X∞
n=0
X
β∈N E(X)
1
n!hTi, Tj, Tk, Tniβqβ. The n = 0 part Φijk(0) gives the small quantum product, that is,
Ti∗ Tj =X
k
X
β∈N E(X)hTi, Tj, TkiβqβTk.
Let f : X 99K X0 be a simple Pr flop. 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 :=X
β∈N E(X)hα1, α2, α3i0,3,βqβ
= (α1.α2.α3) +X
d∈Nhα1, α2, α3id`qd`+X
β6∈Z`hα1, α2, α3iβqβ.
The difference (Fα1.Fα2.Fα3)−(α1.α2.α3) has already been determined.
The next step is to compute the middle term, namely quantum correc- tions coming from the extremal ray ` = [C]. We will see that the three- point functions attached to the extremal ray exactly remedy the defect caused by the classical product. In the end, we will achieve that the re- maining terms are invariant under a simple flop in the sense of analytic continuation over the extended K¨ahler moduli space.
3.3. GW invariants attached to extremal rays
We derive a precise formula for this case.
Theorem 10 For all αi ∈ H2li(X) with 1 ≤ li ≤ r and Pn i=1li = 2r + 1 + (n − 3), there are recursively determined universal constants Nl1,...,ln, which are independent of d, such that for n ≤ 3, N∗≡ 1 and
hα1, . . . , αni0,n,d= (−1)(d−1)(r+1)Nl1,...,lndn−3(α1.hr−l1) · · · (αn.hr−ln).
Equivalently, Z
M¯0,n(Pr,d)
e∗1hl1· · · e∗nhln.e(Ud) = (−1)(d−1)(r+1)Nl1,...,lndn−3. The proof consists of two main steps. The first step is to use the theory on Euler data [13] to compute certain twisted Gromov-Witten invariants of concave bundle spaces. Originally it was used to compute invariants without marked points (i.e. no cohomology insertions), but it works only for critical bundles and does not apply to our case. Yet, through a closer study, the theory of Euler data does lead to the deter- mination of one-point invariants with descendents (i.e. ψ classes).
The second step is to use the divisor relation on the genus zero stable map moduli spaces [9] to reduce the multiple marked points in- variants to the ones with fewer marked points. I will now sketch both steps.
For step 1, we need two moduli spaces other than the original stable map moduli:
Md= ¯M0,0(P1× Pr, (1, d)) π //
ϕ
M¯0,0(Pr, d)
Nd ∼= P(r+1)(d+1)−1
Here Md is the graph space and Nd is the linear sigma model. A point in Nd is denoted by (zis)i=0,...r; s=0,...,d, which corresponds to the map
(w0: w1) 7→ (X
z0sws0w1d−s: · · · :X
zrsws0w1d−s).
The torus T = (C×)r+1 acts on Pr with weight λ0, . . . , λrand C× acts on P1 by t(w0, w1) = (tw0, w1) with weight α, 0. The equivariant cohomology of Nd is given by
HG∗(Nd) = Q[α, λ0, . . . , λr][κ]/Y
i,s(κ − (λi+ sα)), where κ is the equivariant hyperplane class. Let
Qd = ϕ∗π∗eT(Ud) ∈ HG(Nd), Then Lian, Liu and Yau [13] show that
Qd= (−1)(d−1)(r+1)Yd−1
m=1(κ − mα)r+1.
For Id : Pr = N0 → Nd, (a0, · · · , ar) 7→ (a0wd1, · · · , arwd1), the Atiyah-Bott localization theorem implies that
Id∗Qd=Yr
j=0
Yd
m=1(h − λj− mα).e1∗
eT(Ud) α(α + ψ1)
,
where ψ1= cG1(L1) and Li is the i-th cotangent line bundle. Recall τk1(hl1), · · · , τkn(hln)
d:=
Z
M¯0,n(Pr,d)
Yn
i=1ψikie∗ihli
.eT(Ud).
Then we combine the above two formulas of Qd to obtain:
Theorem 11 (One point invariants with ψ class) For l + k = 2r − 1,
τk(hl)
d= (−1)(d−1)(r+1)(−1)k−(r+1) dk+2 Crk+1.
Now we review step 2. Recall the divisor relation of Lee and Pand- haripande [9]: For L ∈ Pic(X) and i 6= j,
e∗iL ∩ [ ¯M0,n(X, β)]virt
= (e∗jL + (β.L)ψj) ∩ [ ¯M0,n(X, β)]virt− X
β1+β2=β
(β1.L)[Di,β1|j,β2]virt
in A∗( ¯M0,n(X, β)). That is, we may switch marked points if we can handle boundary divisors and ψ classes. In fact we can do that and obtain:
Theorem 12 (Two point invariants) The only non-trivial two point invariant (without ψ classes) is given by
hhr, hrid = (−1)(d−1)(r+1)1 d.
Recall for the equivalent form we need to show: For all d ∈ N, Pn
i=1li= 2r + 1 + (n − 3),
hhl1, . . . , hlnid= (−1)(d−1)(r+1)Nl1,...,lndn−3.
For n ≥ 3 and for any 3 markings i, j and k, ψj= [Dik|j]virt, the divisor relation can be re-written as
e∗iL = e∗jL + X
β1+β2=β
((β2.L)[Dik,β1|j,β2]virt− (β1.L)[Di,β1|jk,β2]virt).
This leads to the
Theorem 13 (Final Reduction) The following reduction formula holds for n ≥ 3:
hhl1+1, hl2, hl3, . . .in,d
= hhl1, hl2+1, hl3, . . .in,d
+ dhhl1+l3, hl2, . . .in−1,d− dhhl1, hl2+l3, . . .in−1,d. The desired formula then follows by induction.
3.4. Analytic continuations along extremal rays
Theorem 3.1, together with some algebraic manipulations, implies that the quantum corrections attached to the extremal ray exactly remedy the defect caused by the classical product and the big quantum products re- stricted to exceptional curve classes are invariant under simple ordinary flops. There are Novikov variables qβ involved in these transformations:
F(qβ) = qFβ.
To put the result into perspective, we interpret the change of vari- ables in terms of analytic continuation over the extended complexified K¨ahler moduli space.
The quantum cohomology is parameterized by the complexified K¨ahler class ω = B + iH with qβ= exp(2πi(ω.β)), where B ∈ H1,1
R (X) and H ∈ KX, the K¨ahler cone of X. For a simple Pr flop X 99K X0,F identifies H1,1, A1and the Poincar´e pairing (−, −) on X and X0. Then by applying Theorem 3.1, hα1, α2, α3iX restricted to Z` converges in the region
H+1,1= {ω | (H.`) > 0} ⊃ HR1,1× i KX
and the corresponding geometric series equals
(α1.α2.α3) + (α1.hr−l1)(α2.hr−l2)(α3.hr−l3) e2πi(ω.`) 1 + (−1)re2πi(ω.`).
This is a well-defined analytic function of ω on the whole H1,1, which defines the analytic continuation of hα1, α2, α3iX from H1,1
R × i KX to H1,1.
Similarly, hFα1,Fα2,Fα3iX0 restricted to Z`0 converges in {ω | (H.`0) > 0} = {ω | (H.`) < 0} = H−1,1 ⊃ HR1,1× i KX0. After the change of variable replacing `0 by −` and the identification of (Fαi.h0(r−li)) with (−1)li(αi.hr−li), it equals
(Fα1.Fα2.Fα3) − (α1.hr−l1)(α2.hr−l2)(α3.hr−l3) e−2πi(ω.`) 1 + (−1)re−2πi(ω.`) which is the analytic continuation of the previous one from H+1,1to H−1,1.
This illustrates that the three-point functions attached to the ex- tremal ray exactly remedy the defect caused by the classical product.
For invariance of big quantum product restricted to exceptional curve classes, we need to compare n = 3 + k point invariants with k ≥ 1.
By Theorem 3.1 again, we get
hα1, . . . , αni = Nl1,...,ln(α1.hr−l1) · · · (αn.hr−ln)
q` d
dq`
k (−1)r+1 1 − (−1)r+1q`. Similarly, since (−1)Pli = (−1)k+1, hFα1, . . . ,Fαni equals
(−1)k+1Nl1,...,ln(α1.hr−l1) · · · (αn.hr−ln)
q`0 d
dq`0
k
(−1)r+1 1 − (−1)r+1q`0. Taking into account of
q−` d
dq−` = −q` d
dq` and 1
1 − (−1)r+1q−` = 1 − 1 1 − (−1)r+1q` we get hFα1, . . . ,Fαni = hα1, . . . , αni for all k ≥ 1 (n ≥ 4).
It is 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∈N E(X) are introduced to avoid the convergence issue.
Since KX∩ KX0 = ∅ for non-isomorphic K-equivalent models, the collection 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
that ΦXijk0 can be analytically continued from KX0 to KX and agrees with ΦXijk. Equivalently, Φijk is well-defined on KX∪ KX0 which verifies the functional equation
FΦijk(ω, T ) ∼= Φijk(ω,FT ).
4. Degeneration analysis
To achieve the invariance of big quantum product, non-extremal curve classes need to be analyzed.
4.1. Cohomology reduction to local models
The main purpose of this section is to reduce cohomology classes in general X to cohomology classes in local models.
Given a Prflop f : X 99K X0, the deformations to the normal cone on X is the blowing-up Φ : W → X × A1 along Z × {0}. Wt ∼= X for all t 6= 0 and W0 = Y1∪ Y2 with ji : Yi ,→ W0 the inclusion maps for i = 1, 2. Here Y1 = Y with φ = Φ|Y : Y → X is the blowing-up along Z and Y2 = ˜E = PZ(NZ/X ⊕O) where p = Φ|E˜ : ˜E → Z ⊂ X is the compactified normal bundle. Y ∩ ˜E = E = PZ(NZ/X) is the φ- exceptional divisor which consists of the infinity part of ˜E. Similarly we have Φ0 : W0 → X0× A1 and W00 = Y0∪ ˜E0. By definition of ordinary flops, Y = Y0 and E = E0. In fact ˜E ∼= ˜E0 too, but they are glued into Y in a different manner (up to a twist), thus W06∼= W00.
Since the family W → A1 comes from a trivial family, all coho- mology classes α ∈ H∗(X, Z)⊕n have global liftings and the restriction α(t) on Wt is defined for all t. The class α(0) can be represented by (j1∗α(0), j2∗α(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. Actually 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).
We start with the representative (φ∗α, p∗(α|Z)) for α(0) and the representative (φ0∗Fα, p0∗(Fα|Z0)) for Fα(0). Then we can modify the choices φ∗α and φ0∗Fα by adding suitable classes in E to make them equal. This is possible since
φ∗α − φ0∗Fα ∈ ι1∗H∗(E).
Finally, we can show that for representatives α(0) = (α1, α2) andFα(0) = (α01, α02),
if α1= α01 then Fα2= α02.
Here we must mention that the ordinary flop f induces an ordinary flop f : ˜˜ E 99K ˜E0
on the local model, so the graph closureF of ˜f also gives a correspon- dence of H∗( ˜E) and H∗( ˜E0).
4.2. Degeneration formula
The degeneration formula expresses the absolute invariants of X in terms of the relative invariants of the two smooth pairs (Y1, E) and (Y2, E) stated in §4.1.
We start with the formulation given by J. Li [11]. It reads:
hαiXg,k,β=X
ηCηh
hj1∗α(0)i(YΓ1,E)
1 . hj2∗α(0)i(YΓ2,E)
2
i
0
.
Here for given genus g, number of marked points k and β ∈ N E(X), η = (Γ1, Γ2, I) with I = (IL, IR) runs through all equivalence classes of admissible triples. Namely, each Γiis an admissible graph without edges which consists of vertexes VΓi(connected components), legs LΓi(marked points), roots RΓi (gluing points), attaching map LΓi` RΓi → VΓi, genus function gi : VΓi → N ∪ {0}, degree function βi : VΓi → N E(Yi), ordering of marked points Ii : LΓi → {1, · · · , ki := |LΓi|} and contact order µi: RΓi → N.
The graph Γiis pre-connected in the sense that either vi:= |VΓi| = 1 or RΓi → VΓi is surjective. The two admissible graphs Γ1 and Γ2
glues together along roots via IR : RΓ1 ∼= RΓ2 with µ1 = µ2 under the identification. Two vertexes in the new graph Γ is assigned an edge connecting them whenever they are related via roots. These data satisfy ceratin compatibility identities. Namely k1+ k2= k, the total ordering IL : LΓ1` LΓ2 → {1, · · · , k} preserves the ordering of I1 and I2 and g − 1 =P
v∈VΓ1(g1(v) − 1) +P
v∈VΓ2(g2(v) − 1) + ρ, where ρ = |RΓi| is the number of roots.
The crucial condition is that Γ is connected. In particular, ρ = 0 if and only if that one of the Γi is empty. Also the total degree βi :=
P
v∈VΓiβi(v) ∈ N E(Yi) satisfies the splitting relation φ∗β1+ p∗β2= β.
The constants Cη = m(µ)/|Aut η|, where m(µ) =Q µiand Aut η = { σ ∈ Sρ | ησ = η }. For each η there is a gluing morphism for moduli
spaces of relative stable maps under prescribed constraints:
Φη : ¯MΓ1(Y1, E) ×Eρ M¯Γ2(Y2, E) → ¯Mg,k,β(W/A1), which is finite ´etale of degree |Aut η| onto its image ¯Mη(W0).
Each moduli space has perfect obstruction theory. The virtual mod- uli cycle of ¯Mg,k,β(W/A1) is flat over A1 and its zero fiber is made up by fiber products of those virtual moduli cycles of ¯MΓ1(Y1, E) and M¯Γ2(Y2, E) via Φη. The relative invariants (i = 1, 2) are
hji∗α(0)i(YΓi,E)
i ≡ qi∗ ev∗iji∗α(0) ∩ [MΓi(Yi, E)]virt ∈ H∗(Eρ, Q), where evi: MΓi(Yi, E) → Yiki and qi : MΓi(Yi, E) → Eρ are evaluation maps on marked points and gluing points respectively. This formulation will be used in dealing with examples at the end of this section.
On the other hand, the following (equivalent) numerical form orig- inally obtained by A. Li and Y. Ruan [10] will be used in our proof:
hαiXg,n,β =X
I
X
η∈Ωβ
Cη
D α1
eI, µE•(Y1,E) Γ1
D α2
eI, µE•(Y2,E) Γ2
where {ei} is a basis of H∗(E) with {ei} its dual basis and {eI} forms a basis of H∗(Eρ) with dual basis {eI} where |I| = ρ, eI = ei1⊗ · · · ⊗ eiρ. In this formulation, Γ = (g, n, β, ρ, µ) with µ = (µ1, . . . , µρ) ∈ Nρ a partition of the intersection number (β.E) = |µ| := Pρ
i=1µi. 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 µiin E at the i-th contact point) is
hA | ε, µi(Y,E)Γ :=
Z
[MΓ(Y,E)]virt
e∗YA ∪ e∗Eε
where eY : MΓ(Y, E) → Yn, eE : MΓ(Y, E) → Eρ are evaluation maps on marked points and contact points respectively.
If Γ = `
πΓπ, the relative invariants (with disconnected domain curves)
hA | ε, µi•(Y,E)Γ :=Y
πhA | ε, µi(Y,E)Γπ
are defined to be the product of the connected components.
An admissible triples η = (Γ1, Γ2, Iρ) consists of (possibly discon- nected) topological types
Γi=a|Γi| π=1Γπi
with the same contact order partition µ under the identification Iρ of contact points. The gluing Γ1+IρΓ2has type (g, n, β) and is connected.
We denote by Ω the equivalence class of all admissible triples, also by Ωβ and Ωµ the subset with fixed degree β and fixed contact order µ respectively.
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 β2 as a class in E ⊂ Y . For the n-point function hαiX =P
β∈N E(X)hαiXβ qβ we have hαiX = X
β∈N E(X)
X
η∈Ωβ
X
I
Cηhα1| eI, µi•(YΓ 1,E)
1 hα2| eI, µi•(YΓ 2,E)
2 qφ∗β
=X
µ
X
I
X
η∈Ωµ
Cη
hα1| eI, µi•(YΓ 1,E)
1 qβ1
hα2| eI, µi•(YΓ 2,E)
2 qβ2 .
To simplify the generating series, we consider also absolute invari- ants hαi•X with possibly disconnected domain curves as before. Then by comparing the order of automorphisms,
hαi•X=X
µ
m(µ)X
I
hα1| eI, µi•(Y1,E)hα2| eI, µi•(Y2,E)
where the generating series with possibly disconnected domain curves are
hA | ε, µi•( ˜E,E):= X
Γ; µΓ=µ
1
|Aut Γ|hA | ε, µi•( ˜ΓE,E)qβΓ.
To compare Fhαi•X and hFαi•X0, by the cohomology reduction we may assume that α1 = α01 and α02 =Fα2. By comparing with the similar expression for hFαi•X0, the relative terms for (Y, E) are identical.
It remains to compare
hα2| eI, µi•( ˜E,E) and hFα2| eI, µi•( ˜E0,E). We further split the sum into connected invariants.
hA | ε, µi•( ˜E,E)= X
P ∈P (µ)
Y
π∈P
X
Γπ
1
|Aut µπ|hAπ | επ, µπi( ˜ΓE,E)π qβΓπ where Γπ is a connected part with contact order µπ induced from µ and P (µ) is the set of all partitions P : µ =P
π∈Pµπ.
Notice that only βΓπ can vary in the sum over Γπ and we may denote the generating series of connected relative invariants as sum over β2∈ N E( ˜E). This reduces the problem to hAπ| επ, µπi. We summarize the result as follows.
Proposition 14 To proveFhαiX ∼= hFαiX0, it is enough to show that FhA | ε, µi ∼= hFA | ε, µi.
4.4. Relative invariants to absolute invariants
Inspired by a method of Maulik and Pandharipande [17], we further reduce the relative local cases to the absolute local cases with at most descendent insertions along E.
Proposition 15 For simple ordinary flops ˜E 99K ˜E0, 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).
We apply the deformation to the normal cone for Z ,→ ˜E to get W → A1. Then W0 = Y1∪ Y2 with Y1 ∼= PE(OE(−1, −1) ⊕O) a P1 bundle and Y2∼= ˜E. Denote E0= E = Y1∩ Y2 and E∞∼= E the infinity divisor of Y1.
Given a relative invariant hα1, . . . , αn | ε, µi on ( ˜E, E), the idea is to analyze the degeneration formula for hα1, . . . , αn, τµ1−1ε1, . . . , τµρ−1ερiE˜ and to use induction on the triple (|µ|, n, ρ) in the lexicographical order with ρ in the reverse order. Since ρ ≤ |µ|, it is clear that there are only finitely many triples of lower order. The proposition holds for those cases by the induction hypothesis.
For β = d1` + d2γ ∈ N E( ˜E), (c1( ˜E).β) = d2(c1( ˜E).γ), hence by the virtual dimension counting d2 is uniquely determined for a given generating series with fixed cohomology insertions.
Note that N E(Y1) = Z+δ +Z+γ +Z+¯γ with γ, δ the two line classes in E and ¯γ the fiber class of Y1 and N E(Y2) = Z+` ⊕ Z+γ. A curve class β = d1` + d2γ ∈ N E( ˜E) is split into β1= aδ + bγ + c¯γ ∈ N E(Y1) and β2= d` + eγ ∈ N E(Y2) which satisfy
a, b, c, d, e ≥ 0, a + d = d1, c = d2
and the total contact order condition
e = (β2.E)Y2 = (β1.E)Y1 = −a − b + c.
In particular, e ≤ d2 with e = d2 if and only if that a = b = 0. In this case β1= d2γ and the invariants on (Y¯ 1, E) are fiber class integrals.
Since εi|Z = 0, one may choose the cohomology representative εi(0) = (ι1∗εi, 0). For a general cohomology insertion α ∈ H∗( ˜E), the representative can be chosen to be α(0) = (a, α) for some a.
As before the relative invariants on (Y1, E) can be regarded as con- stants underF. Let (ε1, . . . , ερ) = eI = (ei1, . . . , eiρ). Then
hα1, . . . , αn, τµ1−1ei1, . . . , τµρ−1eiρi• ˜E=X
µ0
m(µ0)×
X
I0
hτµ1−1ei1, . . . , τµρ−1eiρ | eI0, µ0i•(Y1,E)hα1, . . . , αn| eI0, µ0i( ˜E,E)+ R,
where R denotes the remaining terms which either have total contact order smaller than d2 or have number of insertions fewer than n on the ( ˜E, E) side or the invariants on ( ˜E, E) are disconnected ones.
The crucial point is that we can show that the highest order term in the sum consists of the single term
C(µ)hα1, . . . , αn| eI, µi( ˜E,E)
where C(µ) 6= 0. Then the induction hypothesis for R together with the absolute cases with at most descendent insertions along E give us the desired relative case.
4.5. Examples
I would like to illustrate the degeneration formula by treating a slightly more general case which includes simple (r, r0) flips. 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.
The construction of the (r, r0) flip f : X 99K X0 is the same as the Pr flop case.
We will consider simple (r, r0) flips with KX nef. If β = d` then the invariant depends only on Z, α|Z and NZ/X. In particular hαiXg,n,d` = hp∗(α|Z)iEg,n,d`˜ . For other curve classes, we will see that the invariants degenerate cleanly. For odd dimensional classes they must contribute on Y1 since H∗(Y2) has only algebraic classes, for the purpose of comparing GW invariants we thus consider only αi ∈ H2li(X). Because of the divisor axiom, we require also that li≥ 2 for all i.
The following proposition works for more general setup:
Proposition 16 Let φ : Y → X be the blow-up of X along a smooth center Z of dimension r and codimension r0+ 1 with KX nef and r ≤ r0+ 1. Then
(1) Cη6= 0 only if g1= 0, v1= µ = ρ 6= 0 and µ1≡ 1, v2= 1.
