Algebraic Geometry3 (5) (2016) 615–653 doi:10.14231/AG-2016-027
Invariance of quantum rings under ordinary flops II:
A quantum Leray–Hirsch theorem
Yuan-Pin Lee, Hui-Wen Lin and Chin-Lung Wang
Abstract
This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invari- ance of the big quantum rings under ordinary flops of split type. To achieve that, several new ingredients are introduced. One is a quantum Leray–Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D-module of the Dubrovin connection on the base by a Picard–Fuchs system of the toric fibers. Non-split flops as well as further applications of the quantum Leray–Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in part III (Lee, Lin, Qu and Wang, “Invariance of quantum rings under ordinary flops III”, Cambridge Journal of Mathematics, 2016), which reduces the general ordinary flops to the split case solved here.
1. Introduction 1.1 Overview
This paper continues our study of the quantum invariance of genus zero Gromov–Witten theory, up to analytic continuation along the K¨ahler moduli spaces, under ordinary flops over a non- trivial base. The quantum invariance via analytic continuation plays an important role in the study of various Calabi–Yau compactifications in string theory. It is also a potential tool in comparing various birational minimal models in higher-dimensional algebraic geometry. We refer the readers to [LLW10] and Part I of this series [LLW16] for a general introduction.
In Part I, we determine the defect of the cup product under the canonical correspondence [LLW16, Section 2] and show that it is corrected by the small quantum product attached to the extremal ray [LLW16, Section 3]. We then perform various reductions to local models [LLW16, Sections 4 and 5]. The most important consequence of this reduction is that we may assume that our ordinary flops are between two toric fibrations over the same smooth base.
In this paper, we study the local models via various techniques and complete the proof of the quantum invariance of Gromov–Witten theory in genus zero under ordinary flops of split type.
Received 30 July 2014, accepted in final form 1 February 2016.
2010 Mathematics Subject Classification14N35, 14E30.
Keywords: quantum Leray–Hirsch, split ordinary flops, Dubrovin connections, Picard–Fuchs ideal, lifting of QDE, Birkhoff factorization, generalized mirror transform.
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This is, as far as we know, the first result on the quantum invariance under the K-equivalence (crepant transformation) [Wan04,Wan03] where the local structure of the exceptional loci cannot be deformed to any explicit (for example, toric) geometry and the analytic continuation is non- trivial. This is also the first result for which the analytic continuation is established with non- trivial Birkhoff factorization.
Several new ingredients are introduced in the course of the proof. One main technical ingre- dient is the quantum Leray–Hirsch theorem for the local model, which is related to the canonical lift of the quantumD-module from the base to the total space of a (toric) bundle. The techniques developed in this paper are applicable to more general cases and will be discussed in subsequent papers.
Conventions. This paper is strongly correlated with [LLW16], which will be referred to as Part I throughout the paper. All conventions and the notation introduced there carry over to this paper.
1.2 Outline of the contents
1.2.1 On the splitting assumption. We recall the local geometry of an ordinary Pr flop f : X 99K X0 (Part I [LLW16, Section 2.1]). The local geometry of the f -exceptional loci Z ⊂ X and Z0 ⊂ X0 is encoded in a triple (S, F, F0), where S is a smooth variety, and F and F0 are two rank r + 1 vector bundles over S. In Part I [LLW16], we reduce the proof of the invariance of the big quantum ring of any ordinary flop to that of its local model. Therefore, we may assume
X = ˜E = PZ(O(−1) ⊗ F0⊕O) , X0 = ˜E0 = PZ0(O(−1) ⊗ F ⊕ O) ,
where Z ∼= PS(F ) and Z0 ∼= PS(F0) are projective bundles. In particular, X and X0 are toric bundles over the smooth base S. Moreover, proving the invariance of the local model is equivalent to proving the type I quasi-linearity property, namely the invariance for 1-pointed descendent fiber series of the form
h¯t1, . . . , ¯tn−1, τkaξi ,
where ¯ti ∈ H(S) and ξ is the common infinity divisor of X and X0.
To proceed, recall that the descendent Gromov–Witten (GW) invariants are encoded by their generating function, that is, the so-called (big) J function: for τ ∈ H(X),
JX τ, z−1 := 1 + τ
z + X
β,n,µ
qβ n!Tµ
Tµ
z(z − ψ), τ, . . . , τ
X 0,n+1,β
.
The determination of J usually relies on the existence of C×-action. Certain localization data Iβ coming from the stable map moduli spaces are of hypergeometric type. For “good” cases, say c1(X) is semipositive and H(X) is generated by H2, the generating function I(t) = P Iβqβ determines J (τ ) on the small parameter space H0⊕ H2 through the “classical” mirror transform τ = τ (t). For a simple flop, X = Xloc is indeed semi-Fano toric and the classical mirror theorem (of Lian–Liu–Yau and Givental) is sufficient [LLW10]. (It turns out that τ = t and I = J on H0⊕ H2.)
For a general base S with given quantum cohomology ring QH(S), the determination of QH(P ) for a projective bundle P → S is far more involved. To allow fiberwise localization to determine the structure of the GW invariants of Xloc, the bundles F and F0 are then assumed to be split bundles.
In this paper (Part II), we only consider ordinary flops of split type, namely F ∼= Lr i=0Li and F0 ∼=Lr
i=0L0i for some line bundles Li and L0i on S.
1.2.2 Birkhoff factorization and generalized mirror transformation. The splitting assump- tion allows one to apply C×-localization along the fibers of the toric bundle X → S. Using this and other sophisticated technical tools, Brown (and Givental) [Bro14] proved that the hyperge- ometric modification
IX D, ¯t, z, z−1 :=X
β
qβeD/z+(D.β)IβX/S z, z−1ψ¯∗JβS
S ¯t, z−1
lies in Givental’s Lagrangian cone generated by JX(τ, z−1). Here D = t1h + t2ξ, ¯t ∈ H(S) and βS = ¯ψ∗β, and the explicit form of IβX/S is given in Section3.2.
Based on Brown’s theorem, we prove the following result. (See Section 2 for the notation concerning higher derivatives ∂ze.)
Theorem 1.1 (BF/GMT). There is a unique matrix factorization
∂zeI z, z−1 = z∇J z−1B(z) , called the Birkhoff factorization (BF) of I, valid along τ = τ (D, ¯t, q).
The BF can be stated in another way. There is a recursively defined polynomial differential operator P (z, q; ∂) = 1 + O(z) in t1, t2 and ¯t such that
J z−1 = P (z, q; ∂)I z, z−1 .
In other words, P removes the z-polynomial part of I in the NE(X)-adic topology. In this form, the generalized mirror transform (GMT)
τ (D, ¯t, q) = D + ¯t +X
β6=0
qβτβ(D, ¯t)
is the coefficient of z−1 in J = P I.
1.2.3 Hypergeometric modification andD-modules. In principle, knowing the BF, the GMT and the GW invariants on S allows us to calculate all g = 0 invariants on X and X0 by recon- struction. These data are in turn encoded in the I-functions. One might be tempted to prove the F -invariance by comparing IX and IX0. While they look rather symmetric, the defect of the cup product impliesF IX 6= IX0 and the comparison via tracking the defects of the ring isomorphism becomes hopelessly complicated. This can be overcome by studying a more “intrinsic” object:
the cyclic D-module MJ = DJ, where D denotes the ring of differential operators on H with suitable coefficients.
It is well known by the topological recursion relations (TRR) that (z∂µJ ) forms a fundamental solution matrix of the Dubrovin connection: Namely, we have the quantum differential equations (QDE)
z∂µz∂νJ =X
κ
C˜µνκ (t)z∂κJ , where the ˜Cµνκ (t) = P
ιgκι∂µνι3 F0(t) are the structural constants of ∗t. This implies that M is a holonomic D-module of length N = dim H. For I we consider a similar D-module MI =DI.
Theorem1.1 furnishes a change of basis which implies thatMI is also holonomic of length N .
The idea is to go backward: to first find MI and then transform it to MJ. We do not have similar QDE since I does not have enough variables. Instead we construct higher-order Picard–
Fuchs equations 2`I = 0 and 2γI = 0 in divisor variables, with the nice property that “up to analytic continuation” they generateF -invariant ideals:
F 2X` ,2Xγ
∼=
2X`00,2Xγ00 .
1.2.4 The quantum Leray–Hirsch theorem and the conclusion of the proof. We now want to determine MI. While the derivatives along the fiber directions are determined by the Picard–
Fuchs equations, we need to find the derivatives along the base direction. Write ¯t =P¯tiT¯i. This is achieved by lifting the QDE on QH(S), namely
z∂iz∂jJS =X
k
C¯ijk(¯t)z∂kJS,
to a differential system on H(X). A key concept needed for such a lift is the I-minimal lift of a curve class βS ∈ NE(S) to βSI ∈ NE(X). Various lifts of curve classes are discussed in Section3.
See in particular Definition 3.7.
Using the Picard–Fuchs equations and the lifted QDE, we show thatF MIX ∼=MIX0. Theorem 1.2 (Quantum Leray–Hirsch). (1) (I-lift) The quantum differential equation on QH(S) can be lifted to H(X) as
z∂iz∂jI =X
k,βS
qβISe(D·βSI)C¯ij,βk S(¯t)z∂kDβI
S(z)I , where DβI
S(z) is an operator depending only on βSI. Any other lift is related to it modulo the Picard–Fuchs system.
(2) Together with the Picard–Fuchs operators 2` and 2γ, the QDE determine a first-order matrix system under the naive quantization ∂ze (Definition 4.7) of the canonical basis Te (No- tation4.1) of H(X):
z∂a(∂zeI) = (∂zeI)Ca(z, q) , where ta= t1, t2 or ¯ti.
(3) The system has the property that for any fixed βS ∈ NE(S), the coefficients are formal functions in ¯t and polynomial functions in qγet2, q`et1 and f (q`et1). Here the basic rational function
f (q) := q/ 1 − (−1)r+1q
(1.1) is the “origin of analytic continuation” satisfying f (q) + f (q−1) = (−1)r.
(4) The system is F -invariant.
The final step is to go from MI to MJ. From the perspective of D-modules, the BF can be considered as a gauge transformation. The defining property (∂zeI) = (z∇J )B of B can be rephrased as
z∂a(z∇J ) = (z∇J ) ˜Ca, so that
C˜a= (−z∂aB + BCa)B−1 (1.2)
is independent of z.
This formulation has the advantage that all objects in (1.2) are expected to be F -invariant (while I and J are not). It is therefore easier to first establish theF -invariance of the Caand use
it to derive the F -invariance of the BF and GMT. As a consequence, this allows us to deduce the type I quasi-linearity (Proposition 2.11), and hence the invariance of the big quantum rings for local models.
Theorem 1.3 (Quantum invariance). For ordinary flops of split type, the big quantum coho- mology ring is invariant up to analytic continuation.
By the reduction procedure in Part I [LLW16], this is equivalent to the quasi-linearity property of the local models. This completes the outline.
Remark 1.4. Results in this paper had been announced by the authors, in increasing degree of generality, at various conferences during 2008–2012; see, for example, [Lin10, Wan11, LLW12], where more examples are studied. Examples of the quantum Leray–Hirsch theorem are included in Section5. The complete proofs of Theorems1.2 and 1.3were achieved mid-2011.
It might seem possible to prove Theorem 1.3 directly from comparisons of J -functions and Birkhoff factorizations on X and X0. Indeed, we were able to carry this out for various special cases. A mysterious regularization phenomenon appears during such a direct approach. In the appendix we explain how the regularization of certain rational functions leads to the beginning steps of analytic continuations in our context. However, the combinatorial complexity becomes intractable (to us) in the general case. Some examples can be found in the proceedings articles referred to above.
In Part III [LLQW16], the final part of this series, we will develop a quantum splitting principle to remove the splitting assumption in Theorem 1.3. This then completes our study of the quantum invariance under ordinary flops.
2. Birkhoff factorization
In this section, a general framework for calculating the J -function for a split toric bundle is discussed. It relies on a given (partial) section I of the Lagrangian cone generated by J . The process to go from I to J is introduced in a constructive manner, and Theorem1.1will be proved (as the combination of Proposition2.6 and Theorem 2.10).
2.1 Lagrangian cone and the J -function
We start with Givental’s symplectic space reformulation of Gromov–Witten theory arising from the dilaton, string and topological recursion relation. The main references for this section are [Giv04,CG07], with supplements and clarification from [LP,Lee09]. In the following, the under- lying ground ring is the Novikov ring
R =C[NE(X)] .\ All complicated issues on completion are deferred to [LP].
Let H := H(X), H := H[z, z−1]], H+ := H[z] and H− := z−1H[[z−1]]. Let 1 ∈ H be the identity. One can identify H as T∗H+, and this gives a canonical symplectic structure and a vector bundle structure on H.
Let
q(z) =X
µ
∞
X
k=0
qµkTµzk∈ H+
be a general point, where {Tµ} forms a basis of H. In the Gromov–Witten context, the natural coordinates on H+ are t(z) = q(z) + 1z (dilaton shift), with t(ψ) =P
µ,ktµkTµψk serving as the general descendent insertion. Let F0(t) be the generating function of the genus zero descendent Gromov–Witten invariants on X. Since F0 is a function on H+, the 1-form dF0 gives a section of π : H → H+.
Givental’s Lagrangian cone L is defined as the graph of dF0, which is considered as a section of π. By construction it is a Lagrangian subspace. The existence of C∗-action on L is due to the dilaton equation P qµk∂/∂qµkF0 = 2F0. Thus L is a cone with vertex q = 0 (cf. [Giv04, Lee09]).
Let τ =P
µτµTµ∈ H. Define the (big) J -function to be JX τ, z−1 = 1 +τ
z + X
β,n,µ
qβ n!Tµ
Tµ
z(z − ψ), τ, . . . , τ
0,n+1,β
= eτ /z+ X
β6=0,n,µ
qβ
n!eτ1/z+(τ1.β)Tµ
Tµ
z(z − ψ), τ2, . . . , τ2
0,n+1,β
,
(2.1)
where in the second expression τ = τ1+ τ2 with τ1 ∈ H2. The equality follows from the divisor equation for descendent invariants. Furthermore, the string equation for JX says that we can take out the fundamental class 1 from the variable τ to get an overall factor eτ0/z in front of (2.1).
The J -function can be considered as a map from H to zH−. Let Lf = TfL be the tangent space of L at f ∈ L. Let τ ∈ H be embedded into H+ via
H ∼= −1z + H ⊂ H+.
Set Lτ = L(τ,dF0(τ )). Here we list the basic structural results from [Giv04]:
(i) We have zLτ ⊂ Lτ, and so Lτ/zLτ ∼= H+/zH+∼= H has rank N := dim H.
(ii) We have Lτ ∩ L = zLτ, considered as subspaces inside H.
(iii) The subspace Lτ of H is the tangent space at every f ∈ zLτ ⊂ L. Moreover, Tf = Lτ implies f ∈ zLτ. The subspace zLτ is considered as the ruling of the cone.
(iv) The intersection of L and the affine space −1z + zH− is parameterized by its image −1z + H ∼= H via the projection by π. For τ ∈ H,
−zJX τ, −z−1 = −1z + τ + O(1/z) is the function of τ whose graph is the intersection.
(v) The set of all directional derivatives z∂µJX = Tµ+O(1/z) spans an N -dimensional subspace of Lτ, namely Lτ∩ zH−, such that its projection to Lτ/zLτ is an isomorphism.
Note that we have used a convention for the J -function which differs from that of some more recent papers [Giv04,CG07] by a factor z.
Lemma 2.1. The matrix z∇JX = (z ∂µJν) has column vectors z ∂µJX(τ ) that generate the tangent space Lτ of the Lagrangian cone L as an R{z}-module. Here a =P qβaβ(z) ∈ R{z} if aβ(z) ∈ C[z].
Proof. Apply result (v) above to Lτ/zLτ and multiply by zk to get zkLτ/zk+1Lτ.
We see that the germ of L is determined by an N -dimensional submanifold. In this sense, zJX generates L. Indeed, all discussions are applicable to the Gromov–Witten context only as formal germs around the neighborhood of q = −1z.
2.2 Generalized mirror transform for toric bundles
Let ¯p : X → S be a smooth fiber bundle such that H(X) is generated by H(S) and fiber divisors Di as an algebra, in such a way that there is no linear relation among the Di and H2(S). An example of X is a toric bundle over S. Assume that H(X) is a free module over H(S) with finite generators {De:=Q
iDeii}e∈Λ+. Let ¯t := P
st¯sT¯s be a general cohomology class in H(S), which is identified with ¯p∗H(S).
Similarly, denote by D = P tiDi the general fiber divisor. Elements in H(X) can be written as linear combinations of {T(s,e) = ¯TsDe}. Denote the ¯Ts-directional derivative on H(S) by
∂T¯s ≡ ∂¯ts, and denote the multiple derivative by
∂(s,e) := ∂¯ts
Y
i
∂teii.
Note, however, that most of the time z will appear with derivative. For notational convenience, denote the index (s, e) by e. We then set
∂ze ≡ ∂z(s,e):= z∂t¯s
Y
i
z ∂teii = z|e|+1∂(s,e). (2.2) As usual, the Te-directional derivative on H(X) is denoted by ∂e= ∂Te. Here Te is a special choice of the basis Tµ (and ∂µ) of H(X), defined by
Te≡ T(s,e)≡ ¯TsDe, e ∈ Λ+.
The two operators ∂ze and z∂e are by definition very different; nevertheless, they are closely related in the study of quantum cohomology, as we will see below.
Assume that ¯p : X → S is a toric bundle of split type, that is, the toric quotient of a split vector bundle over S. Let JS(¯t, z−1) be the J -function on S. The hypergeometric modification of JS by the ¯p-fibration takes on the form
IX t, D, z, z¯ −1 := X
β∈NE(X)
qβeDz+(D.β)IβX/S z, z−1JβSS ¯t, z−1 , (2.3)
with the relative factor IβX/S, whose explicit form for X = ˜E → S will be given in Section3.2.
The major difficulty which makes IX deviate from JX lies in the fact that in general, positive z-powers may occur in IX. Nevertheless, for each β ∈ NE(X) the power of z in IβX/S(z, z−1) is bounded from above by a constant depending only on β. Thus we may study IX in the space H := H[z, z−1]] over R.
Notice that the I-function is defined only on the subspace ˆt := ¯t + D ∈ H(S) ⊕M
i
CDi ⊂ H(X) . (2.4)
We will use the following theorem by Brown (and Givental).
Theorem 2.2 ([Bro14, Theorem 1]). The vector (−z)IX(ˆt, −z) lies in the Lagrangian cone L of X.
Definition 2.3 (GMT). For each ˆt, the vector zI(ˆt) lies in the subspace Lτ of L. The corre- spondence
ˆt 7→ τ (ˆt) ∈ H(X) ⊗ R
is called the generalized mirror transformation (cf. [CG07,Giv04]).
Remark 2.4. In general τ (ˆt) may be outside the submodule of the Novikov ring R generated by H(S) ⊕L
iCDi. This is in contrast to the (classical) mirror transformation where τ is a trans- formation within (H0(X) ⊕ H2(X))R (small parameter space).
To use Theorem 2.2, we start by outlining the idea behind the following discussions. By the properties of L, Theorem 2.2 implies that I can be obtained from J by applying a certain differential operator in the z∂eto it, with series in z as coefficients. However, what we need is the reverse direction, namely to obtain J from I, which amounts to removing the positive z-powers from I. Note that the I-function has variables only in the subspace H(S) ⊕L
iCDi. Thus a priori the reverse direction does not seem to be possible.
The key idea below is to replace derivatives in the missing directions by higher-order differ- entiations in the fiber divisor variables ti, a process similar to transforming a first-order ordinary differential equation system to a higher-order scalar equation. This is possible since H(X) is generated by the Di as an algebra over H(S).
Lemma 2.5. We have z∂1JX = JX and z∂1IX = IX.
Proof. The first equality is the string equation. For the second one, by definition IX =X
β
qβeD/z+(D·β)IβX/SJβSS(¯t) ,
where IβX/S depends only on z. The differentiation with respect to t0 (dual coordinate of 1) only applies to JβS
S(¯t). Hence the string equation on JβS
S(¯t) concludes the proof.
To avoid cluttered notation, we use I and J to denote the I-function and J -function, respec- tively, when the target space is clear.
Proposition 2.6. (1) The GMT τ = τ (ˆt) satisfies τ (ˆt, q = 0) = ˆt.
(2) Under the basis {Te}e∈Λ+, there exists an invertible formal series B(τ, z) with N × N matrices as coefficients, which is free from cohomology classes, such that
∂zeI ˆt, z, z−1 = z∇J τ, z−1B(τ, z) , (2.5) where (∂zeI) is the N × N matrix with ∂zeI as the column vectors.
Proof. By Theorem 2.2, we have zI ∈ L, hence z∂I ∈ T L = L. Then z(z∂)I ∈ zL ⊂ L and so z∂(z∂)I lies again in L. Inductively, ∂zeI lies in L. The factorization (∂zeI) = (z∇J )B(z) then follows from Lemma 2.1. Also, Lemma2.5 says that the I- and J -functions appear as the first column vectors of (∂zeI) and (z∇J ), respectively. By the R{z}-module structure, it is clear that B does not involve any cohomology classes.
By the definitions of J , I and ∂ze (cf. (2.1), (2.3), (2.2)), it is clear that
∂zeeˆt/z= Teeˆt/z, z∂eet/z = Teet/z (2.6) (t ∈ H(X)). Hence modulo Novikov variables, ∂zeI(ˆt) ≡ Teeˆt/z and z ∂eJ (τ ) ≡ Teeτ /z.
To prove statement (1), note that modulo all qβ we have eˆt/z ≡ X
e∈Λ+
Be,1(z)Teeτ (ˆt)/z. Thus
e(ˆt−τ (ˆt))/z ≡X
e
Be,1(z)Te,
which forces τ (ˆt) ≡ ˆt (and Be,1(z) ≡ δTe,1).
To prove statement (2), notice that by statement (1) and (2.6), we have B(τ, z) ≡ IN ×N modulo Novikov variables, so in particular B is invertible. Notice that in deriving (2.5) we do not need to worry about the sign on “−z” since it appears in both I and J .
Definition 2.7 (BF). The left-hand side of (2.5) involves z and z−1, while the right-hand side is the product of a function of z and a function of z−1. Such a matrix-factorization process is termed a Birkhoff factorization.
Besides its existence and uniqueness, for actual computations it will be important to know how to calculate τ (ˆt) directly or inductively.
Proposition 2.8. There are scalar-valued formal series Ce(ˆt, z) such that J τ, z−1 = X
e∈Λ+
Ce(ˆt, z) ∂zeI ˆt, z, z−1 , (2.7) where Ce≡ δTe,1 modulo Novikov variables.
In particular, τ (ˆt) = ˆt + · · · is determined by the coefficient of 1/z on the right-hand side.
Proof. Proposition2.6implies
z∇J = (∂zeI) B−1.
Take the first column vector on the left-hand side, which is z∇1J = J by Lemma 2.5. One gets expression (2.7) by defining Ce to be the corresponding eth entry of the first column vector of B−1. Modulo the qβ, we have B−1 ≡ IN ×N, hence Ce≡ δTe,1.
Definition 2.9. A differential operator P is of degree Λ+ if P = P
e∈Λ+Ce∂ze for some Ce. Namely, its components are multi-derivatives indexed by Λ+.
Theorem 2.10 (BF/GMT). There is a unique, recursively determined, scalar-valued degree Λ+ differential operator
P (z) = 1 + X
β∈NE(X)\{0}
qβPβ(ti, ¯ts, z; z∂ti, z∂t¯s) ,
with each Pβ polynomial in z, such that P (z)I(ˆt, z, z−1) = 1 + O(1/z).
Moreover,
J τ (ˆt), z−1 = P (z)I ˆt, z, z−1 , with τ (ˆt) determined by the coefficient of 1/z on the right-hand side.
Proof. The operator P (z) is constructed by induction on β ∈ NE(X). We set Pβ = 1 for β = 0.
Suppose that Pβ0 has been constructed for all β0 < β in NE(X). We set P<β(z) =P
β0<βqβ0Pβ0. Let
A1 = zk1qβ X
e∈Λ+
fe(ti, ¯ts)Te (2.8)
be the top z-power term in P<β(z)I. If k1 < 0, then we are done. Otherwise we will remove it by introducing “certain Pβ”. Consider the “naive quantization”
Aˆ1 := zk1qβ X
e∈Λ+
fe(ti, ¯ts)∂ze. (2.9)
In the expression
P<β(z) − ˆA1I = P<β(z)I − ˆA1I , the target term A1 has been removed since
Aˆ1I(q = 0) = ˆA1eˆt/z = A1eˆt/z = A1+ A1O(1/z) .
All the newly created terms either have smaller top z-power or have curve degree qβ00 with β00> β in NE(X). Thus we may keep on removing the new top z-power term A2, which has k2 < k1. Since the process will stop in no more than k1 steps, we simply define Pβ by
qβPβ = − X
16j6k1
Aˆj.
By induction we get P (z) =P
β∈NE(X)qβPβ, which is clearly of degree Λ+.
Now we prove the uniqueness of P (z). Suppose that P1(z) and P2(z) are two such operators.
The difference δ(z) = P1(z) − P2(z) satisfies δ(z)I =:X
β
qβδβI = O(1/z) .
Clearly δ0 = 0. If δβ 6= 0 for some β, then β can be chosen such that δβ0 = 0 for all β0 < β. Let the highest non-zero z-power term of δβ be zkP
eδβ,k,e∂ze. Then qβzkX
e
δβ,k,e∂ze
eˆt/z+ X
β16=0
qβ1Iβ1
+ RI = O(1/z) .
Here R denotes the remaining terms in δ. Note that terms in RI either do not contribute to qβ or have z-power less than k. Thus the only qβ-term is
qβzkX
e
δβ,k,eTeeˆt/z. This is impossible since k > 0 and {Te} is a basis. Thus δ = 0.
Finally, by Lemma2.1, both B and B−1 have entries in R{z}. Thus Proposition2.8provides an operator which satisfies the required properties. By the uniqueness it must coincide with the effectively constructed P (z).
2.3 Reduction to special BF/GMT
Proposition 2.11. Let f : X 99K X0be the projective local model of an ordinary flop with graph correspondence F . Suppose that there are formal lifts τ, τ0 of ˆt in H(X) ⊗ R and H(X0) ⊗ R, respectively, with τ (ˆt), τ0(ˆt) ≡ ˆt modulo Novikov variables in NE(S), and with F τ(ˆt) ∼= τ0(ˆt).
Then
F J(τ(ˆt)) · ξ ∼= J0(τ0(ˆt)) · ξ0 =⇒ F J(ˆt) · ξ ∼= J0(ˆt) · ξ0,
and consequently QH(X) and QH(X0) are analytic continuations of each other underF . Proof. For an induction on the weight w := (βS, d2) ∈ W , suppose that for all w0 < w we have the invariance of any n-point function (except that if β0S = 0, then n > 3). Here we would like to recall that W := (NE( ˜E)/∼) ⊂ NE(S) ⊕ Z is the quotient Mori cone.
By the definition of J in (2.1), for any a ∈ H(X) we may pick up the fiber series over w from the ξaz−(k+2)-component of the assumedF -invariance:
F τn, ψkξa X ∼=τ0n, ψkξ0F a X0. (2.10)
Write τ (ˆt) =P
¯
w∈Wτw¯(ˆt)qw¯. The fiber series is decomposed into a sum of subseries in q` of the form
τw¯1(ˆt), . . . , τw¯n(ˆt), ψkξa X w00q
Pn
j=1w¯j+w00
.
Since P ¯wj + w00 = w, any ¯wj-term with ¯wj 6= 0 leads to w00 < w, whose fiber series is of the form P
igi(q`, ˆt)hi(q`) with gi from Q τw¯j(ˆt) and hi a fiber series over w00. The gi are F - invariant by assumption and the hi are F -invariant by induction, thus the sum of products is alsoF -invariant.
From (2.10) and τ0(ˆt) = ˆt, the remaining fiber series with ¯wj = 0 for all j satisfies F ˆtn, ψkξa X
w ∼=ˆtn, ψkξF a Xw00, which holds for any n, k and a.
Now by Theorem 5.2 (divisorial reconstruction and WDVV reduction) of Part I [LLW16], this implies theF -invariance of all fiber series over w.
Later we will see that for the GMT τ (ˆt) and τ0(ˆt), the lifting condition τ (ˆt) ≡ ˆt modulo NE(S)\{0} (instead of modulo NE(X)\{0}) and the identity F J(τ(ˆt)) · ξ ∼= J0(τ0(ˆt)) · ξ0 hold for split ordinary flops.
3. Hypergeometric modification
From now on we work with a split local Pr flop f : X 99K X0 with bundle data (S, F, F0), where F =
r
M
i=0
Li and F0=
r
M
i=0
L0i.
We study the explicit formulae of the hypergeometric modifications IX and IX0 associated with the double projective bundles X → S and X0 → S, especially the symmetry property between them.
In order to get a better sense of the factor IX/S, it is necessary to have a precise description of the Mori cone first. We then describe the Picard–Fuchs equations associated with the I-function.
3.1 Minimal lift of curve classes and F -effective cone Let C be an irreducible projective curve with ψ : V = Lr
i=0O(µi) → C a split bundle. Let µ = max µi, and denote by ¯ψ : P (V ) → C the associated projective bundle. Let h = c1(OP (V )(1)), let
b = ¯ψ∗[C] · Hr = Hr = hr+ c1(V )hr−1 be the canonical lift of the base curve, and ` be the fiber curve class.
Lemma 3.1. The Mori cone NE(P (V )) is generated by ` and b − µ`.
Proof. Consider V0=O(−µ) ⊗ V = O ⊕ N. Then N is a semi-negative bundle and NE(P (V )) ∼= NE(P (V0)) is generated by ` and the zero section b0 of N → P1. In this case b0 is also the canonical lift b0 = h0r+ c1(V0)h0r−1. From the Euler sequence we know that h0 = h + µp. Hence
b0 = (h + µp)r+
r
X
i=1
(µi− µ)p(h + µp)r+1 = hr+ rµphr−1+
r
X
i=1
(µi− µ)phr−1
= hr+ c1(V )hr−1− µphr−1= b − µ` .
Let ψ : V = Lr
i=0Li → S be a split bundle with ¯ψ : P = P (V ) → S. Since ¯ψ∗: NE(P ) → NE(S) is surjective, for each βS ∈ NE(S) represented by a curve C =P
jnjCj, the determination of ¯ψ∗−1(βS) corresponds to the determination of NE(P (VCj)) for all j. Therefore by Lemma3.1, the minimal lift with respect to this curve decomposition is given by
βP :=X
j
nj( ¯ψ∗[Cj] · Hr− µCj`) = βS− µβS` , with µCj = maxi(Cj · Li) and µ = µβS := P
jnjµCj. As before we identify the canonical lift ψ¯∗βS· Hr with βS. Thus the crucial part is to determine the case of primitive classes. The general case follows from the primitive case by additivity. When there is more than one way to decompose into primitive classes, the minimal lift is obtained by taking the minimal one. Notice that further decomposition leads to a smaller (or equal) lift. Also, there could be more than one minimal lift coming from different (non-comparable) primitive decompositions.
Now we apply the above results to study the effective andF -effective curve classes under the local split ordinary flop f : X 99K X0of type (S, F, F0). Fixing a primitive curve class βS ∈ NE(S), we define
µi:= (βS· Li) , µ0i:= (βS· L0i) .
Let µ = max µi and µ0 = max µ0i. Then by working on an irreducible representation curve C of βS, we get by Lemma 3.1
NE(Z)βS = (βS− µ`) + Z>0` ≡ βZ+ Z>0` , NE(Z0)βS = (βS− µ0`0) + Z>0`0 ≡ βZ0+ Z>0`0.
Now we consider the further lifts of the primitive elements βZ and βZ0 to X. The bundle N ⊕O is of split type with Chern roots −h+L0i and 0 for i = 0, . . . , r. On βZ they take on values µ + µ0i (i = 0, . . . , r) and 0 . (3.1) To determine the minimal lift of βZ in X, we separate it into two cases.
Case (1): µ + µ0 > 0. The greatest number in (3.1) is µ + µ0 and NE(X)βZ = (βZ− (µ + µ0)γ) + Z>0γ . Case (2): µ + µ0 6 0. The greatest number in (3.1) is 0 and
NE(X)βZ = βZ+ Z>0γ . To summarize, we have the following.
Lemma 3.2. Given a primitive class βS ∈ NE(S), we have β = βS+ d` + d2γ ∈ NE(X) if and only if
d > −µ and d2> −ν , (3.2)
where ν = max{µ + µ0, 0}.
Remark 3.3. For the general case βS=P
jnj[Cj], the constants µ and ν are replaced by µ = µβS :=X
j
njµCj and ν = νβS :=X
j
njmax{µCj+ µC0
j, 0} .
Thus a geometric minimal lift βSX ∈ NE(X) for βS ∈ NE(S), with respect to the given primitive decomposition is
βXS = βS− µ` − νγ . (If µCj + µ0C
j > 0 for all j, then ν = µ + µ0.)
The geometric minimal lifts describe NE(X). We will however only need a “generic lift”
(I-minimal lift in Definition3.7) in the study of GW invariants.
Definition 3.4. A class β ∈ N1(X) is F -effective if β ∈ NE(X) and F β ∈ NE(X0).
Proposition 3.5. Let βS ∈ NE(S) be primitive. A class β ∈ NE(X) over βS is F -effective if and only if
d + µ > 0 and d2− d + µ0 > 0 . (3.3) Proof. Let β = βS + d` + d2γ, then F β = βS− d`0+ d2(γ0 + `0) = βS + (d2− d)`0 + d2γ =:
βS+ d0`0 + d02γ0. It is clear that β being F -effective implies both inequalities. Conversely, the two inequalities imply
d2 > d − µ0 > −(µ + µ0) > −ν , hence β ∈ NE(X). Similarly,F β ∈ NE(X0).
3.2 Symmetry for I For F =Lr
i=0Li and F0 =Lr
i=0L0i, the Chern polynomials for F and N ⊕O take on the form fF =Y
ai:=Y
(h + Li) , fN ⊕O = br+1Y
bi := ξY
(ξ − h + L0i) .
For β = βS+ d` + d2γ, we set µi := (Li.βS) and µ0i := (L0i.βS). Then for i = 0, . . . , r we have (ai· β) = d + µi, (bi· β) = d2− d + µ0i and (br+1· β) = d2. Let
λβ = (c1(X/S) · β) = (c1(F ) + c1(F0)) · βS+ (r + 2)d2. (3.4) The relative I-factor is given by
IβX/S := 1 zλβ
Γ 1 + ξ/z Γ 1 + ξ/z + d2
r
Y
i=0
Γ 1 + ai/z Γ 1 + ai/z + µi+ d
Γ 1 + bi/z
Γ 1 + bi/z + µ0i+ d2− d , (3.5) and the hypergeometric modification of ¯p : X → S is
I = I D, ¯t; z, z−1 = X
β∈NE(X)
qβeD/z+(D·β)IβX/SJβS
S(¯t) , (3.6)
where D = t1h + t2ξ is the fiber divisor and ¯t ∈ H(S).
In more explicit terms, for a split projective bundle ¯ψ : P = P (V ) → S, the relative I-factor equals
IβP /S :=
r
Y
i=0
β·(h+Li)
Y
m=1
(h + Li+ mz)−1, (3.7)
IβP /S :=
r
Y
i=0
1
β·(h+Li)
Y
m=1
(h + Li+ mz)
, (3.8)
where the product in m ∈ Z is directed in the sense that
s
Y
m=1
:=
s
Y
m=−∞
/
0
Y
m=−∞
. (3.9)
Thus for each i with β · (h + Li) 6 −1, the corresponding subfactor is understood as occurring in the numerator; furthermore, the numerator must contain the factor h + Li corresponding to m = 0. In general, I is viewed as a Laurent series in z−1 with cohomology-valued coefficients.
By the dimension constraint it in fact has only finite terms.
Remark 3.6. The relative factor comes from the equivariant Euler class of H0(C, TP /S|C) − H1(C, TP /S|C)
at the moduli point [C ∼= P1 → X].
Definition 3.7 (I-minimal lift). Introduce µIβS := max
i {βS· Li} , µ0IβS := max
i {βS· L0i} and
νβIS = maxµIβ
S+ µ0IβS, 0 > 0 . Define the I-minimal lift of βS to be
βSI := βS− µIβ
S` − νβISγ ∈ NE(X) , where βS∈ NE(X) is the canonical lift such that h · βS = 0 = ξ · βS.
Clearly, βSI is an effective class in NE(X), as µIβ
S 6 µβS and νβI
S 6 νβS. When the inequality is strict, the I-minimal lift is more effective than any geometric minimal lift. Nevertheless, it is uniquely defined and we will show that it encodes the information of the hypergeometric modification.
Definition 3.8. Define β to be I-effective, denoted by β ∈ NEI(X), if d > −µIβS and d2 > −νβIS.
It is called F I-effective if β is I-effective and F β is I0-effective. By the same proof as that of Proposition3.5, this is equivalent to
d + µIβS > 0 and d2− d + µ0Iβ
S > 0 .
Lemma 3.9 (Vanishing lemma). If ¯ψ∗β ∈ NE(S) but β 6∈ NE(P ), then IβP /S = 0. In fact, the vanishing statement holds for any β = βS+ d` with d < −µIβ
S.
Proof. We have β · (h + Li) = d + µi 6 d + µIβS < 0 for all i. This implies IβP /S = 0 since it contains the Chern polynomial factor Q
i(h + Li) = 0 in the numerator.
Now IβX/S ≡ IβZ/SIβX/Z is given by
r
Y
i=0 β·ai
Y
m=1
(ai+ mz)−1
r
Y
i=0 β·bi
Y
m=1
(bi+ mz)−1
β·ξ
Y
m=1
(ξ + mz)−1 =: AβBβCβ. (3.10) Although (3.10) makes sense for any β ∈ N1(X), we have the following.
Lemma 3.10. The I-factor IβX/S is non-trivial only if β ∈ NEI(X).
Proof. Indeed, if βS∈ NE(S) but β 6∈ NEI(X), then either d < −µIβ
S and Aβ = 0 by Lemma3.9, or d > −µIβS and we must have d2< −νβI
S 6 0 and all factors in Bβ appear in the numerator:
d2− d + µ0i 6 d2+ µIβS+ µ0IβS 6 d2+ νβIS < 0 . In particular, BβCβ contains the Chern polynomial fN ⊕O = 0.
Remark 3.11. In view of Lemma 3.2, if β is a primitive class, then β ∈ NEI(X) if and only if β ∈ NE(X). Hence the condition β ∈ NEI(X) is the “effective condition that β behaves as a primitive class.” One way to think about this is that the localization calculation of the I-factor is performed on the main component of the stable map moduli space where β is represented by a smooth rational curve.
As far as I is concerned, the I-effective class plays the role of effective class. However, one needs to be careful that the converse of Lemma 3.10 is not true: If β is I-effective, it is still possible to have IβX/S = 0.
The expression (3.10) agrees with (3.5) by taking out the z-factor with m. The total factor is clearly
z−
Pr
i=0ai+Pr+1 i=0bi
·β
= z−c1(X/S)·β. Similarly, for β0 ∈ NE(X0), the I-factor IX
0/S
β0 ≡ IβZ00/SIX
0/Z0
β0 is given by
r
Y
i=0 β0·a0i
Y
m=1
(a0i+ mz)−1
r
Y
i=0 β0·b0i
Y
m=1
(b0i+ mz)−1
β0·ξ0
Y
m=1
(ξ0+ mz)−1=: A0β0Bβ00Cβ00. (3.11) Here a0i= h0+ L0i =F bi and b0i = ξ0− h0+ Li =F ai.
By the invariance of the Poincar´e pairing, (β ·ai) = d+µi= (F β·b0i) and (β ·bi) = d2−d+µ0i = (F β · a0i), and it is clear that all the linear subfactors in IβX/S and IF βX0/S correspond perfectly under Aβ 7→ BF β0 , Bβ 7→ A0F β and Cβ 7→ CF β0 .
However, since the cup product is not preserved under F , in general F Iβ 6= IF β0 . Clearly, any direct comparison of Iβ and IF β0 (without analytic continuations) can make sense only if β is F I-effective. This is the case if the (β · ai) and (β · bi), respectively, are not all negative.
Namely, Aβ and Bβ both contain factors in the denominator.
Lemma 3.12 (Naive quasi-linearity). (1) F Iβ· ξ = IF β0 · ξ0. (2) If d2 := β.ξ < 0, thenF Iβ = IF β0 .
The expressions in statements (1) or (2) are non-trivial only if β isF I-effective.
Proof. Statement (1) follows from the fact that f : X 99K X0 is an isomorphism over the infin- ity divisors E ∼= E. For statement (2), notice that since d2 < 0, the factor Cβ contains ξ in the numerator corresponding to m = 0. Similarly, CF β0 contains ξ0 in the numerator. Hence, statement (2) follows for the same reason as statement (1). The last statement follows from Lemma3.10.
3.3 The Picard–Fuchs system
Now, we return to the BF/GMT constructed in Theorem 2.10 and multiply it by the infinity divisor ξ:
JX(τ (ˆt)) · ξ = P (z)IX(ˆt) · ξ .
