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.

This journal is c Foundation Compositio Mathematica2016. This article is distributed with Open Access under the terms of theCreative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact theFoundation Compositio Mathematica.

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 P^{r} flop
f : X 99K X^{0} (Part I [LLW16, Section 2.1]). The local geometry of the f -exceptional loci Z ⊂ X
and Z^{0} ⊂ X^{0} is encoded in a triple (S, F, F^{0}), where S is a smooth variety, and F and F^{0} 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 = P_{Z}(O(−1) ⊗ F^{0}⊕O) ,
X^{0} = ˜E^{0} = PZ^{0}(O(−1) ⊗ F ⊕ O) ,

where Z ∼= PS(F ) and Z^{0} ∼= PS(F^{0}) are projective bundles. In particular, X and X^{0} 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, τ_{k}aξi ,

where ¯ti ∈ H(S) and ξ is the common infinity divisor of X and X^{0}.

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),

J^{X} τ, z^{−1} := 1 + τ

z + X

β,n,µ

q^{β}
n!T_{µ}

T^{µ}

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 I_{β}
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 H^{2}, the generating function I(t) = P I_{β}q^{β}
determines J (τ ) on the small parameter space H^{0}⊕ H^{2} through the “classical” mirror transform
τ = τ (t). For a simple flop, X = X_{loc} 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
H^{0}⊕ H^{2}.)

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 X_{loc}, the bundles F and F^{0} are then assumed
to be split bundles.

In this paper (Part II), we only consider ordinary flops of split type, namely F ∼= Lr
i=0L_{i}
and F^{0} ∼=Lr

i=0L^{0}_{i} for some line bundles Li and L^{0}_{i} 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

I^{X} D, ¯t, z, z^{−1} :=X

β

q^{β}e^{D/z+(D.β)}I_{β}^{X/S} z, z^{−1}ψ¯^{∗}J_{β}^{S}

S ¯t, z^{−1}

lies in Givental’s Lagrangian cone generated by J^{X}(τ, z^{−1}). Here D = t^{1}h + t^{2}ξ, ¯t ∈ H(S) and
β_{S} = ¯ψ∗β, and the explicit form of I_{β}^{X/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

∂^{ze}I z, z^{−1} = z∇J z^{−1}B(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 t^{1}, t^{2} 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

q^{β}τβ(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 X^{0} by recon-
struction. These data are in turn encoded in the I-functions. One might be tempted to prove the
F -invariance by comparing I^{X} and I^{X}^{0}. While they look rather symmetric, the defect of the cup
product impliesF I^{X} 6= I^{X}^{0} 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˜_{µν}^{κ} (t)z∂κJ ,
where the ˜C_{µν}^{κ} (t) = P

ιg^{κι}∂_{µνι}^{3} 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 2^{X}` ,2^{X}γ

∼=

2^{X}`^{0}^{0},2^{X}γ^{0}^{0} .

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¯t^{i}T¯i. This
is achieved by lifting the QDE on QH(S), namely

z∂_{i}z∂_{j}J^{S} =X

k

C¯_{ij}^{k}(¯t)z∂_{k}J^{S},

to a differential system on H(X). A key concept needed for such a lift is the I-minimal lift of a
curve class β_{S} ∈ NE(S) to β_{S}^{I} ∈ 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 MI^{X} ∼=M_{I}^{X0}.
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

q^{β}^{I}^{S}e^{(D·β}^{S}^{I}^{)}C¯_{ij,β}^{k} _{S}(¯t)z∂kD_{β}^{I}

S(z)I ,
where D_{β}^{I}

S(z) is an operator depending only on β_{S}^{I}. 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 T_{e} (No-
tation4.1) of H(X):

z∂a(∂^{ze}I) = (∂^{ze}I)Ca(z, q) , where t^{a}= t^{1}, t^{2} or ¯t^{i}.

(3) The system has the property that for any fixed β_{S} ∈ NE(S), the coefficients are formal
functions in ¯t and polynomial functions in q^{γ}e^{t}^{2}, q^{`}e^{t}^{1} and f (q^{`}e^{t}^{1}). Here the basic rational
function

f (q) := q/ 1 − (−1)^{r+1}q

(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 (∂^{ze}I) = (z∇J )B of B can be
rephrased as

z∂_{a}(z∇J ) = (z∇J ) ˜C_{a},
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 X^{0}. 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^{−1}H[[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

q^{µ}_{k}Tµz^{k}∈ H_{+}

be a general point, where {T_{µ}} 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,kt^{µ}_{k}Tµψ^{k} serving as the
general descendent insertion. Let F_{0}(t) be the generating function of the genus zero descendent
Gromov–Witten invariants on X. Since F_{0} is a function on H_{+}, the 1-form dF_{0} gives a section
of π : H → H+.

Givental’s Lagrangian cone L is defined as the graph of dF_{0}, 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 q^{µ}_{k}∂/∂q^{µ}_{k}F0 = 2F0. Thus L is a cone with vertex q = 0 (cf. [Giv04,
Lee09]).

Let τ =P

µτ^{µ}T_{µ}∈ H. Define the (big) J -function to be
J^{X} τ, z^{−1} = 1 +τ

z + X

β,n,µ

q^{β}
n!T_{µ}

T^{µ}

z(z − ψ), τ, . . . , τ

0,n+1,β

= e^{τ /z}+ X

β6=0,n,µ

q^{β}

n!e^{τ}^{1}^{/z+(τ}^{1}^{.β)}Tµ

T^{µ}

z(z − ψ), τ2, . . . , τ2

0,n+1,β

,

(2.1)

where in the second expression τ = τ1+ τ2 with τ1 ∈ H^{2}. The equality follows from the divisor
equation for descendent invariants. Furthermore, the string equation for J^{X} says that we can
take out the fundamental class 1 from the variable τ to get an overall factor e^{τ}^{0}^{/z} in front of (2.1).

The J -function can be considered as a map from H to zH−. Let L_{f} = T_{f}L 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,dF}_{0}_{(τ ))}. Here we list the basic structural results from [Giv04]:

(i) We have zLτ ⊂ L_{τ}, and so Lτ/zLτ ∼= H+/zH+∼= H has rank N := dim H.

(ii) We have Lτ ∩ L = zL_{τ}, considered as subspaces inside H.

(iii) The subspace Lτ of H is the tangent space at every f ∈ zLτ ⊂ L. Moreover, T_{f} = 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,

−zJ^{X} 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∂µJ^{X} = Tµ+O(1/z) spans an N -dimensional subspace
of L_{τ}, namely L_{τ}∩ zH_{−}, such that its projection to L_{τ}/zL_{τ} 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∇J^{X} = (z ∂_{µ}J^{ν}) has column vectors z ∂_{µ}J^{X}(τ ) that generate the
tangent space L_{τ} of the Lagrangian cone L as an R{z}-module. Here a =P q^{β}a_{β}(z) ∈ R{z} if
aβ(z) ∈ C[z].

Proof. Apply result (v) above to L_{τ}/zL_{τ} and multiply by z^{k} to get z^{k}L_{τ}/z^{k+1}L_{τ}.

We see that the germ of L is determined by an N -dimensional submanifold. In this sense,
zJ^{X} 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
D_{i} as an algebra, in such a way that there is no linear relation among the D_{i} and H^{2}(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 {D^{e}:=Q

iD^{e}_{i}^{i}}_{e∈Λ}+.
Let ¯t := P

st¯^{s}T¯_{s} be a general cohomology class in H(S), which is identified with ¯p^{∗}H(S).

Similarly, denote by D = P t^{i}D_{i} the general fiber divisor. Elements in H(X) can be written
as linear combinations of {T_{(s,e)} = ¯TsD^{e}}. Denote the ¯Ts-directional derivative on H(S) by

∂T¯s ≡ ∂¯t^{s}, and denote the multiple derivative by

∂^{(s,e)} := ∂¯t^{s}

Y

i

∂_{t}^{e}i^{i}.

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 ∂_{t}^{e}i^{i} = z^{|e|+1}∂^{(s,e)}. (2.2)
As usual, the T_{e}-directional derivative on H(X) is denoted by ∂_{e}= ∂_{T}_{e}. Here T_{e} is a special
choice of the basis Tµ (and ∂µ) of H(X), defined by

Te≡ T_{(s,e)}≡ ¯TsD^{e}, 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 J^{S}(¯t, z^{−1}) be the J -function on S. The hypergeometric modification
of J^{S} by the ¯p-fibration takes on the form

I^{X} t, D, z, z¯ ^{−1} := X

β∈NE(X)

q^{β}e^{D}^{z}^{+(D.β)}I_{β}^{X/S} z, z^{−1}J_{β}^{S}_{S} ¯t, z^{−1} , (2.3)

with the relative factor I_{β}^{X/S}, whose explicit form for X = ˜E → S will be given in Section3.2.

The major difficulty which makes I^{X} deviate from J^{X} lies in the fact that in general, positive
z-powers may occur in I^{X}. Nevertheless, for each β ∈ NE(X) the power of z in I_{β}^{X/S}(z, z^{−1}) is
bounded from above by a constant depending only on β. Thus we may study I^{X} 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)I^{X}(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 L_{τ} 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 (H^{0}(X) ⊕ H^{2}(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 t^{i}, 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 D_{i} as an algebra over H(S).

Lemma 2.5. We have z∂1J^{X} = J^{X} and z∂_{1}I^{X} = I^{X}.

Proof. The first equality is the string equation. For the second one, by definition
I^{X} =X

β

q^{β}e^{D/z+(D·β)}I_{β}^{X/S}J_{β}^{S}_{S}(¯t) ,

where I_{β}^{X/S} depends only on z. The differentiation with respect to t^{0} (dual coordinate of 1) only
applies to J_{β}^{S}

S(¯t). Hence the string equation on J_{β}^{S}

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

∂^{ze}I ˆt, z, z^{−1} = z∇J τ, z^{−1}B(τ, z) , (2.5)
where (∂^{ze}I) is the N × N matrix with ∂^{ze}I 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, ∂^{ze}I lies in L. The factorization (∂^{ze}I) = (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 (∂^{ze}I) 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

∂^{ze}e^{ˆ}^{t/z}= Tee^{ˆ}^{t/z}, z∂ee^{t/z} = Tee^{t/z} (2.6)
(t ∈ H(X)). Hence modulo Novikov variables, ∂^{ze}I(ˆt) ≡ Tee^{ˆ}^{t/z} and z ∂eJ (τ ) ≡ Tee^{τ /z}.

To prove statement (1), note that modulo all q^{β} we have
e^{ˆ}^{t/z} ≡ X

e∈Λ^{+}

B_{e,1}(z)T_{e}e^{τ (ˆ}^{t)/z}.
Thus

e^{(ˆ}^{t−τ (ˆ}^{t))/z} ≡X

e

B_{e,1}(z)T_{e},

which forces τ (ˆt) ≡ ˆt (and B_{e,1}(z) ≡ δ_{T}_{e}_{,1}).

To prove statement (2), notice that by statement (1) and (2.6), we have B(τ, z) ≡ I_{N ×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^{+}

C_{e}(ˆt, z) ∂^{ze}I ˆt, z, z^{−1} , (2.7)
where C_{e}≡ δ_{T}_{e}_{,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 = (∂^{ze}I) B^{−1}.

Take the first column vector on the left-hand side, which is z∇_{1}J = J by Lemma 2.5. One gets
expression (2.7) by defining C_{e} to be the corresponding eth entry of the first column vector
of B^{−1}. Modulo the q^{β}, we have B^{−1} ≡ I_{N ×N}, hence Ce≡ δ_{T}_{e}_{,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}

q^{β}P_{β}(t^{i}, ¯t^{s}, z; z∂_{t}i, z∂t¯^{s}) ,

with each P_{β} 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 P_{β} = 1 for β = 0.

Suppose that Pβ^{0} has been constructed for all β^{0} < β in NE(X). We set P<β(z) =P

β^{0}<βq^{β}^{0}Pβ^{0}.
Let

A_{1} = z^{k}^{1}q^{β} X

e∈Λ^{+}

f^{e}(t^{i}, ¯t^{s})T_{e} (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 P_{β}”. Consider the “naive quantization”

Aˆ_{1} := z^{k}^{1}q^{β} X

e∈Λ^{+}

f^{e}(t^{i}, ¯t^{s})∂^{ze}. (2.9)

In the expression

P_{<β}(z) − ˆA_{1}I = P_{<β}(z)I − ˆA_{1}I ,
the target term A_{1} has been removed since

Aˆ_{1}I(q = 0) = ˆA_{1}e^{ˆ}^{t/z} = A_{1}e^{ˆ}^{t/z} = A_{1}+ A_{1}O(1/z) .

All the newly created terms either have smaller top z-power or have curve degree q^{β}^{00} with β^{00}> β
in NE(X). Thus we may keep on removing the new top z-power term A_{2}, which has k_{2} < k_{1}.
Since the process will stop in no more than k1 steps, we simply define Pβ by

q^{β}P_{β} = − X

16j6k1

Aˆ_{j}.

By induction we get P (z) =P

β∈NE(X)q^{β}P_{β}, 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) = P_{1}(z) − P_{2}(z) satisfies
δ(z)I =:X

β

q^{β}δ_{β}I = O(1/z) .

Clearly δ_{0} = 0. If δ_{β} 6= 0 for some β, then β can be chosen such that δ_{β}^{0} = 0 for all β^{0} < β. Let
the highest non-zero z-power term of δ_{β} be z^{k}P

eδ_{β,k,e}∂^{ze}. Then
q^{β}z^{k}X

e

δ_{β,k,e}∂^{ze}

e^{ˆ}^{t/z}+ X

β16=0

q^{β}^{1}I_{β}_{1}

+ RI = O(1/z) .

Here R denotes the remaining terms in δ. Note that terms in RI either do not contribute to q^{β}
or have z-power less than k. Thus the only q^{β}-term is

q^{β}z^{k}X

e

δ_{β,k,e}Tee^{ˆ}^{t/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 X^{0}be the projective local model of an ordinary flop with graph
correspondence F . Suppose that there are formal lifts τ, τ^{0} of ˆt in H(X) ⊗ R and H(X^{0}) ⊗ R,
respectively, with τ (ˆt), τ^{0}(ˆt) ≡ ˆt modulo Novikov variables in NE(S), and with F τ(ˆt) ∼= τ^{0}(ˆt).

Then

F J(τ(ˆt)) · ξ ∼= J^{0}(τ^{0}(ˆt)) · ξ^{0} =⇒ F J(ˆt) · ξ ∼= J^{0}(ˆt) · ξ^{0},

and consequently QH(X) and QH(X^{0}) are analytic continuations of each other underF .
Proof. For an induction on the weight w := (βS, d2) ∈ W , suppose that for all w^{0} < w we have
the invariance of any n-point function (except that if β^{0}_{S} = 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 τ^{n}, ψ^{k}ξaX ∼=τ^{0n}, ψ^{k}ξ^{0}F a^{X}^{0}. (2.10)

Write τ (ˆt) =P

¯

w∈Wτ_{w}_{¯}(ˆt)q^{w}^{¯}. The fiber series is decomposed into a sum of subseries in q^{`} of
the form

τ_{w}_{¯}_{1}(ˆt), . . . , τw¯n(ˆt), ψ^{k}ξaX
w^{00}q

Pn

j=1w¯j+w^{00}

.

Since P ¯wj + w^{00} = w, any ¯wj-term with ¯wj 6= 0 leads to w^{00} < w, whose fiber series is of
the form P

igi(q^{`}, ˆt)hi(q^{`}) with gi from Q τ_{w}_{¯}_{j}(ˆt) and hi a fiber series over w^{00}. The gi are F -
invariant by assumption and the h_{i} 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 ˆt^{n}, ψ^{k}ξaX

w ∼=ˆt^{n}, ψ^{k}ξF a^{X}_{w}0^{0},
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 τ^{0}(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 ∼= J^{0}(τ^{0}(ˆt)) · ξ^{0} hold
for split ordinary flops.

3. Hypergeometric modification

From now on we work with a split local P^{r} flop f : X 99K X^{0} with bundle data (S, F, F^{0}), where
F =

r

M

i=0

Li and F^{0}=

r

M

i=0

L^{0}_{i}.

We study the explicit formulae of the hypergeometric modifications I^{X} and I^{X}^{0} associated with
the double projective bundles X → S and X^{0} → S, especially the symmetry property between
them.

In order to get a better sense of the factor I^{X/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 µ_{i}, and denote by ¯ψ : P (V ) → C the associated projective bundle. Let h = c_{1}(OP (V )(1)),
let

b = ¯ψ^{∗}[C] · H_{r} = H_{r} = h^{r}+ c_{1}(V )h^{r−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 V^{0}=O(−µ) ⊗ V = O ⊕ N. Then N is a semi-negative bundle and NE(P (V )) ∼=
NE(P (V^{0})) is generated by ` and the zero section b^{0} of N → P^{1}. In this case b^{0} is also the
canonical lift b^{0} = h^{0r}+ c1(V^{0})h^{0r−1}. From the Euler sequence we know that h^{0} = h + µp. Hence

b^{0} = (h + µp)^{r}+

r

X

i=1

(µi− µ)p(h + µp)^{r+1} = h^{r}+ rµph^{r−1}+

r

X

i=1

(µi− µ)ph^{r−1}

= h^{r}+ c_{1}(V )h^{r−1}− µph^{r−1}= b − µ` .

Let ψ : V = Lr

i=0L_{i} → 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}(β_{S}) corresponds to the determination of NE(P (V_{C}_{j})) for all j. Therefore by Lemma3.1,
the minimal lift with respect to this curve decomposition is given by

β^{P} :=X

j

n_{j}( ¯ψ^{∗}[C_{j}] · H_{r}− µ_{C}_{j}`) = β_{S}− µ_{β}_{S}` ,
with µCj = maxi(Cj · L_{i}) and µ = µβS := P

jnjµCj. As before we identify the canonical lift
ψ¯^{∗}β_{S}· H_{r} with β_{S}. 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 X^{0}of type (S, F, F^{0}). Fixing a primitive curve class β_{S} ∈ NE(S),
we define

µ_{i}:= (β_{S}· L_{i}) , µ^{0}_{i}:= (β_{S}· L^{0}_{i}) .

Let µ = max µ_{i} and µ^{0} = max µ^{0}_{i}. Then by working on an irreducible representation curve C
of β_{S}, we get by Lemma 3.1

NE(Z)β_{S} = (βS− µ`) + Z>0` ≡ βZ+ Z>0` ,
NE(Z^{0})_{β}_{S} = (β_{S}− µ^{0}`^{0}) + Z>0`^{0} ≡ β_{Z}^{0}+ Z>0`^{0}.

Now we consider the further lifts of the primitive elements β_{Z} and β_{Z}^{0} to X. The bundle
N ⊕O is of split type with Chern roots −h+L^{0}_{i} and 0 for i = 0, . . . , r. On β_{Z} they take on values
µ + µ^{0}_{i} (i = 0, . . . , r) and 0 . (3.1)
To determine the minimal lift of β_{Z} in X, we separate it into two cases.

Case (1): µ + µ^{0} > 0. The greatest number in (3.1) is µ + µ^{0} and
NE(X)_{β}_{Z} = (β_{Z}− (µ + µ^{0})γ) + Z>0γ .
Case (2): µ + µ^{0} 6 0. The greatest number in (3.1) is 0 and

NE(X)_{β}_{Z} = βZ+ Z>0γ .
To summarize, we have the following.

Lemma 3.2. Given a primitive class βS ∈ NE(S), we have β = β_{S}+ d` + d2γ ∈ NE(X) if and
only if

d > −µ and d2> −ν , (3.2)

where ν = max{µ + µ^{0}, 0}.

Remark 3.3. For the general case β_{S}=P

jn_{j}[C_{j}], the constants µ and ν are replaced by
µ = µ_{β}_{S} :=X

j

njµ_{C}_{j} and ν = ν_{β}_{S} :=X

j

njmax{µ_{C}_{j}+ µ_{C}^{0}

j, 0} .

Thus a geometric minimal lift β_{S}^{X} ∈ NE(X) for β_{S} ∈ NE(S), with respect to the given primitive
decomposition is

β^{X}_{S} = β_{S}− µ` − νγ .
(If µ_{C}_{j} + µ^{0}_{C}

j > 0 for all j, then ν = µ + µ^{0}.)

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(X^{0}).

Proposition 3.5. Let βS ∈ NE(S) be primitive. A class β ∈ NE(X) over β_{S} is F -effective if
and only if

d + µ > 0 and d_{2}− d + µ^{0} > 0 . (3.3)
Proof. Let β = β_{S} + d` + d_{2}γ, then F β = βS− d`^{0}+ d_{2}(γ^{0} + `^{0}) = β_{S} + (d_{2}− d)`^{0} + d_{2}γ =:

β_{S}+ d^{0}`^{0} + d^{0}_{2}γ^{0}. It is clear that β being F -effective implies both inequalities. Conversely, the
two inequalities imply

d_{2} > d − µ^{0} > −(µ + µ^{0}) > −ν ,
hence β ∈ NE(X). Similarly,F β ∈ NE(X^{0}).

3.2 Symmetry for I
For F =L_{r}

i=0L_{i} and F^{0} =L_{r}

i=0L^{0}_{i}, the Chern polynomials for F and N ⊕O take on the form
f_{F} =Y

a_{i}:=Y

(h + L_{i}) , f_{N ⊕}_{O} = b_{r+1}Y

b_{i} := ξY

(ξ − h + L^{0}_{i}) .

For β = β_{S}+ d` + d_{2}γ, we set µ_{i} := (L_{i}.β_{S}) and µ^{0}_{i} := (L^{0}_{i}.β_{S}). Then for i = 0, . . . , r we have
(ai· β) = d + µ_{i}, (bi· β) = d_{2}− d + µ^{0}_{i} and (br+1· β) = d_{2}. Let

λβ = (c1(X/S) · β) = (c1(F ) + c1(F^{0})) · βS+ (r + 2)d2. (3.4)
The relative I-factor is given by

I_{β}^{X/S} := 1
z^{λ}^{β}

Γ 1 + ξ/z Γ 1 + ξ/z + d2

r

Y

i=0

Γ 1 + a_{i}/z
Γ 1 + ai/z + µi+ d

Γ 1 + b_{i}/z

Γ 1 + bi/z + µ^{0}_{i}+ d2− d , (3.5)
and the hypergeometric modification of ¯p : X → S is

I = I D, ¯t; z, z^{−1} = X

β∈NE(X)

q^{β}e^{D/z+(D·β)}I_{β}^{X/S}J_{β}^{S}

S(¯t) , (3.6)

where D = t^{1}h + t^{2}ξ 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

I_{β}^{P /S} :=

r

Y

i=0

β·(h+Li)

Y

m=1

(h + Li+ mz)^{−1}, (3.7)

I_{β}^{P /S} :=

r

Y

i=0

1

β·(h+Li)

Y

m=1

(h + L_{i}+ 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
H^{0}(C, T_{P /S}|_{C}) − H^{1}(C, T_{P /S}|_{C})

at the moduli point [C ∼= P^{1} → X].

Definition 3.7 (I-minimal lift). Introduce
µ^{I}_{β}_{S} := max

i {β_{S}· L_{i}} , µ^{0I}_{β}_{S} := max

i {β_{S}· L^{0}_{i}}
and

ν_{β}^{I}_{S} = maxµ^{I}_{β}

S+ µ^{0I}_{β}_{S}, 0 > 0 .
Define the I-minimal lift of β_{S} to be

β_{S}^{I} := βS− µ^{I}_{β}

S` − ν_{β}^{I}_{S}γ ∈ NE(X) ,
where β_{S}∈ NE(X) is the canonical lift such that h · β_{S} = 0 = ξ · β_{S}.

Clearly, β_{S}^{I} is an effective class in NE(X), as µ^{I}_{β}

S 6 µβ_{S} and ν_{β}^{I}

S 6 νβ_{S}. 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 ∈ NE^{I}(X), if
d > −µ^{I}βS and d2 > −νβ^{I}S.

It is called F I-effective if β is I-effective and F β is I^{0}-effective. By the same proof as that of
Proposition3.5, this is equivalent to

d + µ^{I}_{β}_{S} > 0 and d2− d + µ^{0I}_{β}

S > 0 .

Lemma 3.9 (Vanishing lemma). If ¯ψ∗β ∈ NE(S) but β 6∈ NE(P ), then I_{β}^{P /S} = 0. In fact, the
vanishing statement holds for any β = β_{S}+ d` with d < −µ^{I}_{β}

S.

Proof. We have β · (h + Li) = d + µi 6 d + µ^{I}_{β}_{S} < 0 for all i. This implies I_{β}^{P /S} = 0 since it
contains the Chern polynomial factor Q

i(h + L_{i}) = 0 in the numerator.

Now I_{β}^{X/S} ≡ I_{β}^{Z/S}I_{β}^{X/Z} is given by

r

Y

i=0 β·ai

Y

m=1

(a_{i}+ mz)^{−1}

r

Y

i=0 β·bi

Y

m=1

(b_{i}+ mz)^{−1}

β·ξ

Y

m=1

(ξ + mz)^{−1} =: A_{β}B_{β}C_{β}. (3.10)
Although (3.10) makes sense for any β ∈ N_{1}(X), we have the following.

Lemma 3.10. The I-factor I_{β}^{X/S} is non-trivial only if β ∈ NE^{I}(X).

Proof. Indeed, if β_{S}∈ NE(S) but β 6∈ NE^{I}(X), then either d < −µ^{I}_{β}

S and A_{β} = 0 by Lemma3.9,
or d > −µ^{I}_{β}_{S} and we must have d_{2}< −ν_{β}^{I}

S 6 0 and all factors in Bβ appear in the numerator:

d2− d + µ^{0}_{i} 6 d2+ µ^{I}_{β}_{S}+ µ^{0I}_{β}_{S} 6 d2+ ν_{β}^{I}_{S} < 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 ∈ NE^{I}(X) if and only if
β ∈ NE(X). Hence the condition β ∈ NE^{I}(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 I_{β}^{X/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^{−c}^{1}^{(X/S)·β}.
Similarly, for β^{0} ∈ NE(X^{0}), the I-factor I^{X}

0/S

β^{0} ≡ I_{β}^{Z}0^{0}^{/S}I^{X}

0/Z^{0}

β^{0} is given by

r

Y

i=0
β^{0}·a^{0}_{i}

Y

m=1

(a^{0}_{i}+ mz)^{−1}

r

Y

i=0
β^{0}·b^{0}_{i}

Y

m=1

(b^{0}_{i}+ mz)^{−1}

β^{0}·ξ^{0}

Y

m=1

(ξ^{0}+ mz)^{−1}=: A^{0}_{β}0B_{β}^{0}0C_{β}^{0}0. (3.11)
Here a^{0}_{i}= h^{0}+ L^{0}_{i} =F bi and b^{0}_{i} = ξ^{0}− h^{0}+ Li =F ai.

By the invariance of the Poincar´e pairing, (β ·a_{i}) = d+µ_{i}= (F β·b^{0}_{i}) and (β ·b_{i}) = d_{2}−d+µ^{0}_{i} =
(F β · a^{0}_{i}), and it is clear that all the linear subfactors in I_{β}^{X/S} and I_{F β}^{X}^{0}^{/S} correspond perfectly
under A_{β} 7→ B_{F β}^{0} , B_{β} 7→ A^{0}_{F β} and C_{β} 7→ C_{F β}^{0} .

However, since the cup product is not preserved under F , in general F Iβ 6= I_{F β}^{0} . Clearly,
any direct comparison of I_{β} and I_{F β}^{0} (without analytic continuations) can make sense only if
β is F I-effective. This is the case if the (β · ai) and (β · b_{i}), 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 = I_{F β}^{0} · ξ^{0}.
(2) If d2 := β.ξ < 0, thenF Iβ = I_{F β}^{0} .

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 X^{0} is an isomorphism over the infin-
ity divisors E ∼= E. For statement (2), notice that since d2 < 0, the factor C_{β} contains ξ in
the numerator corresponding to m = 0. Similarly, C_{F β}^{0} contains ξ^{0} 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:

J^{X}(τ (ˆt)) · ξ = P (z)I^{X}(ˆt) · ξ .