A + B theory in conifold transitions of Calabi--Yau threefolds

arXiv:1502.03277v1 [math.AG] 11 Feb 2015

FOR CALABI–YAU THREEFOLDS

YUAN-PIN LEE, HUI-WEN LIN, AND CHIN-LUNG WANG

ABSTRACT. For projective conifold transitions between Calabi-Yau three-folds X and Y, with X close to Y in the moduli, we show that the com-bined information provided by the A model (Gromov–Witten theory in all genera) and B model (variation of Hodge structures) on X determines the corresponding combined information on Y, and vice versa.

CONTENTS

0. Introduction 1

1. Preliminaries of conifold transitions 7 2. Hodge theory and the basic exact sequence 10 3. Gromov–Witten theory and Dubrovin connections 19 4. Periods and Gauss–Manin connections 29 5. From A(X) +B(X)to A(Y) +B(Y) 45 6. From A(Y) +B(Y)to A(X) +B(X) 53 7. Remarks on the basic exact sequence 57

References 59

0. INTRODUCTION

0.1. Statements of results and idea of proofs. Let X be a smooth projective 3-fold. A (projective) conifold transition XրY is a projective degeneration

π : X_→∆

of X to a singular variety ¯X = X_{0 with a finite number of ordinary} dou-ble points (abbreviated as ODPs or nodes) p1,· · · , pk, locally analytically

defined by the equation

x21+x22+x23+x24 =0,

followed by a projective small resolution

ψ : Y _→X.¯

In the process of complex degeneration from X to ¯X, k vanishing cycles Si ∼= S3 with trivial normal bundle collapse to nodes pi. In the process of

“K¨ahler degeneration” from Y to ¯X, the exceptional loci of ψ above each pi

is a smooth rational curve Ci ∼=P1with NCi/Y ∼=OP1(−1) ⊕OP1(−1). (See

Section 1 for details.) We write YցX for the reverse process.

Notice that ψ is a crepant resolution and π is a finite distance degener-ation with respect to the quasi-Hodge metric [37, 38]. A transition of this type (in all dimensions) is called an extremal transition. All known Calabi– Yau 3-folds with the same fundamental group are connected through ex-tremal transitions, of which conifold transitions are the simplest kind. It therefore makes sense to start the investigation with conifold transitions. In this paper we mainly consider conifold transitions among projective Calabi–

Yau threefolds.

We start by studying the changes of the so-called A model and B model under a general projective conifold transition. In the scope of this paper, the A model is the Gromov–Witten theory of all genera; the B model is the variation of Hodge structures (VHS), which is in a sense only the genus zero part of the quantum B model. A conifold transition, or more generally an extremal transition, can be regarded as a finite distance B model degen-eration followed by an inverse of a finite distance A model degendegen-eration. In contrast to the usual birational K equivalence, an extremal transition may be considered as a generalized K equivalence in the sense that ψ is crepant and the degenerating family π preserves sections of canonical bundles.

In general, the conditions for the existence of projective conifold tran-sitions is an unsolved problem except in the case of Calabi–Yau 3-folds, for which we have fairly complete understanding. For the inverse coni-fold transition Y ց X, a celebrated theorem of Friedman [7] (also

Kawa-mata [14] and Tian [36]) states that a small contraction Y → _{X can be}¯ smoothed if and only if there is a totally nontrivial relation between the exceptional curves. (Friedman’s theorem was inspired by Clemens’s ear-lier work [4].) That is, there exist constants ai 6= 0 for all i = 1, . . . , k such

that ∑k_i=1ai[Ci] =0. These are relations among curves[Ci]’s in the kernel of

H2(Y)Z → H2(X)Z. Let µ be the number of independent relations and let A ∈ Mk×µ(Z)be the relation matrix for Ci’s. Therefore, the dimension of

H2(Y)/H2(X) is k−µ. Conversely, Smith, Thomas and Yau proved dual

statement in [34], asserting that for a conifold transition X _րY the k

van-ishing 3-spheres Si must satisfy a totally nontrivial relation ∑ki=1bi[Si] = 0

with bi 6= 0 for all i. They are relations among the vanishing cycles [Si]’s

in VZ := ker(H3(X)Z → H3(X¯)Z). Similarly, let ρ be the number of

inde-pendent relations and B ∈ Mk×ρ(Z)be the relation matrix for Si’s. Thus

dimension of V is k−ρ.

The relation matrices A and B are defined for general conifold transitions regardless the Calabi–Yau assumption on X and Y. It turns out that µ+ρ= k and the following exact sequence holds.

Theorem 0.1(= Theorem 2.9). Under a conifold transition X ր Y of smooth projective threefolds, we have

0→ H2(Y)/H2(X)_−→B Ck _−→At _{V →0}

as a exact sequence of weight two Hodge structures.

We interpret this as a partial exchange of topological information be-tween the excess A model of Y over X (in terms of H2(Y)/H2(X)) and the

excess B model of X over Y in terms of V.

The main goal of this paper is to study the changes of A and B models under a projective conifold transition of Calabi–Yau 3-folds and its inverse. We remark that the Kuranishi spaceM_X¯ is smooth due to the

unobstruct-edness result of Ran, Kawamata and Bogomolov–Tian–Todorov. Our main result is the following theorem, stated in a imprecise form below.

Theorem 0.2. Let XրY be a projective conifold transition of Calabi–Yau three-folds such that[X]is a nearby point of[_X¯]inM_X¯. Then

(A+B)theory of X⇐⇒ (A+B)theory of Y. More precisely,

(1) A(X)is a sub-theory of A(Y).

(2) B(Y)is a sub-theory of B(X).

(3) A(Y)can be reconstructed from a refined A model of X◦ := X\Sk

i=1Si

“linked” by the vanishing spheres.

(4) B(X)can be reconstructed from a refined B model of Y◦ := Y\Sk

i=1Ci

“linked” by the exceptional curves.

The meaning of these slightly obscure statements can not be made com-pletely precise in the limited space here, but will take the entire paper to spell them out. Nevertheless, we will attempt to give a brief explanation below.

(1) is due to Li–Ruan [22]. Even though transitions for A model were intensively studied in the physics literature since 1990’s, the mathemati-cal study started in [22]. In fact, it follows from their degeneration for-mula (cf. Proposition 3.1) that Gromov–Witten invariants on X can be re-constructed from those on Y. This gives (1).

For (2), we note that there are natural identifications of MY with the

boundary of _MX¯ consisting of equisingular deformations, and MX with

MX¯ \Dwhere the discriminant locus D is a central hyperplane arrangement

with axis MY (cf. Section 4, especially Section 4.3.3). Therefore, the VHS

associated to Y can be considered as a sub-system of VHS (with logarithmic singularity) associated to ¯X, which is a regular singular extension of the

VHS associated to X.

With (3), we first have to introduce the “linking data” of the holomorphic curves in X◦, which not only records the curve classes but also how the curve links with the vanishing spheresS

identified with the curve classes in Y by H2(X◦) ∼= H2(Y)(cf. Definition 5.3

and (5.4)). We then proceed to show, by the degeneration argument, that the virtual class of moduli spaces of stable maps to X◦is naturally a disjoint union of pieces labeled by elements of the linking data (c.f. Proposition 5.9): Given β a curve class in X, we can associate to it a set of curve classes γ on

Y, called the liftings of β, so that there is a decomposition [M(X, β)]virt =

∐

γ∈H2(X◦)

[M(X, γ)]virt _∼

∐

[M(Y, γ)]virt.

Furthermore, the Gromov–Witten invariants of the curve class γ in Y is the same as the numbers produced by the component of the virtual class on X labeled by the corresponding linking data (c.f. Proposition 5.9). Thus, the refined A model is really the “linked A model” and the linked A model on X is equivalent to the (usual) A model of Y (for non-extremal curves classes) in all genera. Note that here the vanishing cycles from B(X)plays a key role in reconstructing A(Y).

For (4), the goal is to reconstruct VHS on MX from VHS on MY and

A(Y). The deformation of ¯X is unobstructed. Moreover it is well known

that Def(X¯) ∼= H1_(Y◦_{, T}

Y◦). Even though geometrically the deformation of

Y◦ is obstructed (in the direction transversal toMY), there is a first order

deformation parameterized by H1_(Y◦_{, T}

Y◦)which gives enough initial

con-dition to uniquely determine the degeneration of Hodge bundles onMX¯

nearMY. The most technical result needed in this process is a short exact

sequence

0→V →H3(X) →H3(Y◦) →0

which connects the limiting mixed Hodge structure (MHS) of Schmid on

H3_{(X)and the canonical MHS of Deligne on H}3_(Y◦_{)(c.f. Proposition 6.1).}

Together with the monodromy data associated to the ODPs, which is en-coded in the relation matrix A of the extremal rays on Y, we will be able to determine the VHS on MX nearMY. In the process, we need a slight

extension of Schmid’s nilpotent orbit theorem [32] to degenerations with certain non-normal crossing discriminant loci.

Consider a degeneration of polarized Hodge structures over ∆µ ×M

with discriminant locus D = Sk

i=1Di being a central hyperplane

arrange-ment with axis M. Let N(i)be the nilpotent monodromy around the hyper-plane Di _{= Z(w}

i)and suppose that the monodromy group Γ generated by

N(i)’s is abelian. Let D denote the period domain and ˇD its compact dual.

We prove in Theorem 4.15 that the period map

φ : ∆µ×M\D_→D/Γ takes the following form

φ(r, s) =exp k

∑

where ψ : ∆µ×M →D is holomorphic and horizontal.ˇ

The fact that the monodromy group Γ is abelian for conifold degenera-tions follows from the Picard–Lefschetz formula easily (c.f. (4.5) in Section 4). In particular it applies to degenerations over_MX¯ associated to conifold

transitions XրY of Calabi–Yau 3-folds through ¯X. In general, the abelian

constraint is automatic if µ ≥3 (see Remark 4.16).

In the proof of Theorem 0.2 (4) (Section 6.3), it turns out that the natural coordinates r1,· · · , rµonMX¯ in the directions transversal toMYare given

by periods of independent vanishing cycles Γ1,· · · , Γµ(a basis of V):

rj =

Z

Γ_jΩ,

where Ω is a relative holomorphic 3-form over_MX¯. The horizontal map ψ

corresponds to the refined B model on Y◦, and the linear forms

wi =

Z

Si

Ω=ai1r1+ · · · +aiµrµ, 1≤i≤k

correspond to the k×µ relation matrix A = (aij). Thus the above

factor-ization of φ gives the precise description of the “linking” of B(Y◦)by the exceptional curves Ci’s (viewed as data from A(Y)). This completes the

outline of the proof of Theorem 0.2.

The following comparison of proofs of (3) and (4) might be helpful. In the proof of (3), we calculate the Gromov–Witten invariants associated to the extremal rays in Y, via the multiple cover formulas and relation ma-trix B of the vanishing spheres Si’s in X. As a consequence we determine

the monodromy along the discriminant loci in the K¨ahler moduli (see Sec-tion 3.3). Combined with the refined A model on X◦ we then determine

A(Y). Notice that we actually identify all Gromov–Witten invariants with-out making use of the monodromy or the Dubrovin connection—though at the end we may write down a dual formulation as in Theorem 4.15.

In the proof of (4), the starting point is the Picard–Lefschetz monodromy and then the refined B model on Y◦. Finally they are “linked” via the ex-tended nilpotent orbit theorem (Theorem 4.15). In this approach the corre-sponding invariants are the so called Yukawa couplings, which turn out are derived as consequences (see Proposition 4.18 and Section 4.3.4) and are not used in the proof. Thus, while the general structures on both directions are similar, the technical details and logic of proof are different.

0.2. Motivation and future plans. All known examples of Calabi–Yau 3-folds of the same fundamental group are connected by extremal transitions, and many of them are indeed known to be connected by conifold transi-tions. The famous Reid’s fantasy [29] suggests the possibility that in fact all of them are connected by conifold transitions. Therefore, in order to study

A model or B model of any Calabi–Yau threefold one might “only” needs

one simplest example, which is “easy”. This work is meant to be the first general study in this direction.

Theorem 0.2 above can be interpreted as partial exchange of A and B models under a conifold transition. We hope to be able to answer the fol-lowing intriguing question concerning with “global symmetries” on mod-uli spaces of Calabi–Yau 3-folds in the future: Would this partial exchange of

A and B models lead to “full exchange” when one connects a Calabi–Yau threefold to its mirror via a finite steps of extremal transitions? If so, what is the rela-tion between this full exchange and the one induced by “mirror symmetry”? In

particular, the Fermat quintic and its mirror would be an excellent testing ground as their genus zero A model are both computed in [9, 24] and [19]. To this end, we need to devise a computationally effective way to achieve explicit determination. One missing piece of ingredients in this direction is a blowup formula in the Gromov–Witten theory for conifolds, which we are working on and have had some partial success [18].1The reverse impli-cation is not constructive either. It might be possible to explicitly construct the VHS of X from that of Y◦ via the logarithmic model of degenerating Hodge structure of Steenbrink [35] (and Clemens [5]). The details remain to be worked out.

More speculatively, the mutual determination of A and B models on X and Y leads us to surmise the possibility of a unified “A+B model” which

will be invariant under any extremal transition. For example, the string the-ory only predicts that Calabi–Yau threefolds form an important ingredient of our universe, but fails to tell us which Calabi–Yau threefold we should live in. Should the A+B model be available and proven to be invariant

under any extremal transition, there is no need to choose which universe to live in (at least for the worlds governed by the TQFT).

The first step of achieving this goal is to find a D-module version of the basic exact sequence (Theorem 0.1). On V there is a natural flat con-nection given by the Gauss–Manin concon-nection. H2(Y)/H2(X)is naturally endowed with the Dubrovin connection. Therefore, it is not unreasonable to expect aD-module lift of the basic exact sequence (c.f. Proposition 7.1), which may be heuristically interpreted as

“excess A theory”+“excess B theory”=“trivial”.

We hope to be able to “glued” the flat (log) connections of the excess the-ories to the Dubrovin connection on the A side and the Gauss–Manin con-nection on the B side. This will be a key step in constructing the speculative

A+B theory.

0.3. Outline of the paper. In Section 1 we review the basic geometry of a projective conifold transition.

1_{For (smooth) blowups with complete intersection centers, we have a fairly good} solu-tion in genus zero.

In Section 2, we compute the limiting mixed Hodge structures of the two semistable models associated to the conifold degeneration. Using ingredi-ents from Hodge theory, we derive the basic exact sequence in Theorem 2.9. Section 3 is devoted to some discussions on Gromov–Witten theory un-der a conifold transition. We explain Theorem 0.2 (1) in Section 3.1 and then concentrate on the genus zero Gromov–Witten theory associated to the exceptional curves (extremal rays) of ψ : Y→X.¯

In Section 4, we recall the relevant deformation theory of Calabi–Yau threefold conifolds and extend part of Bryant–Griffiths’s study of periods of smooth Calabi–Yau threefolds to Calabi–Yau conifolds. In doing so, we prove an extension of the nilpotent orbit theorem where the discriminant loci is not a simple normal crossing divisor but a central hyperplane ar-rangement. Using it, we identify the singular part of the period map.

Section 5 finishes the proof of Theorem 0.2 (3). The major new construc-tion in this secconstruc-tion is the definiconstruc-tion of the refined Gromov–Witten invariants on X◦:=X_\Sk

i=1Si. Together with ingredients on extremal ray invariants

from Section 3 we complete the determination of A(Y).

With Section 6 the proof Theorem 0.2 (4) is complete. The major theme in this section is to study the deformation theory on Y◦ := Y\Sk

i=1Ci.

The resulting variations of mixed Hodge structures is what we called the refined B model on Y◦. Together with the extended nilpotent orbit theorem we complete the determination of B(X).

The paper is concluded in Section 7 by two remarks concerning our fu-ture plans on the D-module lift of the basic exact sequence and effective methods to determine Gromov–Witten theory on Y in terms of X.

0.4. Acknowledgements. We are grateful to C.H. Clemens, C.-C. M. Liu, I. Smith, and R. Thomas, for discussions related to this project.

Y.-P. L.’s research is partially funded by the National Science Foundation. H.-W. Lin and C.-L. Wang are both supported by the Ministry of Science and Technology, Taiwan. We are grateful to Taida Institute of Mathematical Sciences (TIMS) for its generous and constant support which makes this long term collaboration possible.

1. PRELIMINARIES OF CONIFOLD TRANSITIONS

In Sections 1–3, all discussions are for any projective conifold transition

without the Calabi–Yau condition, unless otherwise specified. The Calabi–Yau

condition is imposed in Sections 4–6. 1.1. Local geometry.

Definition 1.1. Let X be a smooth projective 3-fold. A (projective) conifold transition XրY is a projective degeneration

of X to a singular variety ¯X = X_{0with a finite number of ordinary double} points (ODPs, nodes, A1 singularities) pi,· · · , pk, followed by a projective

small resolution

ψ : Y →_X.¯

We write YցX for the inverse conifold transition.

Locally analytically, an ordinary double point is defined by the equation (1.1) x21+x22+x23+x24 =0,

or equivalently

uv₋ws=0.

The small resolution ψ can be achieved by blowing up the Weil divisor defined by u=w=0 or by u=s =0, these two choices differ by a flop.

Lemma 1.2. The exceptional locus of ψ above each pi is a smooth rational curve

Ci ∼=P1with the normal bundles

NCi/Y ∼=OP1(−1) ⊕OP1(−1). Topologically, NCi/Yis a trivial rank 4 real vector bundle.

Proof. This follows from the above local description of blowing up. Away

from the isolated singular points pi’s, the Weil divisors are Cartier and the

blowups do nothing. Locally near pi, the Weil divisor is generated by two

functions u and w. The blowup Y ⊂ A4_×P1_{is defined by z0v−z1s} = 0, in addition to uv−ws = 0 defining X, where(z0 : z1)are the coordinates

of P1. Namely we have u w = s v = z0 z1.

It is now easy to see the exceptional locus near pi is isomorphic to P1and

the normal bundle is as described (by the definition of OP1(−1)). It is

topo-logically trivial since all Z/2 Stiefel–Whitney classes wk’s are zero. 

Locally to each node p= pi ∈ X, the transition X¯ րY can be considered

as two different ways of “smoothing” the singularities in ¯X: deformation

leads to Xt and small resolution leads to Y. Topologically, we have seen

that the exceptional loci of ψ are ∐ki=1Ci, a disjoint union of k 2-spheres.

For the deformation, the classical results of Picard, Lefschetz and Milnor state that there are k vanishing 3-spheres Si ∼=S3.

Lemma 1.3. Topologically the normal bundle NSi/Xt ∼= T

∗ Si is a trivial rank 3 real vector bundle.

Proof. From the local description of the singularity (1.1), we have, after

de-gree two base change, the local equation of the family near an ordinary double point: 4

∑

With a simple change of variables yj = e √

−1θ_x

jfor j=1, . . . , 4, the equation

becomes (1.2) 4

∑

Write yjin terms of real coordinates yj = aj+

√

−1bj, (1.2) becomes

(1.3) |~a_|2_{= |}t_|2_{+ |}~b_|2 and ~_a·~_b=0,

where~_{a and}~_{b are two vectors in R}4_{. The set of solutions to (1.3) can be}

identified with T∗Srwith the bundle structure T∗Sr→Srdefined by

(~a,~b) 7→r~a |~a| ∈Sr

where Sr is the 3-sphere with radius r = |t|. The vanishing sphere can

be chosen to be the real locus of the equation of (1.2). Therefore, NSr/Xt

is naturally identified with the cotangent bundle T∗Sr, which is a trivial

bundle since S3 ∼=SU(2)is a Lie group.  Remark 1.4. We see from the above description that the vanishing spheres

are Lagrangian with respect to the natural symplectic structure on T∗S3_.

A theorem of Seidel and Donaldson [33] states that this is true globally, namely the vanishing spheres can be chosen to be Lagrangian with respect to the symplectic structure coming from the K¨ahler structure of Xt.

By Lemma 1.2, the δ neighborhood of the vanishing 3-sphere Sr3in Xtis

homeomorphic to trivial disc bundle S3r×Dδ3. By Lemma 1.2 the r

neighbor-hood of the exceptional 2-sphere Ci = S2δ is D4r ×S2δ, where δ is the radius

defined by 4πδ2 ₌ R

Ciω for the background K¨ahler metric ω. Therefore,

we have the following conclusion.

Corollary 1.5. On the topological level one can go between Y and Xtby surgery

via

∂(S3r×D3δ) =S3r×S2δ =∂(D4r×S2δ).

Remark 1.6 (Orientations on S3). The two choices of orientations on Sr3

in-duces two different surgeries. The resulting manifolds Y and Y′are in gen-eral not even homotopically equivalent. In the complex analytic setting the induced map Y99KY′ is known as an ordinary (Atiyah) flop.

1.2. Global topology. Now we turn to the global topological constraint. Lemma 1.7. Define µ := 1 2(h3(X) −h3(Y)) and ρ := h2(Y) −h2(X) Then, (1.4) µ+ρ=k. Proof. The Euler numbers satisfy

χ(X) −kχ(S3) =χ(Y) −kχ(S2). That is,

2−2h1(X) +2h2(X) −h3(X) =2−2h1(Y) +2h2(Y) −h3(Y) −2k. By the above surgery argument we know that conifold transitions preserve the fundamental group. Therefore,

1

2(h3(X) −h3(Y)) + (h2(Y) −h2(X)) =k.

 Remark 1.8. In the Calabi-Yau case, µ = h2,1(X) −h2,1(Y) = −∆h2,1 _{is the}

lose of complex moduli, and ρ = h1,1(Y) −h1,1(X) = ∆h1,1 _{is the gain of}

K¨ahler moduli. Thus (1.4) is really ∆(h1,1−h2,1) =k= 1 2∆χ.

This might suggest the expression A−B instead of A+B. We use the

latter since it really means a combined (A, B) theory, with the interpretation that A corresponds to Hevand B corresponds to Hodd.

In the next section, we will study the Hodge-theoretic meaning of this sim-ple topological equality.

2. HODGE THEORY AND THE BASIC EXACT SEQUENCE

Convention. In this paper, unless otherwise specified, cohomology groups

are over Q when only topological aspect (including weight filtration) is concerned; they are considered over C when the (mixed) Hodge-theoretic aspect is involved.

All equalities, whenever they make sense in the context of mixed Hodge structure (MHS), hold as equalities for MHS unless otherwise specified. 2.1. Two semistable degenerations. In order to apply Hodge-theoretic tech-niques on the degenerations, we factor the transition XրY as a

composi-tion of two semistable degeneracomposi-tionsX →∆ and_{Y →}∆. The complex degeneration

is the semistable reduction for X → ∆ obtained by a degree two base change X′ → ∆ followed by the blow-up_{X →} X′ _{of all the four} dimen-sional nodes p′_i ∈ X′_{. The special fiber X0 =} Sk

j=0Xj is a simple normal

crossing divisor with ˜

ψ : X0 ∼=Y :˜ =Bl∐k_i=1{pi}X¯ →X¯

being the blow-up at the nodes and with

Xi =Qi ∼= Q⊂P4, i=1, . . . , k

being quadric threefolds. Let X[j]_{be the disjoint union of j+1 intersections}

from Xi’s. Then the only nontrivial terms are

X[0] =Y˜

∐

∐

are the ˜ψ exceptional divisors. The semistable reduction f does not require

the existence of a small resolution of X0.

The K¨ahler degeneration

(2.2) g :Y →∆ is simply the deformations to the normal cone

Y =Bl∐Ci×{0}Y×∆→∆.

The special fiberY0 =Skj=0Yjwith

φ : Y0∼=Y :˜ =Bl∐_ik₌₁{Ci}Y →Y

being the blow-up along the curves Ci’s and

Yi =E˜i ∼= E˜ =PP1(O(−1)2⊕O), i=1, . . . , k.

In this case the only non-trivial terms for Y[j]are

Y[0] =Y˜

∐

∐

is now understood as the infinity divisor (or relative hyperplane section) of

πi : ˜Ei →Ci ∼=P1.

2.2. Mixed Hodge Structure and the Clemens–Schmid exact sequence. We apply the Clemens–Schmid exact sequence to the above two semistable degenerations. A general reference for the background material here is [11]. We will mainly be interested in H≤3, although the computation of H>3is similar.

2.2.1. The cohomology of the central fiber H∗(X0), with its canonical mixed

Hodge structure, is computed from the spectral sequence

E₁p,q(X0) =Hq(X[p])

with the differential d1=δ the combinatorial coboundary operator

δ : Hq(X[p]) →Hq(X[p+1]).

The spectral sequence degenerates at E2 terms. The weight filtration on

H∗(X0)is induced from the following increasing filtration on the spectral

sequence Wm :=Lq≤mE∗,q. Therefore,

GrWm(Hj) =E

j−m,m

2 , GrWm(Hj) =0 for m <0 or m> j.

Since X[j] 6=∅ only when j=0, 1, we have

H0∼= E0,0₂ , H1=∼ E1,0₂ ⊕E0,1₂ , H2∼=E₂1,1⊕E0,2₂ , H3 ∼=E1,2₂ ⊕E₂0,3. The only weight 3 piece is E₂0,3, which can be computed by

δ : E0,3₁ = H3(X[0]) −→E1,3₁ =H3(X[1]).

Since Qi, ˜Eiand Eihave no odd cohomologies, H3(X[1]) =0 and H3(X[1]) =

H3(Y˜). We have thus E₂0,3 = H3(Y˜).

The weight 2 pieces, which is the most essential part, can be computed from the following map

(2.3) H2(X[0]) =H2(Y˜) ⊕ k M i=1 H2(Qi) δ2 −→H2(X[1]) = k M i=1 H2(Ei).

We have E1,2₂ =cok(δ2)and E20,2=ker(δ2).

The weight 1 and weight 0 pieces can be similarly computed. For weight 1 pieces we have

E₂0,1= H1(X[0]) = H1(Y˜) ∼= H1(Y) ∼= H1(X), and E1,1₂ =0. The weight 0 pieces are computed from

δ : H0(X[0]) →H0(X[1])

and we have

E0,0₂ =H0(Y˜) ∼=H0(Y) ∼=H0(X), and E1,0₂ =0.

We summarize these calculations in the following lemma.

Lemma 2.1.

H3_(X0) ∼= H3(Y˜) ⊕cok(δ2),

H2_(X0) ∼=ker(δ2),

H1_(X0) ∼= H1(Y˜) ∼=H1(Y) ∼=H1(X),

In particular, Hj(X0)is pure of weight j for j≤2.

2.2.2. Here we give a dual formulation of (2.3) which will be useful later. Letℓ_,ℓ′_{be the line classes of the two rulings of E}∼₌P1_×P1_{. Then H}2(Q, Z)

is generated by e= [E]as a hyperplane class and e|E = ℓ + ℓ′. The map δ2

in (2.3) is then equivalent to (2.4) ¯δ2: H2(Y˜) −→ k M i=1 H2(Ei)/H2(Qi). Since H2(Y˜) = φ∗H2(Y) ⊕Lk

i=1h[Ei]iand[Ei]|Ei = −(ℓi+ ℓ′i), the second

componentLk

i=1h[Ei]ilies in ker(¯δ2)and ¯δ2factors through

(2.5) φ∗H2(Y) → k M i=1 H2(Ei)/H2(Qi) ∼= k M i=1 hℓi− ℓ′ii

(as Q-spaces). Notice that the quotient is isomorphic toLk

i=1hℓ′iiintegrally.

By reordering we may assume that φ∗ℓi = [Ci] and φ∗[Ci] = ℓi− ℓ′i

(c.f. [16]). The dual of (2.5) then coincides with the fundamental class map

ϑ :

k

M

i=1

h[Ci]i −→ H2(Y).

In general for a Q-linear map ϑ : P→Z, we have

im ϑ∗ ∼= (P/ ker ϑ)∗ ∼= (im ϑ)∗. Thus

(2.6) dimQcok(δ2) +dimQim(ϑ) =k.

We will see in Corollary 2.5 that dim cok δ = µ and dim im ϑ = ρ. This

gives the Hodge theoretic meaning of µ+ρ = k in Lemma 1.7. Further

elaboration of this theme will follow in Theorem 2.9.

2.2.3. On Y0, the computation is similar and a lot easier. The weight 3

piece can be computed by the map

H3(Y[0]) =H3(Y˜) −→H3(Y[1]) =0; the weight 2 piece is similarly computed by the map

H2(Y[0]) =H2(Y˜) ⊕ k M i=1 H2(E˜i) δ′ 2 −→H2(Y[1]) = k M i=1 H2(Ei).

Let h=π∗(pt)and ξ = [E]for

π : ˜E→P1.

Then h|E = ℓ′ and ξ|E = ℓ + ℓ′. In particular the restriction map H2(E˜) →

H2(E)is an isomorphism and hence δ₂′ is surjective. The computation of pieces from weights 1 and 0 is the same as for X0. We have therefore the

Lemma 2.2.

H3(Y0) ∼= H3(Y[0]) ∼= H3(Y˜),

H2(Y0) ∼=ker(δ2′) ∼= H2(Y˜),

H1(Y0) ∼= H1(Y˜) ∼= H1(Y) ∼= H1(X),

H0_(Y0) ∼= H0(Y˜) ∼= H0(Y) ∼= H0(X).

2.2.4. Slightly abusing the notation, we denote by N the monodromy op-erator for both _X and _Y families. N induces the weight filtrations on Schmid’s limiting Hodge structures on H∗(X)and H∗(Y).

Lemma 2.3. We have the following exact sequences (of MHS) for H2and H3of X0andY0:

0→H3_(X0) →H3(X)−→N H3(X) →H3(X0) →0,

0→ H0(X) →H6(X0) →H2(X0) →H2(X)−→N 0,

0→ H3(Y0) →H3(Y)−→N 0,

0→ H0(Y) →H6(Y0) →H2(Y0) →H2(Y)−→N 0,

Proof. These follow from the Clemens–Schmid exact sequence, which is

compatible with the MHS. Note that the monodromy is trivial forY → ∆ since the punctured family is trivial. By Lemma 2.1, we know that H2(X0)

is pure of weight 2. Hence N on H2(X)is also trivial and the Hodge struc-ture does not degenerate.  Remark 2.4. Strictly speaking there are other terms in the first sequence,

namely H1(X) →H5(X0)to the left end and H5(X0) →H5(X)to the right

end. It can be ignored since they induce isomorphisms, as can be checked using MHS on H5(X0). Similar comments apply to the third sequence for

H3(Y). All these vanish if we impose the regularity condition h1(O_{) =0.}

Corollary 2.5. (i) ρ=dim im(ϑ)and µ=dim cok(δ2).

(ii) H3_{(Y) ∼=H}3_(Y

0) ∼= H3(Y[0]) ∼= H3(Y˜) ∼=GrW3 H3(X).

(iii) Denote by K the kernel of the monodromy operator

K :=ker(N : H3(X) →H3(X)).

We have H3_(X

0) ∼= K. More precisely,

GrW3 (H3(X0)) ∼= H3(Y), GrW2 (H3(X0)) ∼=cok(δ2).

Proof. By Lemma 2.1, h2(X0) =dim ker(δ2). It follows from the second and

the fourth exact sequences in Lemma 2.3 that

h2(X) =dim ker(δ2) +1− (k+1).

Rewrite (2.3) as

which implies

dim ker(δ2) +2k=dim cok(δ2) +2k+h2(Y).

Combining these two equations with (2.6), we have

ρ= h2(Y) −h2(X) =k−dim cok(δ2) =dim im(ϑ).

This proves the first equation for ρ in (i).

Combining the first equation in Lemma 2.2 and the third exact sequence in Lemma 2.3, we have

(2.8) H3(Y) ∼= H3(Y0) ∼= H3(Y˜).

(This can also be seen from the geometry of blowing up.) This shows (ii) except the last equality.

By Lemmas 2.3 and 2.1,

K ∼=H3(X0) ∼=H3(Y˜) ⊕cok(δ2) ∼= H3(Y) ⊕cok(δ2),

where the last equality follows from (2.8). This proves (iii).

For the remaining parts of (i) and (ii): From the non-trivial terms of the limiting Hodge diamond, where Hn:= Hn(X)and

H∞p,qHn= F∞pGrW_p+qHn, we have (2.9) H∞2,2H3 N ∼  H∞3,0H3 H∞2,1H3 t t t t t t t t t H∞1,2H3 tttt tttt t H0,3∞ H3 H∞1,1H3,

where H∞3,0H3does not degenerate due to a result in [38] (which holds for

more general degenerations with canonical singularities, and first proved in [37] for the Calabi–Yau case). We conclude that H∞1,1H3 ∼= cok(δ2)and

GrW3 H3(X) ∼= H3(Y). Thus

µ=h2,2∞ H3 =h∞1,1H3 =dim cok(δ2).



2.2.5. We denote the vanishing cycle space V as the Q-vector space gener-ated by vanishing 3-cycles. We first define the abelian group VZfrom

(2.10) 0→VZ→ H3(X, Z) → H3(X, Z¯ ) →0,

and V := VZ⊗ZQ. We note that the exactness on the right holds for any

3-fold isolated singularities.

We will give a further geometric characterization of the defect invariant

Lemma 2.6. (i) H3(X¯) ∼=K∼= H3(X_0). (ii) V∗∼= H∞2,2H3and V ∼= H∞1,1H3=cok(δ2).

Proof. Dualizing (2.10) over Q, we have

0→H3(X¯_{) →}H3(X_{) →}V∗ _→0. The invariant cycle theorem in [1] implies that

H3(X¯) ∼=ker N =K∼= H3(X0).

This proves (i).

Hence we have the canonical isomorphism

V∗ ∼= H∞2,2H3 =F∞2G4WH3(X).

Moreover, the non-degeneracy of the pairing(Nα, β)on GW

4 H3(X)implies

that

H∞1,1H3 = NH∞2,2H3 ∼= (H∞2,2H3)∗ ∼=VC∗∗∼=VC.

This proves (ii).  Remark 2.7. We must be careful in dealing with this isomorphism H∞1,1H3∼=

V. The vanishing cycle space V is defined over Z while H1,1∞ H3is

intrinsi-cally defined only as a complex vector space. In identifying V with H1,1∞ H3,

we used two different duality: Hom_(·, Q), which brings it to the dual space, and the duality under a bilinear pairing (Poincar´e pairing), which stays in the same vector space.

Remark 2.8 (On threefold extremal transitions). Most results in Section 2.2

works for more general geometric contexts. The mixed Hodge diamond (2.9) holds for any 3-folds degenerations with at most canonical singulari-ties [38]. The identification of vanishing cycle space V via (2.10) works for 3–folds with only isolated singularities, hence Lemma 2.6 works for any 3-fold degenerations with isolated canonical singularities.

Later on we will impose the Calabi–Yau condition on all the 3-folds in-volved. If XրY is a terminal transition of Calabi–Yau 3-folds, i.e., X0= X¯

has at most (isolated Gorenstein) terminal singularities, then ¯X has

unob-structed deformations [26]. Moreover, the small resolution Y →X induces¯

an embedding Def(Y) ֒→ Def(X¯) which identifies the limiting/ordinary pure Hodge structures GrW₃ H3_{(X) ∼= H}3_{(Y)as in Corollary 2.5 (iii).}

For conifold transitions all these can be described in explicit terms and more precise structure will be formulated.

2.3. The basic exact sequence. We may combine the four Clemens–Schmid exact sequences into one short exact sequence, which we call the basic exact

sequence, to give the Hodge-theoretic realization of the equality “ρ+µ=k”

Let A= (aij) ∈ Mk×µ(Z)be the relation matrix for Ci’s, i.e., k

∑

aij[Ci] =0, j=1, . . . , µ.

Similarly, let B= (bij) ∈ Mk×ρ(Z)be the relation matrix for Si’s: k

∑

bij[Si] =0, j=1, . . . , ρ.

Theorem 2.9 (Basic exact sequence). The group of real 2-cycles generated by

exceptional curves Ci (vanishing S2cycles) on Y and the group of 3-cycles

gener-ated by[Si](vanishing S3cycles) on X are linked by the following weight 2 exact

sequence 0→ H2(Y)/H2(X)_−→B k M i=1 H2(Ei)/H2(Qi) At −→V _→0.

In particular B=ker Atand A=ker Bt.

Proof. To see this, we use the sequence in (2.7). From the discussions in

Section 2.2.2, we know that cok(δ2) =cok(¯δ2)and (2.7) can be replaced by

(2.11) 0→H2(Y˜)/(ker ¯δ)_−→D

k

M

i=1

H2(Ei)/H2(Qi)−→C cok(δ2) →0.

By Lemma 2.6 (ii), we have cok(δ2) ∼=V. To prove the theorem, we need to

show that

H2(Y˜)/ ker ¯δ∼= H2(Y)/H2(X), and D= B, C= At.

Let us start with making sense of the quotient H2(Y)/H2(X). Again by the version of invariant cycle theorem in [1], we have H2_{(X) = H}2_{(X¯). By}

the blow-up description in Section 1.1, H2_{(X¯)injects to H}2_{(Y)by pullback.}

This defines the embedding

(2.12) ι : H2(X) ֒→H2(Y)

and the quotient H2(Y)/H2(X).

Recast the relation matrix A of the rational curves Ci in the following

form 0→Qµ_−→A Qk∼₌ k M i=1 h[Ci]i S −→im(ϑ) →0

where S = cok(A_{) ∈} Mρ×k is the matrix for ϑ, and im(ϑ)has rank ρ. The

dual sequence reads (2.13) 0→ (im ϑ)∗ ∼= (Qρ)∗_−→(St Qk)∗ ∼= k M i=1 H2(Ei)/H2(Qi) A t −→(Qµ)∗ _→0.

Compare (2.13) with (2.11), we see that(Qµ)∗ _{∼=V. From the discussion in}

Section 2.2.2, we have(im ϑ)∗ _{= H}2_(Y)/H2_(X).

We want to reinterpret the map At : (Qk)∗ _→ V in (2.13). This is a

presentation of V by k generators, denoted by σi, and the relation matrix of

which is given by St. If we show that σican be identified with the vanishing

sphere Si, then(Qµ)∗ =∼ V and B = St = ker At is the relation matrix for

Si’s.

Consider the following topological construction. For any non-trivial in-tegral relation ∑k_i=1ai[Ci] =0, there is a 3-chain θ in Y with

∂θ=

k

∑

aiCi.

Under ψ : Y → X, C¯ i collapses to the node pi. Hence it creates a 3-cycle

¯θ := ψ∗θ ∈ H3(X, Z¯ ), which deforms (lifts) to γ ∈ H3(X, Z) in nearby

fibers. Using the intersection pairing on H3(X, Z), γ then defines an

ele-ment PD(γ)in H3_{(X, Z). Under the restriction to the vanishing cycle space}

V, we get PD(γ) ∈V∗. It remains to show that

(γ.Si) =ai.

Let Uibe a small tubular neighborhood of Si and ˜Ui be the corresponding

tubular neighborhood of Ci, then by Corollary 1.5,

∂Ui∼= ∂(Si3×D3) ∼=S3×S2 ∼=∂(D4×Ci) ∼= ∂ ˜Ui.

Now θi := θ∩U˜igives a homotopy between ai[Ci](in the center of ˜Ui) and

ai[S2](on ∂ ˜Ui). Denote by ι : ∂Ui ֒→ X and ˜ι : ∂ ˜Ui ֒→Y. Then

(γ.Si)X = (γ.ι∗[S3])X = (ι∗γ.[S3])∂Ui = (˜ι∗γ.[S3])∂ ˜Ui

= (ai[S2],[S3])S

3_×S2

=ai.

The proof is complete.  Remark 2.10. As a byproduct, notice that there are precisely k−ρ = µ =

dim V∗independent relations, hence we also see directly that(ai) 7→PD(γ)

establishes a group isomorphism from curve relations among Ci’s to V∗.

Convention. We would like to choose a preferred basis of the vanishing

co-cycles V∗as well as a basis of divisors dual to the space of extremal curves. These notations will fixed henceforth and will be used in later sections.

During the course of the proof of Theorem 2.9 (c.f. Remark 2.10) we es-tablish the correspondence for each column vector Aj = (a1j,· · · , akj)twith

the element PD(γj) ∈V∗, 1≤ j≤µ, characterized by

aij = (γj.Si).

The subspace of H3(X)spanned by these γj’s will be denoted by V′.

Dually, we denote by T1,· · · , Tρ ∈ H2(Y)those divisors which form an

In particular they form an integral basis of H2(Y)/H2(X). Notice that we may choose Tl’s such that Tl corresponds to the l-th column vector of the

matrix B via

bil = (Ci.Tl).

Such a choice is consistent with the basic exact sequence since

(AtB)jl = k

∑

∑

∑

for all j, l. We may also assume that the first ρ×ρ minor of B has full rank.

3. GROMOV–WITTEN THEORY ANDDUBROVIN CONNECTIONS

3.1. Consequences of the degeneration formula for threefolds. Gromov– Witten theory on X can be related to that on Y by the degeneration formula through the two semistable degenerations introduced in Section 2.1.

In the previous section, we have seen that the monodromy actions are trivial on H(X)except H3(X)for which we have

H_inv3 (X) =K ∼=H3(Y) ⊕H∞1,1H3(X) ∼= H3(Y) ⊕V.

There we implicitly have a linear map

(3.1) ι : H_invj (X) →Hj(Y)

as follows. For j=3, it is the projection

H_inv3 (X) ∼= H3(Y) ⊕V→ H3(Y).

For j = 2, it is the embedding defined in (2.12) and j = 4 case is the same as (dual to) j=2 case. For j=0, 1, 5, 6, ι is an isomorphism.

The following is a refinement of a result of Li–Ruan [22]. (See also [23].)

Proposition 3.1. Let X _րY be a projective conifold transition. Given ~_{a∈ (H}≥2

inv(X)/V)⊕n

and a curve class β∈ NE(X) \ {0}, we have

(3.2) h~aiXg,n,β =

∑

hι(~a)iYg,n,γ.

If some component of~_{a lies in H}0_{, then both sides vanish. Furthermore, the RHS}

is a finite sum.

Proof. (3.2) has been proved in [22, 23] under slightly stronger assumptions.

We review its proof with slight refinements as it will be useful in Section 5. A cohomology class a∈ H>_inv2(X)/V can always find an admissible lift to

(ai)ki=0∈ H(Y˜) ⊕

k

M

i=1

H(Qi)

such that ai =0 for all i 6=0. This is the lifting of the cohomology class we

We apply J. Li’s algebraic version of degeneration formula [21, 23] to the complex degeneration (2.1) X Y˜ _∪E Q, where Q = ∐Qi is a disjoint

union of quadrics Qi’s and E := ∑ki=1Ei. One has K_Y˜ = ψ˜∗KX¯ +E. The

topological data(g, n, β)lifts to two admissible triples Γ1on(Y, E˜ )and Γ2

on(Q, E)such that Γ1has curve class ˜γ∈ NE(Y˜), contact order µ = (γ.E˜ ),

and number of contact points ρ. Then

(γ.c˜ 1(Y˜)) = (ψ˜∗γ.c˜ 1(X¯)) − (γ.E˜ ) = (β.c1(X)) −µ.

The virtual dimension (without marked points) is given by

dΓ₁ = (γ.c˜ 1(Y˜)) + (dim X−3)(1−g) +ρ−µ

=dβ+ρ−2µ.

Since we chose the lifting (~ai)k_i=0 of~a to have~ai = 0 for all i 6=0, all

inser-tions contribute to ˜Y. If ρ ₆₌ 0 then ρ−2µ < _{0. This leads to vanishing}

relative GW invariant on(Y, E˜ ). Therefore, ρ must be zero. To summarize, we get (3.3) h~aiXg,n,β =

∑

We note that this equation also holds for ai a divisor by the divisor axiom.

We use a similar argument to compute h~biY

g,n,γ via the K¨ahler

degener-ation (2.2) Y Y˜ ∪E, where ˜˜ E is a disjoint union of ˜Ei (cf. [16,

Theo-rem 4.10]). By the divisor equation we may assume that deg bj ≥ 3 for all

j = 1, . . . , n. We still choose the lifting (~b)k

i=0 of~b such that~bi = 0 for all

i₆₌0. In the lifting γ1on ˜Y and γ2on π : ˜E= ∐iE˜i →∐iCi, we must have

γ = φ∗γ1+π∗γ2. The contact order is given by µ = (γ1.E)which has the

property that µ =0 if and only if γ1 =φ∗γ (and hence γ2=0). If ρ6=0 we

still get

dΓ₁ = dγ+ρ−2µ<dγ

and the invariant is thus zero. This proves that (3.5) h~biYg,n,γ= hφ∗~b|∅i

(_Y,E˜ ) g,n,φ∗_γ,

such that

(3.6) φ_∗γ˜ =γ, γ.E˜ =0, γ˜E˜ =0.

To combine these two degeneration formulas together, we notice that in the K¨ahler degeneration, ˜γ_∈ NE(Y˜)can have contact order µ= (γ.E˜ ) =0 if and only if ˜γ= φ∗γ for some γ∈ NE(Y)(indeed for γ = φ∗γ). Choose˜

~_{b=ι(~a)and the formula in the proposition follows.}

The vanishing statement (of H0insertion) follows from the fundamental class axiom.

Now we proceed to prove the finiteness of the sum. (This part is not stated in [22].) For φ : ˜Y _→ Y being the blow-up along Ci’s, the curve

class γ ∈ NE(Y)contributes a non-trivial invariant in the sum only if φ∗γ

is effective on ˜Y. By combining (2.5), (3.3) and (3.5), the effectivity of φ∗γ

forces the sum to be finite. Equivalently, the condition that φ∗γ is effective

is equivalent to that γ is F-effective under the flop

Y // ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ Y′ ⑦⑦⑦⑦ ⑦⑦⑦⑦ ¯ X

(i.e. effective in Y and in Y′ under the natural correspondence [16]). Recall that under the flop the flopping curve class in Y is mapped to the negative flopping curve in Y′. Therefore, the sum is finite.  Remark 3.2. (i) The phenomena, including finiteness of the sum, were

ob-served in [12] for Calabi–Yau hypersurfaces in weighted projective spaces from the numerical data obtained from the corresponding B model gener-ating function and mirror symmetry.

(ii) For general 3-folds extremal transitions worse than conifolds, the double point degeneration formula does not apply directly. In general, the relative GW invariants will enter the degeneration formula in an essential way and can not be reduced to the absolute GW theory in a simple explicit way as above. For example, in higher dimensions none of the complicated features in the degeneration formula can be avoided.

Corollary 3.3. Gromov–Witten theory on even cohomology GWev_{(X)(of all}

gen-era) can be considered as a proper sub-theory of GWev(Y).

In particular, the big quantum cohomology ring is functorial with respect to ι : Hev(X) →Hev(Y)in (3.1).

Proof. We first note that ι is an injection on Hev. Proposition 3.1 then implies that all Gromov–Witten invariants of X with even classes can be recovered from invariants of Y. The only exception, H0_{, can be treated by the}

funda-mental class axiom. Therefore, in this sense that GWev(X)is a sub-theory of GWev(Y).

In genus zero, however, there is a more precise sense of being a sub-theory via functoriality. Observe that the degeneration formula also holds for β =0. For g=0, this leads to the equality of classical triple product on

Hinv(X)under ι:

(a, b, c)X _{= (ι(a), ι(b), ι(c))}Y_.

Since the Poincar´e pairing on Hev(X)is also preserved under ι, we see that the classical ring structure on Hev(X)are naturally embedded in Hev(Y).

To see the functoriality of the big quantum ring with respect to ι, we note that (ι(a).Ci) = 0 for any a ∈ Hev(X) and for any extremal curve

the insertions must involve only divisors by the virtual dimension count. Hence in the level of generating functions with at least one insertion we also have

∑

∑

Note that the case of H0is not covered in Proposition 3.1, but can be treated by the fundamental class axiom as above.  Remark 3.4. It is clear that the argument and conclusion hold even if some

insertions lie in H3

inv(X)/V ∼=H3(Y)by Proposition 3.1.

The full GW theory is built on the full cohomology superspace H= Hev⊕ Hodd. However, the odd part is not as well-studied in the literature as the even one. In some special cases the difficulty does not occur for elementary reasons.

Lemma 3.5. Let X be a smooth minimal 3-fold (e.g., Calabi–Yau threefold) with H1(X) =0. The non-trivial primary GW invariants are all supported on H2(X). More generally the conclusion holds for any curve class β ∈ NE(X) with c1(X).β≤0 for any 3-fold X with H1(X) =0.

Proof. For n-point invariants, the virtual dimension of Mg,n(X, β)is given

by

c1(X).β+ (dim X−3)(1−g) +n≤n.

Since the appearance of fundamental class in the insertions leads to triv-ial invariants, we must have the algebraic degree deg ai ≥ 1 for all

inser-tions ai, i = 1, . . . , n. Hence in fact we must have deg ai = 1 for all i and

c1(X).β=0. 

Remark 3.6. By the divisor axiom, the primary GW theory for smooth

min-imal 3-folds is then completely reduced to the case without any insertions. 3.2. The even and extremal quantum cohomology. From now on, we re-strict to genus zero theory.

3.2.1. For simplicity we restrict our discussions on insertions s=∑_ǫsǫ_T¯

ǫ ∈

H2_{(X) where ¯T}

ǫ’s form a basis of H2(X). Then the genus zero GW

pre-potential is given by (3.7) F0X(s) = ∞

∑

∑

∑

0,0,β, with formal variables qβ’s. It can be considered as a

function in the “K¨ahler moduli” via identification

qβ =exp 2π√−1(β.ω), where

ω= B+√₋1H∈ KXC := H2(X) + √

the complexified K¨ahler cone of X.

FX

0 (s)almost gives the small quantum cohomology of X. (By Lemma 3.5,

this is the same as big quantum cohomology if X is a Calabi–Yau threefold.) In order to have the full small quantum cohomology ring, we will need to consider s_∈ Hev_{(X)in the first term s}3_{/(3!), which will be called classical,}

topological or cubic terms. Namely, in terms of dual basis notations,

s=s0T¯0+

∑

∑

∑

∑

For simplicity of notation, and without loss of generality, we treat the divi-sor variables first and bring back the other two topological terms when we need to write down the complete Dubrovin connection.

Remark 3.7. The reader might consider all the following discussions are for

the big quantum cohomology of a Calabi–Yau threefolds, since that is the case we will be primarily concerned with in the later sections.

Remark 3.8. In practice, the variables s and ω encode basically equivalent

information: By divisor axiom, qβ always appears in the form qβexp(β, s). If there is no convergence issue then it makes no essential difference to drop out the Novikov variables by setting qβ ≡1 for all β.

3.2.2. Similarly we have F₀Y(t)on H2(Y) × KY

C. Here we use the variable t= s+u ∈H2(Y) =ι(H2(X)) ⊕

ρ

M

l=1

hTli.

Namely we identify s with ι(s)in H2(Y)and write u = ∑ρ_l=1ul_T

l. F0Y can

be analytically continued across those boundary faces ofKY

C which

corre-sponds to flopping contractions. In the case of conifold transitions Y ցX,

the boundary face is preciselyKX

C ⊂K¯YC.

Convention. The following convention of indices on Hev(Y)will be used throughout the rest of this section:

• Lowercase Greek alphabets for indices from the subspace ι(Hev(X));

• lowercase Roman alphabets for indices from the subspace spanned by the divisors Tl’s and exceptional curves Ci’s;

• uppercase Roman alphabets for variables from the total space Hev(Y). For C ∼=P1 _{with twisted bundle N} =O_P1(−1)⊕2_{, the extremal function} is given by the well-known multiple cover formula

EC₀(t) =

_∑

_∑

We also consider the total (global) extremal function EY₀(t):= t 3 3!+ k

∑

where we notice that ECi

0 (t) =E

Ci

0 (u)depends only on u.

Then the degeneration formula is equivalent to the following restriction

F0X(s) − s3 3! =  F0Y(s+u) − (s+u)3 3! −E Y 0(u) + u3 3!    qγ_7→qψ_∗(γ),

or equivalently the restriction of K¨ahler moduli to the boundary faceKX

C.

Notice that the Novikov variables q[Ci]_{’s are subject to the relations: For}

∑k_i=1aij[Ci] = 0 in NE(Y)with A = (aij) ∈ Mk×µ(Z) being the relation

matrix, we define rj(q):=

∏

∏

and force the relation rj(q) = 0 for 1≤ j≤ µ since they vanish trivially on

the K¨ahler moduli.

Summarizing the above discussion, we have

Lemma 3.9. F0Y(s+u) =  F0X(s) +E0Y(u) + 1 3!((s+u) 3_−s3_−u3₎  rj(q)=0, 1≤j≤µ .

Convention. For simplicity of notation, we restrict the Novikov variables

implicitly and drop it from the notation henceforth.

A complete splitting of variables of the pre-potential function F₀Y would imply that the big quantum cohomology QHev(Y)decomposes into two blocks. One piece is identified with QHev(X), and another piece with con-tributions from the extremal rays. However, the classical cup product terms enter into the formula and destroy the complete splitting. Thus the two pieces are not completely independent.

3.2.3. The structural coefficients for QHev(Y)are CPQR=∂3_PQRF0Y. We will

determine them according to the above splitting. For FX

0 (s), the structural coefficients of quantum product are given by

Cǫζι :=∂3ǫζιF0X(s) = (T¯ǫ. ¯Tζ. ¯Tι) +

∑

Recall that B = (bip) with bip = (Ci.Tp) is the relation matrix for the

vanishing 3-spheres. For E₀Y(u), the triple derivatives are

Clmn :=∂3lmnEY0(u) = (Tl.Tm.Tn) + k

∑

∑

∑

∑

∑

is the fundamental rational function with a simple pole at q = 1 with residue−1. (f(q)plays an important role in our study of GW invariants associated to a flopping contraction in [16] and subsequent works.)

However, due to the existence of possible cross terms, Clmn’s do not

sat-isfy the WDVV equations. Indeed, the remaining cross terms are

θ(s+u):= 1

2(s

2_u+su2_{) =} 1

2su

2_.

The first term s2u = 0 since Tl’s are chosen to be orthogonal to NE(X).

Then the only non-trivial mixed triple derivatives are constants (cup prod-uct)

Cǫmn:=∂3ǫmnθ(s+u) = (T¯ǫ.Tm.Tn).

Denote by ¯Tǫ_{∈ H}4_{(X)the dual basis of ¯T}

ǫ’s, and write

Tl, 1≤l≤ρ,

the dual basis of Tl’s. Also ¯T0= T0=1with dual ¯T0= T0the point class.

Remark 3.10. The more canonical choice T(l):=

k

∑

bil[Ci]

is not the dual basis since

(T(l).Tm) = k

∑

∑

This implies that

T(l)=

ρ

∑

m=1

(BtB)lmTm.