TOWARDS A + B THEORY IN CONIFOLD TRANSITIONS FOR CALABI–YAU THREEFOLDS

110 (2018) 495-541

TOWARDS A + B THEORY IN CONIFOLD TRANSITIONS FOR CALABI–YAU THREEFOLDS

Yuan-Pin Lee, Hui-Wen Lin & 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 combined information provided by the A model (Gromov–Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corre- sponding combined information on Y . Similar result holds in the reverse direction when linked with the exceptional curves.

0. Introduction

0.1. Statements of main results. Let X be a smooth projective 3- fold. A (projective) conifold transition X % Y is a projective degenera- tion π : X → ∆ of X to a singular variety ¯ X = X

with a finite number of ordinary double points (abbreviated as ODPs or nodes) p

, . . . , p

, locally analytically defined by the equation

x

+ x

+ x

+ x

= 0,

followed by a projective small resolution ψ : Y → ¯ X. In the process of complex degeneration from X to ¯ X, k vanishing spheres S

∼ = S

with trivial normal bundle collapse to nodes p

. In the process of “K¨ ahler degeneration” from Y to ¯ X, the exceptional loci of ψ above each p

is a smooth rational curve C

∼ = P

with N

∼ = O

(−1)

. We write Y & X for the reverse process.

Notice that ψ is a crepant resolution and π is a finite distance degen- eration with respect to the quasi-Hodge metric [39, 40]. A transition of this type (in all dimensions) is called an extremal transition. In contrast to the usual birational K-equivalence, an extremal transition may be considered as a generalized K-equivalence in the sense that the small resolution ψ is crepant and the degeneration π preserves sections of the canonical bundle. It is generally expected that simply connected Calabi–Yau 3-folds are connected through extremal transitions, of which

Received December 23, 2015.

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conifold transitions are the most fundamental. (This has been exten- sively checked numerically [17].) It is, therefore, a natural starting point of investigation.

We study the changes of the so-called A model and B model under a projective conifold transition. In this paper, the A model is the Gromov–

Witten (GW) 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.

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 good understanding. For the inverse coni- fold transition Y & X, a celebrated theorem of Friedman [8] (see also [15, 38]) states that a small contraction Y → ¯ X can be smoothed if and only if there is a totally nontrivial relation between the exceptional curves. That is, there exist constants a

6= 0 for all i = 1, . . . , k such that P

i=1

a

[C

] = 0. These are relations among curves [C

]’s in the kernel of H

(Y )

→ H

(X)

. Let µ be the number of independent re- lations and let A ∈ M

(Z) be a relation matrix for C

’s, in the sense that the column vectors span all relations. Conversely, for a conifold transition X % Y , Smith, Thomas and Yau proved a dual statement in [36], asserting that the k vanishing 3-spheres S

must satisfy a totally nontrivial relation P

i=1

b

[S

] = 0 in V

:= ker(H

(X)

→ H

( ¯ X)

) with b

6= 0 for all i. Let ρ be the number of independent relations and B ∈ M

(Z) be a relation matrix for S

’s. It turns out that µ + ρ = k [5] and the following exact sequence holds.

Theorem 0.1 (= Theorem 1.14). Under a conifold transition X % Y of smooth projective threefolds, we have an exact sequence of weight two Hodge structures:

(0.1) 0 → H

(Y )/H

(X) −→ C

−→ V → 0.

We interpret this as a partial exchange of topological information between the excess A model of Y /X (in terms of H

(Y )/H

(X)) and the excess B model of X/Y in terms of the space of vanishing cycles V . To study the changes of quantum A and B models under a projective conifold transition of Calabi–Yau 3-folds and its inverse, the first step is to find a D-module version of Theorem 0.1. We state the result below in a suggestive form and leave the precise statement to Theorem 4.1:

Theorem 0.2 (= Theorem 4.1). Via the exact sequence (0.1), the trivial logarithmic connection on (C ⊕ C

)

→ C

induces simultane- ously the logarithmic part of the Gauss–Manin connection on V and the Dubrovin connection on H

(Y )/H

(X).

Note that the Gauss–Manin connection on V determines the excess

B model and Dubrovin connection on H

(Y )/H

(X) determines the

excess A model in genus zero. The logarithmic part of the connection determines the residue connection and, hence, the monodromy. One can interpret Theorem 0.2 heuristically as “excess A theory + excess B theory ∼ trivial”. In other words, the logarithmic parts of two flat connections on excess theories “glues” to form a trivial theory. This gives a strong indication towards a unified A + B theory.

“Globalizing” this result, i.e., going beyond the excess theories, is the next step towards a true A + B theory, which is still beyond immediate reach. Instead we will settle for results on mutual determination in implicit form. Recall that the Kuranishi spaces M

, M

of Calabi–Yau manifolds are unobstructed (the Bogomolov–Tian–Todorov theorem).

For a Calabi–Yau conifold ¯ X, the unobstructedness of M

also holds [15, 38, 27].

Theorem 0.3. Let X % Y be a projective conifold transition of Calabi–Yau threefolds such that [X] is a nearby point of [ ¯ X] in M

. Then

(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 \ S

i=1

S

“linked” by the vanishing spheres in B(X).

(4) B(X) can be reconstructed from a refined B model of Y

:= Y \ S

i=1

C

“linked” by the exceptional curves in A(Y ).

The meaning of these slightly obscure statements will take the entire paper to spell them out. It may be considered as a categorification of Clemens’ identity µ + ρ = k. Here we give only brief explanations.

(1) is mostly due to Li–Ruan, who in [22] pioneered the mathematical study of conifold transitions in GW theory. The proof follows from degeneration arguments and existence of flops (cf. Proposition 2.1).

For (2), we note that there are natural identifications of M

with the boundary of M

consisting of equisingular deformations, and M

with M

\ D where the discriminant locus D is a central hyperplane arrangement with axis M

(cf. §3.3.2). Therefore, the VHS associated to Y can be considered as a sub-VHS system of VMHS associated to X (cf. Corollary ¯ 3.20), which is a regular singular extension of the VHS associated to X.

With (3), we introduce the “linking data” of the holomorphic curves in X

, which not only records the curve classes in X but also how the curve links with the vanishing spheres S

i

S

. The linking data on X can

be identified with the curve classes in Y by H

(X

) ∼ = H

(Y ) (cf. Def-

inition 5.2 and (5.3)). 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 (cf. Proposition 5.6). Furthermore, the Gromov–Witten invariants

in Y is the same as the numbers produced by the component of the virtual class on X labeled by the corresponding linking data. Thus, the refined A model is really the “linked A model” and is equivalent to the (usual) A model of Y (for non-extremal curves classes) in all gen- era. The vanishing cycles from B(X) plays a key role in reconstructing A(Y ).

For (4), the goal is to reconstruct VHS on M

from VHS on M

and A(Y ). The deformation of ¯ X is unobstructed. Moreover, it is well known that Def( ¯ X) ∼ = H

(Y

, T

). Even though the deformation of Y

is obstructed (in the direction transversal to M

), there is a first order deformation parameterized by H

(Y

, T

) which gives enough initial condition to uniquely determine the degeneration of Hodge bundles on M

near M

. A technical result needed in this process is a short exact sequence

0 → V → H

(X) → H

(Y

) → 0,

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

(X) and the canonical MHS of Deligne on H

(Y

) (cf. Proposi- tion 6.1). Together with the monodromy data associated to the ODPs, which is encoded in the relation matrix A of the extremal rays on Y , we will be able to determine the VHS on M

near M

. In the process, an extension of Schmid’s nilpotent orbit theorem [34] to degenerations with certain non-normal crossing discriminant loci is also needed. See Theorem 3.14 for details.

0.2. Motivation and future plans. Our work is inspired by the fa- mous Reid’s fantasy [30], where conifold transitions play a key role in connecting irreducible components of moduli of Calabi–Yau threefolds.

Theorems 0.2 and 0.3 above can be interpreted as the partial exchange of A and B models under a conifold transition. We hope to answer the following intriguing question concerning with “global symmetries” on moduli 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 tran- sitions? If so, what is the relation between this full exchange and the one induced by “mirror symmetry”? To this end, we need to devise a computationally effective way to achieve explicit determination of this partial exchange. 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. (For smooth blowups with complete intersection centers, we have a fairly good solu- tion in genus zero [19].)

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 theory predicts that Calabi–Yau threefolds form an important

ingredient of our universe, but it does not specify which Calabi–Yau threefold we live in. Should the A + B model be available and proven invariant under extremal transitions, one would then have no need to make such a choice.

The first step of achieving this goal is to generalize Theorem 0.2 to the full local theory, including the non-log part of the connections. We note that the excess A model on H

(Y /X) can be extended to the (flat) Dubrovin connection on Y while the excess B model on H

(X/Y ) can be extended to the (flat) Gauss–Manin connection on X. We hope to be able to “glue” the complete A model on Y and the complete B model on X as flat connections on the unified K¨ ahler plus complex moduli.

Acknowledgments. We are grateful to C.H. Clemens, C.-C. M. Liu, M. Rossi, I. Smith and R. Thomas for discussions related to this project, and to the referee for his/her valuable suggestions.

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

The vast literature on extremal transitions makes it impossible to cite all related articles in this area, and we apologize for omission of many important papers which are not directly related to our approach.

1. The basic exact sequence from Hodge theory

In this section, we recall some standard results on the geometry of projective conifold transitions. Definitions and short proofs are mostly spelled out to fix the notations, even when they are well known. Com- bined with well-known tools in Hodge theory, we derive the basic exact sequence, which is surprisingly absent in the vast literature on the coni- fold transitions.

Convention. In §1–2, all discussions are for projective conifold tran- sitions without the Calabi–Yau condition, unless otherwise specified.

The Calabi–Yau condition is imposed in §3–5. Unless otherwise spec- ified, cohomology groups are over Q when only topological aspect (in- cluding weight filtration) is concerned; they are considered over C when the (mixed) Hodge-theoretic aspect is involved. All equalities, when- ever make sense in the context of mixed Hodge structure (MHS), hold as equalities for MHS.

1.1. Preliminaries on conifold transitions. The results here are mostly contained in [5] and are included here for readers’ convenience.

1.1.1. Local geometry. Let X be a smooth projective 3-fold and X %

Y a projective conifold transition through ¯ X with nodes p

, . . . , p

as in

§0.1. Locally analytically, a node (ODP) is defined by the equation (1.1) x

+ x

+ x

+ x

= 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 1.1. The exceptional locus of ψ above each p

is a smooth rational curve C

with N

∼ = O

(−1)

. Topologically, N

is a trivial rank 4 real bundle.

Proof. Away from the isolated singular points p

’s, the Weil divisors are Cartier and the blowups do nothing. Locally near p

, the Weil divisor is generated by two functions u and w. The blowup Y ⊂ A

× P

is defined by z

v − z

s = 0, in addition, to uv − ws = 0 defining X, where (z

: z

) are the coordinates of P

. Namely we have u/w = s/v = z

/z

. It is now easy to see the exceptional locus near p

is isomorphic to P

and the normal bundle is as described (by the definition of O

(−1)). Since oriented R

-bundles on P

∼ = S

are classified by the second Stiefel–

Whitney class w

(via π

(SO(4)) ∼ = Z/2), the last assertion follows

immediately. q.e.d.

Locally to each node p = p

∈ ¯ X, the transition X % Y can be considered as two different ways of “smoothing” the singularities in ¯ X:

deformation leads to X

and small resolution leads to Y . Topologically, we have seen that the exceptional loci of ψ are `

i=1

C

, 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 S

∼ = S

. Lemma 1.2. The normal bundle N

∼ = T

is a trivial rank 3 real bundle.

Proof. From (1.1), after a degree two base change the local equation of the family near an ODP is

X

j=1

x

= t

= |t|

e

√−1θ

.

Let y

= e

√−1θ

x

for j = 1, . . . , 4, the equation leads to

(1.2) X

j=1

y

= |t|

. Write y

in terms of real coordinates y

= a

+ √

−1b

, we have |~a|

=

|t|

+ |~b|

and ~a · ~b = 0, where ~a and ~b are two vectors in R

. The

set of solutions can be identified with T

S

with the bundle structure

T

S

→ S

defined by (~a,~b) 7→ r~a/|~a| ∈ S

where S

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, N

is naturally identified with

the cotangent bundle T

S

, which is a trivial bundle since S

∼ = SU (2)

is a Lie group. q.e.d.

Remark 1.3. The vanishing spheres above are Lagrangian with re- spect to the natural symplectic structure on T

S

. A theorem of Sei- del and Donaldson [35] 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 X

.

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

in X

is diffeomorphic to the trivial disc bundle S

× D

.

By Lemma 1.1 the r neighborhood of the exceptional 2-sphere C

= S

is D

× S

, where δ is the radius defined by 4πδ

= R

Ci

ω for the background K¨ ahler metric ω.

Corollary 1.4. [5, Lemma 1.11] On the topological level one can go between Y and X

by surgery via

∂(S

× D

) = S

× S

= ∂(D

× S

).

Remark 1.5 (Orientations on S

). The two choices of orientations on S

induces two different surgeries. The resulting manifolds Y and Y

are in general not even homotopically equivalent. In the complex analytic setting the induced map Y 99K Y

is known as an ordinary (Atiyah) flop.

1.1.2. Global topology.

Lemma 1.6. Define

µ :=

(h

(X) − h

(Y )) and ρ := h

(Y ) − h

(X).

Then,

(1.3) µ + ρ = k.

Proof. The Euler numbers satisfy

χ(X) − kχ(S

) = χ(Y ) − kχ(S

).

That is,

2 − 2h

(X) + 2h

(X) − h

(X) = 2 − 2h

(Y ) + 2h

(Y ) − h

(Y ) − 2k.

By the above surgery argument we know that conifold transitions pre- serve π

. Therefore,

(h

(X) − h

(Y )) + (h

(Y ) − h

(X)) = k. q.e.d.

Remark 1.7. In the Calabi–Yau case, µ = h

(X) − h

(Y ) =

−∆h

is the lose of complex moduli, and ρ = h

(Y )−h

(X) = ∆h

is the gain of K¨ ahler moduli. Thus, (1.3) is really

∆(h

− h

) = k =

∆χ.

In the following, we study the Hodge-theoretic meaning of (1.3).

1.2. Two semistable degenerations. To apply Hodge-theoretic methods on degenerations, we factor the transition X % Y as a com- position of two semistable degenerations X → ∆ and Y → ∆.

The complex degeneration

f : X → ∆

is the semistable reduction of X → ∆ obtained by a degree two base change X

→ ∆ followed by the blow-up X → X

of all the four dimen- sional nodes p

∈ X

. The special fiber X

= S

j=0

X

is a simple normal crossing divisor with

ψ : X ˜

∼ = ˜ Y := Bl

i=1{pi}

X → ¯ ¯ X being the blow-up at the nodes and with

X

= Q

∼ = Q ⊂ P

, i = 1, . . . , k

being quadric threefolds. Let X

be the disjoint union of j + 1 inter- sections from X

’s. Then the only nontrivial terms are X

= ˜ Y `

i

Q

and X

= `

i

E

where E

= ˜ Y ∩ Q

∼ = P

× P

are the ˜ ψ exceptional divisors. The semistable reduction f does not require the existence of a small resolution of X

.

The K¨ ahler degeneration

g : Y → ∆

is simply the deformations to the normal cone Y = Bl

Y × ∆ →

∆. The special fiber Y

= S

j=0

Y

with φ : Y

∼ = ˜ Y := Bl

i=1{Ci}

Y → Y being the blow-up along the curves C

’s and

Y

= ˜ E

∼ = ˜ E := P

( O(−1)

⊕ O), i = 1, . . . , k.

In this case the only non-trivial terms for Y

are Y

= ˜ Y `

i

E ˜

and Y

= `

i

E

where E

= ˜ Y ∩ ˜ E

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

: ˜ E

→ C

∼ = P

.

1.3. Mixed Hodge structure and the Clemens–Schmid exact sequence. We now apply the Clemens–Schmid exact sequence [6] to the above two semistable degenerations. A general reference is [11]. We will mainly be interested in H

. The computation of H

is similar.

1.3.1. The cohomology of H

(X

), with its canonical mixed Hodge structure, is computed from the spectral sequence E

(X

) = Ω

(X

) with d

= d, the de Rham differential, and then

E

(X

) = H

(X

),

with d

= δ being the combinatorial coboundary operator δ : H

(X

) → H

(X

).

The spectral sequence degenerates at E

terms.

The weight filtration on H

(X

) is induced from the increasing filtra- tion on the spectral sequence W

:= L

q≤m

E

. Therefore, Gr

(H

) = E

, Gr

(H

) = 0 for m < 0 or m > j.

Since X

6= ∅ only when j = 0, 1, we have

H

∼ = E

, H

∼ = E

⊕E

, H

∼ = E

⊕E

, H

∼ = E

⊕E

. The only weight 3 piece is E

, which can be computed by

δ : E

= H

(X

) −→ E

= H

(X

).

Since Q

, ˜ E

and E

have no odd cohomologies, H

(X

) = 0 and H

(X

) = H

( ˜ Y ). We have, thus, E

= H

( ˜ Y ).

The weight 2 pieces, which is the most essential part, is computed from

(1.4)

H

(X

) = H

( ˜ Y ) ⊕ M

i=1

H

(Q

) −→ H

(X

) = M

i=1

H

(E

).

We have E

= cok(δ

) and E

= ker(δ

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

E

= H

(X

) = H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X),

and E

= 0. The weight 0 pieces are computed from δ : H

(X

) → H

(X

) and we have E

= H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X), and E

= 0.

We summarize these calculations as

Lemma 1.8. There are isomorphisms of MHS:

H

(X

) ∼ = H

( ˜ Y ) ⊕ cok(δ

), H

(X

) ∼ = ker(δ

),

H

(X

) ∼ = H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X), H

(X

) ∼ = H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X).

In particular, H

(X

) is pure of weight j for j ≤ 2.

1.3.2. Here we give a dual formulation of (1.4) which will be useful later. Let `, `

be the line classes of the two rulings of E ∼ = P

×P

. Then H

(Q, Z) is generated by e = [E] as a hyperplane class and e|

= ` + `

. The map δ

in (1.4) is then equivalent to

(1.5) δ ¯

: H

( ˜ Y ) −→ M

i=1

H

(E

)/H

(Q

).

Since H

( ˜ Y ) = φ

H

(Y ) ⊕ L

i=1

h[E

]i and [E

]|

= −(`

+ `

), the second component L

i=1

h[E

]i lies in ker(¯ δ

) and ¯ δ

factors through (1.6) φ

H

(Y ) → M

i=1

H

(E

)/H

(Q

) ∼ = M

i=1

h`

− `

i

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

i=1

h`

i inte- grally.

By reordering we may assume that φ

`

= [C

] and φ

[C

] = `

− `

(cf. [18]). The dual of (1.6) then coincides with the fundamental class map

ϑ : M

i=1

h[C

]i −→ H

(Y ).

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

∼ = (P/ ker ϑ)

∼ = (im ϑ)

. Thus,

(1.7) dim

cok(δ

) + dim

im(ϑ) = k.

We will see in Corollary 1.11 that dim cok δ = µ and dim im ϑ = ρ.

This gives the Hodge theoretic meaning of µ + ρ = k in Lemma 1.6.

Further elaboration of this theme will follow in Theorem 1.14.

1.3.3. On Y

, the computation is similar and a lot easier. The weight 3 piece can be computed by the map H

(Y

) = H

( ˜ Y ) −→ H

(Y

) = 0;

the weight 2 piece is similarly computed by the map H

(Y

) = H

( ˜ Y ) ⊕ M

i=1

H

( ˜ E

)

0

−→ H

(Y

) = M

i=1

H

(E

).

Let h = π

(pt) and ξ = [E] for π : ˜ E → P

. Then h|

= `

and ξ|

= ` + `

. In particular, the restriction map H

( ˜ E) → H

(E) is an isomorphism and, hence, δ

is surjective. The computation of pieces from weights 1 and 0 is the same as for X

. We have, therefore, the following lemma.

Lemma 1.9. There are isomorphisms of MHS:

H

(Y

) ∼ = H

(Y

) ∼ = H

( ˜ Y ), H

(Y

) ∼ = ker(δ

) ∼ = H

( ˜ Y ),

H

(Y

) ∼ = H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X), H

(Y

) ∼ = H

( ˜ Y ) ∼ = H

(Y ) ∼ = H

(X).

1.3.4. We denote by N the monodromy operator for both X and Y families. The map N induces the unique monodromy weight filtrations W on H

(X) which, together with the limiting Hodge filtration F

, leads to Schmid’s limiting MHS [34, 37]. That is,

0 ⊂ W

⊂ W

⊂ · · · ⊂ W

⊂ W

= H

(X) such that N W

⊂ W

and for ` ≥ 0,

(1.8) N

: G

∼ = G

on graded pieces. The induced filtration F

G

:= F

∩W

/F

∩W

defines a pure Hodge structure of weight k on G

. Similar constructions

apply to H

(Y ) as well.

Lemma 1.10. We have the following exact sequences (of MHS) for H

and H

:

0 → H

(X

) →H

(X) −→ H

(X) → H

(X

) → 0, 0 → H

(X) → H

(X

) → H

(X

) →H

(X) −→ 0,

0 → H

(Y

) →H

(Y ) −→ 0,

0 → H

(Y ) → H

(Y

) → H

(Y

) →H

(Y ) −→ 0.

Proof. These follow from the Clemens–Schmid exact sequence, which is compatible with the MHS. The other terms in the first sequence, namely H

(X) → H

(X

) to the left end and H

(X

) → H

(X) to the right end, can be ignored since they induce isomorphisms, as can be checked using MHS on H

(X

). Similar comments apply to the third sequence for H

(Y ).

Note that the monodromy is trivial for Y → ∆ since the punctured family is trivial. For the second sequence, by Lemma 1.8, we know that H

(X

) is pure of weight 2. Hence, N on H

(X) is also trivial and the Hodge structure does not degenerate. Indeed, if N 6= 0 then ker N contains some part of weight ≤ 2 by (1.8). q.e.d.

Corollary 1.11. (i) ρ = dim im(ϑ) and µ = dim cok(δ

).

(ii) H

(Y ) ∼ = H

(Y

) ∼ = H

(Y

) ∼ = H

( ˜ Y ) ∼ = Gr

H

(X).

(iii) Denote by K := ker(N : H

(X) → H

(X)). Then H

(X

) ∼ = K. More precisely, Gr

(H

(X

)) ∼ = H

(Y ) and Gr

(H

(X

)) ∼ = cok(δ

).

Proof. By Lemma 1.8, h

(X

) = dim ker(δ

). It follows from the second and the fourth exact sequences in Lemma 1.10 that h

(X) = dim ker(δ

) + 1 − (k + 1). Rewrite (1.4) as

(1.9) 0 → ker(δ

) → H

(X

) −→ H

(X

) → cok(δ

) → 0, which implies dim ker(δ

) + 2k = dim cok(δ

) + 2k + h

(Y ).

Combining these two equations with (1.7), we have ρ = h

(Y ) − h

(X) = k − dim cok(δ

) = dim im(ϑ). This proves the first equation for ρ in (i).

Combining the first equation in Lemma 1.9 and the third exact se- quence in Lemma 1.10, we have

(1.10) H

(Y ) ∼ = H

(Y

) ∼ = H

( ˜ Y ).

This shows (ii) except the last equality.

By Lemmas 1.10 and 1.8, K ∼ = H

(X

) ∼ = H

( ˜ Y )⊕cok(δ

) ∼ = H

(Y )⊕

cok(δ

), where the last equality follows from (1.10). This proves (iii).

For the remaining parts of (i) and (ii), we investigate the non-trivial terms of the limiting mixed Hodge diamond for H

:= H

(X):

(1.11)

H

H

∼ N



H

H

H

H

H

H

H

H

,

H

H

where H

H

= F

Gr

H

. The space H

(X) does not degen- erate by [40] (which holds for degenerations with canonical singulari- ties, and first proved in [39] for the Calabi–Yau case). We conclude that H

H

∼ = cok(δ

) and Gr

H

(X) ∼ = H

(Y ). By definition µ =

1

2

(h

(X) − h

(Y )), hence, µ = h

H

= h

H

= dim cok(δ

). q.e.d.

1.3.5. We denote the vanishing cycle space V as the Q-vector space generated by vanishing 3-cycles. We first define the abelian group V

from

(1.12) 0 → V

→ H

(X, Z) → H

( ¯ X, Z) → 0,

and V := V

⊗

Q. The sequence (1.12) arises from the homology Mayer–Vietoris sequence and the surjectivity on the right hand side follows from the fact that H

( `

S

, Z) = 0.

Lemma 1.12. Denote by H

:= H

(X).

(i) H

( ¯ X) ∼ = K ∼ = H

(X

) ∼ = W

H

.

(ii) V

∼ = H

H

and V ∼ = H

H

= cok(δ

) via Poincar´ e pairing.

Proof. Dualizing (1.12 ) over Q, we have

0 → H

( ¯ X) → H

(X) → V

→ 0.

The invariant cycle theorem in [1] then implies that H

( ¯ X) ∼ = ker N = K ∼ = H

(X

). This proves (i).

Hence, we have the canonical isomorphism

V

∼ = H

(X)/H

( ¯ X) = G

H

= F

G

H

= H

H

.

Moreover, the non-degeneracy of the pairing (α, N β) on G

H

implies H

H

= N H

H

∼ = (H

H

)

∼ = V

∼ = V

.

This proves (ii). q.e.d.

Remark 1.13 (On threefold extremal transitions). Most results in

§1.3 works for more general geometric contexts. The mixed Hodge dia-

mond (1.11) holds for any 3-folds degenerations with at most canonical

singularities [40]. The identification of vanishing cycle space V via

(1.12) works for 3–folds with only isolated (hypersurface) singularities.

Indeed, the exactness on the RHS holds for degenerations X → ∆ such that X is smooth and X

has only isolated singularities. This follows from Milnor’s theorem that the vanishing cycle has the homotopy type of a bouquet of middle dimensional spheres [26, Theorem 6.5]. Hence, Lemma 1.12 works for any 3-fold degenerations with isolated hypersur- face canonical singularities.

Later on we will impose the Calabi–Yau condition on all the 3-folds involved. If X % Y is a terminal transition of Calabi–Yau 3-folds, i.e., X

= ¯ X has at most (isolated Gorenstein) terminal singularities, then X has unobstructed deformations [27]. Moreover, the small resolution ¯ Y → ¯ X induces an embedding Def(Y ) ,→ Def( ¯ X) which identifies the limiting/ordinary pure Hodge structures Gr

H

(X) ∼ = H

(Y ) as in Corollary 1.11 (iii).

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

1.4. 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 “ρ + µ = k”

in Lemma 1.6.

Let A = (a

) ∈ M

(Z) be a relation matrix for C

’s, i.e., X

i=1

a

[C

] = 0, j = 1, . . . , µ

give all relations of the curves classes [C

]’s. Similarly, let B = (b

) ∈ M

(Z) be a relation matrix for S

’s:

X

i=1

b

[S

] = 0, j = 1, . . . , ρ.

Theorem 1.14 (Basic exact sequence). The group of 2-cycles gener- ated by exceptional curves C

(vanishing S

cycles) on Y and the group of 3-cycles generated by [S

] (vanishing S

cycles) on X are linked by the following weight 2 exact sequence

0 → H

(Y )/H

(X) −→

M

i=1

H

(E

)/H

(Q

) −→ V → 0.

In particular, B = ker A

and A = ker B

.

Proof. From §1.3.2, cok(δ

) = cok(¯ δ

) and (1.9) can be replaced by (1.13)

0 → H

( ˜ Y )/(ker ¯ δ) −→

M

i=1

H

(E

)/H

(Q

) −→ cok(δ

) → 0.

By Lemma 1.12 (ii), we have cok(δ

) ∼ = V . To prove the theorem, we

need to show that H

( ˜ Y )/ ker ¯ δ ∼ = H

(Y )/H

(X), and D = B, C = A

.

By the invariant cycle theorem [1], H

(X) ∼ = H

( ¯ X). Since H

( ¯ X) injects to H

(Y ) by pullback, this defines the embedding

ι : H

(X) ,→ H

(Y ), and the quotient H

(Y )/H

(X).

Recast the relation matrix A of the rational curves C

in 0 → Q

−→ Q

∼ = M

i=1

h[C

]i −→ im(ϑ) → 0,

where S = cok(A) ∈ M

is the matrix for ϑ, and im(ϑ) has rank ρ.

The dual sequence reads (1.14)

0 → (im ϑ)

∼ = (Q

)

−→(Q

)

∼ = M

i=1

H

(E

)/H

(Q

) −→(Q

)

→ 0.

Compare (1.14) with (1.13 ), we see that (Q

)

∼ = V . From the discussion in §1.3.2, we have (im ϑ)

= H

(Y )/H

(X).

We want to reinterpret the map A

: (Q

)

→ V in (1.14). This is a presentation of V by k generators, denoted by σ

, and the relation matrix of which is given by S

. If we show that σ

can be identified with S

, then (Q

)

∼ = V and B = S

= ker A

is the relation matrix for S

’s.

Consider the following topological construction. For any non-trivial integral relation P

i=1

a

[C

] = 0, there is a 3-chain θ in Y with ∂θ = P

i=1

a

C

. Under ψ : Y → ¯ X, C

collapses to the node p

. Hence, it creates a 3-cycle ¯ θ := ψ

θ ∈ H

( ¯ X, Z), which deforms (lifts) to γ ∈ H

(X, Z) in nearby fibers by the surjectivity in ( 1.12). Using the intersection pairing on H

(X, Z), γ then defines an element PD(γ) in H

(X, Z). Under the restriction V , we get PD(γ) ∈ V

.

It remains to show that (γ.S

) = a

. Let U

be a small tubular neighborhood of S

and ˜ U

be the corresponding tubular neighborhood of C

, then by Corollary 1.4,

∂U

∼ = ∂(S

× D

) ∼ = S

× S

∼ = ∂(D

× C

) ∼ = ∂ ˜ U

.

Now θ

:= θ ∩ ˜ U

gives a homotopy between a

[C

] (in the center of ˜ U

) and a

pt × [S

] (on ∂ ˜ U

). Denote by ι : ∂U

,→ X and ˜ ι : ∂ ˜ U

,→ Y . Then

(γ.S

)

= (γ.ι

[S

])

= (ι

γ.[S

])

= (˜ ι

γ.[S

])

= (a

[S

], [S

])

= a

.

The proof is complete. q.e.d.

Remark 1.15. We would like to choose a preferred basis of the van- ishing cocycles 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 proof of Theorem 1.14, we establish the correspondence

between A

= (a

, . . . , a

)

and PD(γ

) ∈ V

, 1 ≤ j ≤ µ, characterized

by a

= (γ

.S

). The subspace of H

(X) spanned by γ

’s is denoted by V

.

Dually, we denote by T

, . . . , T

∈ H

(Y ) those divisors which form an integral basis of the lattice in H

(Y ) dual (orthogonal) to H

(X) ⊂ H

(Y ). In particular, they form an integral basis of H

(Y )/H

(X). We choose T

’s such that T

corresponds to the l-th column vector of the matrix B via b

= (C

.T

). Such a choice is consistent with the basic exact sequence since

(A

B)

= X

i=1

a

b

= X

i=1

a

(C

.T

) =  X a

[C

]



.T

= 0, for all j, l. We may also assume that the first ρ × ρ minor of B has full rank.

2. Gromov–Witten theory and Dubrovin connections In §2.1 the A model A(X) is shown to be a sub-theory of A(Y ). We then move on to study the genus 0 excess A model on Y /X associated to the extremal curve classes in §2.2. As a consequence the (nilpotent) monodromy is calculated in terms of the relation matrix B at the end of §2.3.

2.1. Consequences of the degeneration formula for threefolds.

The Gromov–Witten theory on X can be related to that on Y by the degeneration formula through the two semistable degenerations intro- duced in §1.2.

In the previous section, we see that the monodromy acts trivially on H(X) \ H

(X) and we have

H

(X) = K ∼ = H

(Y ) ⊕ H

H

(X) ∼ = H

(Y ) ⊕ V.

There we implicitly have a linear map

(2.1) ι : H

(X) → H

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

H

(X) ∼ = H

(Y ) ⊕ V → H

(Y ).

For j = 2, it is the embedding defined before and the case j = 4 is the same as (dual to) the 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 2.1. Let X % Y be a projective conifold transition.

Given ~a ∈ (H

(X)/V )

and a curve class β ∈ N E(X) \ {0}, we have (2.2) h~ai

= X

ψ∗(γ)=β

hι(~a)i

.

If some component of ~a lies in H

, then both sides vanish. Furthermore,

the RHS is a finite sum.

Proof. A slightly weaker version of (2.2) has been proved in [22, 23].

We review its proof with slight refinements as it will be useful in §5.

We follow the setup and argument in [18, §4] closely. By [18, §4.2], a cohomology class a ∈ H

(X)/V can always find a lift to

(a

)

∈ H( ˜ Y ) ⊕ M

i=1

H(Q

),

such that a

= 0 for all i 6= 0. We apply J. Li’s algebraic version of degeneration formula [21, 23 ] to the complex degeneration X ˜ Y ∪

Q, where

Q := a

Q

is a disjoint union of quadrics Q

’s and E := X

i=1

E

.

One has K

= ˜ ψ

K

+ E. The topological data (g, n, β) lifts to two admissible triples Γ

on ( ˜ Y , E) and Γ

on (Q, E) such that Γ

has curve class ˜ γ ∈ N E( ˜ Y ), contact order µ = (˜ γ.E), and number of contact points ρ. Then

(˜ γ.c

( ˜ Y )) = ( ˜ ψ

˜ γ.c

( ¯ X)) − (˜ γ.E) = (β.c

(X)) − µ.

The virtual dimension (without marked points) is given by d

= (˜ γ.c

( ˜ Y )) + (dim X − 3)(1 − g) + ρ − µ = d

+ ρ − 2µ, where d

is the virtual dimension of the absolute invariant with curve class β (without marked points). Since we chose the lifting (~a

)

of ~a to have ~a

= 0 for all i 6= 0, all insertions 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

(2.3) h~ai

= X

ψ˜∗(˜γ)=β

h~a

| ∅i

, such that

(2.4) ψ ˜

˜ γ = β, γ.E = 0, ˜ γ ˜

= 0.

Formula (2.3) also holds for a

a divisor by the divisor axiom.

We use a similar argument to compute h~bi

via the K¨ ahler degen- eration Y ˜ Y ∪ ˜ E, where ˜ E is a disjoint union of ˜ E

(cf. [18, Theo- rem 4.10]). By the divisor equation we may assume that deg b

≥ 3 for all j = 1, . . . , n. We choose the lifting (~b)

of ~b such that ~b

= 0 for all i 6= 0. In the lifting γ

on ˜ Y and γ

on π : ˜ E = `

i

E ˜

→ `

i

C

, we

must have γ = φ

γ

+ π

γ

. The contact order is given by µ = (γ

.E)

which has the property that µ = 0 if and only if γ

= φ

γ (and, hence,