5.3.1. Analysis of the moduli of stable maps to the degenerating families. Here we recall some results in J. Li’s study of degeneration formula [20, 21]. Given a projective flat family over a curve
π : W →A1
such that π is smooth away from 0 ∈ B and the central fiber W0 = Y1∪Y2
has only double point singularity with D := Y1∩Y2 a smooth (but not necessarily connected) divisor, Li constructed a moduli stack in [20]
M(W, Γ) →A1
which has a perfect obstruction theory and hence a virtual fundamental class [M(W, Γ)]virt in [21]. The following properties will be useful to us.
(The notations are slightly changed.)
(1) For every 06=t ∈A1, one has
M(W, Γ)t = M(X, β), [M(W, Γ)]virtt = [M(X, β)]virt
where M(X, β)is the corresponding moduli of (absolute) stable maps.
(2) For the central fiber, the perfect obstruction theory on M(W, Γ) in-duces a perfect obstruction theory on M(W0, Γ)and
[M(W0, Γ)]virt= [M(W, Γ)]virt∩π−1(0) is a virtual divisor of[M(W, Γ)]virt.
(3) M(W0, Γ)and its virtual class are related to the relative moduli and their virtual classes. For each admissible triple (consisting of gluing data) ǫ, there is a ”gluing map”
Φǫ: M(Y1, D; Γ1) ×DρM(Y2, D; Γ2) →M(W0, Γ), inducing the relation between the virtual cycles [M(W0, Γ)]virt=
∑
ǫ
mǫΦǫ∗∆! [M(Y1, D; Γ1)]virt× [M(Y2, D; Γ2)]virt , where
∆ : Dρ →Dρ×Dρ
is the diagonal morphism and mǫis a rational number (multiplicity divided by the degree of Φǫ).
5.3.2. Decomposition of M(W0, Γ). We will study the properties of M(W0, Γ) and their virtual fundamental classes in the setting of Section 3.1. A com-prehensive comparison of the curve classes in X, Y and ˜Y is collected in the following diagram.
It is easy to see that there is a unique lifting ˜γ of γ satisfying (3.4). From this and the degeneration analysis we have the following lemma.
Lemma 5.6.
[M(Y, γ)]virt ∼ [M(Y, D; ˜˜ γ)]virt,
where∼stands for ”homotopy equivalence”. 2 They define the same GW invari-ants.
Because of this lemma, we will sometimes abuse the notation and identify [M(Y, D; ˜˜ γ)]virtwith[M(Y, γ)]virt.
2If π can be extended to a family over P1, then the two cycles are rationally equivalent.
Lemma 5.7. In the case of complex degeneration in Section 3.1, images of Φγ˜ for different ˜γ are disjoint from each other.
Proof. This follows from Li’s study of the corresponding moduli stacks. In this special case of ρ = 0, for any element in M(W0, Γ)there is only one way to split it into two ”relative maps” (with one of them being empty).
We note that this is not true in general, when there are more than one way of splitting of the maps to the central fiber. As discussed before, given β6=0, if ˜γ and ˜γ′ both satisfy (3.4), in partic-ular they are non-exceptional for ˜ψ : ˜Y→ X, we have¯
γ˜−γ˜′ =
∑
i
ai(ℓi− ℓ′i), whereℓi andℓ′
i are the ˜ψ exceptional curve classes (two rulings) in Ei. By Proposition 3.1, there are only finitely many nonzero ai.
For each ˜γ above, there is a unique γ in Y, which is non-extremal for ψ : Y →X, such that ˜¯ γ satisfies (3.6).
Corollary 5.8. Given β 6= 0 a curve class in X, we can associate to it sets of non- ˜ψ-exceptional curve classes ˜γ and γ discussed above. Then
[M(X, β)]virt∼
∑
˜ γ
[M(Y, D; ˜˜ γ)]virt∼
∑
γ
[M(Y, γ)]virt,
where ∼ stands for the homotopy equivalence and the summations are over the above sets.
Notice that the conclusion holds for any projective small resolution Y of ¯X.
Proof. This follows from (3.3), (3.5) and the above discussions. Recall in Section 5.2 we have the identification of the linking data in (5.4) H2(Y◦) =H2(Y) =H2(X◦) =H2(X\D) =H2(X¯ \X¯sing) where
X\
k
[
i=1
Si =: X◦∼ M ∼Y◦ :=Y\
k
[
i=1
Ci
and D is a tubular neighborhood of the union of vanishing S3’s D=[
i
Di ∼=[
i
S3×D3.
Therefore, a curve class γ ∈ H2(Y)can be identified as a ”curve class” in X◦ ∼X¯ \X¯sing, with the latter a quasi-projective variety. Therefore, we can think of γ as a curve class in X◦.
Proposition 5.9. For Xtwith t∈A1very small in the degenerating family π :X →A1,
we have a decomposition of the virtual class [M(Xt, β)]virt into a finite disjoint union of cycles
[M(Xt, β)]virt =
∐
γ∈H2(X◦)
[M(Xt, γ)]virt, where
[M(Y, γ)]virt∼ [M(Xt, γ)]virt∈ Avdim M(Xt, β) is a cycle class corresponding to the linking data γ of Xt.
Proof. By the construction of the virtual class of the family π, we know that the virtual classes for Xt and for X0 are restrictions of that forX. Lemma 5.7 tells us that at t= 0, the virtual class decomposes into a disjoint union.
By semicontinuity of connected components, we conclude that the virtual classes for Xtremain disconnected with (at least) the same number of con-nected components labeled by γ∈ H2(X◦). We call the numbers defined by[M(Xt, γ)]virtthe refined GW numbers of X◦with linking data γ.
Corollary 5.10. The refined GW numbers of X◦ with linking data γ are the same as the GW invariants of Y with curve class γ, where γ is interpreted in two ways via (5.4).
This corollary shows that A(X) +B(X)classical ⇒ A(Y).
Remark 5.11. According to Fukaya [8], if one allows the deformation of the almost complex structures J, the pseudo-holomorphic curves in a Calabi–
Yau threefold do not intersect any number of given Lagrangian S3for generic J. Those J’s for which some pseudo-holomorphic curves intersect some vanishing S3 form a codimension 1 walls in the space of almost complex structures. These walls divide the space of almost complex structures into chambers. When one moves from one chamber to another, the wall cross-ing effect consists of countcross-ing pseudo-holomorphic disks with boundaries on the vanishing S3. That is, the difference between counting of pseudo-holomorphic curves with J in one chamber and that with J in another is accountable by pseudo-holomorphic disk counting.
The results above, in particular Proposition 5.9 can be interpreted in the following way. If we know that the (integrable) complex structures in our moduli lie in (the interior of) the chambers, then the curve classes will never intersect the union of the vanishing S3’s. Therefore, the moduli of stable maps with a fixed β has a natural partition into disjoint unions of those with curve classes γ∈ H2(X◦). Even though we do not know if this holds in general, Proposition 5.9 says that this still holds at the level of virtual classes when[X]is sufficiently close to[X¯]in the moduli. Once we move far away from[X¯], the wall crossing is possible. Thus, the refined GW numbers for(X◦, γ)are not symplectic invariants (with respect to X). In a work in progress [18], we plan to prove a blowup formula for genus zero which will cover any smooth blowups and some singular cases as well. That blowup
will give A(X) +B(X)classical ⇒ A(Y), removing the constraint that [X] must be sufficiently close to[X¯].
6. FROMA(Y) +B(Y)TO A(X) +B(X) 6.1. Overview.
6.1.1. A(Y) ⇒ A(X). As is explained in Section 3, A(X)is a sub-theory of A(Y). Indeed, A(X)is obtained from A(Y)by setting all extremal ray invariants to be zero, in addition to “reducing the linking data” γ∈NE(Y) to β∈ NE(X).
6.1.2. A(Y) +B(Y) ⇒ B(X). We have seen earlier that B(Y)can be con-sidered as a sub-theory of B(X). In this section, we will show that B(Y), together with the knowledge of extremal curves SiCI ⊂ Y uniquely de-termines B(X). More precisely, we will show that the “Hodge filtration”
underlying the variation of MHS of the quasi-projective Y◦ = Y\SiCi on the first jet space ofMY ⊂ MX¯ can be lifted uniquely to the Hodge filtra-tion underlying the degenerating VHS of X. Furthermore, the informafiltra-tion of the Gauss–Manin up to the first jet is sufficient to uniquely single out the VHS of X.
In the next subsection, we start with a statement of compatibility of MHS which is needed in our discussion. After that we will give a proof showing the unique determination. As in our implication of B(X) +A(X) ⇒ A(Y) in Section 5, our A(Y) +B(Y) ⇒ B(X)implication is not constructive. It seems likely that a constructive recipe of this determination can be worked out by detailed analysis on the logarithmic model of degeneration of Hodge bundles by Steenbrink in [35] (see also [5]).
6.2. Compatibility of the mixed Hodge structures. Recall from Section 4.1 thatMX¯ is smooth and containsMXas an open subscheme with “bound-ary”MX¯ \ MX∼=MY. Set
U :=Y◦ =Y\
k
[
i=1
Ci ∼=X¯◦ =X¯ \X¯sing where ¯Xsing =Ski=1{pi}.
To construct the VHS with logarithmic degeneration onMX¯ nearMY, we start with the following lifting property.
Proposition 6.1. There is a short exact sequence of mixed Hodge structures (6.1) 0→V→ H3(X) →H3(U) →0,
where H3(X)is equipped with the limiting MHS of Schmid, V ∼= H1,1∞ H3(X), and H3(U)is equipped with the canonical mixed Hodge structure of Deligne.
In particular, F3H3(X) ∼=F3H3(U)and F2H3(X) ∼=F2H3(U).
Proof. In the topological level, the short exact sequence (6.1) is equivalent to the defining sequence of the vanishing cycle space (2.10). Indeed, since X is nonsingular, H3(X) ∼= H3(X)by Poincar´e duality. Also,
(6.2) H3(X¯) =H3(X, p¯ ) ∼=H3(Y, E˜ ) ∼= H3(Y˜\E) =H3(U) by the excision theorem and Lefschetz duality.
Now we consider the mixed Hodge structures. Since U is smooth quasi-projective, it is well know that the canonical mixed Hodge structure on H3(U)has its Hodge diamond supported on the upper triangular part, i.e., with weights ≥ 3. Or equivalently, the MHS on H3(X¯) has weights≤ 3 by duality in (6.2). The crucial point is that Lefschetz duality is compatible with mixed Hodge structures, as stated in Lemma 6.2 below. Hence the short exact sequence (6.1) follows from Lemma 2.6 which is essentially the invariant cycle theorem.
Notice that V ∼= H1,1∞ H3(X)by Lemma 2.6 (ii). In particular, the isomor-phisms on Fi for i=3, 2 follows immediately by applying Fi to (6.1). Lemma 6.2. Let Y be an n dimensional complex projective variety, i : Z ֒→ Y a closed subvariety with smooth complement j : U ֒→ Y where U := Y\Z. Then the Lefschetz duality
Hi(Y, Z) ∼=H2n−i(U) is compatible with the canonical mixed Hodge structures.
This is well known in mixed Hodge theory, though we are not able to locate an exact reference in the literature. For the readers’ convenience we include a proof which is communicated to us by M. de Caltaldo.
Proof. We will make use of the structural theorem of Saito on mixed Hodge modules (MHM) [31, Theorem 0.1] which says that there is a correspon-dence between the derived categories of MHM and that of perverse sheaves (c.f. Axiom A in 14.1.1 of Peters and Steenbrink’s book [28]).
There is a triangle in the derived category of constructible sheaves j!j!QY →QY→i∗i∗QY.
This triangle gives maps of MHS:
Hi(Y, Z) →Hi(Y) →Hi(Z) with
Hi(Y, Z) =Hi(Y, j!j!QY).
In fact, the MHS of Hi(Y, Z)can be defined by the RHS from Saito’s theory, since j!j!QYis a complex of MHM.
Dualizing the above setup, we have
(6.3) Hi(Y, Z) = Hi(Y, j!j!QY)∗,
where the LHS of (6.3) having MHS for the same reason as above and compatibly with taking dual as MHS. Furthermore, the RHS of (6.3) is
H−c i(Y, j∗j∗ωY)by Verdier duality, where ωY is the Verdier dualizing com-plex. Due to the compactness of Y we have
H−ci(Y, j∗j∗ωY) = H−i(Y, j∗j∗ωY) =H−i(U, ωU)
= HiBM(U) =H2n−i(U),
where HBM is the Borel–Moore homology. Since every step above is com-patible with MHM, it shows that the Lefschetz duality is comcom-patible with
the MHS.
6.3. Conclusion of the proof. We now apply the above result to our set-ting. We have on ¯X (cf. [27])
· · ·HX1¯sing(ΘX¯) →H1(ΘX¯) →H1(U, TU) →H2X¯sing(ΘX¯) → · · · . Since each pi is a hypersurface singularity, we have depth Opi = 3. Using this fact, Schlessinger showed that
H1p(ΘX¯) =0 and H2p(ΘX¯) ∼=
k
M
i=1
Cpi. Putting these together, we have
(6.4) 0→H1(ΘX¯) →H1(U, TU) →H2X¯sing(ΘX¯) → · · · .
Since ¯X is a Calabi–Yau 3-fold with only ODPs, its deformation theory is unobstructed by the T1-lifting property [14]. Comparing (6.4) with (4.1) we see that
Def(X¯) ∼= H1(U, TU). Similarly, on Y we have
· · ·H1Z(TY) → H1(TY) → H1(U, TU) →HZ2(TY) →H2(TY) → · · ·. Recall that Y is smooth Calabi–Yau and we have H1Z(TY) =0. Thus
Def(Y) =H1(TY) ⊂H1(U, TU) ∼=Def(X¯).
MYis a natural submanifold of MX¯. Write I := IMY as the ideal sheaf of MY ⊂ MX¯.
Since H2(U, TU) 6= 0, the deformation of U could be obstructed. Nev-ertheless, the first-order deformation of U exists and is parameterized by H1(U, TU) ⊃Def(Y). Therefore, we have the following smooth family
π : U→ Z1:=ZMX¯(I2) ⊃ MY,
whereZ1 =ZMX¯(I2)stands for the nonreduced subscheme ofMX¯ defined by the ideal sheaf I2. NamelyZ1is the first jet extension ofMYinMX¯.
Now we may complete the construction of VHS overMX¯ near the bound-ary loci MY ֒→ MX¯. The Gauss–Manin connection for a smooth family over non-reduced base was constructed in [13]. For our smooth family π : U → Z1, it is defined by the integral lattice H3(U, Z) ⊂ H3(U, C). Since U is only quasi-projective, the Gauss–Manin connection underlies
VMHS instead of VHS. By Proposition 6.1, we have WiH3(U) =0 for i≤2, W3⊂W4with GrW3 H3(U) ∼= H3(Y), and GrW4 H3(U) ∼=V∗.
The Hodge filtration of the locally system F0 = H3(U, C)has the fol-lowing structure: F• = {F3 ⊂ F2 ⊂ F1 ⊂ F0}which satisfies the Griffiths transversality. Since KU ∼= OU and H0(U, KU) ∼= H0(Y, KY) ∼= C, F3 is a line bundle overZ1spanned by a nowhere vanishing relative holomorphic 3-form Ω ∈ Ω3U/Z1. Near the moduli point [Y] ∈ Z1, F2 is then spanned by Ω and v(Ω) where v runs through a basis of H1(U, TU). Notice that v(Ω) ∈W3precisely when v∈H1(Y, TY).
By Proposition 6.1, the partial Hodge filtration F3 ⊂ F2 on H3(U)over Z1lifts uniquely to a filtration ˜F3⊂ F˜2on H3(X)overZ1with ˜F3 ∼=F3and F˜2 ∼=F2. The complete lifting ˜F• is then uniquely determined since
F˜1= (F˜3)⊥
by the first Hodge–Riemann bilinear relation on H3(X). Alternatively, ˜F1 is spanned by ˜F2and v(F˜2)for v runs through a basis of H1(U, TU).
Now ˜F• overZ1uniquely determines a horizontal map Z1→ D.ˇ
Since it has maximal tangent dimension h1(U, TU) = h1(X, TX), it deter-mines uniquely the maximal horizontal slice
ψ :M → Dˇ
with M ∼= MX¯ locally nearMY. According to Theorem 4.15, namely an extension of Schmid’s nilpotent orbit theorem, under the coordinates t = (r, s), the period map
φ :Z× ∼=MX= MX¯\[ki=1Di → D/Γ is then given by
φ(r, s) =exp
k
∑
i=1
log wi
2π√
−1N
(i)
! ψ(r, s)
where Γ is the monodromy group generated by the local monodromy N(i) around the divisor Di defined by wi = ∑µj=1aijrj = 0 (c.f. (4.7)). Since N(i) is determined by the Picard–Lefschetz formula (Lemma 4.14), we see that the period map φ is completely determined by the relation matrix A of the extremal curves Ci’s. (The period map gives the desired VHS, with degenerations, overZ×.) This completes the proof that refined B model on Y\Z=U determines the B model on X.
Remark 6.3. Bryant and Griffiths reformulate the VHS for Calabi–Yau three-folds in terms of Legendre subvarieties in P(H3(X))It might be possible to show that ˜F• over Z1 uniquely determines a Legendre subvariety inside P(H3(X))which coincides withMX¯.
7. REMARKS ON THE BASIC EXACT SEQUENCE
For a conifold transition of Calabi–Yau 3-folds X ր Y, we have shown that the combined information of A model and B model of X determines the corresponding information on Y, and vice versa. However, the effective computational method for such a determination has not been addressed much besides the vanishing/extremal invariants.
In this final section of the paper we make two remarks concerning the quantum aspects of the basic exact sequence. The aim is to shed some light on the possible directions to achieve such an effective computational method, especially on the implication A(X) +B(X) ⇒ A(Y).
7.1. Local transitions between A(Y)andB(X). The basic exact sequence in Theorem 2.9 provides a Hodge theoretic realization of the numerical identity µ+ρ = k. Now H2(Y)/H2(X) ⊗C ∼= Cρ is naturally the pa-rameter space of the extremal Gromov–Witten invariants of the K¨ahler de-generation ψ : Y →X, and V¯ ∗⊗C∼=Cµis naturally the parameter space of periods of vanishing cycles of the complex degeneration from X to ¯X. Both of them are equipped with flat connections induced from the Dubrovin (resp. Gauss–Manin) connection over their tangent bundles. Thus it is nat-ural to ask if there is aDmodule lift of the basic exact sequence.
We rewrite the basic exact sequence in the form Ck
HC2(Y)/H2C(X) ∼=Cρ
B
♦ 77♦
♦♦
♦♦
♦♦
♦♦
♦♦
♦
VC∗ ∼=Cµ
A
cc❍❍❍❍❍❍❍❍❍❍
with AtB=0. This simply means that Ckis an orthogonal direct sum of the two subspaces im(A)and im(B). Let A = [A1,· · · , Aµ], B = [B1,· · · , Bρ], and consider the invertible matrix
S= (sij):= [A, B] ∈Mk×k(Z), namely sij = aij for 1≤j≤µ and siµ+j =bij for 1≤ j≤ρ.
Denote the standard basis of Ckby e1,· · · , ekwith coordinates y1,· · · , yk. Let e1,· · · , ekbe the dual basis on(Ck)∨. We consider the standard (trivial) logarithmic connection on the bundle Ck⊕ (Ck)∨over Ck defined by
(7.1) ∇ =d+1
z
k
∑
i=1
dyi
yi ⊗ (ei⊗e∗i),
where z is a parameter. It is a direct sum of k copies of its one dimensional version. We will show that the principal (logarithmic) part of the Dubrovin connection over Cρ (c.f. (3.8)) as well as the Gauss–Manin connection on Cµ (c.f. (4.11)) are all induced from this standard logarithmic connection through the embeddings defined by B and A respectively.
Recall the basis T1,· · · , Tρ of Cρ with coordinates u1,· · · , uρ, and the Denote by P the orthogonal projection
P : Ck⊕ (Ck)∨ →Cρ⊕ (Cρ)∨.
We compare it with the one obtained in (3.8), (3.9) and (3.11):
∇zTlTm = −1z the discussion in the following proposition:
Proposition 7.1. Let XրY be a projective conifold transition through ¯X with k ordinary double points. Let the bundle Ck⊕ (Ck)∨ over Ck be equipped with the standard logarithmic connection defined in (7.1). Then
(1) The connection induced from the embedding B : Cρ → Ck defined by the relation matrix of vanishing 3 spheres for the degeneration from X to ¯X gives rise to the logarithmic part of the Dubrovin connection on H2(Y)/H2(X).
(2) The connection induced from the embedding A : Cµ → Ck defined by the relation matrix of extremal rational curves for the small contraction Y → X gives rise to the logarithmic part of the Gauss–Manin connection¯ on V∗, where V is the space of vanishing 3-cycles.
Part (1) has just been proved. The proof for (2) is similar (by setting z=2π√
−1 and wi =yi, c.f. (4.11)) and is omitted. We remark that the two subspaces B(Cρ)and A(Cµ)are indeed defined over Q and orthogonal to each other, hence A and B determine each other up to choices of basis.
7.2. Speculation for globalization. Our proof for A(X) +B(X) ⇒ A(Y) in Section 5.3 is not constructive. Here we discuss briefly an idea developed in a forthcoming work to attack the problem for genus zero theory [18].
We have seen that the Dubrovin connection on H2(Y)/H2(X)is deter-mined by the relation matrix B of vanishing spheres. Consider the diagram
H2(Y)/H2(X)oo H2(Y)
H2(X),
OO
and regard it as the cohomology realization of the small contraction Sk
i=1Ci //Y
ψ¯
¯ X.
Since ¯X is singular and not an orbifold, the Gromov–Witten theory on ¯X is so far undefined in the literature. Nevertheless, in the current situa-tion, according to the principle of deformation invariance we may treat it as GW(X), which is given. Now the picture looks very similar to the quantum Leray–Hirsch theorem for projective (or toric) bundles proved in [17] despite the fact that ¯ψ is a birational contraction/crepant blowup in-stead of a bundle morphism. However, in the cohomology level it looks just like a bundle. Thus it is reasonable to believe that the idea of quantum Leray–Hirsch principle can also be applied to such a situation.
To see how the B model of X enters the picture, we mention only the observation that ¯ψ : Y → X can be realized as the blow-up along certain¯ Weil divisors in ¯X, and those Weil divisors can in fact be constructed from the relation matrix B of the k vanishing spheres Si’s.
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Y.-P. LEE: DEPARTMENT OFMATHEMATICS, UNIVERSITY OFUTAH, SALTLAKECITY, UTAH84112-0090, U.S.A.
E-mail address:[email protected]
H.-W. LIN: DEPARTMENT OFMATHEMATICS ANDTAIDAINSTITUTE FORMATHEMATI
-CALSCIENCES(TIMS), NATIONALTAIWANUNIVERSITY, TAIPEI10617, TAIWAN
E-mail address:[email protected]
C.-L. WANG: DEPARTMENT OFMATHEMATICS, CENTER FORADVANCEDSTUDIES IN
THEORETICALSCIENCES(CASTS),ANDTAIDAINSTITUTE FORMATHEMATICALSCIENCES
(TIMS), NATIONALTAIWANUNIVERSITY, TAIPEI10617, TAIWAN
E-mail address:[email protected]