STABLE PAIRS ON NODAL K3 FIBRATIONSAMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Abstract. We study Pandharipande-Thomas’s stable pair theory on
K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of
Kawai-Yoshioka’s formula for the Euler characteristics of moduli spacesof stable pairs on K3 surfaces and Noether-Lefschetz numbers of the
fibration. Moreover, we investigate the relation of these invariants withthe perverse (non-commutative) stable pair invariants of the K3 fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of
2-dimensional Gieseker semistable sheaves supported on the fibers.

Contents
1. Introduction
2. Stable pairs on smooth K3 fibrations
3. Stable pairs on nodal K3 fibrations
4. Wall-crossing formula for K3 fibrations
References

1
6
17
29
36

1. Introduction
1.1. Overview. The Fourier-Mukai transform is an equivalence of derived
categories of coherent sheaves of algebraic varieties, which was first studied
for dual pairs of abelian varieties in the seminal paper of Mukai [Muk81].
When a Calabi-Yau 3-fold admits an elliptic fibration, its relative FourierMukai transform turned out to be an effective tool to prove the correspondence between one dimensional sheaf theory and two dimensional sheaf theory [ACRY00], [AYCR01]. In particular, it provides a recipe to relate twokinds of invariants: Pandharipande-Thomas stable pair invariants [PT09]
which count curves in Calabi-Yau 3-folds, and Donaldson-Thomas invariants which count two dimensional semistable sheaves on them [Tho00].
The above correspondence is also important in string theory. The relative Fourier-Mukai transform is a mathematical interpretation of relative Tduality in string theory, which was used by physicists [KMW12, MNVW98,
BDF07] to discuss the correspondence between the BPS theory of D2 branes
wrapped on the base and the D4 branes which also wrap the elliptic fibers.
The PT stable pairs and the two dimensional semistable sheaves are the12

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

mathematical analogues of the D6-D2-D0 branes, and the D4 branes respectively, and the relative T-duality provides a recipe to relate to the BPS state
counting of these D-brane systems.
In a more general setting, without assuming that the threefold admits
an elliptic fibration, we can still ask the question regarding the correspondence of PT stable pair theory and the DT theory of two dimensional
semistable sheaves (i.e. D6-D2-D0 BPS theory and D4 theory in string
theory). This question shapes the main motivation behind the current articles. In [Tod13a], the third author derived the formula relating the low
degree terms of the generating series of the latter invariants with those of
the former invariants, using the wall-crossing technique and some ideas fromstring theory [DM]. In [GS13], the first and the second authors studied thegenerating series of DT invariants counting two dimensional stable sheaves
on K3 fibered threefolds with possibly nodal fibers, and proved their modularity property.
The purpose of this article is to obtain an analogue of the results in [GS13]
for PT stable pair theory on K3 fibered threefolds. Namely we investigate
the modularity property of the stable pair invariants on K3 fibrations. We
first treat the case of a smooth K3 fibration, and relate the stable pair invariants with the generating series satisfying the modularity property. When
the K3 fibration has possibly nodal fibers, we relate their stable pair invariants with those on smooth K3 fibrations via conifold transition. In a moregeneral situation, we also prove a formula relating stable pair invariants with
the DT invariants counting two dimensional semistable sheaves, where thelatter invariants are expected to have the modular invariance property.
1.2. Main result I: smooth K3 fibration. Let X be a smooth projective
threefold over C. For β ∈ H2 (X, Z) and n ∈ Z, the moduli space of stablepairs Pn (X, β) and the associated invariants were introduced by Pandharipande and Thomas [PT09]. The moduli space Pn (X, β) parametrizes pairs(1)s : OX → F

where F is a pure 1-dimensional sheaf on X with [F ] = β and χ(F ) = n,
and the cokernel of s is 0-dimensional. They constructed a perfect obstruction theory on Pn (X, β) (see Theorem 2.1) by interpreting it as the moduli
space of two term complexes (1) in the derived category. When the virtual
dimension of Pn (X, β) is zero, the stable pair invariant Pn,β is defined bytaking the degree of the virtual cycle obtained from this obstruction theory.
We assume that X admits a morphism(2)

π: X → C

onto a smooth projective curve C, whose generic fiber is a smooth K3 surface. The morphism π is called a K3 fibration. We always take the curveclass β to be in the kernel of π∗ denoted by H2 (X, Z)π . The virtual dimension of Pn (X, β) is zero for such curve classes1. We define the following
1Note that X is not required to be a Calabi-Yau 3-fold here.

STABLE PAIRS ON NODAL K3 FIBRATIONS

3

generating series:(3)

PT(X)β =X

X

Pn,β q n , PT(X) =PT(X)β tβ .

β∈H2 (X,Z)πn∈Z

In Section 2, we study the case when π is a smooth morphism. The majority of this section (Subsections 2.1-2.6) is devoted to the case where β
is an irreducible curve class. In this case, we relate Pn,β with the Euler
characteristics of the moduli space of stable pairs on a nonsingular K3 surface S and the Noether-Lefschetz numbers N Lπh,β of the fibration having
modular properties [MP13, Bor99, Bor98, KM90]. We denote by Pn (K3, h)
the moduli space of stable pairs on a K3 surface containing an irreducible
curve class γ satisfying γ 2 = 2h − 2. It is known that the topological typeof Pn (K3, h) only depends on (n, h) (cf. [MPT10]).
Theorem 1. (Theorem 2.16) Let π : X → C be a smooth K3 fibration, andβ ∈ H2 (X, Z)π be an irreducible curve class. Then we have the following
formula:PT(X)β =∞ X∞X(−1)n−1 χ(Pn (K3, h)) · N Lπh,β q n .

h=0 n=1−h

The proof of Theorem 1 involves studying the restriction of the obstruction theory constructed in [PT09] to isolated and non-isolated componentsof Pn (X, β). The Euler characteristics χ(Pn (K3, h)) in Theorem 1 can beread off from Kawai-Yoshioka’s formula having modular properties [KY00,
Theorem 5.80]:∞ X∞X(−1)n−1 χ(Pn (K3, h))y n q h =h=0 n=1−h

(4)

−


√

1
−y − √
−y

−2 Y
∞
n=1

1
.
(1 − q n )20 (1 + yq n )2 (1 + y −1 q n )2

In Subsection 2.7, we study a special analog of Theorem 1 in which theclass β ∈ H2 (X, Z)π is allowed to be reducible.
1.3. Main result II: nodal K3 fibration. We next study the case that
the K3 fibration (2) has a finite number of nodal fibers. In this case, we calla morphism (2) a nodal K3 fibration. For a nodal K3 fibration, we find aconifold transition formula, which relate the stable pair invariants on nodal
K3 fibrations with those on smooth K3 fibrations. The following result isbased on assuming some foundation of stable pair theory of algebraic spacesthat has not been established yet (see Remarks 3.3 and 3.4):
Theorem 2. (Theorem 3.11) Let π : X → C be a nodal K3 fibration. Thene →Ce and morphisms
there is a smooth K3 fibration2 π
e: X
h



e → X0 → Xe
: X
2Here the situation is a little more general than (2), and we allow X
e to be a proper
algebraic space.

4

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

where h is a small resolution and  is a double cover, such that the following
formula holds:
e
∗

(5)

e
PT(X)
= PT(X)2 .
ePTh (X)e is the subseries ofHere e∗ is the variable change tβ 7→ te∗ β , and PTh (X)e of stable pair invariants of curve classes βe with h∗ βe = 0.
PT(X)Let us consider the invariant Pn,β with an irreducible curve class β on anodal K3 fibration π : X → C. The above result leads us to working withthe stable pair invariants Pñ,β̃ for possibly reducible classes β̃ on a smoothe →C
e lifting β. In Subsection 3.6, we present an approachK3 fibration X
to compute the invariants Pñ,β̃ by studying the relationship between theinvariants Pñ,β̃ and the invariants of the moduli spaces of perverse (noncommutative) stable pairs.
1.4. Main result III: product expansion formula. Finally let X be asmooth projective Calabi-Yau 3-fold, which admits a K3 fibration π : X →P1 such that every fiber is an integral scheme. In Section 4, using the wallcrossing techniques mostly developed by the third author [Tod13b, Tod10a,
Tod10b, Tod12a, Tod12b, Tod], we express the generating series (3) in termsof the generalized Donaldson-Thomas invariantsJ(r, β, n) ∈ Q(6)which count semistable sheaves E supported on the fibers of π satisfying(7)ch(E) = (0, r[F ], β, n) ∈ H 0 (X) ⊕ H 2 (X) ⊕ H 4 (X) ⊕ H 6 (X).

Here [F ] is the class of a fiber of π in X. The invariants (6) are defined by
the method of Joyce-Song [JS12]. These invariants were studied by the firstand second authors in [GS13] when there are no strictly semistable sheaves
F satisfying (7). The modularity property of the invariants (6) holds in thesituation of [GS13], and we expect such a property in general. The following
result gives a complete answer on describing the relationship between stable
pair invariants and DT type invariants counting two dimensional sheaves:
Theorem 3. (Theorem 4.4) We have the following formula

n+2r
Y
PT(X) =
exp (−1)n−1 J(r, β, r + n)q n tβr≥0,β>0,n≥0

·

Y


n+2r
exp (−1)n−1 J(r, β, r + n)q −n tβ.

r>0,β>0,n>0

The above result is proved by generalizing the argument in [Tod12b] showing a similar result when X = S × C for a K3 surface S. Namely, we apply
Joyce-Song’s wall-crossing formula [JS12] in a certain space of weak stability
conditions. As the arguments are the almost same as in [Tod12b] except a
few modifications, we will just outline its proof in Section 4.

STABLE PAIRS ON NODAL K3 FIBRATIONS

5

1.5. Summary. The main purpose of the current article is to find the connection between the vertices of the following triangle:
Modular forms
•
2

•
PTX/C

1

3

•
DTX/C

Here by PTX/C and DTX/C we mean the fiberwise stable pair and Donaldson-Thomas theories of the Calabi-Yau K3 fibration X → C discussed
above. As mentioned earlier, the connection 1 ↔ 3 is motivated by T duality and we provide a complete answer to it in Section 4. The connection 3 ↔ 2 is motivated by S-duality, and the first two authors haveprovided partial results on it in [GS13]. In Section 2 and Section 3 of thispaper, we provide partial results on the connection between 1 ↔ 2. More recently, Pandharipande-Thomas [PT14] studied the connection 1 ↔ 2 in moregenerality by a different method (degeneration method) and for a different
purpose. A refined (motivic) version of connection 1 ↔ 2 is conjectured
in [KKP]. We hope the combination of the established bridges 1 ↔ 2 and1 ↔ 3, sheds some lights on the connection 3 ↔ 2 in full generality, which
we hope we can work out in the future.
1.6. Acknowledgment. We would like to thank Jan Manschot and Richard
Thomas for helpful discussions. The first author was partially supported by
NSF grant DMS-1406788. The second author would like to thank MIT and
the Institute for the Physics and Mathematics of the Universe (IPMU) for
hospitality and providing the opportunity to discuss about this project during his visits. Special thanks to Max Planck Institut für Mathematik for
hospitality during the second author’s stay in Bonn. The second and third
authors are supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. The third author is also supported
by Grant-in Aid for Scientific Research grant (No. 26287002) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
1.7. Notation and convention. In this paper, all the varieties or schemes
are defined over C. For a morphism π : X → Y of schemes and coherent
sheaves E, F on X, we write the sheaf π∗ HomX (E, F ) as Homπ (E, F ).
Its right derived functor is denoted by RHomπ (E, F ), which coincides withRπ∗ RHom(E, F ). If π is smooth, we denote by ωπ the relative canonicalline bundle on X. For a scheme X, we denote by L•X the cotangent complex
of X.

6

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

2. Stable pairs on smooth K3 fibrations
2.1. Stable pairs on K3 fibrations. Let X be a smooth projective threefold over C. By definition, a K3 fibration is a morphism
π: X → C
onto a smooth projective curve C whose generic fiber is a smooth K3 surface.
If π is a smooth morphism, then it is called a smooth K3 fibration. We set
H2 (X, Z)π := {β ∈ H2 (X, Z) : π∗ β = 0}.
We always take curve classes contained in the above subgroup of H2 (X, Z).
An element β ∈ H2 (X, Z) is called irreducible if it is not written as a sum
β1 + β2 for non-zero effective curve classes β1 , β2 .
For β ∈ H2 (X, Z)π , we study the moduli space of stable pairs Pn (X, β)in the sense of [PT09]. It parametrizes the pairss : OX → F(8)where F is a pure 1-dimensional sheaf on X with [F ] = β and χ(F ) = n, andthe cokernel of s is 0-dimensional. Here [F ] is the fundamental homology
class determined by F , which is the Poincare dual of ch2 (F ). For simplicity
we set P = Pn (X, β), and denote byI• = (OX×P → F)(9)

the universal pairs, which we interpret as an object in Db (X × P). Notethat we have the exact triangle on X × P
I• → OX×P → F → I• [1].

(10)
Let

πP : X × P → P, πX : X × P → X
be the natural projections. In [PT09], Pn (X, β) is equipped with a perfectobstruction theory by studying the deformations of the two term complex(8) in the derived category:
Theorem 2.1. [PT09] There is a perfect obstruction theory over P givenby the following morphism in the derived category:E • := RHomπP (I• , I• ⊗ ωπP )0 [2] → L•P .
Here (−)0 means taking the traceless part.
Since ωX · β = 0 for β ∈ H2 (X, Z)π , the Riemann-Roch calculation easily
implies that the virtual dimension of P is zero. The stable pair invariants
are then defined byZ
Pn,β :=
1.
[Pn (X,β)]vir

We define the generating series PT(X)β , PT(X) to be
XX(11)
PT(X)β :=Pn,β q n , PT(X) :=PT(X)β tβ .
n∈Z

β∈H2 (X,Z)πSTABLE PAIRS ON NODAL K3 FIBRATIONS

7

2.2. Stable pairs on K3 surfaces. Let S be a smooth projective K3
surface over C. For γ ∈ H2 (S, Z), we can similarly define the moduli spaceof stable pairs Pn (S, γ) on S. We review several results on the moduli space
Pn (S, γ). In [PT10], Pandharipande and Thomas identify Pn (S, γ) by arelative Hilbert scheme of points:
Proposition 2.2. [PT10, Proposition B.8] The moduli space of stable pairs
2
Pn (S, γ) on S is isomorphic to the relative Hilbert scheme Hilbn+γ /2 (C/M)
where M is the moduli space of pure one dimensional subschemes of S inclass γ and C denotes the universal curve over S × M.
We next consider the case that γ is irreducible. In this case, the following
result is proved in [MPT10]:
Proposition 2.3. [MPT10, Proposition 5] If γ is irreducible, the moduli
space Pn (S, γ) is non-singular of dimension n + γ 2 + 1. It depends only
upon (n, γ 2 ) up to deformation equivalence.
By the above proposition, we may write
Pn (S, h) := Pn (S, γ)(12)

for an irreducible curve class γ ∈ H2 (S, Z) with γ 2 = 2h − 2. We sometimes
write Pn (K3, h) if we do not want to specify a particular K3 surface withthis property. The generating series of χ(Pn (K3, h)) is computed by KawaiYoshioka’s formula (4).
2.3. Stable pairs with irreducible curve classes. In this subsection, we
assume that β ∈ H2 (X, Z)π is an irreducible curve class. Let P = Pn (X, β)be the moduli space of stable pairs, and consider the perfect obstruction
theory E • → L•P in Theorem 2.1. We have the following proposition:
Proposition 2.4. In the above situation, we have the canonical isomorphism∼=

H1 (E •∨ ) → HomπP (I• , F ⊗ ωπP )∨ .
Proof. We write ω = ωπP for simplicity. Applying RHomπP (I• , − ⊗ ω) tothe exact triangle (10), we obtain the following exact triangle on P:
RHomπP (I• , F ⊗ ω) → RHomπP (I• , I• ⊗ ω)[1] → RHomπP (I• , ω)[1].
By the above triangle and the natural morphism
RπP∗ ω → RHomπP (I• , I• ⊗ ω)
we can form the following commutative diagram of vertical and horizontal
exact triangles:(13)

RπP∗ ω[1]=/ RπP∗ ω[1]



RHomπP (I• , F ⊗ ω) / RHomπP (I• , I• ⊗ ω)[1] / RHomπP (I• , ω)[1]


id





RHomπP (I• , F ⊗ ω) / RHomπP (I• , I• ⊗ ω)0 [1] / RHomπP (F, ω)[2]8AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

After dualizing, the bottom row of the diagram gives the exact triangle
RHomπP (O, F)[1] → RHomπP (I• , I• )0 [2] → RHomπP (F, I• )[3].
The 0-th cohomology of the sequence above gives the exact sequencegExt2πP (I• , I• )0 →− Ext3πP (F, I• ) → Ext2πP (O, F)
Note that the first term in the sequence above is H1 (E •∨ ). Since the fibersof πP |Supp(F) are one dimensional, we obtain Ext2πP (O, F) = 0 and hence themap g is surjective. Now we show that the map g is also injective. It isenough to prove this over any closed point p ∈ P. Here we will use part ofthe strategy of the proof of [PT09, Proposition 4.4]. Over the point p andafter dualizing and taking cohomology, the diagram (13) gives the following
diagram of the vertical and horizontal exact sequences0

Ext2 (F, I • )=Ext (F, I • )2/ Ext2 (I • , I • )0

/ H 1 (F )gp/ Ext2 (OX , I • )/ Ext2 (I • , I • )H 2 (OX )


0

=

'
/ Ext3 (F, I • )tr/ H 2 (OX )0where the map gp is induced by the map g described above. As in [PT09,
gp
Proposition 4.4], in order to show that Ext2 (I • , I • )0 −→ Ext3 (F, I • ) is injective, it is enough to show that the composition(14)

Ext1 (F, F ) → Ext2 (F, I • ) → H 1 (F )is surjective. Here the first map is induced by the exact triangle I • → OX →F for the stable pair I • = (OX → F ).
Suppose that F is supported on a curve D ⊂ X which is irreducible by
our assumption. We claim the inclusion and the isomorphism(15)

Ext1 (F, F ) ⊇ H 1 (Hom(F, F )) ∼= H 1 (OD ).

The first inclusion follows from the local to global spectral sequence. As fortr
the second isomorphism, let us consider the trace map Hom(F, F ) −→ OD .
The above map is obviously surjective, and the stability of pairs implies
that its kernel is at most zero dimensional. Therefore we have the second
isomorphism in (15). On the other hand, using the fact that the cokernelsof the map OD →− F defining the stable pair is zero dimensional, we canssee that H 1 (OD ) →− H 1 (F ) is surjective. Hence combined with (15), we
conclude that the map (14) is also surjective.

STABLE PAIRS ON NODAL K3 FIBRATIONS

9

Now we have proved that the map g is an isomorphism:∼=g : H1 (E •∨ ) → Ext3πP (F, I• ).
The desired result follows from the above isomorphism and the relative duality Ext3πP (F, I• ) ∼= HomπP (I• , F ⊗ ωπP )∨ .
We next distinguish two kinds of components of P. We use the following
lemma:
Lemma 2.5. Suppose that β is irreducible. Then for any (OX → F ) ∈Pn (X, β), there exists a unique point p ∈ C such that F is scheme theoretically supported on π −1 (p).
Proof. Since β is irreducible, the sheaf F is set theoretically supported on
π −1 (p) for some p ∈ C. Let mp ⊂ OC,p be the maximal ideal, and takes ∈ mp . Then the sheaf homomorphism ·s : F → F must be a zero map as
β is irreducible and F is pure. Hence F is an Oπ−1 (p) -module.

The map sending (OX → F ) to p ∈ C in Lemma 2.5 defines the morphismρ : P → C.
Definition 2.6. We call a connected component Pc of P = Pn (X, β) a type
I component if ρ(Pc ) = C. Otherwise we call Pc a type II component.
Below we denote by PI ⊂ P the union of type I connected components,
and PII ⊂ P the union of type II connected components. We set
iS : S := X ×C P ,→ X × P.
We have the Cartesian square:
S

(16)
ρS



C

iS


/ X ×P(π,ρ)/ C × C.

Here the bottom arrow is the diagonal. The universal one dimensionalsheaves F in (9) is written as iS∗ G for a sheaf G on S. We denote byI•S the universal pairs on S:
I•S := (OS → G).
0 and π 0 the compositions of i with π
Also we denote by πPS
P and πX
X
respectively, i.e. they are projections:
00πP: S → P, πX: S → X.

By [KT14, Proposition 3.4] and the standard techniques of [HT10], the type I(resp. II) components can be equipped with a relative (absolute) obstruction
theory:Theorem 2.7. (i) There exists a ρ-relative perfect obstruction theory overPI given byGI• := RHomπP0 (G, I•S ⊗ ωπP0 )|PI [2] → L•PI /C .
(ii) There exists a perfect obstruction theory over PII given by•:= RHomπ0 (G, I•S ⊗ ωπ0 )|PII [2] → L•PII .
GII
P
P

10

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

2.4. Type I component. Now we assume that π : X → C is a smooth K3
fibration and β an irreducible curve class. Here we investigate the contributions of the type I components to the virtual classes. We have the following
statement:
Proposition 2.8. [PT10, Proposition C.2 and Lemma C.7] The type I components PI is non-singular of dimension n + 2g, where g is the arithmetic
genus of the support of a stable pair in PI . Moreover, we have the isomorphism of the tangent bundles:
TP ∼= Homπ (I• , F)|P , TP /C ∼= Homπ0 (I• , G)|P .
P

I

I

I

P

S

I

Let K be the line bundle on C given by
K := π∗ ωX/C .
Proposition 2.9. We have the exact sequence of vector bundles on PI :0 → ρ∗ K∨ →HomπP (I• , F ⊗ ωπP )∨ |PI→ HomπP0 (I•S , G ⊗ ωπP0 )∨ |PI ⊗ ρ∗ TC → 0.
Proof. Since F = iS∗ G, we have the exact triangle
G(−S) → Li∗S I• → I•S .

(17)

We apply RHomπP0 (−, G ⊗ i∗S ωπP ) to the above exact triangle. By taking
the cohomologies and restricting to PI , we get the exact sequence
0 → HomπP0 (I•S , G ⊗ i∗S ωπP )|PI → HomπP (I• , F ⊗ ωπP )|PI
→ HomπP0 (G(−S), G ⊗ i∗S ωπP )|PI .
By Proposition 2.8, all the sheaves in the above sequences are vector bundles.
We investigate the fibers of the vector bundles in the sequence above. By
the fiberwise stability of G, the bundle HomπP0 (G(−S), G ⊗ i∗S ωπP )|PI is aline bundle on PI . Therefore by Proposition 2.8 and the definition of type
I component, the above sequence must be a short exact sequence. After
dualizing, we obtain the short exact sequence
0 → Ext2π0 (G, G)|PI → HomπP (I• , F ⊗ ωπP )∨ |PI
P→ HomπP0 (I•S , G ⊗ i∗S ωπP )∨ |PI → 0.
Here the first term is obtained by applying the Grothendieck duality andthe adjunction formula ωπP0 ∼= i∗S ωπP ⊗ OS (S). Now we haveExt2π0 (G, G)|PI ∼= R2 HomπP0 (G, G)|PIP

0 O |which is identified with R2 πP∗
S PI via the trace map using the fiberwise
stability of G again. By the Grothendieck duality and the adjunction for0
mula again and noting that ωπP0 = πX∗ ωX/C , we obtain
00
R2 πP∗
OS |PI ∼ωπP0 )∨ |PI ∼= (R0 πP∗
= ρ∗ K∨ .
Finally, again using the adjunction and noting OS (S) ∼= ρ∗ TC by the diagram (16), we have the isomorphism
Homπ0 (I• , G ⊗ i∗ ωπ )∨ |P ∼= Homπ0 (I• , G ⊗ ωπ0 )∨ |P ⊗ ρ∗ TC .
P

S

S

P

I

P

S

Therefore we obtain the desired exact sequence.

P

I



STABLE PAIRS ON NODAL K3 FIBRATIONS

11

Using the above proposition, we have the following statement:
Proposition 2.10. The restriction of the virtual class of the obstruction
theory E • in Theorem 2.1 to the type I components PI is given by[PI , E • ]vir = [PI ] ∩ c1 (ρ∗ K∨ ) ∪ ctop (ΩPI /C ) .
Proof. By Propositions 2.4 and 2.8, PI is smooth with the obstruction sheaf
HomπP (I• , F ⊗ ωπP )∨ |PI . Using the short exact sequence in Proposition 2.9,
we can write(18)

ctop (HomπP (I• , F ⊗ ωπP )∨ |PI )= c1 (ρ∗ K∨ ) ∪ ctop (HomπP0 (I•S , G ⊗ ωπP0 )∨ |PI ⊗ ρ∗ TC ).

By the following lemma and using the fact that the intersection product ofany two classes on A1Q (C) is zero, we can see that onlyc1 (ρ∗ K∨ ) ∪ ctop (HomπP0 (I•S , G)∨ |PI ) = c1 (ρ∗ K∨ ) ∪ ctop (ΩPI /C )contributes to the formula (18). Here the above identity is due to Proposition 2.8. Therefore, the desired equality follows from [BF97, Proposition
5.6].

Lemma 2.11. Suppose that P0 ⊂ PI is a type I component of dimension l.
Then, we have the following relation in A∗Q (P0 ):ctop (HomπP0 (I•S , G ⊗ ωπP0 )∨ ) = cl (HomπP0 (I•S , G)∨ ) + A · ρ∗ (B)001
for some A ∈ Al−1
Q (P0 ) and B ∈ AQ (C).

Proof. Since ωX/C is fiberwise trivial, it must be the pullback of a line bundle0∗ ω∗∗
M on C. So we can write ωπP0 = πX
X/C = πP0 ◦ ρ M . The Grothendieck0
Riemann-Roch formula gives:
X

1
j
•
j
∨
ch
(−1) Extπ0 (IS , G ⊗ ωπP0 )
P0

j=0
0
= πP∗



0



ch(I•S ) · ch(G)∨ · ch(ωπP0 )∨ · td(X π ×ρ P0 )0∗= (1 − ρ c1 (M )) · ch

X
1
j=0

(−1) Extjπ0 (I•S , G))∨
P
j


.

0

By (C.6) in the proof of [PT10, Proposition C.2], the fibers of the sheaves
Ext1π0 (I•S , G ⊗ ωπP0 )∨P0

0and Ext1π0 (I•S , G)∨P0

at any closed point IS• = (OS → G) ∈ P0 is naturally identified with
Ext1 (IS• , G) ∼= H 2 (OS )for a K3 fiber S ⊂ X. Therefore, both sheaves are line bundles pulled backfrom the base curve C, thus their Chern characters are the pull backs of theclasses from A∗Q (C). The lemma is then proven by an inductive argument
on n.


12

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

2.5. Type II component. Let Pc ⊂ PII be a type II component. Then
there exists a point p ∈ C such that ρ(Pc ) = p. Let us set S = π −1 (p)and i : S ,→ X is the inclusion. Then there exists unique γ ∈ H2 (S, Z) withi∗ γ = β and a closed embedding
Pn (S, γ) ⊂ Pc .

(19)

Definition 2.12. A type II component Pc is called isolated if the embedding(19) is an isomorphism.
Remark 2.13. Since Pn (S, γ) is non-singular by Proposition 2.3, the embedding (19) is an isomorphism if and only if (19) induces isomorphisms oftangent spaces at every points in Pn (S, γ).
We have the following lemma:
Lemma 2.14. Suppose that any component of PII is isolated. Then we havethe isomorphism of vector bundles:∼=ΩPII → HomπP (I, F ⊗ ωπP )∨ |PII .
Proof. By the definition of type II components and Remark 2.13, the following morphism is an isomorphism∼=

TPII ∼= HomπP0 (IS , G)|PII → HomπP (I, F)|PII .
Here the first isomorphism is due to Theorem 2.7 (ii). Since ωX restricted to
any K3 fiber is trivial, and for any component Pc ⊂ PII , F|Pc ×X is supported
on Pc × S for some K3 fiber S ⊂ X, we have the isomorphismF ⊗ ωπ |P ×X ∼= F|P ×X .
P

II

II

Therefore we obtain the lemma.



By the above lemma and Proposition 2.4, the contributions to virtual
classes from the isolated type II components are given as follows:
Corollary 2.15. In the situation of Lemma 2.14, the restriction of thevirtual class of the obstruction theory E • in Theorem 2.1 to the type II components PII is given by[PII , E • ]vir = [PII ] ∩ ctop (ΩPII ).
2.6. Generating series of stable pair invariants. Following [MP13], let(20)

V = R2 π∗ (Z) → C

be the rank 22 local system determined by the K3 fibration π. We denoteby Vc the fiber of (20) at c ∈ C. Let HV denote the π-relative moduli spaceof Hodge structures as in [MP13, Section 1.4] 3, i.e. there is a map
HV → C
such that each fiber HVc at c ∈ C is the moduli space of weight two Hodge
structures on Vc ⊗ C = H 2 (S, C) for S = π −1 (c). There exists a section map(21)

σ : C → HV

3In [MP13, Section 1.4] this is denoted by MV .

STABLE PAIRS ON NODAL K3 FIBRATIONS

13

which is determined by the Hodge structures of the fibers of π:(22)σ(c) = [H 2,0 (S) ⊂ H 2 (S, C)] ∈ HVc , S = π −1 (c).

For c ∈ C, an irreducible class β ∈ H 2 (X, Z)π and h ∈ Z, we define
Vc (h, β) = {0 6= γ ∈ Vc | γ 2 = 2h − 2, i∗ γ = β}.
Also let Bc (h, β) ⊆ Vc (h, β) be the subset of γ ∈ Vc (h, β) where γ is a (1, 1)class on S = π −1 (c). Then Bc (h, β) is finite by [MP13, Proposition 1]. The
subset[B(h, β) :=Bc (h, β) ⊂ Vc∈C

can be decomposed intoB(h, β) = BI (h, β)aBII (h, β)

where the first component defines a finite local system  : BI (h, β) → C,
and the second component is an isolated set. Let P be the connectedcomponent of the stable pair moduli space P = Pn (X, β) corresponding to
the local system , and letZPn, =
1[P ,E • ]virbe the contribution of this component to Pn,β . Note that P is a type I
component of Pn (X, β) in the sense of Definition 2.6. By Proposition 2.10,
we have(23)[P , E • ]vir = [P ] ∩ c1 (ρ∗ K∨ ) ∪ ctop (ΩP/C ) .
On the other hand, suppose that α ∈ BII (h, β) supported on the fiber S isa result of a transversal intersection (i.e. with intersection multiplicity 1) ofσ(C) with a Noether-Lefschetz divisor. Then the corresponding connectedcomponent Pα of Pn (X, β) is an isolated type II component in the sense of
Definition 2.6, and it is isomorphic to Pn (S, h) given in (12). By Corollary
2.15, we have
(24)

[Pα , E • ]vir = [Pα ] ∩ ctop (ΩPα ).

We defineZPn,α =

1[Pα ,E • ]vir

to be the contribution to Pn,β of this component. For any integer h ∈ Z,
the Noether-Lefschetz number
N Lπh,β ∈ Z
was defined in [MP13] by intersecting σ(C) with the π-relative NoetherLefschetz divisor in HV associated to h and β. Informally, N Lπh,β is thenumber of the fibers S of π for which there exists a (1, 1) class γ ∈ H 2 (S, Z)
such thatγ 2 = 2h − 2 and i∗ γ = β.
The following theorem expresses the stable pair invariants of X in termsof the Euler characteristics of of the moduli spaces of stable pairs on thefibers and the Noether-Lefschetz numbers:

14

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Theorem 2.16. For a smooth K3 fibration π : X → C and an irreducibleclass β ∈ H2 (X, Z)π , we have
∞ X∞
X
PT(X)β =(−1)n−1 χ(Pn (K3, h)) · N Lπh,β q n .
h=0 n=1−hHere, the Euler characteristics χ(Pn (K3, h)) are determined by Kawai-Yoshioka’s
formula (4).
Proof. The proof follows the same ideas as the proof of [MP13, Theorem 1].
We compare the contributions of BI and BII to Pn,β and the NoetherLefschetz numbers. Suppose that  is a local system giving rise to BI (h, β)
as above. Then, we can writeZPn, =
ctop (ΩP/C ) · c1 (ρ∗ K∨ )PZ
= (−1)n−1 χ(Pn (K3, h)) ·c1 (∗ K∨ ).
BI (h,β)By virtue of (22), it is shown in [MP13, Theorem 1] that the integrationZ
c1 (∗ K∨ )BI (h,β)gives the contribution of BI (h, β) to N Lπh,β .
Using the deformation invariance of the stable pair invariants, we may
assume that any α ∈ BII (h, β) is a result of a transversal intersection ofσ(C) with a Noether-Lefschetz divisor4. As a result, the contribution of α
to N Lπh,β is exactly 1, and moreover, the corresponding component Pα is an
isolated type II component. Therefore by Corollary 2.15, we haveZ
Pn,α =
ctop (ΩPα ) = (−1)n−1 χ(Pn (K3, h)).
PαThe above arguments prove the desired identity.



2.7. Reducible curve classes. In this section we study a special analog ofTheorem 2.16 in which the class β ∈ H2 (X, Z)π is allowed to be reducible.
Let Pc be an isolated type II component of Pn (X, β). This means that
we have an isomorphism Pc ∼= Pn (S, γ) where β = i∗ γ and i : S ,→ X is aK3 fiber of X. Here, γ is no longer needed to be irreducible, but we always
assume that5(25)

H 1 (L) = 0 for any line bundle on S with c1 (L) = γ.

4Here as in [MP13, pg. 23], one may need to make a local holomorphic perturbation of

the section σ to make it transversal to the Noether-Lefschetz divisor (i.e. with the local
intersection multiplicity 1). The K3 fibration after perturbation will be over an analytic
curve, but the stable pairs under consideration will remain on the K3 fibers, which are
algebraic. The corresponding type II component of the moduli space of stable pairs isalways algebraic and compact and lies over the interior of a small (analytic) open set in
the curve that is being perturbed. So the usual deformation invariance of the intersection
numbers gives the deformation invariance of the contribution of the stable pair invariantsof this type II component.
5This is the case for example if γ is big and nef (see [Huy16, Proposition 3.1]).

STABLE PAIRS ON NODAL K3 FIBRATIONS

15

As γ is not an irreducible class, the moduli space Pn (S, γ) may be singularand have several irreducible components. However we have the following
lemma:
Lemma 2.17. Suppose that γ satisfies (25), then the dimension of each
irreducible component of Pn (S, γ) does not exceed v := n + γ 2 + 1.
Proof. In the notation of Proposition 2.2, the dimension of M is γ 2 /2 + 1,
and for a fixed curve D in class γ the dimension of the components of theHilbert scheme are at most n + γ 2 /2. These two claims prove the lemma.
The first claim follows because in this case M is the linear system of curvesin class γ on S satisfying (25). The second claim can be verified by analyzing
the fibers of the Hilbert to Chow morphism from the Hilbert scheme of kpoints on D
HC : Hilbk (D) → Symk (D).
We know that the dimension of the punctual Hilbert scheme of a points
supported on a fixed point of S is equal to a − 1 (see [Bri77]). Since D ⊂ S
the dimension of the punctual Hilbert scheme of a points supported on a
fixed point of D does not exceed a − 1. Now given a d-dimensional diagonal∆d ⊂ Symk (D) corresponding to the partition k = a1 + · · · + ad and a point−1
pP∈ ∆d , by what we said above, the dimension of HC (p) does not exceedi (ai − 1) = k − d. From this the claim follows.
By [KT14, Proposition 3.4] (also see Theorem 2.7 (ii)), the component
Pc = Pn (S, γ) is equipped with a perfect obstruction theoryG • := RHomπP0 (G, I•S ⊗ ωπP0 )|Pc [2] → L•Pc .
Note that the obstruction sheaf H1 (G •∨ ) of G • admits a surjection to thetrivial vector bundle OPc , given by the natural surjection(26)

Hom(IS• , G[1])  Ext2S (G, G)  C

at the fiber of IS• = (OS → G) ∈ Pc . The second morphism of (26) is dual to
C·id ⊂ Hom(G, G). By removing the trivial factor OPc from the obstruction
theory G • , Kool-Thomas [KT14] constructs a v-dimensional reduced virtualcycle[Pn (S, γ)]red ∈ Av (Pn (S, γ)).
Here v is given in Lemma 2.17. Let E • be the obstruction theory in Theorem 2.1 restricted to Pc . We define the following element in the K-group
V = −E •∨ + G •∨ + OPc ∈ K(Pc ).
Then V is of constant rank v.
Proposition 2.18. Suppose that γ satisfies (25) and v is as in Lemma 2.17,
then we have the following identity:(27)[Pc , E • ]vir = cv (V) ∩ [Pn (S, γ)]red .

Proof. The obstruction theories E • and G • are related by the following two
natural exact triangles:(28)

RHomπP (I• , F) → RHomπP (I• , I• )0 [1] → RHomπP (F, OX×P )[2],16AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

and
(29)

RHomπP0 (I•S , G) → RHomπP (I• , F) → RHomπP0 (G(−S), G).

Here the triangle (28) is obtained similarly to the bottom triangle of the
diagram (13), replacing ω by OX×P . The triangle (29) follows from applying
RHomπP0 (−, G) to the triangle (17). The complexes E •∨ and G •∨ can be
identified in the derived category with 2-term complexes
E0 → E1

and G0 → G1

of vector bundles. Their K-group classes are then respectively E0 − E1 and
G0 − G1 . We define the element V 0 of the K-group by
V 0 = E1 − E0 − (G1 − G0 ).
Then we have V = V 0 + OPc . By the exact triangles (28), (29) above,
the fiber of V 0 over a closed point (OS → G) ∈ Pc is naturally given by
χ(G) − χ(G, G).
Using these facts and noting that the summation of OPc does not affectthe total Chern class, the result of [Sie04, Theorem 4.6] implies(30)[Pc , E • ]vir = {c(E1 − E0 ) ∩ cF (Pc )}0= {(c(V) ∪ c(G1 − G0 )) ∩ cF (Pn (S, γ))}0 .

Here {−}r means the r-dimensional part, and cF denotes Fulton’s canonicalclass. Similarly, we have[Pn (S, γ)]red = {c(G1 − G0 ) ∩ cF (Pn (S, γ))}v .
By the discussion in [Sie04, Section 4.1], we know that{c(G1 − G0 ) ∩ cF (Pn (S, γ))}r = 0(31)

for r < v − 1. It is also 0 for r = v − 1 because as mentioned earlier the
obstruction theory G • contains a trivial factor, hence [Pc , G • ]vir = 0. The
vanishing (31) also holds for r > v because dim Pn (S, γ) ≤ v by Lemma 2.17.
Therefore by the dimension reason, we obtain the desired identity (27). 
Corollary 2.19. In the situation of Proposition 2.18, if γ is irreducible then[Pn (S, γ)] = [Pn (S, γ)]red ,cv (V) ∩ [Pn (S, γ)]red = cv (ΩPn (S,γ) ) ∩ [Pn (S, γ)].
Proof. In this case by Proposition 2.3, Pn (S, γ) is smooth of dimension v,
which is the same as the virtual dimension of [Pn (S, γ)]red . This proves thefirst equality. The second equality follows by noting that both sides give
[Pc , E • ]vir by Proposition 2.18 and Corollary 2.15.
Corollary 2.20. In the situation of Proposition 2.18, suppose that there isa smooth deformation of S to a K3 surface in which γ becomes irreducible.
ThenZ
cv (V) = (−1)v χ(Pn (K3, h)),[Pn (S,γ)]red

where h = γ 2 /2 + 1.

STABLE PAIRS ON NODAL K3 FIBRATIONS

17

Proof. Since we are working with an isolated type II component, the virtual
bundle V depends on S only. To see this first note that by Verdier duality
applied to the closed immersion S ×Pc → X ×Pc and that ωS = OS ∼= ωX |S ,we have
RHomπP (F, OX×P ) ∼= RHomπP0 (G, OS×P )[−1].
Using this and the exact triangles (28) and (29), it is clear that the K-groupclass [RHomπP (I• , I• )0 ] only depends on the data on S, and hence the sameis true for V. The corollary now follows from Corollary 2.19 and deformation
invariance of the reduced virtual cycle for (1, 1) classes.
In the following special situation we can prove an analog of Theorem 2.16
for possibly reducible classes β:Theorem 2.21. For a smooth K3 fibration π : X → C suppose that theclass β ∈ H2 (X, Z)π is such that Pn (X, β) consists of only isolated type II
components. For any such component Pn (S, γ) as above, suppose that γ
satisfies the condition (25). Then,(32)

Pn,β =∞
X(−1)n−1 χ(Pn (K3, h)) · N Lπh,β .

h=0Proof. Since all the components of Pn (X, β) are isolated type II, for any
such component Pn (S, γ), we necessarily have γ = γ1 + · · · + γr where γi ’s
are distinct irreducible classes6. In particular, γ is a primitive class andhence one can find a deformation of S as in Corollary 2.20. This together
with similar argument as in the proof of Theorem 2.16 give the result. 
Remark 2.22. The right hand side of the formula (32) can be obtained by
Kawai-Yoshioka’s formula (4).
3. Stable pairs on nodal K3 fibrations
3.1. Nodal K3 fibrations. In this section, we aim to prove the compatibility condition for stable pair invariants via conifold transitions. Let X bea smooth projective 3-fold, and π be a K3 fibration
π:X→C
onto a smooth projective curve C. A K3 fibration π is called a nodal K3
fibration if the singularities of the fibers of π are at worst ordinary double
point (ODP) singularities.
Example 3.1. Let X ⊂ P3 × P1 be a generic hyperplane section of bidegree(4, 2), and π the composition
π : X ⊂ P3 × P1 → P1
where the second morphism is the projection. Then π is a nodal K3 fibration.
6If any of γ ’s is not reduced then one could have a stable pair whose support infinitesi
imally thickens outside of the supporting K3 fiber S contradicting the assumption that
all the components of Pn (X, β) are isolated type II.

18

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

For any β ∈ H2 (X, Z)π , the stable pair invariants Pn,β are defined as
before byZ
1,
Pn,β =[Pn (X,β)]vir

where [Pn (X, β)]vir is the 0-dimensional virtual cycle associated to the obstruction theory in Theorem 2.1. We will study the generating series PT(X)β ,PT(X) given by (11).
Remark 3.2. When the singularities of fibers of π are more general type ofrational double points (RDP), then the study of stable pair invariants may
be reduced to the nodal case by using the deformation invariance of stable
pair invariants.
3.2. Conifold transition. Let
s1 , . . . , sk ∈ X, c1 , · · · , ck0 ∈ C
are the singular points of the fibers of π : X → C, the points in C over which
the fibers have singularities, respectively. By our assumption, each si ∈ X
is an ordinary double point (nodal) singularity for any i in the fiber of π. If
k 0 is even, define c0 = c1 and if k 0 is odd, define c0 to be an arbitrary pointof C distinct from c1 , . . . , ck0 . Lete→C
:Cbe the double cover of C branched over the points c0 , c1 , . . . , ck0 . It canbe seen that X0 := ∗ (X) is a threefold with the conifold singularities. Let
e → X0 be its small resolution with the exceptional nonsingular rational
h: X
curvese h(ei ) = si
(33)
e1 , · · · , ek ⊂ X,
e → C
e the induced morphism. In general, the small resolution
and π
e : X
e
X may not be a projective variety, but is realized as an algebraic spaceby [Art74]. As a summary, we have the commutative diagram
e


(34)

h

e
X

 

π
e

$
/X

/ X0

e
C

π





/ C.



e is isomorphic to OP1 (−1) ⊕ OP1 (−1) [Ati58].
The normal bundle of ei in X
et → C be a double cover of C branched at k + 2{k/2}Moreover, let t : C
7
e0 = C.
e Define Xt = ∗ (X), so wegeneric points of C when t 6= 0, and set Cthave the Cartesian square
/X

Xt


et
C


t



π

/ C.

7{k/2} = 0 if k is even, and {k/2} = 1/2 if k is odd.

STABLE PAIRS ON NODAL K3 FIBRATIONS

19

e and Xt which differ by the
Our plan is to relate stable pair invariants of X
conifold transitions. As in GW theory [LY06], [LR01], this can be doneusing degeneration techniques.
Remark 3.3. We will not pursue the foundation of moduli theory of stable
e The Hilbert schemes of curves onpair invariants on the algebraic space X.
algebraic spaces are realized as algebraic spaces [Art69, Corollary 6.2], andthe similar argument may be applied to construct the moduli spaces of stablee as algebraic spaces. Also in the situation that any stable pair onpairs on Xe
X is scheme theoretically supported on a K3 fiber, then by Proposition 2.2,
the moduli space of stable pairs can alternatively be constructed as a relative Hilbert scheme of points over the universal curve of the relative linear
system. Hence it is an algebraic space by [Art69, Corollary 6.2].
e is an algebraic space,
Remark 3.4. If the moduli space of stable pairs on X
then the arguments in [BF97] show the existence of the zero dimensional
virtual fundamental class. Below we will also assume that Li-Wu’s degeneration formula [LW15] works for stable pair invariants on algebraic spaces.
Of course, we don’t have to address the foundational matters in Remark 3.3
e to be projective.
and this remark if we can choose the small resolution X
3.3. Relative stable pair invariants. In order to apply the degeneration
technique, we will use the notion of relative stable pair theory given by
Li-Wu [LW15].
Definition 3.5. Let W be a smooth projective threefold and D ⊂ W asmooth divisor. For β ∈ H2 (W, Z) and n ∈ Z, we denote by Pn (W/D, β)
the moduli stack of relative stable pairs (OW → F ) on W satisfying [F ] = βand χ(F ) = n, in the sense of [LW15].
There is an open substack of Pn (W/D, β) whose C-points correspondto pairs (OW → F ) such that F is supported on W and normal to D, i.e.
T or1OX (F, OD ) = 0. A C-point of the boundary corresponds to an admissible
stable pair (cf. [LW15, Definition 4.8]) supported on an n-step degenerationof (W [n], D[n]). By the relativity of stable pairs, the restriction map defines
the morphism
ev : Pn (W/D, β) → Hilb(D, |η|).
Given a cohomology weighted partition η with respect to a basis of H ∗ (D, Q),
we can associate a cohomology classCη ∈ H ∗ (Hilb(D, |η|), Q)
which forms a basis of H ∗ (Hilb(D, |η|), Q) called Nakajima basis (cf. [Nak99]).
If we choose a basis of H ∗ (D, Q) which is self-dual with respect to the
Poincaré pairing, then for each cohomology weighted partition η, there is adual partition η ∨ such thatZ(−1)|η|−l(η)δν,η∨
Cη ∪ Cν =a(η)
Hilb(D,|η|)20

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

for any cohomology weighted partition ν with |ν| = |η|. Here l(η) is thelength of the partition η and a(η) is defined byYa(η) =
ηi |Aut(η)|.
i

Definition 3.6. The relative stable pair invariant (without insertions) isdefined byZ
Pn,β (W/D)η :=ev∗ Cη .
[Pn (W/D,β)]vir

The virtual dimension of Pn (W/D, β) is given by c1 (W ) · β. Therefore
the above invariant is zero unless
c1 (W ) · β = deg Cη ≥ 0.
We define the following generating seriesX
PT(W/D)β,η =Pn,β (W/D)η q nnX

PT(W/D)η =PT(W/D)β,η tβ .

βWe drop η or D from the notation if respectively |η| = 0 or D = ∅.
3.4. Degeneration formula. Let Y be the threefold obtained by blowing
up X0 at all the conifold points, with the exceptional divisors
D := D1 t · · · t Dk , Di = P1 × P1 .
Note that we have the factorizationghe→
f: Y →X
X0such that g is the blowing up at all the exceptional loci (33) of h. We use
e and Xt to respectively
degenerations of the threefolds X
(35)

[

Y

k
a

P1

and Y

D1 ,...,Dk i=1

[

k
a

P2 .

D1 ,...,Dk i=1

Here P1 is given by
P1 ∼= P(OP1 ⊕ OP1 (−1) ⊕ OP1 (−1))
and P2 is a smooth quadric hypersurface in P4 .
The first degeneration in (35) is the degeneration to the normal cone
[Ful98] in which Di ⊂ Y is attached to the divisor at infinity H1 = P(OP1 (−1)2 )in the i-th copy of P1 . The second degeneration is called the semi-stablereduction of a conifold degeneration [LY06] in which Di ⊂ Y is attached toa smooth hyperplane section H2 in the i-th copy of P2 . We denote by
X1 → A 1 ,X2 → A1

the total spaces of the first and second degenerations above. Let L be an
ample line bundle on X. We define line bundles Lt on Xt to be
Lt = ∗t (L)where we set 0 := .

STABLE PAIRS ON NODAL K3 FIBRATIONS

21

Now we apply the degeneration formula of stable pair invariants withrespect to the above degenerations. Suppose for simplicity that there is only
one critical locus for the fibration X → C, i.e. k = 1. By the degeneratione Z)πeformula, we obtain the following identities for β ∈ H2 (X,
(36)

e β=
PT(X)
X
(−1)|η|−l(η) a(η)
PT(P1 /H1 )β2 ,η∨PT(Y /D1 )β1 ,η
q |η|η,β +β =β
1

2

and
(37)

PT(Xt )β =X(−1)|η|−l(η) a(η)
PT(P2 /H2 )β2 ,η∨ .
PT(Y /D1 )β1 ,η
q |η|η,β +β =β12Here the sum β1 + β2 = β is an equality in H2 (X1 ) and H2 (X2 ) respectively.
Proposition 3.7. The degeneration formula (36) implies that
X
e β=
e β .
PT(X)
PT(Y /D)f ! β · PT(X)12eβ1 ∈H2 (X0 ), β2 ∈H2 (X)h! β1 +β2 =β, h∗ β2 =0

Proof. The degeneration formula of relative rank one DT invariants for theblow-up at (−1, −1)-curves is worked out by Hu-Li [HL12], and we apply
the same argument. By the agreement of the virtual dimensions, it is provedin [HL12, Theorem 4.2] that a non-zero term of the RHS of (36) satisfies
0 = |η| = β1 · D1 = β2 · H1 .
This implies that β1 is written as h! β10 for some β10 ∈ H2 (X0 ), and β2 is amultiple of the class of the curve e ⊂ P1 given by the embeddingOP1 ⊂ OP1 ⊕ OP1 (−1) ⊕ OP1 (−1)
into the first factor. The curve e is a (−1, −1)-curve which does not intersectwith H1 . The contributions of the relative stable pairs on P1 with curve
e with curve classclass m[e] is identified with the stable pair invariants on X
8m[e1 ] . Hence we obtain the desired result for k = 1. The case of k > 1 issimilarly discussed.

Proposition 3.8. For any d ∈ Z>0 , the degeneration formula (37) implies
thatX
X
PT(Xt )β =PT(Y /D)f ! β 0 .
β∈H2 (Xt )Lt ·β=dβ 0 ∈H2 (X0 )L0 ·β 0 =d

Proof. By the agreement of the virtual dimensions, a non-zero term of the
RHS of (37) satisfies
c1 (Y ) · β1 = −D1 · β1 = deg Cη ≥ 0.
8This follows by a parallel argument as (4.4) in the proof of [HL12, Theorem 4.2].

22

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

We also have the following compatibility condition
D1 · β1 = H2 · β2 = |η| ≥ 0.
The above inequalities imply that D1 · β1 = |η| = 0 and H2 · β2 = 0. The
first equality implies that β1 is written as f ! β 0 for some β 0 ∈ H2 (X0 ). The
second equality implies that β2 = 0, since H2 is an ample divisor in P2 . By
rearranging the sum, we obtain the desired formula for k = 1. The case ofk > 1 is similarly discussed.
e Now we choose k 0 generic3.5. Relation between PT(X) and PT(X).
fibers S1 , . . . , Sk0 of X → C. By our assumption Si is a K3 surface. Let
Xi = Si × P1 . Then Xi is a smooth K3-fibration over P1 . We identify thesurface Si with the divisor Si × {0} in Xi .
Lemma 3.9. For any curve class β contained in fibers of Xi → P1 , we havePn,β (Xi /Si ) = 0.
Proof. By (21) in [MPT10] the cup product map, ∪β : H 1 (TSi ) → H 2 (OSi )is surjective, and hence by the proof of [KT14, Theorem 2.7] the obstruction
theory of Pn (Xi /Si , β) contains a trivial factor which implies the vanishingof the invariants9.
We set S to be the disjoint union of Si for 1 ≤ i ≤ k 0 .
Lemma 3.10. Pn,β (X/S) = Pn,β (X).
Proof. Use the degeneration formula for the degeneration of X into
[ a
X
Xi .
i

S1 ,...,Sr

The vanishing of Pn,β (Xi /Si ) from Lemma 3.9 proves the claim.



We use Lemma 3.10 to relate the PT invariants of X to Xt . To achieveethis we use the degeneration of Xt obtained by degenerating its base C
to two copies of C by attaching two copies of X along the generic fibers
S1 , . . . , Sk0 . The degeneration formula then implies that
X
X
X
X
PT(Xt )β =PT(X/S)β1(38)
PT(X/S)β2
Lt ·β=dd1 +d2 =d L·β1 =d1

L·β2 =d2

e be the generating series defined byfor each d ∈ Z>0 . Let PTh (X)Xe :=e n tβ(39)
PTh (X)Pn,β (X)qh∗ β=0=Y(1 − (−q)n t[ei ] )n .

1≤i≤k,n≥1

Here the second identity follows from the computation of stable pair invarie → X be the natural morphism
ants on a (−1, −1)-curve [NN11]. Let e: Xgiven in the diagram (34). Combined with the results in the previous subsection, we obtain the following result:
9The proof of [KT14, Theorem 2.7] is given for absolute geometries but a parallel
argument applies to the relative geometry here.

STABLE PAIRS ON NODAL K3 FIBRATIONS

23

Theorem 3.11. We have the formula
e
∗

(40)

e
PT(X)
= PT(X)2 .
ePTh (X)

Here e∗ is the variable change tβ 7→ te∗ β .
Proof. For each d ∈ Z>0 , Proposition 3.8, Lemma 3.10 and (38) imply
X
X
PT(X)β1 · PT(X)β2 .
PT(Y /D)f ! β =β1 ,β2 ∈H2 (X)L·β1 +L·β2 =d

β∈H2 (X0 )L·∗ β=dSince the above formula holds for any L, we haveX!
(41)
PT(Y /D)f ! β th β = PT(X)2 .
e∗β∈H2 (X0 )On the other hand, the LHS of (41) coincides with the LHS of (40) by
Proposition 3.7. Therefore we obtain the result.
Remark 3.12. The variable change tβ 7→ te∗ β only makes sense after taking
the quotient series in the LHS of (40). Otherwise there are infinite numbersof contributions to the coefficients.
Remark 3.13. The LHS of (40) is known to be independent of a choice ofa small resolution [HL12], [Tod13b], [Cal]. The formula (40) is consistent
with this fact.
In an irreducible curve class case, we have the following corollary:
Corollary 3.14. For an irreducible curve class β ∈ H2 (X, Z)π , we have thefollowing formula:Pe eXh∗ e=0 PT(X)β+e
e
∗
(42)
= 2 PT(X)β .
ePTh (X)ee∗ β=βe Z) are irreducible curve classes, giving lifts of β.
Here βe ∈ H2 (X,
3.6. Conjectural relation to perverse (non-commutative) stable pairtheory. In this subsection, we introduce perverse (or non-commutative) stable pair theory, and propose a conjectural relationship between the quotiente PTh (X)e in Theorem 3.11 and the generating series of theseries PT(X)/perverse (non-commutative) stable pair invariants. We first recall the hearte associated to the small resolutionof the perverse t-structure on Db Coh(X)ee be theh : X → X0 , introduced by Bridgeland [Bri02]. Let C ⊂ Db Coh(X)full subcategory defined by
e : Rh∗ E = 0}.
C = {E ∈ Db Coh(X)e induces a t-structure
By [Bri02], the standard t-structure on Db Coh(X)≤0≥0e 0 ) to be(C , C ) on C. We define Per(X/XRh∗ E ∈ Coh(X0 ),bee
Per(X/X0 ) = E ∈ D Coh(X) :.
Hom(C <0 , E) = Hom(E, C >0 ) = 0

24

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODAe 0 ) is the heart of a bounded t-structure on Db Coh(X),eThe category Per(X/Xehence it is an abelian category. Note that OX ∈ Per(X/X0 ) by definition.
e 0 ), Per0 (X/X
e 0 ) be the subcategories of Per(X/X
e 0 ), defined
Let Per≤1 (X/X
bye 0 ) = {E ∈ Per(X/X
e 0 ) : dim h(Supp(E)) ≤ 1},
Per≤1 (X/X
e 0 ) = {E ∈ Per(X/X
e 0 ) : dim h(Supp(E)) = 0}.
Per0 (X/X
e 0 ) and Per0 (X/X
e 0 ) are closed under quoIt is easy to see that Per≤1 (X/Xetients and subobjects in Per(X/X0 ). In particular, they are abelian subcate 0 ).
egories of Per(X/X
e → X0 consists of data
Definition 3.15. A perverse stable pair for h : X(F, s)(43)e 0 ),F ∈ Per≤1 (X/X

s : OXe → F

satisfying the following conditions:e 0 ), F ) = 0.
• Hom(Per0 (X/X
e 0 ) is an object in Per0 (X/X
e 0 ).
• The cokernel of s in Per(X/X
The above definition coincides with the usual definition of stable pairs if
e → X0 is an isomorphism (i.e. there is no nodal fiber for π : X → C).
h: X
Also the above perverse stable pairs are related to the non-commutative
e were a projective variety.
version of stable pairs as follows. Suppose that X
eThen, Van den Bergh [dB04] shows that there is a vector bundle E0 on X
such that E = OXe ⊕ E0 gives a derived equivalence∼e → Db Coh(AX )Φ = Rh∗ RHom(E, ∗) : Db Coh(X)0where AX0 = h∗ End(E) is the sheaf of non-commutative algebras on X0 . Thee 0 ) and Coh(AX ).
equivalence Φ restricts to an equivalence between Per(X/X
0
Let U = Φ(OXe ) ∈ Coh(AX0 ) be the local projective generator of AX0 . Let
Coh≤1 (AX0 ),Coh0 (AX0 )be the subcategories of Coh(AX0 ), consisting of E ∈ Coh(AX0 ) whose support as OX0 -module has dimension ≤ 1, 0 respectively. It is easy to seethate 0 )) = Coh≤1 (AX ),Φ(Per≤1 (X/X0e 0 )) = Coh0 (AX ).
Φ(Per0 (X/X
0
Under the equivalence Φ, the data (43) is equivalent to the data (F 0 , s0 )(44)

F 0 ∈ Coh≤1 (AX0 ),s0 : U → F 0such that Hom(Coh0 (AX0 ), F 0 ) = 0 and the cokernel of s in Coh(AX0 ) is anobject in Coh0 (AX0 ).
The pair (44) is a non-commutative analogue of stable pairs. However
e may not be projective, and the sheaf of nonunfortunately, our 3-fold X
commutative algebras AX0 only exists at formal neighborhoods at each point

STABLE PAIRS ON NODAL K3 FIBRATIONS

25

in X0 . Because of the absence of the global sheaf of non-commutative algebras AX0 , we formulate our non-commutative stable pairs as a morphism inthe perverse heart as in (43).
e Z)πe and n ∈ Z, let
For β ∈ H2 (X,e β)Pnper (X,(45)

be the moduli space of perverse stable pairs (OXe → F ) with [F ] = β andχ(F ) = n. We expect that (see Remark 3.16) the moduli space (45) exists
as a proper algebraic space, with a perfect obstruction theory with virtual
dimension zero given by
(46)

•
.
Eper
:= RHomπP (I•per , I•per ⊗ ωπ )0 [2] → L•P per (X,β)
enHere I•per is the total complex associated to the universal morphism→ Fper .
s : OX×P
per een (X,β)Remark 3.16. Here we assumed that (45) is a proper algebraic space withperfect obstruction theory (46). We have not checked all the details but webelieve that they should all follow from the similar arguments in the existing
literatures. First the argument in [Tod13b, Section 6.4, Step 2] should showthat (45) is an algebraic space of finite type. The properness should follow
from the arguments in [Lo13]. Also the existence of the perfect obstruction
theory (46) should follow from the argument in [HT10] by noting that thecomplexes associated to perverse stable pairs in Definition 3.15 are simple
objects of the derived category as they are stable with respect to some weakstability condition (see Remark 3.18).
By taking the integration, we obtain the invariantZ
perPn,β =
1.
vire[Pnper (X,β)]e β and PTper (X)e to beWe define the generating series PTper (X)
X
X per
e =
e β tβ .
e β=PTper (X)Pn,β q n , PTper (X)
PT(X)
n∈Z

e
e π
β∈H2 (X,Z)

Conjecture 3.17. We have the equality
(47)

e
PT(X)
e
= PTper (X).
h e
PT (X)

Remark 3.18. The conjectural equality (47) is motivated by the third author’s work [Tod13b]. Indeed the equality (47) holds true for the Euler characteristic version, i.e. replace all the invariants by the topological Euler
characteristic of the moduli spaces. This follows immediately from the proofof [Tod13b, Theorem 5.8], and the fact that σξ(0) -semistable objects in thenotation of [Tod13b, Theorem 5.8] consist of total complexes associated toperverse stable pairs in Definition 3.15. (The latter fact can be easily proved
using the same argument of [Tod13b, Proposition 5.5]). Moreover, if our 3e were projective Calabi-Yau 3-fold, then the equality (47) follows fromfold Xe is no longer
the results of [Tod13b], [Cal]. However, since our 3-fold X

26

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Calabi-Yau, we need a new technique of wall-crossing to show the inequality(47).
Remark 3.19. If Conjecture 3.17 is true, then Theorem 3.11 implies thate = PT(X)2 .
(48)
e∗ PTper (X)Hence we can reduce the computation of PT(X) for a nodal K3 fibratione for a smooth K3 fibration X
e → C.
e
X → C to the computation of PTper (X)
3.7. Computation of perverse stable pair invariants. Finally in thissection, we aim at computing perverse stable pair invariants. We prove thefollowing lemma:
e → X0 such
Lemma 3.20. Let (OXe → F ) be a perverse stable pair for X
that h∗ [F ] is an irreducible curve class in X0 . Then we have(1) Hom(F, F ) = C.
e is a K3 fiber and Supp(F ) is the(2) Supp(F ) ⊂ S where i : S ,→ X
scheme theoretic support of the complex F .
Proof. (1) It is enough to show that any non-zero morphism u : F → F ine 0 ) is an isomorphism. Let us take a factorization in Per(X/X
e 0)
Per(X/XF  Im u ,→ F.
e \ Ex(h). Suppose first that u|U = 0. Then im u is a non-zero
Let U = Xe 0 ), which contradicts to Definition 3.15. Hence we have
object in Per0 (X/X
u|U 6= 0. Since the object F |U is a pure one dimensional sheaf whose scheme
theoretic support is irreducible, the morphism u|U is an injective morphisme 0 ) is non-zero, thenin Coh(U ). This implies that, if Ker(u) ∈ Per(X/X
e 0 ). This contradicts to Definition 3.15, hence
it is an object in Per0 (X/X
Ker(u) = 0.
e 0 ). It remains to
Now we have proved that u is injective in Per(X/X
e 0 ) vanishes. Since Rh∗ takes Per(X/X
e 0 ) toshow that Cok(u) ∈ Per(X/X
Coh(X0 ), we have the exact sequence in Coh(X0 )0 → Rh∗ F → Rh∗ F → Rh∗ Cok(u) → 0.
The above sequence implies that Rh∗ Cok(u) = 0, hence Cok(u) ∈ C. It
follows that, by [dB04, Proposition 3.5.8], the object Cok(u) is isomorphic tothe direct sum of the sheaves Oei (−1). Therefore by taking the cohomology
long exact sequence associated to 0 → F → F → Cok(u) → 0, and notinge 0 )), we obtain the exact
that Hi (F ) = 0 for i 6= 0, −1 (because F ∈ Per(X/X
sequence of sheaves
0 → H0 (F ) → H0 (F ) → Cok(u) → 0.
Therefore we have Cok(u) = 0.
(2) By part (1), F is necessarily supported (set theoretically) on only
one fiber S. Tensoring the natural short exact sequence 0 → OXe (−S) →OXe → i∗ OS → 0 by F , and noting that F ⊗ OXe (−S) ∼= F , we get the exact∗∗∼
triangle F → F → i∗ Li F . Since Li F =6 0, the first map in the exact
triangle cannot be an isomorphism, and hence it must be zero map by part(1) of the lemma. The exact triangle then implies that i∗ Li∗ F ∼= F ⊕ F [1],

STABLE PAIRS ON NODAL K3 FIBRATIONS

27

and so in particular Supp(F ) = Supp(Li∗ F ). Part (2) now follows from
[Huy06, Lemma 3.29].
e → C,
e let us take a curve class
For the smooth K3 fibration π
e:X
(49)

e Z)πe ,β = βe + mei ∈ H2 (X,e Z) and m ∈ Z≥0 . Since βe is anfor some irreducible curve class βe ∈ H2 (X,irreducible class Lemma 3.20 applies for the perverse stable pairs (OXe → F )such that [F ] = β. Motivated by the result of this lemma we assume thefollowing conjecture:10
e → X0Conjecture 3.21. Let (OXe → F ) be a perverse stable pair for X
such that [F ] = β. Using the notation of Lemma 3.20 Part (2), there exists
a complex G on S such that F = i∗ G.
Assuming Conjecture 3.21, there is a natural morphismee β) → C.
ρ : Pnper (X,Also similarly to Definition 2.6, we can define the type I and type II come β). For a point p ∈ C,
e suppose that S :=ponents PIper , PIIper of Pnper (X,ie and γ is a curve class on S with i∗ γ = β, then we denote theπe−1 (p) ,−→X−1
fiber ρ (p) of ρ restricted to a type I or an isolated type II component byPnper (S, h) where h := γ 2 /2 + 1. Note that by the choice of β one can finda smooth deformation S 0 of S in which γ becomes irreducible, and hence a
perverse stable pair on S 0 in this class is the same as a usual stable pair.
From this and Proposition 3.22, one can see that Pnper (S, h) is deformationequivalent to Pn (K3, h). We have the analogs of Propositions 2.4, 2.8, 2.10
and Corollary 2.15 that can be summarized as follows:
Proposition 3.22. The type I components and isolated type II componentse β) are smooth.
of the moduli space of perverse stable pairs P per (X,Proof. We first prove the proposition for isolated type II components. The
proof is along the lines of the proof of [PT10, Proposition C.2]. We show
• =that the obstruction space vanishes for any given perverse stable pair Iperse We will(OXe →− F ) with F = i∗ G is supported on a K3 fiber i : S ,→ X.
•
slightly abuse the notation and use the same symbol Iper for the perversee As in the proof of [ibid], thestable pair on S and its pushforward on X.
obstruction space is the kernel of v in the following natural exact sequence:u•
Hom(Iper, G) → Ext1 (G, G) −→ Ext1 (O, G)v•→ Ext1 (Iper, G) −→ Ext2 (G, G)0 → 0.

Lemma 3.20, and Serre duality implies that Ext2 (G, G)0 = 0, so to provethe claim we need to show that the map u is surjective. Let S0 := h(S),p : S0 → Spec C be the structure morphism, andG0 := Rh∗ G ∈ Coh(S0 ),10This is an analog of Lemma 2.5.

D0 := Supp(G0 ).

28

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

We can write RΓ(G) = Rp∗ G0 and hence
Ext1 (O, G) ∼= H 1 (G0 ).
str− G, we get theOn the other hand, applying Rh∗ to RHom(G, G) −→ OS →
chain of mapsstr0Rh∗ RHom(G, G) → RHom(G0 , G0 ) −→ OS0 −→
G0 .

Clearly, s0 has a zero dimensional cokernel, and the trace map
Hom(G0 , G0 ) → OD0
is surjective with at most a zero dimensional kernel, and therefore s0 ◦ tr
induces the surjection H 1 (Hom(G0 , G0 )) → H 1 (G0 ). By the local to global
spectral sequence
H 1 (H0 (Rh∗ RHom(G, G))) ⊆ Ext1 (G, G),and hence u is surjective as desired.
The proof of proposition for the type I components essentially follows thesame analysis as above but this time carried out relative to the base curveC.
Proposition 3.23. Suppose that any component of PIIper is isolated. Then
we have• vir[PIIper , Eper
] = [PIIper ] ∩ ctop (ΩP per ),
II


• vir
e ∨ ) ∪ ctop (Ω per e ) ,[PIper , Eper
] = [PIper ] ∩ c1 (ρ∗ K
P /C
I

e=π
where Ke∗ ωX/e C
e.
Proof. The same as the proofs of Proposition 2.4 and Corollary 2.15 in whichs• = (O →the stable pair I • is replaced with the perverse stable pair Ipere − F ).
X
The surjectivity Ext1 (F, F) → Ext1 (OXe , F ) = H 1 (F ) which is one of theimportant requirements can be obtained as the proof of surjectivity of themap u in Proposition 3.22.

Now we provide an analog of Theorem 2.16 for perverse stable pair invariants:
e → C,e suppose that β is
Theorem 3.24. For the smooth K3 fibration πe:Xgiven as in (49), and assume Conjecture 3.21. Then we havee =PTper (X)
β

∞ X
∞
X

e
(−1)n−1 χ(Pnper (S, h)) · N Lπh,βqn,

h=0 n=1−hewhere S is a K3 fiber of X.

Remark 3.25. By the discussion before Proposition 3.22, we can see that
the Euler characteristics χ(Pnper (S, h)) = χ(Pn (K3, h)) in Theorem 3.24 canbe read off from Kawai-Yoshioka’s formula (4).
Remark 3.26. If the conjectural relation (48) holds, then by Theorem 3.24,
it enables us to compute stable pair invariants for the nodal K3 fibration
X → C with irreducible curve classes.

STABLE PAIRS ON NODAL K3 FIBRATIONS

29

4. Wall-crossing formula for K3 fibrations
In this section, we study the stable pair invariants on K3 fibrations froma different approach. We first relate the stable pair invariants on a K3
fibration (with possibly singular fibers) to the generalized DT invariants
counting torsion sheaves supported on fibers of the K3 fibration. This isachieved by an argument similar to [Tod12b], where a similar problem wasstudied for the trivial K3 fibration S × C → C for a K3 surface S. The
key ingredient was the wall-crossing formula of DT type invariants in thederived category [JS12], [KS]. The latter invariants are calculated in [GS13]
for nodal K3 fibrations, and we apply its result to calculate the stable pairinvariants on nodal K3 fibrations with irreducible curve classes. For a nonirreducible curve classes, the two approaches give different descriptions of
the stable pair invariants on nodal K3 fibrations. This gives a non-trivial
relationship between the generalized DT invariants on nodal K3 fibrations
and the stable pair invariants on smooth K3 fibrations, which may havesome applications to the study of the generalized DT invariants.
4.1. Stability conditions for K3 fibrations. Let X be a smooth projective Calabi-Yau 3-fold over C, i.e.
KX = 0,H 1 (X, OX ) = 0.

We assume that there is a morphism
π : X → P1
such that every scheme-theoretic fiber is an integral scheme. Note that ageneric fiber of π is a smooth algebraic K3 surface. An example is given in
Example 3.1. Let Cohπ (X) be the subcategory of E ∈ Coh(X) such that
Supp(E) is contained in the fibers of π. We consider its bounded derived
category
D0 := Db Cohπ (X).
Note that for any object E ∈ D0 , we havepch(E) tdX = (0, r[F ], β, n)(50)∈ H 0 (X) ⊕ H 2 (X) ⊕ H 4 (X) ⊕ H 6 (X).

Here [F ] is a fiber class of π, and r, n ∈ Z. Under the Poincaré duality, theelement β is regarded as an element in H2 (X, Z)π . By setting
Γ0 := Z ⊕ H2 (X, Z)π ⊕ Z
the expression (50) gives a group homomorphism(51)

cl0 : K(D0 ) → Γ0 ,E 7→ (r, β, n).

Let ω be an ample divisor on X. Using the map (51), we define the slope
function µω on Cohπ (X) byZµω (E) =ω · β/r ∈ Q ∪ {∞}.
X

30

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Here cl0 (E) = (r, β, n) and µω (E) = ∞ if r = 0, i.e. E is one or zero dimensional sheaf. The slope function µω defines the µω -stability on Cohπ (X) inan obvious way. We define the following full subcategories in Cohπ (X):Tω := hF : µω -semistable with µω (F ) > 0iex
Fω := hF : µω -semistable with µω (F ) ≤ 0iex .
Here h∗iex is the extension closure of ∗, i.e. the smallest extension closed
subcategory which contains ∗. The above pair (Tω , Fω ) determines a torsion
pair (cf. [HRS96]) on Cohπ (X). We have the associated tiltingBω := hFω , Tω [−1]iex ⊂ Db Cohπ (X).
The subcategory Bω is the heart of a bounded t-structure of Db Cohπ (X),hence in particular it is an abelian category. Let Zω,0 : Γ0 → C be the grouphomomorphism defined byZ
Zω,0 (v) :=e−iω vX√r= n − (ω|Xp )2 − (ω · β) −1.
2
Here we have written v = (r, β, n), and Xp = π −1 (p) for a closed point p ∈ C.
Below we assume the familiarity of Bridgeland stability conditions [Bri07]
on triangulated categories.
(52)

Lemma 4.1. The pairs
σtω,0 := (Ztω,0 , Bω ),t>0determine Bridgeland stability conditions on D0 .
Proof. The proof is almost the same as in the K3 surface case in [Bri08,
Proposition 7.1]. However, we need to take a little care since there may bepossible singular fibers of π. For non-zero E ∈ Bω , let us show the following
property(53)

Ztω,0 (E) ∈ {r exp(iπφ) : r > 0, φ ∈ (0, 1]}.

By our construction of Bω , we have Im Ztω,0 (E) ≥ 0. If Im Ztω,0 (E) = 0,then H1 (E) is zero dimensional and H0 (E) is a µω -semistable sheaf withµω (H0 (E)) = 0. Therefore by (52), it is enough to show that for any µω stable sheaf F ∈ Cohπ (X) with cl0 (F ) = (r, β, n) and ω · β = 0, we haven ≤ 0. Since F is µω -stable, it is an OXp -module for some p ∈ P1 . If Xp
is smooth, then the inequality n ≤ 0 this is a consequence of BogomolovGieseker inequality. In general, since Xp is an integral scheme, the structure
sheaf OXp is a µω -stable sheaf with µω (OXp ) = 0. By the µω -stability of F ,if r ≥ 2, we have Hom(OXp , F ) = Hom(F, OXp ) = 0. By the Serre duality
and the Riemann-Roch theorem, we obtain(54)

χ(F ) = r + n ≤ 0.

If r = 1, we may assume that Hom(F, OXp ) = 0. Indeed if otherwise, thenF is a ideal sheaf of OXp , hence the inequality n ≤ 0. Then we also have
Hom(OXp , F ) = 0. Indeed if there is a non-zero morphism OXp → F , thenit must be injective and the cokernel must be a zero dimensional sheaf. But
as Ext1Xp (Ox , OXp ) = 0 for any x ∈ Xp , the sheaf F must contain a zero

STABLE PAIRS ON NODAL K3 FIBRATIONS

31

dimensional sheaf, which contradicts to the µω -stability of F . Therefore
Hom(F, OXp ) = Hom(OXp , F ) = 0, and the Riemann-Roch computation(54) shows n ≤ 0.
We finally check the Harder-Narasimhan property. By following the arguments as in [Bri08, Proposition 7.1], we finally arrive at the exact sequencesof sheaves for i ≥ 1
0 → Q → H−1 (Ei ) → H0 (Li ) → 0.
Here we have used the same notation of [Bri08, Proposition 7.1]. The sheaf
Q is a pure two dimensional sheaf and H0 (Li ) is a zero dimensional sheaf.
A difference from [Bri08, Proposition 7.1] is that our sheaf Q is no longer atorsion free sheaf. However by pushing forward the above sequence by thegeneric projection Supp(Q) → P2 , we arrive at the same situation in [Bri08,
Proposition 7.1]. Therefore the Harder-Narasimhan property also holds. 
4.2. Weak stability conditions on D. Let D be the triangulated subcategory of Db Coh(X) defined by
D := hπ ∗ Pic(P1 ), Cohπ (X)itr ⊂ Db Coh(X).
Here h∗itr is the triangulated closure, i.e. the smallest triangulated subcategory which contains ∗. We construct weak stability conditions on D in thesense of [Tod10a], following the same argument of [Tod12b, Subsection 3.3].
We refer to [Tod10a, Section 2] for details on the space of weak stability conditions on triangulated categories. Let Aω be the subcategory of D definedbyAω := hπ ∗ Pic(P1 ), Bω iex ⊂ D.
One can check that, using the same argument of [Tod12b, Proposition 2.9],
the subcategory Aω is the heart of a bounded t-structure on D. In particular
it is an abelian category. We define the group homomorphism
cl : K(D) → Γ := Z ⊕ Γ0by cl |K(D0 ) = (0, cl0 ) andcl(π ∗ L) = (1, deg L, 0, 0)
for L ∈ Pic(P1 ). We take a filtration Γ• of Γ by
0 = Γ−1 ⊂ Γ0 ⊂ Γ1 := Γ
where the second inclusion is given by v 7→ (0, v). We define the element
Zω ∈

1
Y

Hom(Γi /Γi−1 , C)

i=0

to be the following:√Zω,1 (R) := R −1, R ∈ Γ1 /Γ0 = Z
ZZω,0 (v) :=e−iω v, v ∈ Γ0 .
X

32AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Lemma 4.2. The pairsσtω := (Ztω , Aω ),t ∈ R>0determine weak stability conditions on D with respect to the filtration Γ• onΓ.
Proof. Using Lemma 4.1 instead of [Tod12b, Lemma 3.3], the same argumentof [Tod12b, Lemma 3.4] is applied without any major modification.

4.3. Donaldson-Thomas type invariants. Let M be a C-scheme. In thepaper [Beh09], Behrend associated a canonical constructible function ν onit, called Behrend function. The ν-weighted Euler characteristicZX(55)
ν dχ :=m · χ(ν −1 (m))
Mm∈Z

is called the DT-type invariants. This is due to the fact by Behrend [Beh09]
that, if M is proper and admits a symmetric perfect obstruction theory, we
have the identity
Z
Z
1=
ν dχ.
[M ]virM

On the other hand, the naive Euler characteristic χ(M ) is called the Euler
characteristic version of the invariant (55).
Since we assumed that X is a smooth projective Calabi-Yau 3-fold, theperfect obstruction theory in Theorem 2.1 is symmetric, hence we haveZ
Pn,β =ν dχ.
Pn (X,β)We can similarly define DT type invariants on X using the Behrend function. For (r, β, n) ∈ Γ0 , let Mω (r, β, n) be the moduli stack of ω-Gieseker
stable sheaves E ∈ Cohπ (X) with cl0 (E) = (r, β, n). If any closed point ofMω (r, β, n) is stable, then it is a C∗ -gerbe over a projective scheme, hence
the following integration makes sense:
Z(56)
J(r, β, n) =ν dχ.
Mω (r,β,n)In general, there may be a closed point of Mω (r, β, n) corresponding toa strictly semistable sheaf. In such a case, the invariant (56) is definedin [JS12], [KS], as the generalized DT invariant, and takes its value in Q. It
is defined as the weighted Euler number of the ‘logarithm’ of Mω (r, β, n) in
the motivic Hall algebra of Cohπ (X). An Euler characteristic version of theinvariant (56) for the trivial K3 fibration S × C → C for a K3 surface S isavailable in [Tod12b, Definition 4.23].
Lemma 4.3. The invariant (56) is independent of a choice of ω.
Proof. Since the Euler pairing on D0 is trivial, the same argument of [Tod12b,
Lemma 4.16] is applied.
STABLE PAIRS ON NODAL K3 FIBRATIONS

33

4.4. Wall-crossing formula. The following is the main result in this section:
Theorem 4.4. We have the following formula

n+2r
Y
PT(X) =
exp (−1)n−1 J(r, β, r + n)q n tβr≥0,β>0,n≥0

·

(57)

Y


n+2r
.
exp (−1)n−1 J(r, β, r + n)q −n tβr>0,β>0,n>0

Proof. The Euler characteristic version of the above result is proved when
X is a trivial fibration in [Tod12b]. The proof proceeds along with thesame argument of [Tod12b] without any major modification, using the derived category version of [JS12, Theorem 5.18] proved in [Tod, Theorem 2.8]
instead of [Joy08, Theorem 6.28]. We only give an outline of the proof.
Step 1. First wall-crossing.
By the result of [Bri11], [Tod] (also see [Tod10b, Theorem 1.3], [Tod12a,
Theorem 3.11]), we have the following formula(58)

PT(X) =Y

exp (−1)n−1 J(0, β, n)q n tβnβ>0,n>0X

L(β, n)q n tβ  .

β,nThe above formula is obtained by applying the wall-crossing formula of DT
type invariants in the category of perverse coherent sheaves on Db Coh(X),which are certain two term complexes of coherent sheaves. The invariant L(β, n) counts perverse coherent sheaves E ∈ Db Coh(X), which aresemistable with respect to a certain self dual weak stability condition on
the perverse heart. (It was denoted by µiω -limit stability in [Tod10b, Section 3], and Zω,1/2 -stability in [Tod12a, Section 5].) The object E satisfies
the numerical condition
ch(E) = (1, 0, −β, −n) ∈ H ∗ (X, Q).
The precise definition of the invariant L(β, n) is available in [Tod12a, Definition 5.5], where it is denoted by Ln,β . When X = S × P1 for a K3 surface
S, its Euler characteristic version is available in [Tod12b, Subsection 4.6].
Step 2. Generating series of DT type invariants.
For (r, β, n) ∈ Γ0 , let Mtω (r, β, n) be the moduli stack of σtω -semistableobjects E ∈ Aω satisfying
cl(E) = (1, −r, −β, −n) ∈ Γ.
Similarly to [Tod12b, Lemma 4.13], the above moduli stack is realized asa constructible subset of an Artin stack M locally of finite type. Similarly to the previous subsection and [Tod12b, Definition 4.10], it defines the(generalized) DT type invariantZ
DTtω (r, β, n) :=ν dχ ∈ Q.
Mtω (r,β,n)34AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

We form the following generating series
X
DTtω (r, β, n)q n tβ sr .
DTtω (X) :=(r,β,n)∈Γ0

We consider the behavior of the above generating series for t  0 and
0 < t  1. We have the following proposition:
Proposition 4.5. We have the following formulasX
lim DTtω (X) =L(β, n)q n tβ sr
t→∞

r,β,n

lim DTtω (X) = lim

t→+0

t→+0

X

DTtω (r, β, 0)tβ sr .

r,βProof. The same proof of [Tod12b, Proposition 4.16] is applied without any
major modification.

Step 3. Wall-crossing in Aω .
We investigate the difference of the generating series DTtω (X) for t =t0 + 0 and t = t0 − 0 for a fixed t0 ∈ R>0 . We have the following result:
Proposition 4.6. We have the following formulalim DTtω (X) = lim DTtω (X)t→t0 +0

t→t0 −0·Y(r)(n+2r)
exp (−1)n−1 J(r, β, r + n)q n tβ sr.

β>0,n= 21 rt20 ω|2Xp

Here (r) = 1 if r > 0, (r) = −1 if r < 0 and (r) = 0 if r = 0.
Proof. The same proof of [Tod12b, Theorem 5.1] is applied by using theresults of [JS12], [Tod] instead of [Joy08]. We just give an outline of theproof. For v ∈ Γ0 , let Mtω,0 (v) be the moduli stack of σtω,0 -semistableobjects E ∈ Bω satisfying cl0 (E) = v. Similarly to the invariant J(v), we
have the generalized DT type invariant
Z
Jtω (v) :=ν dχ ∈ Q
Mtω,0 (v)which counts σtω,0 -semistable objects E ∈ Bω with cl0 (E) = v. The wallcrossing formula [JS12] describes the difference of the two limiting series
DTtω±0√ (X) in terms of the invariants Jtω (v), where v ∈ Γ0 satisfies Ztω,0 (v) ∈R>0 −1. If we write v = (r, β, n), the latter condition implies that1n = rt20 ω|2Xp .
2
On the other hand, the proof of [Tod12b, Corollary 4.27] shows that the invariant Jtω (v) is independent of t, and coincides with J(v). Now by applying
the wall-crossing formula in [JS12], and computing the relevant combinatorial coefficients as in [Tod10b], [Tod10a], we obtain the result.

By combining the above result with Proposition 4.5, we have the following
corollary (cf. [Tod12b, Corollary 5.2]):

STABLE PAIRS ON NODAL K3 FIBRATIONS

35

Corollary 4.7. We have the following formula:X
L(β, n)q n tβ sr(r,β,n)∈Γ0=

(r)(n+2r)d· DT(X)
exp (−1)n−1 J(r, β, r + n)q n tβ sr

Y
β>0,rn>0dwhere DT(X)
is defined to bed:= lim
DT(X)

t→+0

X

DTtω (r, β, 0)tβ sr(r,β,n)∈Γ0

Step 4. Further wall-crossing.
d
The final step is to decompose the series DT(X)
into a product which
involves the invariants J(v). Following [Tod12b, Subsection 3.6], letAω (1/2) ⊂ Aω
be the subcategory given as the extension closure of Ztω -semistable objects
E ∈ Aω for 0 < t  1, satisfying
πlim arg Ztω (E) = .
t→+02
Note that any object E ∈ Aω which contributes to the invariant DTtω (r, β, 0)is an object in Aω (1/2). Similarly to [Tod12b, Subsection 3.7], we are able
to construct a one parameter family of weak stability conditions(Zbω,θ , Aω (1/2)), θ ∈ (0, 1)
on Aω (1/2). Moreover by [Tod12b, Proposition 3.17], an object E ∈ Aω (1/2)
bω,1/2 -semistable,
contributes to the invariant DTtω (r, β, 0) if and only if it is Z
btω,θ -semistable forand an object E ∈ Aω (1/2) with rank(E) = 1 is Z∗
1
0 < θ  1 if and only if it is an object in π Pic(P ). Applying the wallcrossing formula, we obtain
2r X

Y
d
DT(X)
=
exp −J(r, β, r)tβ sr·sr .
r>0,β>0r∈Z

The above formula is obtained by the same argument of [Tod12b, Proposition 5.3] without any major modification. Combined with (58), Corollary 4.7, and taking the s0 -term, we obtain the desired identity (57).

4.5. Irreducible curve class case. When the curve class β is irreducible,
we have the following corollary of Theorem 4.4:
Corollary 4.8. Suppose that β ∈ H2 (X, Z)π is irreducible. Then for n ≥ 0,
we haveXPn,β =(−1)n−1 (n + 2r)J(r, β, r + n)
r≥0

(59)

P−n,β =

Xr>0(−1)n−1 (n + 2r)J(r, β, r + n).

36

AMIN GHOLAMPOUR AND ARTAN SHESHMANI AND YUKINOBU TODA

Remark 4.9. Note that when the data (r, β, n) is chosen so that there does
not exist any strictly semistable sheaf, then the invariants J(r, β, n) appearing on the right hand side of (59) coincide with the invariants computed in[GS13, Section 2].

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amingh@math.umd.edu
University of MarylandCollege Park, MD 20742-4015, USA
artan@mit.edu
Massachusetts Institute of Technology (MIT), Department of Mathematics
Building 2, Room 2-247, 77 Massachusetts Avenue, Cambridge, MA,
USA 02139-4307
yukinobu.toda@ipmu.jp
Institute for the Physics and Mathematics of the Universe
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan