Stable pairs on nodal fibrations
We study Pandharipande-Thomas’s stable pair theory on 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 spaces of stable pairs on surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the fibration. In the case that the 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.
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 [Mu1]. When a Calabi-Yau 3-fold admits an elliptic fibration, its relative Fourier-Mukai transform turned out to be an effective tool to prove the correspondence between one dimensional sheaf theory and two dimensional sheaf theory [FM-mirror], [T-dual]. In particular, it provides a recipe to relate two kinds of invariants: Pandharipande-Thomas stable pair invariants [a17] which count curves in Calabi-Yau 3-folds, and Donaldson-Thomas invariants which count two dimensional semistable sheaves on them [a20].
The above correspondence is also important in string theory. The relative Fourier-Mukai transform is a mathematical interpretation of relative T-duality in string theory, which was used by physicists [Klemm:2012sx, E-string, Black-T-dual] 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 the 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 [TodBg], 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 from string theory [DM]. In [G-S], the first and the second authors studied the generating series of DT invariants counting two dimensional stable sheaves on 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 [G-S] for PT stable pair theory on fibered threefolds. Namely we investigate the modularity property of the stable pair invariants on fibrations. We first treat the case of a smooth fibration, and relate the stable pair invariants with the generating series satisfying the modularity property. When the fibration has possibly nodal fibers, we relate their stable pair invariants with those on smooth fibrations via conifold transition. In a more general situation, we also prove a formula relating stable pair invariants with the DT invariants counting two dimensional semistable sheaves, where the latter invariants are expected to have the modular invariance property.
1.2. Main result I: smooth fibration
Let be a smooth projective threefold over . For and , the moduli space of stable pairs and the associated invariants were introduced by Pandharipande and Thomas [a17]. The moduli space parametrizes pairs
where is a pure 1-dimensional sheaf on with and , and the cokernel of is 0-dimensional. They constructed a perfect obstruction theory on (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 is zero, the stable pair invariant is defined by taking the degree of the virtual cycle obtained from this obstruction theory.
We assume that admits a morphism
onto a smooth projective curve , whose generic fiber is a smooth surface. The morphism is called a K3 fibration. We always take the curve class to be in the kernel of denoted by . The virtual dimension of is zero for such curve classes111Note that is not required to be a Calabi-Yau 3-fold here.. We define the following generating series:
In Section 2, we study the case when is a smooth morphism. The majority of this section (Subsections 2.1-2.6) is devoted to the case where is an irreducible curve class. In this case, we relate with the Euler characteristics of the moduli space of stable pairs on a nonsingular surface and the Noether-Lefschetz numbers of the fibration having modular properties [a90, a116, a125, a117]. We denote by the moduli space of stable pairs on a surface containing an irreducible curve class satisfying . It is known that the topological type of only depends on (cf. [MPT]).
(Theorem LABEL:thm:main_formula) Let be a smooth fibration, and be an irreducible curve class. Then we have the following formula:
The proof of Theorem 1 involves studying the restriction of the obstruction theory constructed in [a17] to isolated and non-isolated components of . The Euler characteristics in Theorem 1 can be read off from Kawai-Yoshioka’s formula having modular properties [a79, Theorem 5.80]:
In Subsection LABEL:sec:reducible, we study a special analog of Theorem 1 in which the class is allowed to be reducible.
1.3. Main result II: nodal fibration
We next study the case that the fibration (2) has a finite number of nodal fibers. In this case, we call a morphism (2) a nodal K3 fibration. For a nodal fibration, we find a conifold transition formula, which relate the stable pair invariants on nodal fibrations with those on smooth fibrations. The following result is based on assuming some foundation of stable pair theory of algebraic spaces that has not been established yet (see Remarks LABEL:rem:nonproj and LABEL:alg-space-degen):
(Theorem LABEL:thm:formula:relation) Let be a nodal fibration. Then there is a smooth fibration222Here the situation is a little more general than (2), and we allow to be a proper algebraic space. and morphisms
where is a small resolution and is a double cover, such that the following formula holds:
Here is the variable change , and is the subseries of of stable pair invariants of curve classes with .
Let us consider the invariant with an irreducible curve class on a nodal fibration . The above result leads us to working with the stable pair invariants for possibly reducible classes on a smooth fibration lifting . In Subsection LABEL:sec:noncomm, we present an approach to compute the invariants by studying the relationship between the invariants and the invariants of the moduli spaces of perverse (non-commutative) stable pairs.
1.4. Main result III: product expansion formula
Finally let be a smooth projective Calabi-Yau 3-fold, which admits a fibration such that every fiber is an integral scheme. In Section LABEL:sec:wall, using the wall-crossing techniques mostly developed by the third author [Tcurve2, Tcurve1, Tolim2, Tsurvey, TodK3, TodHall], we express the generating series (3) in terms of the generalized Donaldson-Thomas invariants
which count semistable sheaves supported on the fibers of satisfying
Here is the class of a fiber of in . The invariants (6) are defined by the method of Joyce-Song [JS]. These invariants were studied by the first and second authors in [G-S] when there are no strictly semistable sheaves satisfying (7). The modularity property of the invariants (6) holds in the situation of [G-S], 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 LABEL:thm:WCF) We have the following formula
The above result is proved by generalizing the argument in [TodK3] showing a similar result when for a surface . Namely, we apply Joyce-Song’s wall-crossing formula [JS] in a certain space of weak stability conditions. As the arguments are the almost same as in [TodK3] except a few modifications, we will just outline its proof in Section LABEL:sec:wall.
The main purpose of the current article is to find the connection between the vertices of the following triangle:
Here by and we mean the fiberwise stable pair and Donaldson-Thomas theories of the Calabi-Yau fibration discussed above. As mentioned earlier, the connection is motivated by -duality and we provide a complete answer to it in Section LABEL:sec:wall. The connection is motivated by -duality, and the first two authors have provided partial results on it in [G-S]. In Section 2 and Section LABEL:sec:nodalK3 of this paper, we provide partial results on the connection between . More recently, Pandharipande-Thomas [KKVPT] studied the connection in more generality by a different method (degeneration method) and for a different purpose. A refined (motivic) version of connection is conjectured in [KKP]. We hope the combination of the established bridges and , sheds some lights on the connection in full generality, which we hope we can work out in the future.
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 . For a morphism of schemes and coherent sheaves , on , we write the sheaf as . Its right derived functor is denoted by , which coincides with . If is smooth, we denote by the relative canonical line bundle on . For a scheme , we denote by the cotangent complex of .
2. Stable pairs on smooth fibrations
2.1. Stable pairs on fibrations
Let be a smooth projective threefold over . By definition, a K3 fibration is a morphism
onto a smooth projective curve whose generic fiber is a smooth surface. If is a smooth morphism, then it is called a smooth K3 fibration. We set
We always take curve classes contained in the above subgroup of . An element is called irreducible if it is not written as a sum for non-zero effective curve classes .
For , we study the moduli space of stable pairs in the sense of [a17]. It parametrizes the pairs
where is a pure 1-dimensional sheaf on with and , and the cokernel of is 0-dimensional. Here is the fundamental homology class determined by , which is the Poincare dual of . For simplicity we set , and denote by
the universal pairs, which we interpret as an object in . Note that we have the exact triangle on
be the natural projections. In [a17], is equipped with a perfect obstruction theory by studying the deformations of the two term complex (8) in the derived category:
[a17] There is a perfect obstruction theory over given by the following morphism in the derived category:
Here means taking the traceless part.
Since for , the Riemann-Roch calculation easily implies that the virtual dimension of is zero. The stable pair invariants are then defined by
We define the generating series , to be
2.2. Stable pairs on surfaces
Let be a smooth projective surface over . For , we can similarly define the moduli space of stable pairs on . We review several results on the moduli space . In [a44], Pandharipande and Thomas identify by a relative Hilbert scheme of points:
[a44, Proposition B.8] The moduli space of stable pairs on is isomorphic to the relative Hilbert scheme where is the moduli space of pure one dimensional subschemes of in class and denotes the universal curve over .∎
We next consider the case that is irreducible. In this case, the following result is proved in [MPT]:
[MPT, Proposition 5] If is irreducible, the moduli space is non-singular of dimension . It depends only upon up to deformation equivalence.
By the above proposition, we may write
for an irreducible curve class with . We sometimes write if we do not want to specify a particular surface with this property. The generating series of is computed by Kawai-Yoshioka’s formula (1.2).
2.3. Stable pairs with irreducible curve classes
In this subsection, we assume that is an irreducible curve class. Let be the moduli space of stable pairs, and consider the perfect obstruction theory in Theorem 2.1. We have the following proposition:
In the above situation, we have the canonical isomorphism
We write for simplicity. Applying to the exact triangle (10), we obtain the following exact triangle on :
By the above triangle and the natural morphism
we can form the following commutative diagram of vertical and horizontal exact triangles: