Multiple cover formula of generalized DT invariants II: Jacobian localizations
The generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. This conjecture is equivalent to Pandharipande-Thomas’s strong rationality conjecture on the generating series of stable pair invariants, and its local version is enough to prove. In this paper, using Jacobian localizations and parabolic stable pair invariants introduced in the previous paper, we reduce the conjectural multiple cover formula for local curves with at worst nodal singularities to the case of local trees of smooth rational curves.
This paper is a sequel of the author’s previous paper [Todpara], and we study the conjectural multiple cover formula of generalized Donaldson-Thomas (DT) invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds. Our main result is to reduce the multiple cover formula for local curves with at worst nodal singularities to that for local trees of . The latter case is easier to study, and we actually prove the multiple cover formula in some cases using our main result. The idea consists of twofold: using the notion of parabolic stable pairs introduced in [Todpara], and the localizations with respect to the actions of Jacobian groups on the moduli spaces of parabolic stable pairs.
1.1 Conjectural multiple cover formula
Let be a smooth projective Calabi-Yau 3-fold over , i.e.
the generalized DT invariant is introduced by Joyce-Song [JS], Kontsevich-Soibelman [K-S],
The invariant (1) counts one dimensional semistable sheaves on satisfying
(cf. Subsection 2.3.) The above invariant is expected to satisfy the following multiple cover conjecture:
[JS, Conjecture 6.20], [Tsurvey, Conjecture 6.3] We have the following formula,
The motivation of the above conjecture is that it is equivalent to Pandharipande-Thomas’s (PT) strong rationality conjecture [PT]. (See [Tsurvey, Theorem 6.4].) The PT strong rationality conjecture claims the product expansion formula (called Gopakumar-Vafa form) of the generating series of rank one DT type invariants, which should be true if we believe GW/DT correspondence [MNOP].
There is also a local version of the invariant (1) and its conjectural multiple cover formula. Namely for a one cycle on , we can associate the invariant,
which counts one dimensional semistable sheaves on satisfying
where the second equality is an equality as a one cycle. The above local invariant is also expected to satisfy the multiple cover formula,
1.2 Main result
Let be as before, a one cycle on and the support of . The invariant can be shown to be zero if there is an irreducible component of whose geometric genus is bigger than or equal to one. (cf. Lemma 2.11.) Therefore in discussing the formula (2), we may assume that is a rational curve, i.e. the normalization of is a disjoint union of . The simple cases are , or is a tree of . The main result of this paper is to show that, when has at worst nodal singularities, then the formula (2) follows from the same formula for local trees of . More precisely, suppose that is a rational curve with at worst nodal singularities, and
a sufficiently small analytic neighborhood of in . We consider data,
where is a reduced curve, is a three dimensional complex manifold and is a local immersion. The above data is called a cyclic neighborhood if it is given as a composition of cyclic coverings of . (See Definition LABEL:def:cyclic for more precise definition.) For any one cycle on supported on , we can similarly construct the invariant
(cf. Subsection LABEL:moduli:cyclic.) Our main result is as follows:
[Theorem LABEL:thm:main:cov] Let be a smooth projective Calabi-Yau 3-fold over , a reduced rational curve with at worst nodal singularities, and a one cycle on supported on . Suppose that for any cyclic neighborhood with a tree of , the following conditions hold:
The moduli stack of one dimensional semistable sheaves on is locally written as a critical locus of some holomorphic function on a complex manifold up to some group action. (cf. Conjecture LABEL:conj:crit.)
For any one cycle on with , the invariant satisfies the formula
Then the invariant satisfies the formula (2).
There are several situations in which the cyclic neighborhood satisfies the assumptions in Theorem 1.2, e.g. is a chain of super rigid rational curves in . Roughly speaking, we will give the following applications in Section LABEL:sec:apply:
If is supported on an irreducible rational curve with one node, or a circle of , we explicitly compute the invariant . (cf. Theorem LABEL:thm:typeI.)
We prove the local multiple cover formula of if for an irreducible rational curve with at worst nodal singularities, and is a prime number. (cf. Theorem LABEL:prop:prime.)
We give some evidence of the conjecture in [TodK3, Conjecture 1.3] on the Euler characteristic invariants of local K3 surfaces. (cf. Theorem LABEL:thm:K3.)
The first and the second applications will be given under a certain assumption on an analytic neighborhood of a one cycle . (cf. Definition LABEL:def:rigid:surface.)
1.3 Idea for a local curve with one node
Here we explain the idea of the proof of Theorem 1.2 in a simple example. Let
be an irreducible rational curve with one node . Suppose that a one cycle on is supported on . Then for any analytic neighborhood as in (3), the Jacobian group acts on the moduli space which defines . If we take to be homotopically equivalent to , then is considered to be a subgroup of . So we would like to apply -localization on the invariant . In order to see this, we need to find -fixed semistable sheaves on supported on .
If we take as above, then we have
Hence if we take the universal covering space of ,
then admits a -action, and it contains the universal cover of denoted by . A key observation is that a stable sheaf on supported on is -fixed if and only if it is a push-forward of some sheaf on supported on , which is unique up to -action on .
The universal cover is described in the following way. Let
be the normalization and the preimage at the node . We take an infinite number of copies of , denoted by
Then is an infinite chain of smooth rational curves,
where and are attached along and . (See Figure 1.)
For instance, let us look at the invariant . By the above argument, we may expect the formula,
Now by the assumptions in Theorem 1.2, we obtain
The above localization argument also implies , and it is also easy to see for . Thus we obtain
which is nothing but the desired formula (2) for . This picture is quite similar to the multiple cover formula for genus zero Gromov-Witten invariants of a local nodal curve with one node [BKL].
1.4 Parabolic stable pairs
In the previous subsection, we explained the idea of the multiple cover formula in a simple example. However it is not obvious to realize the story there directly, especially the formula (5) seems to be hard to deduce. The issue is that, since the definition of involves Joyce’s log stack function [Joy4], denoted by in Subsection 2.3, the above localization argument seems to be very hard to apply. Namely, we have to compare the contribution of on the -fixed points with that on the universal cover. But to do this, we also have to ‘localize’ the product structure on the Hall algebra, which seems to require a new technique. In order to overcome this technical difficulty, we use the idea of parabolic stable pairs, introduced in the previous paper [Todpara]. By definition, a parabolic stable pair consists of a pair,
where is a one dimensional semistable sheaf on , is a fixed divisor in , satisfying a certain stability condition. (cf. Definition 2.7.) In [Todpara], we constructed invariants counting parabolic stable pairs, and showed that Conjecture 1.1 is equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants. The moduli space of parabolic stable pairs is a scheme, (not a stack,) and also acts on the moduli space of (local) parabolic stable pairs. There is no technical difficulty in applying -localizations to parabolic stable pair invariants, and the arguments similar to the previous subsection work for parabolic stable pairs.
When the one cycle on is supported on a nodal curve which has more than one nodes, then its universal covering space is much more complicated. Instead of taking the universal cover, we take cyclic neighborhoods and proceed the induction argument. Combining the above ideas, (Jacobian localizations, parabolic stable pairs, induction via cyclic neighborhoods,) we are able to prove Theorem 1.2.
The author is grateful to Richard Thomas, Jacopo Stoppa for valuable discussions on the subject of this paper, and Kentaro Nagao for pointing out the reference [Bergh]. The author would like to thank the Isaac Newton Institute and its program ‘Moduli Spaces’, during which a part of this work was done. This work is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan. This work is also supported by Grant-in Aid for Scientific Research grant (22684002), and partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
2 Multiple cover formula of generalized DT invariants
In this section, we recall (generalized) DT invariants on Calabi-Yau 3-folds and the conjectural multiple cover formula. In what follows, is a smooth projective Calabi-Yau 3-fold over , i.e.
We fix an ample line bundle and set . Below we say a coherent sheaf on -dimensional if the support of is -dimensional.
2.1 Semistable sheaves
Let us recall the notion of one dimensional -semistable sheaves on . They are defined by the notion of slope: for a one dimensional coherent sheaf , its slope is defined by
Here is the holomorphic Euler characteristic of and is the fundamental one cycle associated to , defined by
In the above sum, runs all the codimension two points in .
A one dimensional coherent sheaf on is -(semi)stable if for any subsheaf , we have the inequality,
Note that any one dimensional -semistable sheaf is pure, i.e. there is no zero dimensional subsheaf in . Also we say that is strictly -semistable if is -semistable but not -stable. For the detail of (semi)stable sheaves, see [Hu].
2.2 DT invariants
Let us take data,
The (generalized) DT invariant is the -valued invariant,
counting one dimensional -semistable sheaves on satisfying
Here by an abuse of notation, we denote by the homology class of the one cycle (6).
The invariant (8) is defined in the following way. Let
be the coarse moduli space of one dimensional -semistable sheaves on satisfying (9). There are some criterions for the moduli space (10) to be fine. For instance suppose that the following condition holds:
e.g. . Then there is no strictly -semistable sheaf on satisfying (9), and (10) is a fine projective scheme over . In this case, the moduli space (10) carries a symmetric perfect obstruction theory, hence the zero dimensional virtual cycle [Thom].
If the condition (11) holds, then we define to be
Another way to define is to use Behrend’s constructible function [Beh]. Recall that for any -scheme , Behrend constructs a canonical constructible function,
such that if carries a symmetric perfect obstruction theory, then we have
Hence by using the Behrend function on , the invariant (8) can also be also expressed as
2.3 Generalized DT invariants
In a general choice of (7), the condition (11) may not hold, and there may be strictly -semistable sheaves satisfying (9). In this case, the invariant (8) is one of generalized DT invariants introduced by Joyce-Song [JS] and Kontsevich-Soibelman [K-S]. It requires sophisticated techniques on Hall algebras of coherent sheaves to define them, and we need some more preparations for this. Since we will not need the detail of the definition of (8) in a general case, we just give a rough explanation.
A strictly -semistable sheaf has non-trivial automorphisms, and we need to involve the contributions of the automorphism groups with the invariant (8). For this purpose, we need to work with the moduli stack,
The Behrend functions on -schemes naturally extend to constructible functions on Artin stacks of finite type over . (cf. [JS, Proposition 4.4].) However the stack (13) may have stabilizer groups whose Euler characteristic are zero, e.g. . Hence the integration of the Behrend function (12), replacing by , does not make sense. The idea of the definition of generalized DT invariant is that, instead of working with the stack (13), we should work with the ‘logarithm’ of (13) in the Hall algebra of coherent sheaves, denoted by .
The algebra is, as a -vector space, spanned by the isomorphism classes of symbols,
Here is an Artin stack of finite type with affine geometric stabilizers, and is the stack of all the coherent sheaves on . There is an associative -product on based on Ringel Hall algebras. For the detail, see [Joy2, Theorem 5.2].
The stack (13) is considered to be an element of , by regarding it as an open substack of ,
The ‘logarithm’ of , denoted by , is defined by the rule,
for any in a certain completion of the algebra . In other words, is given by
The above sum is easily shown to be a finite sum.
The important fact is that is supported on ‘virtual indecomposable sheaves’. Roughly speaking this implies that, modulo some relations in , the element is written as
where , are quasi-projective varieties on which act trivially. The invariant (8) is then defined by the weighted Euler characteristics of , weighted by the Behrend function on pulled back by . Namely, is defined by
Here we need to change the sign due to the appearance of the trivial -action.
We have skipped lots of details in the above definition of (8). For more detail, we refer [JS]. Also see [Tsurvey, Section 4] for a more direct explanation.
In priori, we need to choose an ample divisor to define . However it can be shown that does not depend on a choice of . (cf. [JS, Theorem 6.16].)
2.4 Local generalized DT invariants
There is also a local version of (generalized) DT invariant, which we explain below. Let us fix a reduced curve in ,
with irreducible components . Then a one cycle on supported on is identified with an element of ,
corresponding to -stable sheaves satisfying
Here is an equality as a one cycle on . Then the local DT invariant is defined by
Here is the Behrend function on restricted to . We remark that may not coincide with the Behrend function on .
Even if does not satisfy the condition (11), we can similarly define the local generalized DT invariant,
parameterizing one dimensional -semistable sheaves on satisfying (16). We can similarly take the logarithm of the substack (19) in the Hall algebra , and the invariant (18) is defined by integrating the Behrend function on over it. See [Todpara, Subsection 4.4] for some more detail. Similarly to , the local invariant also does not depend on . (cf. Remark 2.3.)
2.5 Multiple cover formula
The above invariant is introduced by Katz [Katz] as a sheaf theoretic definition of genus zero Gopakumar-Vafa invariant. On the other hand if does not satisfy the condition (11), then may not be an integer and hence does not coincide with . However the invariants for are conjectured to be related to via the multiple cover formula:
[JS, Conjecture 6.20], [Tsurvey, Conjecture 6.3] We have the following formula,
In [Tsurvey, Theorem 6.4], it is shown that the above conjecture is equivalent to Pandharipande-Thomas’s strong rationality conjecture [PT, Conjecture 3.14]. We refer [Tsurvey, Section 6] for discussions on strong rationality conjecture and its relation to Conjecture 2.4.
For a reduced curve , and , we have the local (generalized) DT invariants as in (18). The local version of the above conjecture is also similarly formulated:
[Todpara, Conjecture 4.13] For and , we have the formula,
As shown in [Todpara, Corollary 4.18], the local multiple cover formula is enough to show the global multiple cover formula:
As we discussed in the Introduction, our purpose is to study Conjecture 2.5 in terms of Jacobian localizations and parabolic stable pair invariants, which we recall in the next subsection.
2.6 (Local) parabolic stable pair theory
The notion of parabolic stable pairs is introduced in [Todpara]. It is determined by fixing a divisor,
for some . In what follows, we say a one cycle on intersects with transversally if it satisfies . Equivalently, any irreducible component in is not contained in .
For a fixed divisor on as above, a parabolic stable pair is defined to be a pair
such that the following conditions are satisfied.
The sheaf is a one dimensional -semistable sheaf on .
The one cycle intersects with transversally.
For any surjection with , we have
The moduli space of parabolic stable pairs satisfying , is denoted by
By [Todpara, Theorem 2.10], if satisfies an additional condition given in [Todpara, Lemma 2.9], then the moduli space (23) is a projective scheme even if does not satisfy the condition (11). In the case that does not satisfy the condition in [Todpara, Lemma 2.9], the moduli space (23) is at least a quasi-projective variety. (cf. [Todpara, Remark 2.13].)
Suppose that a reduced one dimensional subscheme satisfies . Then for any with , we have the subscheme,
corresponding to parabolic stable pairs with supported on , as a one cycle on and .
be the Behrend’s constructible function [Beh] on . The local parabolic stable pair invariant is defined in the following way.
For , we define to be
Here as in the local DT theory, we use the Behrend function on , not on , to define the local invariant.
2.7 Multiple cover formula via parabolic stable pairs
In [Todpara], we established a relationship between (local) parabolic stable pair invariants and (local) generalized DT invariants. As a result, conjectures in Subsection 2.5 can be translated into a formula relating (local) parabolic stable pair invariants and (local) DT invariants, which are both integer valued.
Let be a reduced curve, with irreducible components , which intersects with transversally. As in Definition 2.8, we have the local parabolic stable pair invariants w.r.t. . For each , we set the generating series to be
Here is defined by
[Todpara, Proposition 4.5] We have the formula (21) for any with if and only if the following formula holds,
Note that is written as
Then we should have the formula,
2.8 Jacobian actions on the moduli space of parabolic stable pairs
In this subsection, we discuss Jacobian actions on the moduil space of parabolic stable pairs.
Let be a reduced curve, and a divisor which intersects with transversally. Let us take
and set . Let be a complex analytic neighborhood of in ,
Then we have the analytic open subset of the moduli space (10),
corresponding to -semistable one dimensional sheaves with .
Let be the group of line bundles on , whose restriction to any projective curve in has degree zero. Then we have the action of on via
for and . The -action preserves the closed subscheme,
where the LHS is given in (15).
Let us consider parabolic stable pairs w.r.t. the divisor as above. Similarly, we have the analytic open subspace,
corresponding to parabolic stable pairs with . Let be the group defined by
Note that the forgetting map is surjective if is a sufficiently small analytic neighborhood of . The group acts on via
where is the image of by the isomorphism,
The above isomorphism makes sense since is supported on . Obviously the action (30) preserves the closed subspace,
By Chow’s theorem, the complex analytic spaces , are regarded as the moduli spaces of -semistable sheaves, parabolic stable pairs on in an analytic sense respectively. Hence the above , -actions make sense.
2.9 Local multiple cover formula in simple cases
Finally in this section, we discuss some situations in which the formula (21) is easily proved. Let be as in the previous subsection. In the following lemma, which is partially obtained in [JS, Proposition 6.19], we reduce the problem to the case that has only rational irreducible components.
Let be the irreducible components of , and take
Suppose that there is such that and the geometric genus of is is bigger than or equal to one. Then for any , we have . In particular, the formula (21) holds.
Let us consider the action on as in the previous subsection. Note that any point is represented by an -semistable sheaf which is a direct sum of -stable sheaves. If is fixed by the action of , then we have . For the normalization , we have
Taking the determinant of the both sides, we have
for some . Given , there is only a finite number of possibilities for the above , say . Since is a complex torus of positive dimension, we can find a subgroup
which does not pass through any -torsion points for . On the other hand, we have the composition of the pull-backs
Since is a sufficiently small analytic neighborhood of , an argument similar to Subsection 3.2 below shows that both of the arrows in (33) are surjective. Furthermore, the same argument also easily shows that there is a subgroup which restricts to the subgroup (32) under the restriction (33). Then the action of on restricted to is free, hence the same localization argument of [JS, Proposition 6.19] shows the vanishing . ∎
Next we discuss the case that the class is primitive, i.e. is not a multiple of some other element of .
Suppose that is primitive. Then does not depend on . In particular, the formula (21) holds.
Let be the category of coherent sheaves on supported on . We first generalize -stability to twisted stability on . Let be a sufficiently small analytic neighborhood, and take an element
such that is ample. For a one dimensional sheaf , we set to be
Similarly to Definition 2.1, we have the notion of -stability on , called twisted stability. As in the case of -stability, we can construct the moduli stack parameterizing -semistable objects with and , and the generalized DT invariant defined by the above moduli stack. The same argument of [JS, Theorem 6.16] shows that the resulting invariant does not depend on a choice of and , thus coincides with .
Let be the irreducible components of , and set . Since is primitive, we have . Hence we can find such that . Let us take divisors on such that , and set . (This is possible since is taken to be a sufficiently small analytic neighborhood of in .) Then we have the isomorphism of stacks,
given by . Since the generalized DT invariants do not depend on and , the above isomorphism of stacks immediately implies for all . ∎
3 Cyclic covers of nodal rational curves
Let be a smooth projective Calabi-Yau 3-fold over . In what follows, we fix a connected reduced curve and an embedding,
satisfying the following conditions.
Any irreducible component of has geometric genus zero. (We call such a curve as a rational curve.)
The curve has at worst nodal singularities.
Note that for our purpose, we can always assume the first condition by Lemma 2.11. The geometric genus of is defined by
When , each irreducible component of is , and the dual graph of is simply connected. (See Subsection 3.1 below.) In this case, we say is a tree of . Below we assume that .
We also fix an ample divisor in which is smooth, connected, and intersects with transversally at non-singular points of .
3.1 Jacobian group of
We first recall the description of the Jacobian group of a nodal curve . Suppose that has -nodes and has -irreducible components. Let us take the normalization of ,
We have the exact sequence of sheaves,
By the long exact sequence of cohomologies, we obtain the isomorphism,
In particular the arithmetic genus satisfies
Combining the above argument with the standard exact sequence,
we can easily see that generates as a -vector space,
Hence we have the isomorphisms,
We can interpret the above isomorphism in terms of the dual graph associated to , determined in the following way:
The vertices and edges of correspond to irreducible components of and nodal points of respectively.
For an edge corresponding to a nodal point , it connects vertices , if the corresponding irreducible components , satisfies . (Note that the case of corresponds to a self node.)
Then is a connected graph satisfying . We can interpret (35) as the isomorphism,
The isomorphism (36) can be constructed in the following way. For an oriented loop in , we choose an edge so that is still connected. Let be a nodal point corresponding to the edge . Let be vertices in connected by so that starts from and ends at . Let , be the irreducible components of which correspond to , respectively. We partially normalize at , and obtain ,
For , let be the irreducible component of which is mapped to by . The preimage of by is denoted by
(In case of , i.e. is a self node, we need to fix a correspondence between an orientation of and a numbering of two points .) For , we glue the trivial line bundle on by the isomorphism at ,
The above gluing procedure produces a line bundle on . The resulting line bundle is independent of a choice of , and denoted by . The isomorphism (36) is given by sending to .
3.2 Jacobian group of an analytic neighborhood of
Let be as in the previous subsection. We take an analytic open neighborhood of in ,
If we take sufficiently small so that it is homotopically equivalent to , we have the commutative diagram of exact sequences,