Stable pair invariants on Calabi-Yau threefolds containing \mathbb{P}^{2}

# Stable pair invariants on Calabi-Yau threefolds containing P2

## Abstract.

We relate Pandharipande-Thomas stable pair invariants on Calabi-Yau 3-folds containing the projective plane with those on the derived equivalent orbifolds via wall-crossing method. The difference is described by generalized Donaldson-Thomas invariants counting semistable sheaves on the local projective plane, whose generating series form theta type series for indefinite lattices. Our result also derives non-trivial constraints among stable pair invariants on such Calabi-Yau 3-folds caused by Seidel-Thomas twist.

## 1. Introduction

### 1.1. Motivation

It is an important subject to count algebraic curves on Calabi-Yau 3-folds, or more generally on CY3 orbifolds1, in connection with string theory. So far at least three curve counting theories have been proposed and studied: Gromov-Witten (GW) theory [BGW], Donaldson-Thomas (DT) theory [Thom], [MNOP] and Pandharipande-Thomas (PT) stable pair theory [PT]. It was conjectured, and proved in many cases, that these theories are equivalent: the equivalence of DT and PT theories was proved in [BrH], [Tcurve1], [StTh] using Hall algebras, and the equivalence of GW and PT theories was proved by Pandharipande-Pixton [PP] for many Calabi-Yau 3-folds including quintic 3-folds using degenerations and torus localizations.

On the other hand, the derived category of coherent sheaves on a Calabi-Yau 3-fold is also an important mathematical subject, due to its role in Kontsevich’s Homological mirror symmetry conjecture [Kon]. It was suggested by Pandharipande-Thomas [PT] that the derived category also plays a crucial role in curve counting, as their stable pair invariants count two term complexes

 (OX\lx@stackrels→F)∈DbCoh(X)

where is a pure one dimensional sheaf and is surjective in dimension one. In this paper, we concern how symmetries in the derived categories affect stable pair invariants. More precisely, we are interested in the following questions:

###### Question 1.1.
1. How stable pair invariants on two Calabi-Yau 3-folds or orbifolds are related, if they have equivalent derived categories ?

2. How stable pair invariants on a Calabi-Yau 3-fold are constrained, due to the presence of non-trivial autoequivalences of the derived category ?

The purpose of this paper is to study Question 1.1 for stable pair invariants on Calabi-Yau 3-folds which contain , and their derived equivalent CY3 orbifolds . Our results include new kinds of progress on Question 1.1: (i) relation of stable pair invariants on and , where does not satisfy the Hard-Lefschetz (HL) condition2 (ii) constraints of stable pair invariants on caused by Seidel-Thomas twist [ST]. The relation of our work with the existing works will be discussed in Subsection 1.3.

### 1.2. Main result

Let be a smooth projective Calabi-Yau 3-fold which contains a divisor

 P2≅D⊂X.

We have two phenomena related to (i) and (ii) in Question 1.1:

1. The divisor is contracted by a birational morphism to an orbifold singularity with type . The associated smooth Deligne-Mumford stack is derived equivalent to

 Φ:DbCoh(Y)\lx@stackrel∼→DbCoh(X).
2. The object is a spherical object in , and we have the associated autoequivalence called Seidel-Thomas twist [ST]

 STOD:DbCoh(X)\lx@stackrel∼→DbCoh(X).

Contrary to the 3-fold flop case as in [Tcurve2], [Cala], curves on , may be transformed to objects with two dimensional supports under the equivalence , , respectively. In order to deal with this issue, we also involve generalized DT invariants [JS], [K-S]

 (1) DT(r,c,m)∈Q

on the non-compact Calabi-Yau 3-fold . The invariant (1) counts semistable sheaves on satisfying

 rank(π∗E)=r, c1(π∗E)=c, ch2(π∗E)=m.

The following is a rough statement of our main result:

###### Theorem 1.2.

(Theorem LABEL:thm:main, Theorem LABEL:thm:const) Assuming Conjecture 1.3 below, we have the following:

1. The stable pair invariants on are described as explicit polynomials of stable pair invariants on and generalized DT invariants (1) on .

2. If there is with , then there exist explicit polynomial relations among stable pair invariants on and generalized DT invariants (1) on caused by .

The result of Theorem 1.2 (i) in particular derives a recursion formula of stable pair invariants on with curve classes proportional to for a line (in other words stable pair invariants on ), whose coefficients involve the invariants (1) (cf. Corollary LABEL:cor:recursion). The result of Theorem 1.2 (ii) implies a stronger statement: the stable pair invariants on with curve classes satisfying are described in terms of those with curve classes for , with coefficients involving (1) (cf. Remark LABEL:rmk:form).

In the previous paper [TodS3], the author proved a recursion formula of the generating series of the invariants (1) with in terms of theta type series for indefinite lattices. It is also possible to describe the invariants (1) with in terms of stable pair invariants on with curve classes proportional to (cf. Lemma LABEL:rankzero). These results imply that, in principle, one can compute the relations of stable pair invariants concerning Question 1.1 for the derived equivalences and . The resulting formulas in Theorem LABEL:thm:main, Theorem LABEL:thm:const are complicated, and we leave it a future work to give a more conceptual understanding of our result.

We should mention that the result of Theorem 1.2 is still conditional to the following conjecture, which was also assumed in the author’s previous work [Tcurve2].

###### Conjecture 1.3.

Let be the moduli stack of objects satisfying . For , let be a maximal reductive subgroup of . Then there is a -invariant analytic open neighborhood of in , a -invariant holomorphic function with , and a smooth morphism of complex analytic stacks

 Φ:[{df=0}/G]→M

of relative dimension .

The above conjecture has been a technical obstruction to generalize Joyce-Song’s wall-crossing formula of DT invariants [JS] for coherent sheaves to the derived category. It was proved for by Joyce-Song [JS], and announced by Behrend-Getzler. There exist more recent progress toward it, which will be reviewed in the next subsection. Without assuming Conjecture 1.3, we can prove Euler characteristic version of Theorem 1.2 (i.e. results for the naive Euler characteristics of stable pair moduli spaces), as stated in Subsection LABEL:subsec:Ever.

### 1.3. Related works

In [Tcurve2], [Cala], the flop transformation formula of stable pair invariants was obtained from the categorical viewpoint, giving an answer to Question 1.1 (i) for birational Calabi-Yau 3-folds. In the orbifold case, let be a Calabi-Yau 3-fold with Gorenstein quotient singularities and its crepant resolution. Under the HL condition on the associated Deligne-Mumford stack , Bryan-Cadman-Young [BCY] formulated a conjectural relationship between DT invariants on and those on . Combined with the DT/PT correspondence [BrH], [Tcurve1], [StTh] on , and Bayer’s announced work [BaDTPT] on it for CY3 orbifolds with HL condition, we have a conjectural answer to Question 1.1 (i) in this situation. The conjecture in [BCY] is still open, but some progress toward it is obtained in [Cala2], [BrSt], [DR].

In the above HL case, the resulting formula should be described by a product formula of the generating series of stable pair invariants. In our situation of Theorem 1.2, the stack does not satisfy the HL condition, and it seems unlikely that the results are formulated as product formulas of the generating series. From the categorical viewpoint, the main difference from the HL case is the non-triviality of the Euler pairings between objects supported on the fibers of . Due to this non-triviality, the combinatorics of the wall-crossing becomes complicated, and it seems hard to understand the result in terms of the generating series. In any case, we hope that the result of Theorem 1.2 would give a hint toward a generalization of the conjecture in [BCY] without the HL condition.

There exist few works concerning Question 1.1 (ii) so far. We can say that the rationality of the generating series of stable pair invariants, conjectured in [PT] and proved in [Tolim2],  [BrH], is interpreted to be an answer to Question 1.1 (ii) for the derived dualizing functor. Also the automorphic property of sheaf counting invariants on local K3 surfaces under Hodge isometries, together with product expansion of the generating series of stable pair invariants on them [TodK3] in terms of the former invariants, is interpreted to be an answer to Question 1.1 (ii) for autoequivalences of K3 surfaces [Tst3], [TodK3]. The result of Theorem 1.2 (ii) provides a further example of such a phenomena.

In GW theory, an analogue of Question 1.1 (i) has been one of the central themes. Since birational Calabi-Yau 3-folds or orbifolds should be derived equivalent (cf. [Br1], [BKR], [Kawlog]), Question 1.1 (i) for GW theory is related to the analytic continuation problem of quantum cohomologies discussed in [Yong], [BrGr], [CIT]. Also we expect that Question 1.1 (ii) is related to the modularity problem of partition functions of GW invariants, as the action of autoequivalences on the derived category should correspond to the monodromy action under the mirror symmetry. We refer to [OP], [MRS] for the works on the modularity in GW theory.

In recent years, we have seen progress toward an algebraic version of Conjecture 1.3 using derived algebraic geometry. By the work of Pantev-Ton-Vaquie-Vezzosi [PTVV], the stack is shown to be a derived stack with a -shifted symplectic structure. Using this fact, B. Bassat-Brav-Bussi-Joyce [BBBJ] showed that has Zariski locally an atlas which is written as a critical locus of a certain algebraic function. Still this is not enough to conclude Conjecture 1.3. However under the assumption that is Zariski locally written as a quotient stack of the form for some complex scheme , Bussi [Bussi, Theorem 4.3] showed a result which is very similar to Conjecture 1.3. Indeed her result implies relevant Behrend function identities for objects in , which are enough for our applications. At this moment, the author does not know how to eliminate the local quotient stack assumption, nor prove it in the situations we are interested in.

### 1.4. Ideas behind the proof of Theorem 1.2

The proof of Theorem 1.2 follows from wall-crossing argument in the space of weak stability conditions, as in the author’s previous papers [Tcurve1], [Tcurve2], [TodK3]. In order to explain the argument, we first recall Bayer-Macri’s description of the space of Bridgeland stability conditions on in [BaMa]. They showed that the double quotient stack of by the actions of and the additive group contains the parameter space of the mirror family of . The latter space has three special points: large volume limit, conifold point and orbifold point (cf. Figure 1). Near the large volume limit, the semistable objects consist of (essentially) Gieseker semistable sheaves on . At the orbifold point, the semistable objects consist of representations of the McKay quiver under derived McKay correspondence [BKR]. By taking a path connecting the orbifold point with the large volume limit, one can relate representations of the McKay quiver with semistable sheaves on by wall-crossing phenomena: there is a finite number of walls on the above path such that the set of semistable objects are constant on the interval, but jump at walls.

Let us return to our global situation. In the situation of Theorem 1.2, we define the following triangulated category

 DX/Y:=⟨OX,DbCoh≤1(X/Y)⟩tr.

Here is the category of coherent sheaves on which are at most one dimensional outside . Our strategy is to construct a path similar to Figure 1 in the space of weak stability conditions on

 σt∈StabΓ∙(DX/Y), t∈R≥0.

The above one parameter family is an analogue of the path in Figure 1, i.e. corresponds to the large volume limit, and corresponds to the orbifold point. We show that the rank one -stable objects for consist of objects of the form

 (2) OX(rD)⊗(OX→F)

for and a stable pair on . We also show that the rank one -stable objects consist of objects of the form

 (3) Φ(OY→F)

for a stable pair on . Then we can relate objects (2), (3) by wall-crossing phenomena. If we assume Conjecture 1.3, then Joyce-Song’s wall-crossing formula [JS] is applied in our setting. It relates stable pair invariants on with those on together with the invariants (1), giving Theorem 1.2 (i).

We now explain the idea of Theorem 1.2 (ii). It follows from a general principle explained in [TGep, Section 1]. In general, suppose that there is a stability condition on the derived category of a Calabi-Yau 3-fold, which has a symmetric property with respect to an autoequivalence in a certain sense. In [TGep], such a stability condition was called Gepner type with respect to . Let be the DT type invariant (if it exists) counting -semistable objects with numerical class . The Gepner type property of would yield

 (4) DTτ(v)=DTτ(Θ∗v).

On the other hand, one may relate both sides of (4) with classical DT invariants counting sheaves or curves by wall-crossing. Combined with the identity (4), one may obtain non-trivial constraints among classical DT invariants caused by .

In Figure 1, the orbifold point is known to be Gepner type with respect to . Since the weak stability condition on is an analogue of the orbifold point, one expects that the above general philosophy may be applied to obtain constraints among stable pair invariants on caused by . In our situation, the equivalence does not preserve , so is not Gepner type in a strict sense. However one can prove that takes -stable objects to similar stable objects in another triangulated category

 DLX/Y:=⟨L,DbCoh≤1(X/Y)⟩tr.

Namely there also exists a one parameter family of weak stability conditions on such that -stable objects and -stable objects coincide under the equivalence . We then apply the similar wall-crossing formula in from to for . It implies another description of stable pair invariants on in terms of those on and the invariants (1). By comparing it with the result of Theorem 1.2 (i), we obtain the constraints in Theorem 1.2 (ii).

### 1.5. Plan of the paper

In Section 2, we recall derived equivalences concerning Calabi-Yau 3-folds containing , and fix some notation. In Section LABEL:sec:Stable, we recall stable pair invariants, generalized DT invariants on local , and their properties. In Section LABEL:sec:weak, we construct a one parameter family of weak stability conditions on the triangulated category . In Section LABEL:sec:Comp, we describe the wall-crossing phenomena with respect to our weak stability conditions, and prove Theorem 1.2.

### 1.6. Acknowledgment

The author would like to thank Arend Bayer for explaining his announced work [BaDTPT]. 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 (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 stacks are defined over . For a -dimensional variety , we denote by the even part of the singular cohomologies of , and write its element as for . We sometimes abbreviate and just write , as , . For a triangulated category and a set of objects in , we denote by the triangulated closure, i.e. the smallest triangulated category of which contains . Also is the extension closure, i.e. the smallest extension closed subcategory in which contains . For a Deligne-Mumford stack , we denote by the abelian category of coherent sheaves on . For , we denote by the subcategory of objects satisfying . If , we write the subscript ’’ just as ’’, e.g. write as , etc. For , we denote by its -th cohomology.

## 2. Derived category of Calabi-Yau 3-folds containing P2

### 2.1. Geometry of Calabi-Yau 3-folds containing P2

Let be a smooth projective Calabi-Yau 3-fold, i.e.

 KX=0, H1(X,OX)=0.

We always assume that there is a closed embedding

 i:P2↪X

whose image we denote by . There exist several examples of such Calabi-Yau 3-folds, as follows:

###### Example 2.1.

Let be the hypersurface in given by

 X={y1(4∑i=1x4i)+y24∏i=1xi=0}.

Here are homogeneous coordinates of and are those of . Then is a smooth Calabi-Yau 3-fold which contains planes for each .

###### Example 2.2.

Let be a plane, a smooth hypersurface of degree seven, such that the divisor is a normal crossing divisor. Let be the double cover branched along . Then has -singularities along the pull-back of . Let

 X→Z

be the blow-up along the singular locus. Then is a smooth Calabi-Yau 3-fold, and the pull-back of contains as an irreducible component. See [Kapu, Section 4.2].

###### Example 2.3.

Let be the elliptic curve which admits an automorphism of order . The group acts on diagonally. Then the crepant resolution

 X→(E×E×E)/G

is a smooth Calabi-Yau 3-fold which contains 27 planes. See [Bea, Section 2].

Let be an ample divisor on and a line. We note that

 D⋅l=−3, D2=−3[l], D3=9.

The divisor is nef and big on . By the basepoint free theorem, some multiple of it gives a birational morphism

 (5) f:X→Y

which contacts a divisor to a point . It is well-known that3

 (6) ˆOY,p≅C[[x,y,z]]G.

Here acts on via weight . Since is a quotient singularity, we have the associated smooth Deligne-Mumford stack4

 (7) g:Y→Y

whose coarse moduli space is isomorphic to . The diagram

### Footnotes

1. In this paper, an orbifold means a smooth Deligne-Mumford stack.
2. The HL condition on a CY3 orbifold is equivalent to that the crepant resolution of its coarse moduli space has at most one dimensional fibers.
3. For example, the same argument of [Mori, (3.3.5)] shows the isomorphism (6).
4. We refer to [Kawlog, Definition 2.1] for the construction of . We note that satisfies the condition in [Kawlog, Definition 2.1] since is a quotient singularity.
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