Generating functions of stable pair invariants via wall-crossings in derived categories
aa r X i v : . [ m a t h . AG ] M a y Generating functions of stable pair invariants viawall-crossings in derived categories
Yukinobu Toda
Abstract
The notion of limit stability on Calabi-Yau 3-folds is introduced by the author to constructan approximation of Bridgeland-Douglas stability conditions at the large volume limit. Ithas also turned out that the wall-crossing phenomena of limit stable objects seem relevant tothe rationality conjecture of the generating functions of Pandharipande-Thomas invariants.In this article, we shall make it clear how wall-crossing formula of the counting invariants oflimit stable objects solves the above conjecture.
A theory of curve counting on Calabi-Yau 3-folds is interesting in both algebraic geometryand string theory. Now there are three such theories, called Gromov-Witten (GW) theory,Donaldson-Thomas (DT) theory, and Pandharipande-Thomas (PT) theory. Conjecturally thesetheories are equivalent in terms of generating functions, however we also need a conjecturalrationality property of those functions for DT-theory and PT-theory, to formulate that equiva-lence. The purpose of this article is to interpret the rationality conjecture for PT-theory fromthe viewpoint of wall-crossing phenomena in derived categories of coherent sheaves.
First of all, let us recall the conjectural GW-DT-PT correspondences on curve counting theories.Suppose that X is a smooth projective Calabi-Yau 3-fold over C , i.e. there is a nowhere vanishingholomorphic 3-form on X . For g ≥ β ∈ H ( X, Z ), the GW-invariant N g,β is defined bythe integration of the virtual class, N g,β = Z [ M g ( X,β )] vir ∈ Q , where M g ( X, β ) is the moduli stack of stable maps f : C → X with g ( C ) = g and f ∗ [ C ] = β .The GW-potential is given by the following generating function, Z GW = exp X g,β =0 N g,β λ g − v β . For n ∈ Z and β ∈ H ( X, Z ), let I n ( X, β ) be the Hilbert scheme of 1-dimensional subschemes Z ⊂ X satisfying [ Z ] = β, χ ( O Z ) = n. I n ( X, β ) is obtained by viewing it as a moduli space of ideal sheaves,and the
DT-invariant I n,β is defined by I n,β = Z [ I n ( X,β )] vir ∈ Z . The generating function of the reduced DT-theory is Z ′ DT = X n,β I n,β q n v β / X n I n, q n . The theory of stable pairs and their counting invariants are introduced and studied by Pandhari-pande and Thomas [19], [20], [21] to give a geometric interpretation of the reduced DT-theory.By definition, a stable pair is data (
F, s ), s : O X −→ F, where F is a pure one dimensional sheaf on X , and s is a morphism with a zero dimensionalcokernel. For β ∈ H ( X, Z ) and n ∈ Z , the moduli space of stable pairs ( F, s ) with[ F ] = β, χ ( F ) = n, is constructed in [19], denoted by P n ( X, β ). The obstruction theory on P n ( X, β ) is obtained byviewing stable pairs (
F, s ) as two term complexes, · · · −→ −→ O X s −→ F −→ −→ · · · . (1)The PT-invariant P n,β is defined by P n,β = Z [ P n ( X,β )] vir ∈ Z . The corresponding generating function is Z PT = X n,β P n,β q v v β . The functions Z GW , Z ′ DT and Z PT are conjecturally equal after suitable variable change. Inorder to state this, we need the following conjecture, called rationality conjecture . Conjecture 1.1. [18, Conjecture 2], [19, Conjecture 3.2]
For a fixed β , the generatingseries I β ( q ) = X n ∈ Z I n,β q n / X n ∈ Z I n, q n , P β ( q ) = X n ∈ Z P n,β q n , are Laurent expansions of rational functions of q , invariant under q ↔ /q . The above conjecture is solved for I β ( q ) when X is a toric local Calabi-Yau 3-fold [18], andfor P β ( q ) when β is an irreducible curve class [21]. Now we can state the conjectural GW-DT-PT-correspondences. Conjecture 1.2. [18, Conjecture 3], [19, Conjecture 3.3]
After the variable change q = − e iλ , we have Z GW = Z ′ DT = Z PT . The variable change q = − e iλ is well-defined by Conjecture 1.1. I ⊂ O X are objects in D b ( X ), where D b ( X ) is the bounded derivedcategory of coherent sheaves on X . We can also interpret stable pairs ( F, s ) as objects in D b ( X )by viewing them as two term complexes (1). As discussed in [19, Secction 3], the equality Z ′ DT = Z PT should be interpreted as a wall-crossing formula for counting invariants in thecategory D b ( X ). The purpose of this article is to show that Conjecture 1.1 is also interpreted asa wall-crossing formula in D b ( X ), using the method of limit stability [23] together with Joyce’sworks [9], [10], [11], [14]. The notion of limit stability on a Calabi-Yau 3-fold X is introduced in [23] to construct anapproximation of Bridgeland-Douglas stability conditions [4], [6], [7] on D b ( X ) at the largevolume limit. It is a certain stability condition on the category of perverse coherent sheaves A p ⊂ D b ( X ) , in the sense of Bezrukavnikov [3] and Kashiwara [15]. (See Definition 3.2.) An element σ ∈ A ( X ) C determines σ -limit (semi)stable objects in A p , where A ( X ) C is the complexified amplecone, A ( X ) C = { B + iω ∈ H ( X, C ) | ω is an ample class } . It has also turned out in [23] that the objects (1) appear as σ -limit stable objects for some σ ∈ A ( X ) C , thus studying stable pairs and limit stable objects are closely related. The objects E given by (1) satisfy(ch ( E ) , ch ( E ) , ch ( E ) , ch ( E )) = ( − , , β, n ) , det E = O X , (2)for some β and n . Under the above observation, we have constructed in [23] the moduli spaceof σ -limit stable objects E ∈ A p satisfying (2) as an algebraic space of finite type, denoted by L σn ( X, β ). Using that moduli space, the counting invariant of limit stable objects L n,β ( σ ) ∈ Z (3)is also defined in [23] as a weighted Euler characteristic with respect to Behrend’s constructiblefunction [2], and (3) coincides with the integration of the virtual class if L σn ( X, β ) is a projectivevariety. A particular choice of σ yields an equality L n,β ( σ ) = P n,β , however L n,β ( σ ) becomesdifferent from P n,β if we deform σ . As discussed in [23, Section 4], a transformation formula ofthe invariants L n,β ( σ ) under change of σ seems relevant to solving Conjecture 1.1 for PT-theory. In this article, we shall proceed the above idea further, using D. Joyce’s works [9], [10], [11],[14] on counting invariants of semistable objects on abelian categories and their wall-crossingformulas. We will make it clear how such a formula for counting invariants of objects in A p implies Conjecture 1.1 for PT-theory. Unfortunately we are unable to solve Conjecture 1.1 at thismoment, as Joyce’s theory is applied only for the motivic invariants (e.g. Euler characteristic)of the moduli spaces, so they do not involve virtual classes. On the other hand, the invariant P n,β coincides with the Euler characteristic of P n ( X, β ) (up to sign), P eun,β := e ( P n ( X, β )) ∈ Z , P n ( X, β ) is non-singular. In general P n,β is written as a weighted Euler characteristic withrespect to Behrend’s constructible function [2], so P n,β resembles P eun,β in this sense. So insteadof solving Conjecture 1.1, we shall show the motivic version of Conjecture 1.1, i.e. the rationalityof the generating series, P euβ ( q ) = X n ∈ Z P eun,β q n . The limit stability does not work well to combine Joyce’s works, so we will introduce the notionof µ σ -limit stability for σ ∈ A ( X ) C , which is a coarse version of σ -limit stability. Then we willintroduce the Joyce type invariants, (cf. Definition 4.1, Remark 4.2, ) L eun,β ∈ Q , N eun,β ∈ Q . Roughly speaking, L eun,β , (resp. N eun,β ) is the “Euler characteristic” of the moduli stack of µ iω -limit semistable objects E ∈ A p with det E = O X , (resp. one dimensional ω -Gieseker semistablesheaves F , ) satisfyingch( E ) = ( − , , β, n ) , (resp. ch( F ) = (0 , , β, n ) . )We will consider the generating series, L euβ ( q ) = X n ∈ Z L eun,β q n , N euβ ( q ) = X n ≥ nN eun,β q n . It will turn out that L euβ ( q ) is a polynomial of q ± , N euβ ( q ) is the Laurent expansion of a rationalfunction of q , and they are invariant under q ↔ /q . (cf. Lemma 4.5, Lemma 4.6.) Some-what surprisingly, Joyce’s wall-crossing formula yields the following equality of those generatingfunctions. Theorem 1.3. [Theorem 4.7]
We have the following equality of the generating series, X β P euβ ( q ) v β = X β L euβ ( q ) v β · exp X β N euβ ( q ) v β . (4)As a corollary, we have the following. Corollary 1.4. [Corollary 4.8]
The generating series P euβ ( q ) is the Laurent expansion of arational function of q , invariant under q ↔ /q . The series Z PT also should have a decomposition such as (4). In Problem 4.18 we willaddress a certain technical problem on the Ringel-Hall Lie algebra of A p , which enables us todecompose Z PT and solve Conjecture 1.1 for PT-theory. As a conclusion, we have obtained aconceptual understanding of the rationality conjecture and DT-PT correspondences in termsof wall-crossing phenomena in the derived category, and they have been reduced to showing arather technical problem, namely a compatibility of Ringel-Hall Lie algebra structure of A p withtaking virtual classes via Behrend’s constructible functions. The author thanks R. Thomas, R. Pandharipande for valuable comments, and D. Joyce for thecomment on Problem 4.18. This work is supported by World Premier International ResearchCenter InitiativeiWPI Initiative), MEXT, Japan.4 .5 Convention
All the varieties and schemes are defined over C . For a variety X , the category of coherentsheaves on X is denoted by Coh( X ). We say E ∈ Coh( X ) is d -dimensional if dim Supp( E ) = d . This section is devoted to review Joyce’s works [9], [10], [11], [14] on counting invariants ofsemistable objects on abelian categories. We discuss in a general framework rather than workingwith the category of perverse coherent sheaves A p , which we will introduce in the next section. We begin with a generality of (weak) stability conditions on abelian categories. Let A be a C -linear abelian category, and K ( A ) its Grothendieck group. We put the same assumption asin [14], i.e. Hom( E, F ), Ext ( E, F ) for any
E, F ∈ A are finite dimensional C -vector spaces, andcompositions Ext i ( E, F ) × Ext j ( F, G ) → Ext i + j ( E, G ) for i, j, i + j = 0 , A = mod A for a finite dimensional algebra A , or A = Coh( X ) for a projective variety X . In the first case, the group K ( A ) is finitelygenerated, but this is not true in the latter case. So instead we fix a quotient space, N ( A ) := K ( A ) / ≡ , for some equivalence relation ≡ such that a class [ E ] ∈ N ( A ) is non-zero for any 0 = E ∈ A .For instance if A = Coh( X ), an equivalence relation ≡ can be taken by E ≡ E ↔ χ ( E , F ) = χ ( E , F ) for any F ∈ A , (5)where χ ( E, F ) is defined by χ ( E, F ) = X i ∈ Z ( − i dim Ext i ( E, F ) . (6)Then N ( A ) is embedded into H ∗ ( X, Q ), and it is a finitely generated Z -module. The closedpositive cone and the positive cone of A are defined by C ( A ) := im( A → K ( A ) → N ( A )) ,C ( A ) := C ( A ) \ { } , respectively. For a subcategory B ⊂ A , we shall use the notation C ( B ) := im( B → C ( A )) ⊂ C ( A ), etc. For an object E ∈ A , its class is denoted by [ E ] ∈ C ( A ), or we omit [ ] if there is noconfusion. Let ( T, (cid:23) ) be a totally ordered set. Definition 2.1. A weak stability function is a map, Z : C ( A ) −→ T, such that if E, F, G ∈ C ( A ) satisfies E = F + G , we have either Z ( E ) (cid:22) Z ( F ) (cid:22) Z ( G ) , or Z ( E ) (cid:23) Z ( F ) (cid:23) Z ( G ) . stability function if, for E, F, G as above, we have either Z ( E ) ≺ Z ( F ) ≺ Z ( G ) , or Z ( E ) ≻ Z ( F ) ≻ Z ( G ) , or Z ( E ) = Z ( F ) = Z ( G ) . Given a weak stability function, we can define the set of (semi)stable objects.
Definition 2.2.
Let Z : C ( A ) → T be a weak stability function. An object E ∈ A is called Z-(semi)stable if for any nonzero subobject F ⊂ E , we have Z ( F ) ≺ Z ( E/F ) , (resp. Z ( F ) (cid:22) Z ( E/F ) . )The notion of (weak) stability conditions is defined as follows. Definition 2.3.
A (weak) stability function Z : C ( A ) → T is a (weak) stability condition if forany object E ∈ A , there is a filtration0 = E ⊂ E ⊂ · · · ⊂ E n = E, (7)such that each subquotient F i = E i /E i − is Z -semistable with Z ( F ) ≻ Z ( F ) ≻ · · · ≻ Z ( F n ) . It is easy to see that the filtration (7) is unique up to an isomorphism, if exists. The filtration(7) is called a
Harder-Narasimhan filtration . Here we give some examples.
Example 2.4. (i) For an abelian category A , let W : N ( A ) → C be a group homomorphismsuch that for any E ∈ A \ { } , we have W ( E ) ∈ H := { r exp( iπφ ) | < φ ≤ } . For instance if A = mod A for a finite dimensional C -algebra A , the positive cone C ( A ) isspanned by finite number of simple objects S , · · · , S n ∈ A , and such W is obtained by choosingthe image of [ S i ] ∈ C ( A ) for 1 ≤ i ≤ n under W . We set ( T, (cid:23) ) = ((0 , , ≥ ), and Z : C ( A ) ∋ E π Im log Z ( E ) ∈ T. Then Z is a stability condition on A . This is Bridgeland’s approach of stability conditions [4].(ii) Let X be a smooth projective surface and set A = Coh( X ). Let ω be an ample divisoron X . For E ∈ Coh( X ) we set µ ω ( E ) = ( c ( E ) · ω rk( E ) if E is not torsion. ∞ if E is torsion.Then the map C ( A ) ∋ E µ ω ( E ) ∈ Q ∪ {∞} is a weak stability condition on A , but not astability condition on A . Remark 2.5.
Here we mention that a theory of stability conditions on triangulated categoriesis developed by Bridgeland [4], motivated by M. Douglas’s Π-stability [6], [7]. For a triangulatedcategory D , Bridgeland’s stability condition consists of ( W, A ), where A ⊂ D is the heart of abounded t-structure on D , and W is a group homomorphism K ( A ) → C , as in Example 2.4 (i).Especially W determines a stability condition on the abelian category A . He then shows thatthe set of “good” stability conditions form a complex manifold Stab( D ). Although Bridgeland’stheory is quite powerful, we shall study in this paper more general notion of (weak) stabilityconditions, which is used in Joyce’s works. 6 .2 Ringel-Hall algebras In this subsection, we introduce the algebra H ( A ) associated to an abelian category A , whosedetails are seen in [10]. Let Z : C ( A ) → T be a weak stability function. At this moment, we putthe following assumption. Assumption 2.6. • A is noetherian and Z -artinian. • There is an Artin stack of locally finite type
Obj ( A ) , which parameterizes objects E ∈ A . • For v ∈ C ( A ) , let M v ( Z ) ⊂ Obj ( A ) be the substack of Z -semistable objects E ∈ A with [ E ] = v . Then M v ( Z ) is an open substack of Obj ( A ) , and it is of finite type. Here we say A is Z - artinian if there is no infinite sequence · · · ⊂ E n ⊂ E n − ⊂ · · · ⊂ E ⊂ E , such that E i +1 = E i and Z ( E i +1 ) (cid:23) Z ( E i /E i +1 ) for any i . The first condition of Assumption 2.6ensures the existence of Harder-Narasimhan filtrations, hence Z is a weak stability condition,by the same argument of [22, Theorem 2]. In order to state the second assumption, we need toknow about the notion algebraic families of objects and morphisms in A . This notion is obviousif A = Coh( X ) for a variety X , but in general we need some additional extra data, which isgiven in [9, Assumptions 7.1, 8.1]. For the introduction of Artin stacks, one can consult [16].For instance, Assumption 2.6 is satisfied when A = mod A for a finite dimensional C -algebra A ,and Z is given as in Example 2.4 (i).For a variety Y , recall that the Grothendieck ring of varieties over Y is defined by K (Var /Y ) = M ( X,ρ ) Z [( X, ρ )] / ∼ , where X is a variety with a morphism ρ : X → Y , and equivalence relations are given by[( X, ρ )] ∼ [( X † , ρ | X † )] + [( X \ X † , ρ | X \ X † )] , where X † is a closed subvariety of X . Taking the fiber products over Y , there is a naturalproduct on K (Var /Y ), [( X, ρ )] · [( X ′ , ρ ′ )] = [( X × Y X ′ , ρ ◦ p )] , (8)where p is the projection X × Y X ′ → X .In order to introduce H ( A ), let us introduce the notion of Grothendieck rings of Artin stacks. Definition 2.7. [13]
Let Y be an Artin stack of locally finite type over C . Define the Q -vectorspace K (St / Y ) to be K (St / Y ) := M ( X ,ρ ) Q [( X , ρ )] / ∼ , where ( X , ρ ) is a pair such that X is an Artin C -stack of finite type with affine geometricstabilizers, and ρ : X → Y is a 1-morphism. The relations ∼ are given by[( X , ρ )] ∼ [( X † , ρ | X † )] + [( X \ X † , ρ | X \X † )] , for closed substacks X † ⊂ X . 7gain taking the fiber products over Y gives a product · on K (St / Y ),[( X , ρ )] · [( X ′ , ρ ′ )] = [( X × Y X ′ , ρ ◦ p )] , (9)where p is the projection X × Y X ′ → X . Definition 2.8. [10]
Let A be an abelian category satisfying the second condition of Assump-tion 2.6. We define the Q -vector space H ( A ) to be H ( A ) := K (St / Obj ( A )) . The vector space H ( A ) is graded by v ∈ C ( A ), H ( A ) = M v ∈ C ( A ) H v ( A ) , H v ( A ) := K (St / Obj v ( A )) , where Obj v ( A ) is the stack of objects E ∈ A with [ E ] = v . There is an associative multiplication ∗ on H ( A ), based on Ringel-Hall algebras , which differs from the product (9). Let Ex ( A ) be themoduli stack of exact sequences 0 → E → E → E → A . It is shown in [9, Theorem 8.2]that Ex ( A ) is an Artin stack of locally finite type over C . We have the following 1-morphisms, p i : Ex ( A ) ∋ (0 → E → E → E → E i ∈ Obj ( A ) , for i = 1 , ,
3. Take f i = [( X i , ρ i )] ∈ H ( A ) for i = 1 ,
2. We have the following diagram,( p , p ) ∗ ( X × X ) u / / (cid:15) (cid:15) Ex ( A ) p / / ( p ,p ) (cid:15) (cid:15) Obj ( A ) X × X ρ ,ρ ) / / Obj ( A ) × Obj ( A ) . Here the left diagram is a Cartesian diagram.
Definition 2.9.
We define the ∗ -product f ∗ f by f ∗ f = [(( p , p ) ∗ ( X × X ) , p ◦ u )] . It is shown in [10, Theorem 5.2] that ∗ is associative and H ( A ) is a Q -algebra with identity[0 ֒ → Obj ( A )]. Remark 2.10.
Our algebra H ( A ) is denoted by SF( Obj ( A )) in Joyce’s paper [10], and anelement of SF( Obj ( A )) is called a stack function on Obj ( A ). There is another version of Ringel-Hall type algebra discussed in [10], defined as the set of constructible functions on Obj ( A ),denoted by CF( Obj ( A )) in [10]. Although most of the readers might be more familiar withconstructible functions than stack functions, we use the latter one since we want to apply [10,Theorem 6.12] which is formulated only for stack functions. δ v ( Z ) , ǫ v ( Z ) Let Z : C ( A ) → T be a weak stability condition, satisfying Assumption 2.6. For an Artinsubstack i : M ֒ → Obj ( A ), we write the element [( M , i )] ∈ H ( A ) as [ M ֒ → Obj ( A )].8 efinition 2.11. For v ∈ C ( A ), we define δ v ( Z ) , ǫ v ( Z ) ∈ H ( A ) to be δ v ( Z ) = [ M v ( Z ) ֒ → Obj ( A )] ∈ H v ( A ) ,ǫ v ( Z ) = X l ≥ , v i ∈ C ( A ) , v + ··· v l = v,Z ( v i )= Z ( v ) , ≤ i ≤ l ( − l − l δ v ( Z ) ∗ · · · ∗ δ v l ( Z ) ∈ H v ( A ) . (10)Under Assumption 2.6, the sum (10) is a finite sum, (see [11, Proposition 4.9],) hence ǫ v ( Z )is an element of H ( A ). In [11, Theorem 8.7], Joyce shows that ǫ v ( Z ) is an element of a certainLie subalgebra of H ( A ), called Ringel-Hall Lie algebra, ǫ v ( Z ) ∈ G ( A ) ⊂ H ( A ) . (11)The Lie algebra G ( A ) is denoted by SF indal ( Obj ( A )) in Joyce’s paper [10]. If we work over theHall-type algebra CF( Obj ( A )), (see Remark 2.10,) the corresponding Lie algebra CF ind ( Obj ( A ))is the set of constructible functions on Obj ( A ), supported on indecomposable objects. One mightexpect, as an analogue for H ( A ), that an element [( X , ρ )] is contained in G ( A ) if the image of ρ in Obj ( A ) is supported on indecomposable objects. However Joyce suggests that this definitionis not the best analogue, and he introduces the notion of “virtual indecomposable objects”, anddefines SF indal ( Obj ( A )) in [10, Definition 5.13] as the set of stack functions supported on virtualindecomposable objects. We omit the precise definition of G ( A ) here, as we will not use this.The Lie algebra G ( A ) also has the decomposition, G ( A ) = M v ∈ C ( A ) G v ( A ) , G v ( A ) := H v ( A ) ∩ G ( A ) , and ǫ v ( Z ) is an element of G v ( A ). The conceptual meaning of the definition of ǫ v ( Z ) is thatthey are “logarithms” of δ v ( Z ), i.e. for t ∈ T , we have formally X v ∈ Z − ( t ) ǫ v ( Z ) = log X v ∈ Z − ( t ) δ v ( Z ) . Also see [5] for more arguments on the elements ǫ v ( Z ). δ v ( Z ) , ǫ v ( Z ) The descriptions of the variations of the elements δ v ( Z ), ǫ v ( Z ) under change of Z are investigatedin [14]. Let us briefly recall the main idea of [14, Theorem 5.2] in this subsection. We firstintroduce the following definition. Definition 2.12.
Let
Z, Z ′ : C ( A ) → T be weak stability conditions and take v ∈ C ( A ). We say Z ′ dominates Z with respect to v if for v , v ∈ C ≤ v ( A ), Z ( v ) (cid:23) Z ( v ) implies Z ′ ( v ) (cid:23) Z ′ ( v ).Here C ≤ v ( A ) is defined by C ≤ v ( A ) = { v ′ ∈ C ( A ) | there is v ′′ ∈ C ( A ) with v ′ + v ′′ = v } . (12)The first step is to show the following theorem.9 heorem 2.13. [10, Theorem 5.11] For weak stability conditions
Z, Z ′ : C ( A ) → T satisfyingAssumption 2.6, suppose that Z ′ dominates Z with respect to v . Then we have δ v ( Z ′ ) = X l ≥ , v i ∈ C ( A ) , v + ··· + v l = v,Z ( v i ) ≻ Z ( v i − ) , Z ′ ( v i )= Z ′ ( v ) , ≤ i ≤ l δ v ( Z ) ∗ · · · ∗ δ v l ( Z ) . (13) The sum (13) may not be a finite sum, but it converges in the sense of [14, Definition 2.16]Proof.
We just explain the idea of the proof. For the full proof, see [14, Theorem 5.11]. For a Z ′ -semistable object E ∈ A with [ E ] = v , there is a Harder-Narasimhan filtration with respectto Z , i.e. there is a unique filtration0 = E ⊂ E ⊂ · · · ⊂ E l = E, (14)such that each F i = E i /E i − is Z -semistable with Z ( F i ) ≻ Z ( F i − ). Since Z ′ dominates Z withrespect to v , and each class [ F i ] ∈ C ( A ) is contained in C ≤ v ( A ), we have Z ′ ( F i ) (cid:23) Z ′ ( F i − ).Hence Z ′ -semistability of E implies Z ′ ( F i ) = Z ′ ( F i − ).Conversely for an object E ∈ A , suppose that there is a filtration (14) such that F i = E i /E i − is Z -semistable with Z ( F i ) ≻ Z ( F i − ) and Z ′ ( F i ) = Z ′ ( F i − ) for all i . Since Z ′ dominates Z with respect to v , the object F i is also Z ′ -semistable, hence E is Z ′ -semistable.As a consequence, an object E ∈ A with [ E ] = v is Z ′ -semistable if and only if there is aunique filtration (14) such that each F i = E i /E i − is Z -semistable and v i = [ F i ] ∈ C ( A ) for i = 1 , · · · , l satisfy v + · · · + v l = v, (15) Z ( v ) ≻ Z ( v ) ≻ · · · ≻ Z ( v l ) ,Z ′ ( v ) = Z ′ ( v ) = · · · = Z ′ ( v l ) . This observation is expressed as (13) in terms of the algebra H ( A ).We omit the definition of the convergence [14, Definition 2.16] here, as we will only treat thecases that the relevant sums have only finitely many terms. The next step is to invert (13), andgive the formula, δ v ( Z ) = X l ≥ , v i ∈ C ( A ) , v + ··· + v l = v, Z ′ ( v i )= Z ′ ( v ) ,Z ( v + ··· + v i ) ≻ Z ( v i +1 + ··· v l ) , ≤ i ≤ l δ v ( Z ′ ) ∗ · · · ∗ δ v l ( Z ′ ) . (16)The proof is provided in [14, Theorem 5.12]. The sum (16) may not converge in the senseof [14, Definition 2.16], but if we impose the assumption that the change from Z to Z ′ is locallyfinite , ( we omit the definition of the local finiteness, see [14, Definition 5.1],) then the sum (16)converges.Finally for two weak stability conditions Z , Z ′ , consider the following situation.( ♠ ) : there are weak stability conditions Z = Z , Z , · · · , Z m = Z ′ , W , · · · , W m − satisfyingAssumption 2.6, such that W i dominates Z i , Z i +1 w.r.t. v, and all changes from Z i to W i , W i − are locally finite.Then in principle one can express δ v ( Z ′ ) in terms of δ v ( Z ) in the algebra H ( A ), by applyingthe formulas (13), (16) successively. The transformation coefficients are determined purelycombinatory, and they are given as follows. 10 efinition 2.14. [14, Definition 4.2] Take v , · · · , v l ∈ C ( A ) and weak stability conditions Z, Z ′ : C ( A ) → T . Suppose that for each i = 1 , · · · , l −
1, we have either (17) or (18), Z ( v i ) (cid:22) Z ( v i +1 ) and Z ′ ( v + · · · + v i ) ≻ Z ′ ( v i +1 + · · · + v l ) , (17) Z ( v i ) ≻ Z ( v i +1 ) and Z ′ ( v + · · · + v i ) (cid:22) Z ′ ( v i +1 + · · · + v l ) . (18)Then define S ( { v , · · · , v l } , Z, Z ′ ) to be ( − r , where r is the number of i = 1 , · · · , l − S ( { v , · · · , v l } , Z, Z ′ ) = 0.We have the following formula. Theorem 2.15. [14, Theorem 5.2]
Under the situation ( ♠ ) , we have δ v ( Z ′ ) = X l ≥ , v i ∈ C ( A ) ,v + ··· + v l = v S ( { v , · · · , v l } , Z, Z ′ ) δ v ( Z ) ∗ · · · ∗ δ v l ( Z ) . (19) The sum (19) converges in the sense of [14, Definition 2.16].
Remark 2.16.
Our condition “ ∗ ′ dominates ∗ w.r.t. v ” in Definition 2.12 is weaker than Joyce’scondition “ ∗ ′ dominates ∗ ” given in [14, Definition 3.16], and [14, Theorem 5.2] is formulatedusing the latter condition. However if we want to know (19) for a fixed v ∈ C ( A ), it is enoughto assume “ ∗ ′ dominates ∗ w.r.t. v ” in ( ♠ ), since all the v i in the sum (19) are contained in C ≤ v ( A ).The relationship between ǫ v ( Z ′ ) and ǫ v ( Z ) is deduced from (10), (19), and inverting (10), δ v ( Z ) = X l ≥ , v i ∈ C ( A ) , v + ··· + v l = v,Z ( v i )= Z ( v ) , ≤ i ≤ l l ! ǫ v ( Z ) ∗ · · · ∗ ǫ v l ( Z ) . (20)The proof of (20) is provided in [11, Theorem 8.2]. The transformation coefficients are given asfollows. Definition 2.17. [14, Definition 4.4]
For v , · · · , v l ∈ C ( A ), we define U ( { v , · · · , v l } , Z, Z ′ ) ∈ Q to be U ( { v , · · · , v l } , Z, Z ′ ) = X ≤ m ′ ≤ m ≤ l X surjective ψ : { , ··· ,l }→{ , ··· ,m } , i ≤ j imply ψ ( i ) ≤ ψ ( j )surjective ξ : { , ··· ,m }→{ , ··· ,m ′ } , i ≤ j imply ξ ( i ) ≤ ξ ( j ) ,ψ and ξ satisfy ( ♦ ) m ′ Y a =1 S ( { w i } i ∈ ξ − ( a ) , Z, Z ′ ) · ( − m ′ m ′ · m Y b =1 | ψ − ( b ) | ! . (21)Here the condition ( ♦ ) is as follows.( ♦ ) : For 1 ≤ i, j ≤ l with ψ ( i ) = ψ ( j ) , we have Z ( v i ) = Z ( v j ) , and for 1 ≤ i, j ≤ m ′ , we have Z ′ ( X k ∈ ψ − ξ − ( i ) v k ) = Z ′ ( X k ∈ ψ − ξ − ( j ) v k ) . Also w i for 1 ≤ i ≤ m is defined as w i = X j ∈ ψ − ( i ) v j ∈ C ( A ) . heorem 2.18. [14, Theorem 5.2] In the situation ( ♠ ) , the following holds. ǫ v ( Z ′ ) = X l ≥ , v i ∈ C ( A ) ,v + ··· + v l = v U ( { v , · · · , v l } , Z, Z ′ ) ǫ v ( Z ) ∗ · · · ∗ ǫ v l ( Z ) . (22) The sum (22) converges in the sense of [14, Definition 2.16].
Remark 2.19.
It is possible to rewrite (22) by a Q -linear combination of multiple commutatorsof ǫ v i ( Z ) such as [[ · · · [[ ǫ v ( Z ) , ǫ v ( Z )] , ǫ v ( Z )] , · · · ] , ǫ v l ( Z )] , so (22) is an equality in G ( A ), rather than in H ( A ). The proof of this fact is given in [14,Theorem 5.4]. As a final step, we integrate the elements ǫ v ( Z ) ∈ G v ( A ) to give Q -valued invariants, andestablish the transformation formula of these invariants. Let us recall that a motivic invariantis a ring homomorphism Υ : K (Var / Spec C ) −→ Λ , where Λ is a Q -algebra and a ring structure on K (Var / Spec C ) is given by (8). In order tosimplify the arguments, we only consider the special case that Λ = Q ( t ) andΥ([ Y ]) = X i ( − i dim H i ( Y, C ) t i , where Y is a smooth projective variety. Since K (Var / Spec C ) is generated by [ Y ] for smoothprojective varieties Y , the above data uniquely determines Υ. In this situation, there is a uniqueextension of Υ, Υ ′ : K (St / Spec C ) −→ Q ( t ) , such that if G is a special algebraic group acting on Y , we have (cf. [13, Theorem 4.9])Υ ′ ([ Y /G ]) = Υ([ Y ]) / Υ([ G ]) . Here an algebraic group is special if every principle G -bundle is locally trivial in Zariski topology.In what follows, we assume that A satisfies the following condition.( ⋆ ) : there is an anti-symmetric biadditive-paring χ : N ( A ) × N ( A ) → Z such that for any E, F ∈ A , we have χ ( E, F ) = dim Hom(
E, F ) − dim Ext ( E, F )+dim Ext ( F, E ) − dim Hom( F, E ) . For instance if A = Coh( X ) for a smooth projective Calabi-Yau 3-fold X , the usual Euler pairing(6) descends to the pairing on N ( A ), which satisfies ( ⋆ ) by Serre duality. Using the pairing χ ,we can define the following Lie algebra. Definition 2.20.
For an abelian category A satisfying ( ⋆ ), we define the Lie algebra g ( A ) tobe the Q -vector space, g ( A ) := M v ∈N ( A ) Q c v , with its Lie-brackets given by [ c v , c v ′ ] = χ ( v, v ′ ) c v + v ′ .12et Π v be the composition,Π v : G v ( A ) ⊂ H ( A ) π ∗ −→ K (St / Spec C ) Υ ′ −→ Q ( t ) , where the map π ∗ sends [( X , ρ )] to [( X , π ◦ ρ )] and π : Obj ( A ) → Spec C is the structure mor-phism. It is shown in [14, Section 6.2] that for ǫ ∈ G v ( A ), the rational function Π v ( ǫ ) ∈ Q ( t )has a pole at t = 1 at most order one. Hence the following definition makes sense,Θ v ( ǫ ) = ( t − v ( ǫ ) | t =1 ∈ Q . (23) Definition 2.21.
We define the invariant J v ( Z ) ∈ Q by J v ( Z ) = Θ v ( ǫ v ( Z )) ∈ Q . Remark 2.22.
If all the Z -semistable objects E ∈ A with [ E ] = v are in fact Z -stable, andtheir moduli problem is represented by a scheme, then ǫ v ( Z ) is written as [ M v ( Z ) / G m ] for ascheme M v ( Z ). Here G m is acting on M v ( Z ) trivially. In this case J v ( Z ) equals to the Eulercharacteristic of M v ( Z ). Note that the factor Υ([ G m ]) = t − E ) ∼ = G m .We have the following theorem. Theorem 2.23. [10, Theorem 6.12]
The map,
Θ : G ( A ) = M v ∈ C ( A ) G v ( A ) ∋ { ǫ v } v X v ∈ C ( A ) Θ v ( ǫ v ) c v ∈ g ( A ) , (24) is a Lie algebra homomorphism. Since (22) is a relationship in the Lie algebra G ( A ), we can obtain the relationship between J v ( Z ′ ) and J v ( Z ) by applying Θ. The result is as follows. Theorem 2.24. [14, Theorem 6.28, Equation (130)]
In the situation of ( ♠ ) , assume thatthere are only finitely many terms in (22). Applying Θ to (22) yields the formula, J v ( Z ′ ) = X l ≥ , v i ∈ C ( A ) ,v + ··· + v l = v X Γ is a connected, simply connectedgraph with vertex { , ··· ,l } , i •→ j • implies i For objects E, F ∈ A Z ≺ t and a morphism f : E → F , it is called a strictmonomorphism if f is injective in A and Coker( f ) ∈ A Z ≺ t . Similarly f is called a strictepimorphism if f is surjective in A and Ker( f ) ∈ A Z ≺ t .For v ∈ C ( A Z ≺ t ), we set C ≤ v ( A Z ≺ t ) = { v ′ ∈ C ( A Z ≺ t ) | there is v ′′ ∈ C ( A Z ≺ t ) with v ′ + v ′′ = v } . (26)For Z, t, v as above, we put the following assumption instead of Assumption 2.6. Assumption 2.26. • The category A Z ≺ t is noetherian and artinian with respect to strictmonomorphisms. • There is an Artin stack of locally finite type Obj ( A ) , which parameterizes objects E ∈ A . • For any v ′ ∈ C ≤ v ( A Z ≺ t ) , the substack M v ′ ( Z ) ⊂ Obj ( A ) is an open substack, and it is offinite type. We also modify the dominant conditions. Definition 2.27. Let Z, Z ′ : C ( A ) → T be weak stability conditions. For t ∈ T and v ∈ C ( A Z ≺ t ), we say Z ′ dominates Z with respect to ( v, t ) if the following holds. • We have A Z (cid:23) t = A Z ′ (cid:23) t and A Z ≺ t = A Z ′ ≺ t . • For v , v ∈ C ≤ v ( A Z ≺ t ), if Z ( v ) (cid:23) Z ( v ) then Z ′ ( v ) (cid:23) Z ′ ( v ).For two weak stability conditions Z , Z ′ , we consider the following situation.( ♠ ′ ) : there are weak stability conditions Z = Z , Z , · · · , Z m = Z ′ , W , · · · , W m − such that W i dominates Z i , Z i +1 w.r.t. ( v, t ) , all changes from Z i to W i , W i − are locally finite,and Z i , W i satisfy Assumption 2.26 for ( v, t ) . We have the following generalization of Theorem 2.24. Theorem 2.28. For weak stability conditions Z , Z ′ on A , t ∈ T and v ∈ C ( A Z ≺ t ) , supposethat the condition ( ♠ ′ ) holds. Then the equation (22) with each v i ∈ C ( A Z ≺ t ) holds. If there areonly finitely many terms in (22) with v i ∈ C ( A Z ≺ t ) , then (25) holds, with each v i ∈ C ( A Z ≺ t ) .Proof. First suppose that Z ′ dominates Z with respect to ( v, t ), and check that (13) holdswith each v i ∈ C ( A Z ≺ t ). Let E ∈ A be Z ′ -semistable with [ E ] = v . As in the proof ofTheorem 2.13, we have a unique filtration (14). Let F i = E i /E i − , v i = [ F i ] ∈ C ( A ). Since v ∈ C ( A Z ≺ t ) = C ( A Z ′ ≺ t ), we have E ∈ A Z ′ ≺ t = A Z ≺ t . Hence we have Z ( v i ) ≺ t , and v i ∈ C ( A Z ≺ t ) follows.Conversely given a filtration (14), suppose that each F i is Z -semistable with v i ∈ C ( A Z ≺ t ).Then F i ∈ A Z ≺ t = A Z ′ ≺ t , hence F i is also Z ′ -semistable as Z ′ dominates Z w.r.t. ( v, t ).14s a summary, an object E ∈ A Z ≺ t is Z -semistable if and only if there is a filtration (14),satisfying (15) with each v i ∈ C ( A Z ≺ t ). Then the same proof of [14, Theorem 5.11] works andgives the formula (13) with each v i ∈ C ( A Z ≺ t ). Note that to state the formula (13), it is enoughto assume that M v ′ ( Z ) ⊂ Obj ( A ) is open and of finite type for any v ′ ∈ C ≤ v ( A Z ≺ t ).By the same idea, we can also show the formulas (16), (19), (20), (22) hold with each v i ∈ C ( A Z ≺ t ). We leave the readers to follow Joyce’s work and that the same proofs are appliedin this case. µ -limit stability In this section, we recall the notion of limit stability on a Calabi-Yau 3-fold X introduced in [23],and also introduce the notion of µ -limit stability. Below we always assume that X is a projectivecomplex 3-fold with a trivial canonical class, i.e. X is a Calabi-Yau 3-fold. We denote by D b ( X )the bounded derived category of Coh( X ), and K ( X ) the Grothendieck group of Coh( X ). The numerical Grothendieck group of Coh( X ) is given by N ( X ) = K ( X ) / ≡ , where the numerical equivalence relation ≡ is given by (5). Note that if A ⊂ D b ( X ) is the heartof a bounded t-structure on D b ( X ), then the group N ( A ) = K ( A ) / ≡ , where ≡ is (5) as above,coincides with N ( X ). So we always regard C ( A ) as a subset of N ( X ).Let us fix notation of the numerical classes of curves on X . An element β ∈ H ( X, Z ) iscalled an effective class if there is a one dimensional subscheme C ⊂ X such that β is thePoincar´e dual of the fundamental cycle of C . We set C ( X ), C ( X ) as C ( X ) := { β ∈ H ( X, Z ) | β is an effective class } ,C ( X ) := C ( X ) ∪ { } . The limit stability introduced in [23] is a stability condition on the category of perverse coherentsheaves A p in the sense of Bezrukavnikov [3] and Kashiwara [15], and it is also one of thepolynomial stability conditions introduced by Bayer [1] independently. In order to introduce A p , let us define the subcategories (Coh ≤ ( X ) , Coh ≥ ( X )) of Coh( X ), as follows. Definition 3.1. We define the pair of subcategories (Coh ≤ ( X ) , Coh ≥ ( X )) to beCoh ≤ ( X ) := { E ∈ Coh( X ) | dim Supp( E ) ≤ } , Coh ≥ ( X ) := { E ∈ Coh( X ) | Hom(Coh ≤ ( X ) , E ) = 0 } . The category A p is defined as follows. Definition 3.2. We define the subcategory A p ⊂ D b ( X ) to be A p := (cid:26) E ∈ D b ( X ) : H − ( E ) ∈ Coh ≥ ( X ) , H ( E ) ∈ Coh ≤ ( X ) , and H i ( E ) = 0 for i = − , . (cid:27) . It is easy to see that (Coh ≤ ( X ) , Coh ≥ ( X )) determines a torsion theory on Coh( X ), and A p is the corresponding tilting. (cf. [23, Definition 2.9, Lemma 2.10].) Therefore A p is the heartof a bounded t-structure on D b ( X ), in particular A p is an abelian category.15ext recall that the complexified ample cone is defined by A ( X ) C = { B + iω ∈ H ( X, C ) | ω is an ample class } . Given σ = B + iω ∈ A ( X ) C , one can define the map Z σ : K ( X ) → C , Z σ : K ( X ) ∋ E Z e − ( B + iω ) ch( E ) p td X ∈ C . Explicitly we have Z σ ( E ) = (cid:18) − v B ( E ) + 12 ω v B ( E ) (cid:19) + (cid:18) ωv B ( E ) − ω v B ( E ) (cid:19) i, (27)where v Bi ∈ H i ( X, R ) for 0 ≤ i ≤ e − B ch( E ) p td X = ( v B ( E ) , v B ( E ) , v B ( E ) , v B ( E )) ∈ H even ( X, R ) . For σ m = B + imω for m ∈ R , one can show the following: for each non-zero object E ∈ A p ,one has Z σ m ( E ) ∈ (cid:26) r exp( iπφ ) : r > , < φ < (cid:27) , (28)for m ≫ 0. (See [23, Lemma 2.20].) Hence the phase of E is well-defined for m ≫ φ σ m ( E ) = 1 π Im log Z σ m ( E ) ∈ (cid:18) , (cid:19) . Definition 3.3. [14, Definition 2.21] An object E ∈ A p is σ - limit (semi)stable if for anynon-zero subobject F ⊂ E in A p , we have φ σ m ( F ) < φ σ m ( E ) , (resp. φ σ m ( F ) ≤ φ σ m ( E ) , ) (29)for m ≫ T be the one valuablefunction field R ( m ). We define the total order on R ( m ) to be f ( m ) (cid:23) g ( m ) def ↔ f ( m ) ≥ g ( m ) for m ≫ . Note that we have Im e − πi/ Z σ m ( E ) > , (30)for m ≫ m , thus thefollowing map is well-defined. Z Tσ : C ( A p ) ∋ E Re e − πi/ Z σ m ( E )Im e − πi/ Z σ m ( E ) ∈ T. Lemma 3.4. The map Z Tσ is a stability condition on A p , and an object E ∈ A p is Z Tσ -(semi)stable if and only if E is σ -limit (semi)stable.Proof. Since (30) holds, it is obvious that for v , v ∈ C ( A p ), the inequality φ σ m ( v ) ≤ φ σ m ( v )holds for m ≫ Z Tσ ( v ) (cid:22) Z Tσ ( v ) holds in T . Hence Z Tσ is a stability function,and the latter statement also follows. The existence of Harder-Narasimhan filtrations for limitstability is proved in [23, Theorem 2.28], hence Z Tσ is a stability condition.16 .2 µ -limit stability In this subsection, we introduce a weak stability condition on A p , which we call µ -limit stability .Let us introduce the following notation. Definition 3.5. Let f = P di =0 a i ( σ ) m i be a polynomial such that each coefficient a i ( σ ) is a R -valued function on A ( X ) C , and a d ( σ ) 0. We define f † = a d ( σ ) m d .By the formula (27), Re Z σ m ( E ) and Im Z σ m ( E ) are written as polynomials of m whosecoefficients are R -valued functions on A ( X ) C . Thus the following makes sense, Z † σ m ( E ) := (Re Z σ m ( E )) † + (Im Z σ m ( E )) † i. The same argument of [23, Lemma 2.20] shows that Z † σ m ( E ) ∈ (cid:26) r exp( iπφ ) : r > , < φ < (cid:27) , for m ≫ 0. Hence as before we can define the following map, Z µ σ : C ( A p ) ∋ E Re e − πi/ Z † σ m ( E )Im e − πi/ Z † σ m ( E ) ∈ T. We have the following lemma. Lemma 3.6. The map Z µ σ is a weak stability function on A p .Proof. Since the operation f f † is just taking the initial term of the polynomials, we havethe implication Z Tσ ( E ) ≻ Z Tσ ( F ) ⇒ Z µ σ ( E ) (cid:23) Z µ σ ( F ) , (31)for E, F ∈ C ( A p ). Hence Z µ σ is a weak stability function.It is easy to see that for 0 = E ∈ A p , one has Z µ σ ( E ) → − m → ∞ . To seethat Z µ σ is a weak stability condition, we introduce a pair of subcategories in A p . (cf. [23,subsection 2.3].) Definition 3.7. We define ( A p , A p / ) to be A p = { E ∈ A p | dim Supp H ( E ) = 0 and H − ( E ) is a torsion sheaf } , A p / = { E ∈ A p | Hom( F, E ) = 0 for any F ∈ A p } . It is shown in [23, Lemma 2.16] that ( A p , A p / ) determines a torsion theory on A p . Weshall use the same notation of “strict monomorphisms”, “strict epimorphisms” in A pi as inDefinition 2.25, i.e. we replace A Z ≺ t by A pi to define them. We have the following. Lemma 3.8. An object E ∈ A p is Z µ σ -semistable with Z µ σ ( E ) → − (resp. ) for m → ∞ ifand only if E ∈ A p / , (resp. A p ,) and for any strict monomorphism = F ֒ → E in A p / , (resp. A p ,) one has Z µ σ ( F ) (cid:22) Z µ σ ( E/F ) .Proof. The proof is same as in [23, Lemma 2.26] for the limit stability, by noting that Z µ σ ( E ) → − , (resp. Z µ σ ( E ) → , )for E ∈ A p / , (resp. E ∈ A p ,) and m → ∞ . 17e also have the following. Lemma 3.9. The weak stability function Z µ σ is a weak stability condition.Proof. The existence of Harder-Narasimhan filtrations follows from the same argument for thelimit stability. The proof of [23, Theorem 2.28] also works in this case, by noting Lemma 3.8.We say E ∈ A p is µ σ -limit (semi)stable if it is (semi)stable in Z µ σ -weak stability. To explainthis notation, let us recall that the (usual) µ -stability is defined by cutting off the lower degreeterms of the reduced Hilbert polynomials. In this sense, our µ σ -limit stability resembles to µ -stability, as we also cut off the lower degree terms of the polynomial Z σ m ( ∗ ). By (31), we havethe following implications, µ σ -limit stable ⇒ σ -limit stable ⇒ σ -limit semistable ⇒ µ σ -limit semistable . Remark 3.10. For 0 = E ∈ A p , let φ † σ m ( E ) be φ † σ m ( E ) = 1 π Im log Z † σ m ( E ) ∈ (cid:18) , (cid:19) . Then obviously an object E ∈ A p is µ σ -limit (semi)stable if and only if for any non-zero subobject F ⊂ E , one has φ † σ m ( F ) < φ † σ m ( E/F ) , (resp. φ † σ m ( F ) ≤ φ † σ m ( E/F ) , )for m ≫ Remark 3.11. Let us take F ∈ Coh ≤ ( X ). In this case we have Z † σ m ( F ) = Z σ m ( F ), andCoh ≤ ( X ) ⊂ A p is closed under taking subobjects and quotients. So F is µ σ -limit (semi)stableif and only it is σ -limit (semi)stable. On the other hand for σ = B + iω ∈ A ( X ) C , let µ σ ( F ) ∈ Q be µ σ ( F ) = ch ( F ) − B ch ( F ) ω ch ( F ) ∈ Q . (32)As in [23, Example 2.24 (ii)], the object F is σ -limit (semi)stable if and only if for any subsheaf0 = F ′ ⊂ F we have µ σ ( F ′ ) < µ σ ( F ), (resp. µ σ ( F ′ ) ≤ µ σ ( F ),) i.e. F is a ( B, ω )-twisted(semi)stable sheaf. If B = kω for k ∈ R , then F is σ -limit (semi)stable if and only if F is a ω -Gieseker (semi)stable sheaf, and this notion does not depend on k . Remark 3.12. By Lemma 3.8, it is obvious that A pZ µσ ≺ = A p / , A pZ µσ (cid:23) = A p , (33)in the notation of subsection 2.6. Since A p / and A p are of finite length with respect to strictmonomorphisms and strict epimorphisms, (cf. [23, Lemma 2.19],) the categories A pZ µσ ≺ and A pZ µσ (cid:23) also have such properties. 18 .3 Characterization of µ -limit semistable objects Take v ∈ C ( A p ) satisfying(ch ( v ) , ch ( v ) , ch ( v ) , ch ( v )) = ( − , , β, n ) , (34)for some β ∈ H ( X, Q ) and n ∈ H ( X, Q ) ∼ = Q . In this subsection we give a characterization of µ -limit semistable objects E ∈ A p of numerical type v . i.e. [ E ] = v ∈ C ( A p ). Note that suchobjects satisfy Z µ σ ( E ) → − m → ∞ , hence we have E ∈ A p / , v ∈ C ( A p / ) . Also if such E exists, the classes β , n are contained in C ( X ), H ( X, Z ) ∼ = Z respectively, by [23,Remark 3.3]. We have the following proposition, whose corresponding result for limit stabilityis seen in [23, Section 3]. Proposition 3.13. Take σ = B + iω ∈ A ( X ) C . For an object E ∈ A p / of numerical type v , itis µ σ -limit (semi)stable if and only if the following conditions hold.(a) For any pure one dimensional sheaf G = 0 which admits a strict epimorphism E ։ G in A p / , one has µ σ ( G ) > − Bω ω . (cid:18) resp. µ σ ( G ) ≥ − Bω ω . (cid:19) (35) (b) For any pure one dimensional sheaf F = 0 which admits a strict monomorphism F ֒ → E in A p / , one has µ σ ( F ) < − Bω ω . (cid:18) resp. µ σ ( F ) ≤ − Bω ω . (cid:19) (36) Proof. By Lemma 3.8 and applying the same argument of [23, Lemma 3.4], an object E ∈ A p / of numerical type v is Z µ σ -limit semistable if and only if(a’) For any pure one dimensional sheaf G = 0 which admits an exact sequence 0 → F → E → G → A p / , one has Z µ σ ( F ) (cid:22) Z µ σ ( G ).(b’) For any pure one dimensional sheaf F = 0 which admits an exact sequence 0 → F → E → G → A p / , one has Z µ σ ( F ) (cid:22) Z µ σ ( G ).By Lemma 3.14 below, the conditions (a’), (b’) are equivalent to (a), (b) respectively. Lemma 3.14. Take v , v ∈ C ( A p / ) with ch( v ) = ( − , , β , n ) , ch( v ) = (0 , , β , n ) , and β = 0 . Then Z µ σ ( v ) (cid:22) Z µ σ ( v ) , (resp. Z µ σ ( v ) (cid:23) Z µ σ ( v ) ,) if and only if µ σ ( v ) ≥ − Bω ω , ( resp. µ σ ( v ) ≤ − Bω ω . ) (37) If σ = kω + iω for k ∈ R , (37) is equivalent to k ≥ − µ iω ( v ) , ( resp. k ≤ − µ iω ( v ) . ) (38)19 roof. An easy computation shows, Z † σ m ( v ) = 12 m ω B + 16 m ω i,Z † σ m ( v ) = − n + Bβ + mωβ i. (39)Since ωβ > 0, we have Z µ σ ( v ) (cid:22) Z µ σ ( v ) ⇔ − m ω B/ m ω / (cid:22) µ σ ( v ) m ⇔ µ σ ( v ) ≥ − ω Bω . If σ = kω + iω with k ∈ R , then µ σ ( v ) = µ iω ( v ) − k and − Bω /ω = − k . Hence (37) isequivalent to (38). µ -limit semistable objects In this subsection, we establish a moduli theory of µ -limit semistable objects. In [23, Theo-rem 1.1], a moduli theory of σ -limit stable objects is studied, and the resulting moduli space isan algebraic subspace of Inaba’s algebraic space [8]. Since we need a moduli theory not onlyfor stable objects but also semistable objects, the resulting space should not be an algebraicspace in general, but an Artin stack. So instead of working with Inaba’s algebraic space, we useLieblich’s algebraic stack of objects E ∈ D b ( X ), satisfying the condition,Ext iX ( E, E ) = 0 , for all i < , (40)which we denote by M . More precisely, the stack M is defined by the 2-functor, M : (Sch / C ) −→ (groupoid) , which takes a C -scheme S to the groupoid M ( S ), whose objects consist of relatively perfectobject E ∈ D b ( X × S ) such that E s satisfies (40) for any closed point s ∈ S . Lieblich [17] showsthe following. Theorem 3.15. ( [17]) The 2-functor M is an Artin stack of locally finite type. For v ∈ C ( A p / ) as in (34), we consider a moduli problem of µ -limit semistable objects ofnumerical type v ′ ∈ C ≤ v ( A p / ), where C ≤ v ( A p / ) is given in (26). First we show the following. Lemma 3.16. For any v ′ ∈ C ≤ v ( A p / ) , we have one of the following. • There is β ′ ∈ C ( X ) and n ′ ∈ Z such that ch( v ′ ) = ( − , , β ′ , n ′ ) . • We have ch( v ′ ) = ( − , , , . • There is β ′ ∈ C ( X ) and n ′ ∈ Z such that ch( v ′ ) = (0 , , β ′ , n ′ ) .Proof. For v ′ ∈ C ≤ v ( A p / ), let v ′′ = v − v ′ ∈ C ( A p / ). Since ch ( v ) = − 1, we have ch ( v ′ ) = 0or ch ( v ′ ) = − 1. Suppose that ch ( v ′ ) = 0 and take E ∈ A p / with [ E ] = v ′ . Then H − ( E )must be torsion, hence H − ( E ) = 0 since E ∈ A p / . Therefore E is a non-zero one dimensionalsheaf, thus ch( v ′ ) = (0 , , β ′ , n ′ ) for some β ′ ∈ C ( X ) and n ′ ∈ Z .20n the latter case, we have ch ( v ′′ ) = 0 thus ch ( v ′′ ) = (0 , , β ′′ , n ′′ ) for some β ′′ ∈ C ( X )and n ′′ ∈ Z . Therefore ch ( v ′ ) = ( − , , β ′ , n ′ ) for some β ′ ∈ C ( X ) and n ′ ∈ Z . If β ′ = 0, then H − ( E ) is a line bundle and H ( E ) is a zero dimensional sheaf, by [23, Lemma 3.2]. Thus E is isomorphic to a direct sum of H − ( E )[1] and H ( E ), which contradicts to E ∈ A p / unless n ′ = dim H ( E ) = 0.For σ = B + iω ∈ A ( X ) C and v ′ ∈ C ≤ v ( A p / ), let us consider the following (abstract) stacks, M v ′ ( Z µ σ ) ⊂ Obj ( A p ) ⊂ M , where Obj ( A p ) is the stack of objects E ∈ A p , and M v ′ ( Z µ σ ) is the stack of µ σ -limit semistableobjects E ∈ A p / of numerical type v ′ . We have the following. Proposition 3.17. The substacks Obj ( A p ) and M v ′ ( Z µ σ ) are open substacks of M , hence theyare Artin stacks of locally finite type. Moreover M v ′ ( Z µ σ ) is of finite type.Proof. The openness of Obj ( A p ) ⊂ M follows from [23, Lemma 3.14]. Let us take v ′ ∈ C ≤ v ( A p / ). Suppose first that ch( v ′ ) = (0 , , β ′ , n ′ ) for β ′ ∈ C ( X ) and n ′ ∈ Z . Then any µ σ -limit semistable object of numerical type v ′ is a ( B, ω )-twisted semistable sheaf. (cf. Re-mark 3.11.) Then it is well-known that M v ′ ( Z µ σ ) is open in M , and it is of finite type. (cf. [24,Proposition 3.9].)Next suppose that ch( v ′ ) = ( − , , β ′ , n ′ ). In this case the claim for M v ′ ( Z µ σ ) follows fromthe straightforward adaptation of the argument of [23, Section 3]. In fact using Lemma 3.13, wecan show the boundedness of µ σ -limit semistable objects of numerical type v ′ , and destabilizingobjects in a family of objects in A p / , along with the same arguments of [23, Proposition 3.13,Lemma 3.15]. Then the same proof of [23, Theorem 3.20] works to show the openness of M v ′ ( Z µ σ ) ⊂ M . The boundedness of relevant µ σ -limit semistable objects implies that M v ′ ( Z µ σ )is of finite type. µ -limit semistable objects The notion of stable pairs and their counting invariants are introduced by Pandharipande andThomas [19] to interpret the reduced Donaldson-Thomas theory geometrically. In [23, Sec-tion 4], the relationship between PT-invariants and counting invariants of limit stable objectsare discussed. In this subsection we state the similar result for µ -limit semistable objects. Sincethe proofs are straightforward adaptation of the arguments in [23, Section 4], we again leave thereaders to check the detail. First let us recall the definition of stable pairs. Definition 3.18. A stable pair on a Calabi-Yau 3-fold X is data ( F, s ), where F is a pure onedimensional sheaf on X , and s is a morphism s : O X −→ F, whose cokernel is a zero dimensional sheaf.To simplify the notation, we also include the pair ( F = 0 , s = 0) in the definition of stablepairs. For a stable pair ( F, s ), we have the associated two term complex, I • = ( O X s −→ F ) ∈ D b ( X ) , (41)21here F is located in degree zero. Note that the object I • satisfies I • ∈ A p / , det I • = O X , ch( I • ) = ( − , , β, n ) , for β = ch ( F ), n = ch ( F ). By abuse of notation, we also call an object (41) as a stable pair.In [19], the moduli space of stable pairs is constructed as a projective variety, and denoted by P n ( X, β ), P n ( X, β ) := { ( F, s ) | ( F, s ) is a stable pair with (ch ( F ) , ch ( F )) = ( β, n ) } . Let Obj ( A p ) be the closed fiber at the point [ O X ] ∈ Pic( X ) of the following morphism,det : Obj ( A p ) ∋ E det E ∈ Pic( X ) . Definition 3.19. For β ∈ C ( X ), n ∈ Z , and σ ∈ A ( X ) C , define L µ σ n ( X, β ) to be L µ σ n ( X, β ) := M v ( Z µ σ ) ∩ Obj ( A p ) , (42)where v ∈ C ( A p / ) satisfies ch( E ) = ( − , , β, n ).Note that L µ σ n ( X, β ) is the moduli stack of µ σ -limit semistable objects E ∈ A p with det E = O X and [ E ] = v . We shall compare L µ σ n ( X, β ) and P n ( X, β ), when σ is written as σ = kω + iω with k ∈ R . For β ∈ C ( X ), we set C ≤ β ( X ) := { β ′ ∈ C ( X ) | β − β ′ ∈ C ( X ) } , (43) C ≤ β ( X ) := C ≤ β ( X ) \ { } . Definition 3.20. For β ∈ C ( X ), we define m ( β ) as follows. If β = 0, we set m ( β ) = 0.Otherwise m ( β ) is m ( β ) := min { ch ( O C ) | C ⊂ X satisfies dim C = 1 , [ C ] ∈ C ≤ β ( X ) } . It is well-known that C ≤ β ( X ) is a finite set and m ( β ) > −∞ , whose proofs are seen in [23,Lemma 3.9, Lemma 3.10]. Thus Definition 3.20 makes sense. For β ∈ C ( X ) and n ∈ Z , wedefine µ n,β ∈ Q to be µ n,β := max (cid:26) n − m ( β − β ′ ) ωβ ′ : β ′ ∈ C ≤ β ( X ) (cid:27) . (44)The following is µ -stability version of [23, Theorem 4.7]. Theorem 3.21. Let σ = kω + iω for k ∈ R . We have L µ σ n ( X, β ) ∼ = [ P n ( X, β ) / G m ] , if k < − µ n,β / , (45) L µ σ n ( X, β ) ∼ = [ P − n ( X, β ) / G m ] , if k > µ − n,β / . (46) Here G m is acting on P ± n ( X, β ) trivially. roof. The same proof of [23, Theorem 4.7] shows that, if k < − µ n,β / 2, then E ∈ A p / is µ σ -limit semistable if and only if E is isomorphic to a stable pair (41). Note that [23, Lemma 4.6]is crucial in [23, Theorem 4.7], and in our case Proposition 3.13 and (38) are applied insteadof [23, Lemma 4.6]. Thus the C -valued points of L µ σ n ( X, β ) and P n ( X, β ) are identified.The existence of a universal stable pair on X × P n ( X, β ) (cf. [19, Section 2]) yields a 1-morphism P n ( X, β ) → L µ σ n ( X, β ), which descends to[ P n ( X, β ) / G m ] −→ L µ σ n ( X, β ) . (47)Since any E ∈ P n ( X, β ) is τ -limit stable for some τ by [23, Theorem 4.7], we have Hom( E, E ) = C and Aut( E ) = G m . Therefore (47) is an equivalence of groupoids on C -valued points. As provedin [19, Theorem 2.7], the stack [ P n ( X, β ) / G m ] is considered as an open substack of Obj ( A p ).By Proposition 3.17, L µ σ n ( X, β ) is also open in Obj ( A p ), hence (47) gives an isomorphism ofArtin stacks. The isomorphism (46) is also similarly proved. In this section, we combine the arguments in the previous sections to show the rationality ofthe generating functions of stable pair invariants. As in the previous section, X is a projectiveCalabi-Yau 3-fold, A p ⊂ D b ( X ) is the heart of a perverse t-structure on D b ( X ). µ -limit stable objects In this subsection, we construct counting invariants of µ -limit semistable objects. Take v ∈ C ( A p / ) which satisfies (34) with β ∈ C ( X ) and n ∈ Z . As in subsection 2.3, there is aRingel-Hall Lie-algebra G ( A p ) ⊂ H ( A p ) and the elements, ǫ v ′ ( Z µ σ ) ∈ G v ′ ( A p ) ⊂ H ( A p ) , for any v ′ ∈ C ≤ v ( A p / ) and σ ∈ A ( X ) C . Here we have used Theorem 3.17 which ensures theexistence of H ( A p ) and ǫ v ′ ( Z µ σ ). We also use the following map on H ( A p ) to construct thecounting invariants, Ξ : G ( A p ) ∋ f f · [ Obj ( A p ) ֒ → Obj ( A p )] ∈ G ( A p ) . (48)The product · is given by (9). In the following, we use the notation of subsection 2.5. Definition 4.1. For β ∈ C ( X ) and n ∈ Z , we define P eun,β ∈ Z , L eun,β ( σ ) ∈ Q and N eun,β ( σ ) ∈ Q to be P eun,β := e ( P n ( X, β )) ,L eun,β ( σ ) := Θ v Ξ ǫ v ( Z µ σ ) , where ch( v ) = ( − , , β, n ) ,N eun,β ( σ ) := Θ v ′ ǫ v ′ ( Z µ σ ) = J v ′ ( Z µ σ ) , where ch( v ′ ) = (0 , , β, n ) . Here e ( ∗ ) is the topological Euler characteristic.For simplicity we set L eun,β := L n,β ( iω ) , N eun,β := N n,β ( iω ) . The subscript ∗ eu means “Euler characteristic ” of the moduli spaces. emark 4.2. Suppose that σ = kω + iω for k ∈ R . Then we have N eun,β ( σ ) = N eun,β , by noting Remark 3.11. Also if k < − µ n,β / 2, then Theorem 3.21 and Remark 2.22 imply L eun,β ( σ ) = P eun,β . Let us recall that for a fixed β , the moduli space P n ( X, β ) is empty for a sufficiently negative n . (See [19].) So we can take N ( β ) ∈ Z such that P eun,β = 0 for n < N ( β ) . (49)In particular the series P euβ ( q ) is a Laurent polynomial of q . µ -limit semistable objects In this subsection, we study the generating functions of the invariants given in Definition 4.1.Below we fix an ample divisor ω on X and only consider the case σ = kω + iω for k ∈ R . For σ = kω + iω , we set σ ∨ = − kω + iω . We have the following symmetry for the invariants L eun,β ( σ )and N eun,β . Lemma 4.3. (i) We have the equalities, L eun,β ( σ ) = L eu − n,β ( σ ∨ ) , N eun,β = N eu − n,β . (50) (ii) For d : = ω · β , we have N eun + d,β = N eun,β for any n ∈ Z .Proof. (i) Let D : D b ( X ) → D b ( X ) op be the dualizing functor, D ( E ) = R H om ( E, O X [2]) . The functor D induces an isomorphism of rings, D : H ( A p ) ∼ −→ H ( D ( A p )) op . (51)On the other hand, the functor D preserves the subcategory A p / ⊂ D b ( X ) by [23, Lemma 2.18].Moreover the same argument of [23, Lemma 2.27] shows that an object E ∈ A p / is µ σ -limitsemistable if and only if D ( E ) ∈ A p / is µ σ ∨ -limit semistable. Hence the map (51) takes δ v ′ ( Z µ σ )to δ v ′∨ ( Z µ σ ∨ ) for any v ′ ∈ C ≤ v ( A p / ). Here if v ′ is given by ch( v ′ ) = ( r, , β ′ , n ′ ) for r = 0 or − v ′ ∨ is given by ch( v ′ ∨ ) = ( r, , β ′ , − n ′ ). Hence (50) follows by the definitions of L eun,β ( σ )and N eun,β .(ii) Let us take an ample line bundle L ∈ Pic( X ) with c ( L ) = ω . The equivalence ⊗L : A p →A p induces an isomorphism of algebras, ⊗L : H ( A p ) −→ H ( A p ) . (52)On the other hand, it is easy to see that an object E ∈ A p / is µ σ -limit semistable if andonly if E ⊗ L is µ σ -limit semistable. Thus the map (52) takes δ v ′ ( Z µ σ ) to δ v ′′ ( Z µ σ ), wherech( v ′ ) = (0 , , β, n ) and ch( v ′′ ) = (0 , , β, n + d ). Hence we can conclude N eun + d,β = N eun,β .24ext we show the following finiteness result. Lemma 4.4. For a fixed β ∈ C ( X ) and σ = kω + iω , the set { n ∈ Z | L eun,β ( σ ) = 0 } (53) is a finite set.Proof. If β = 0, then L eun,β ( σ ) = 0 unless n = 0 by Lemma 3.16. Suppose that β = 0, and take aninteger N ( β ) as in (49). Assume that L eun,β ( σ ) = 0 for n < N ( β ). Then at least the moduli stack L σn ( X, β ) is non-empty, hence we must have k ≥ − µ n,β by Theorem 3.21. By the definition of µ n,β , there is β ′ ∈ C ≤ β ( X ) such that k ≥ − n − m ( β − β ′ )2 ωβ ′ . Hence we have either n ≥ − k ( ωβ ′ ) + m ( β − β ′ ) , or n ≥ N ( β ) . Thus the set (53) is bounded below. Since L eun,β ( σ ) = L eu − n,β ( σ ∨ ) by Lemma 4.3, the set (53) isalso bounded above. Lemma 4.5. For a fixed β ∈ C ( X ) , the generating series L euβ ( q ) = X n ∈ Z L eun,β q n (54) is a polynomial of q ± , hence a rational function of q , invariant under q ↔ /q .Proof. By Lemma 4.4, the series L euβ ( q ) is a polynomial of q ± . Since L eun,β = L eun,β ( iω ) = L eu − n,β ( iω ∨ ) = L eu − n,β ( iω ) = L eu − n,β , by Lemma 4.3, the polynomial L euβ ( q ) is invariant under q ↔ /q . Lemma 4.6. The generating series N euβ ( q ) = X n ≥ nN eun,β q n , is the Laurent expansion of a rational function of q , invariant under q ↔ /q .Proof. Let d ∈ Z be as in Lemma 4.3. Applying Lemma 4.3, we have N euβ ( q )= d − X j =0 X m ≥ ( dm + j ) N euj,β q dm + j = 12 d − X j =1 X m ≥ n ( dm + j ) N euj,β q dm + j + ( dm + d − j ) N eud − j,β q dm + d − j o + X m ≥ dmN eu ,β q dm = d − X j =1 N euj,β X m ≥ n ( dm + j ) q dm + j + ( dm + d − j ) q dm + d − j o + N eu ,β X m ≥ dmq dm 25e can calculate as X m ≥ n ( dm + j ) q dm + j + ( dm + d − j ) q dm + d − j o = ( d − j )( q j + d + q d − j ) + j ( q j + q d − j )(1 − q d ) , (55)and X m ≥ dmq dm = dq d (1 − q d ) . (56)Then the assertion follows since (55), (56) are rational functions of q , invariant under q ↔ /q .For a fixed β ∈ C ( X ), we consider the following generating series, P euβ ( q ) = X n ∈ Z P eun,β q n . Now we state our main theorem in this paper. Theorem 4.7. We have the following equality of the generating series, X β P euβ ( q ) v β = X β L euβ ( q ) v β · exp X β N euβ ( q ) v β . (57)Combining Theorem 4.7 with Lemma 4.5 and Lemma 4.6, we obtain the following. Corollary 4.8. The generating series P euβ ( q ) is the Laurent expansion of a rational function of q , invariant under q ↔ /q . The proof of Theorem 4.7 will be given in subsection 4.5 below. L eun,β ( σ ) In this subsection, we investigate the transformation formula of our invariants L eun,β ( σ ) underchange of σ = kω + iω . For β ∈ C ( X ), we set S ( β ) ⊂ R as S ( β ) := (cid:26) m ωβ ′ | β ′ ∈ C ≤ β ( X ) , m ∈ Z (cid:27) ⊂ R . Note that S ( β ) is a discrete subset in R because C ≤ β ( X ) is a finite set. For k ∈ S ( β ), let C ± ⊂ R \ S ( β ) be the connected components such that C − ⊂ R 0. We have the following. Lemma 4.9. Take v ∈ C ( A p / ) as in (34) and ∈ T = R ( m ) . The weak stability condition Z µ σ dominates Z µ σ ± with respect to ( v, .Proof. Take v , v ∈ C ≤ v ( A p / ), and suppose that Z µ σ ± ( v ) (cid:22) Z µ σ ± ( v ). We want to show that Z µ σ ( v ) (cid:22) Z µ σ ( v ) . (58)By Lemma 3.16, we have ch( v i ) = ( r i , , β i , n i ) with r i = 0 or − 1. If r = r = − 1, it is easy tosee that Z µ σ ( v ) = Z µ σ ( v ) for any σ . If r = r = 0, (58) follows easily from (39). If r = r ,(58) follows from Lemma 3.14. Then the assertion follows by noting Remark 3.12.26 emark 4.10. Lemma 4.9 is not true for the limit stability. This is the reason why we use µ -limit stability rather than the limit stability.For simplicity, we fix k ∈ R < and write Z = Z µ σ for σ = kω + iω with k < , Z ′ = Z µ iω . (59)Applying the results in the previous sections, we obtain the following proposition. Proposition 4.11. For v ∈ C ( A p / ) as in (34), and Z, Z ′ as in (59) w.r.t. ( v, , we have thefollowing. ǫ v ( Z ′ ) = X l ≥ , v i ∈ C ( A p / ) ,v + ··· + v l = v U ( { v , · · · , v l } , Z, Z ′ ) ǫ v ( Z ) ∗ · · · ∗ ǫ v l ( Z ) . (60) The sum (60) has only finitely many non-zero terms.Proof. By Proposition 3.17 and Remark 3.12, Z, Z ′ satisfy Assumption 2.26 w.r.t. ( v, ♠ ′ ) is also satisfied with respect to ( v, { v i } li =1 ⊂ C ( A p / ) satisfy the following by Lemma 3.16.there is a unique 1 ≤ e ≤ l such that ch( v i ) = (0 , , β i , n i ) for i = e (61)with β i ∈ C ( X ) , n i ∈ Z , and ch( v e ) = ( − , , β e , n e ) with β e ∈ C ( X ) , n e ∈ Z . Let us see that (60) is a finite sum. For simplicity we write as µ i := µ iω ( v i ) = n i β i ω ∈ Q , if v i ∈ C ( A p / ) satisfies ch( v i ) = (0 , , β i , n i ). Lemma 4.12. Take v , · · · , v l ∈ C ( A p / ) such that ch( v i ) = (0 , , β i , n i ) for all i with β i ∈ C ( X ) and n i ∈ Z . For Z , Z ′ as in (59), we have S ( { v , · · · , v l } , Z, Z ′ ) = (cid:26) , n = 1 , , n ≥ . Proof. Note that we have Z ( v i ) (cid:22) Z ( v j ) ⇔ µ i ≤ µ j ⇔ Z ′ ( v i ) (cid:22) Z ′ ( v j ) . (62)Then one can check that the same proof of [9, Theorem 4.5] is applied. Lemma 4.13. Take v , · · · , v l ∈ C ( A p / ) as in (61). For Z , Z ′ as in (59), S ( { v , · · · , v l } , Z, Z ′ ) is non-zero only if < µ ≤ µ ≤ · · · ≤ µ e − ≤ − k > µ e +1 > µ e +2 > · · · > µ l ≥ . (63)Moreover in this case, we have S ( { v , · · · , v l } , Z, Z ′ ) = ( − e − .27 roof. Note that for v i , v j with i, j = e , the same implications (62) hold. Also for i = e , we have Z ( v i ) (cid:22) Z ( v e ) ⇔ µ i ≤ − k,Z ′ ( v i ) (cid:22) Z ′ ( v e ) ⇔ µ i ≤ . Suppose that S ( { v , · · · , v l } , Z, Z ′ ) = 0. We say 1 < i < l is of type A (resp. B) if the followingholds, Z ( v i − ) ≻ Z ( v i ) (cid:22) Z ( v i +1 ) , (resp. Z ( v i − ) (cid:22) Z ( v i ) ≻ Z ( v i +1 ) . )If 1 < i ≤ e − Z ′ ( v + · · · + v i − ) (cid:22) Z ′ ( v i + · · · + v l ) ,Z ′ ( v + · · · + v i ) ≻ Z ′ ( v i +1 + · · · + v l ) . Hence we have µ iω ( v + · · · + v i − ) ≤ ,µ iω ( v + · · · + v i − + v i ) > , which implies µ i > 0. Similarly if i is of type B , we have µ i < ≤ i ≤ e − i . We assume i is of type A , hence µ i > 0. We have Z ( v ) ≻ · · · ≻ Z ( v i − ) ≻ Z ( v i ) (cid:22) Z ( v i +1 ) · · · , thus µ > · · · > µ i > Z ′ ( v ) (cid:22) Z ′ ( v + · · · + v l ), thus µ ≤ 0. This is a contradiction, so there is no 1 ≤ i ≤ e − ≤ i ≤ e − B .By the above argument, one of (64) or (65) holds. Z ( v ) ≻ Z ( v ) ≻ · · · ≻ Z ( v e − ) ≻ Z ( v e ) , (64) Z ( v ) (cid:22) Z ( v ) (cid:22) · · · (cid:22) Z ( v e − ) (cid:22) Z ( v e ) . (65)Assume by a contradiction that (64) holds. Then Z ′ ( v ) (cid:22) Z ′ ( v + · · · + v l ), thus (64) implies0 ≥ µ > µ > · · · > µ e − > − k. (66)The inequality (66) does not occur since we took k < 0. Therefore we must have (65).A similar argument for v e +1 , · · · , v l shows Z ( v ) (cid:22) · · · (cid:22) Z ( v e ) ≻ Z ( v e +1 ) ≻ · · · ≻ Z ( v l ) . (67)Obviously (67) together with S ( { v , · · · , v l } , Z, Z ′ ) = 0 imply (63). Proposition 4.14. (i) Take v , · · · , v l ∈ C ( A p / ) as in (61), and let Z , Z ′ be as in (59). Then U ( { v , · · · , v l } , Z, Z ′ ) is non-zero only if ≤ µ ≤ µ ≤ · · · ≤ µ e − ≤ − k ≥ µ e +1 ≥ · · · ≥ µ l ≥ . (68)28 ii) In the same situation of (i), suppose that U ( { v , · · · , v l } , Z, Z ′ ) Q i = e n i is non-zero. Thenwe have U ( { v , · · · , v l } , Z, Z ′ ) = X ≤ m ≤ l X surjective ψ : { , ··· ,l }→{ , ··· ,m } ,i ≤ j implies ψ ( i ) ≤ ψ ( j ) , and satisfies ( ) ( − ψ ( e ) − m Y b =1 | ψ − ( b ) | ! . (69) Here ψ satisfies the following. For i, j < e with ψ ( i ) = ψ ( j ) , we have µ i = µ j , if ψ ( i ) = ψ ( e ) then µ i = − k, (70)and for e < i, j, we have ψ ( i ) = ψ ( j ) if and only if µ i = µ j . (iii) The formula (60) is a finite sum.Proof. (i) Suppose that U ( { v , · · · , v l } , Z, Z ′ ) = 0 and take ψ : { , · · · , l } → { , · · · , m } , ξ : { , · · · , m } → { , · · · , m ′ } , (71)as in (21). Let us set c = ξψ ( e ) ∈ { , · · · , m ′ } and take a ∈ { , · · · , m ′ } with a = c . ByLemma 4.12, the set ξ − ( a ) consists of one element, say b ∈ { , · · · , m } . Then by the definitionof U ( { v , · · · , v l } , Z, Z ′ ), we have µ i = µ j for i, j ∈ ψ − ( b ) and Z ′ ( X i ∈ ψ − ( b ) v i ) = Z ′ ( X i ∈ ψ − ξ − ( c ) v i ) . (72)The condition (72) implies µ iω ( P i ∈ ψ − ( b ) v i ) = 0, hence µ i = 0 for any i ∈ ψ − ( b ), i.e. µ i = 0for any i / ∈ ψ − ξ − ( c ). By Lemma 4.13, we must have (68).(ii) Suppose that U ( { v , · · · , v l } , Z, Z ′ ) Y i = e n i = 0 , (73)and take ψ : { , · · · , l } → { , · · · , m } , ξ : { , · · · , m } → { , · · · , m ′ } as in (i). Then the proofof (i) shows that (73) is non-zero only if m ′ = 1. Then (69) follows from the definition of U ( { v , · · · , v l } , Z, Z ′ ) and Lemma 4.13.(iii) Since v , · · · , v l satisfy (61), the number l is bounded, and there is only finite number ofpossibilities for β i = ch ( v i ). Hence we may fix l and β , · · · , β l . Then the values n i = ch ( v i )have only finite number of possibilities by (68).Now we have the wall-crossing formula of the invariants L eun,β ( σ ). Proposition 4.15. For σ = kω + iω with k < , β ∈ C ( X ) and n ∈ Z , we have the followingformula, L eun,β = X l ≥ , ≤ e ≤ l, β i ∈ C ( X ) for i = e, β e ∈ C ( X ) , n i ∈ Z ,β + ··· + β l = β, n + ··· + n l = n, µ i = n i /β i ω satisfy0 <µ ≤ µ ≤···≤ µ e − ≤− k ≥ µ e +1 ≥···≥ µ l > X ≤ m ≤ l, surjective ψ : { , ··· ,l }→{ , ··· ,m } ,i ≤ j implies ψ ( i ) ≤ ψ ( j ) , and satisfies ( ) (cid:18) − (cid:19) l − m Y b =1 ( − ψ ( e ) − e | ψ − ( b ) | ! Y i = e n i N eun i ,β i L eun e ,β e ( σ ) . (74)29 roof. Let v ∈ C ( A p / ) be as in (34), and Z , Z ′ be as in (59). Applying Ξ given in (48) to (60),we obtainΞ ǫ v ( Z ′ ) = X l ≥ , ≤ e ≤ l X v i ∈ C ( A p / ) , v + ··· + v l = v, ch ( v i )=0 for i = e, ch ( v e )= − U ( { v , · · · , v l } , Z, Z ′ ) ǫ v ( Z ) ∗ · · · ∗ Ξ ǫ v e ( Z ) ∗ · · · ∗ ǫ v l ( Z ) . (75)Note that for i = e , the element ǫ v i ( Z ) ∈ H ( A p ) is supported on Obj ( A p ) ⊂ Obj ( A p ), henceΞ( ǫ v i ( Z ) ∗ ǫ ) = ǫ v i ( Z ) ∗ Ξ( ǫ ) follows for any ǫ ∈ H ( A p ). Thus (75) follows from (60). Also wenote that (75) is a finite sum by Proposition 4.14 (iii). Hence applying Θ given in (24) and usingthe same argument of Theorem 2.24, we obtain L eun,β = X l ≥ , v i ∈ C ( A p / ) ,v + ··· + v l = v X Γ is a connected, simply connectedgraph with vertex { , ··· ,l } , i •→ j • implies i 12 ( n − N ( β ′ )) , k < − µ n,β ′ for any β ′ ∈ C ≤ β ( X ) . (77)In this particular choice of k , we have the following formula. Proposition 4.16. If k satisfies (77), then (74) implies the following. L eun,β = X l ≥ , ≤ e ≤ l, ≤ t ≤ e − , ≤ s ≤ l − e, m For a fixed l , we have X ≤ l ≤ m, surjective ψ : { , ··· ,l }→{ , ··· ,m } ,i ≤ j implies ψ ( i ) ≤ ψ ( j ) m Y b =1 ( − l − m | ψ − ( b ) | ! = 1 l ! . (81) Proof. The proof is elementary, and this is a special case of [14, Proposition 4.9].31 .5 Proof of Theorem 4.7 We finally give a proof of Theorem 4.7. Proof. For a fixed data l ≥ 1, 1 ≤ e ≤ l , β i ∈ C ( X ) ( i = e ) and β e ∈ C ( X ), we set F e ( β , · · · , β l ) = X n ∈ Z X ≤ t ≤ e − , ≤ s ≤ l − e, m 1, 1 ≤ e ≤ l , κ : I → C ( X ), κ : I → C ( X ) and β e ∈ C ( X ), and consider the last sum of (83). If we alsofix bijections λ ′ : { , · · · , e − } → I and λ ′ : { e + 1 , · · · , l } → I , then the choices of λ , λ in(83) correspond to the elements of the symmetric groups γ ∈ S e − , γ ′ ∈ S l − e respectively. Letus rewrite β i = κ λ ′ ( i ) for 1 ≤ i ≤ e − β i = κ λ ′ ( i ) for e + 1 ≤ i ≤ l . Then we have X λ : { , ··· ,e − } ∼ → I ,λ : { e +1 , ··· ,l } ∼ → I F e ( κ λ (1) , · · · , κ λ ( e − , β e , κ λ ( e + 1) , · · · , κ λ ( l ))= X γ ∈ S e − γ ′ ∈ S l − e F e ( β γ (1) , · · · , β γ ( e − , β e , β γ ′ ( e +1) , · · · , β γ ′ ( l ) )= X ≤ t ≤ e − , ≤ s ≤ l − e, m Does there exist a map Θ ′ : G ( A p ) −→ g ( A p ) , such that the following conditions hold? • For v ∈ C ( A p ) , suppose that M v ( Z µ σ ) is written as [ M/ G m ] for a scheme M . Then Θ ′ ( ǫ v ( Z µ σ )) = X n ∈ Z n Θ([[ χ − M ( n ) / G m ] ֒ → Obj ( A p )]) . • For v , v ∈ C ( A p ) , we have [Θ ′ ( ǫ v ( Z )) , Θ ′ ( ǫ v ( Z ))] = ( − χ ( v ,v ) Θ ′ [ ǫ v ( Z ) , ǫ v ( Z )] . (88)There should be sign change in (88), because χ M = ( − dim M on a smooth variety M . Weare unable to solve Problem 4.18 at this moment, but the techniques given in this paper yieldthe following. Theorem 4.19. Suppose that Problem 4.18 is true. Then Conjecture 1.1 is true for PT-theory.Proof. It is enough to work over the invariants, defined by Θ ′ . As a modification of Definition 4.1,let us define L n,β ( σ ), N n,β ( σ ) to be L n,β ( σ ) := Θ ′ Ξ ǫ v ( Z µ σ ) , where ch( v ) = ( − , , β, n ) ,N n,β ( σ ) := Θ ′ ǫ v ( Z µ σ ) , where ch( v ) = (0 , , β, n ) . Then (88) yields a similar wall-crossing formula for L n,β ( σ ), and L n,β ( σ ) = ( − dim Pic( X ) P n,β , for σ = kω + iω with k < − µ n,β / 2. Therefore the same proof of Theorem 4.7 works, and we havethe similar expansion of the generating series Z PT as in (57). Then Conjecture 1.1 for P β ( q )follows as a corollary. The formulation of Problem 4.18 is taught to the author by D. Joyce. eferences [1] A. Bayer. Polynomial Bridgeland stability conditions and the large volume limit. preprint .math.AG/0712.1083.[2] K. Behrend. Donaldson-Thomas invariants via microlocal geometry. Ann. of Math (toappear) . math.AG/0507523.[3] R. Bezrukavnikov. Perverse coherent sheaves (after Deligne). preprint . math.AG/0005152.[4] T. Bridgeland. Stability conditions on triangulated categories. Ann. of Math , Vol. 166, pp.317–345, 2007.[5] T. Bridgeland and V. Toledano Laredo. Stability conditions and Stokes factors. preprint .math.AG/0801.3974.[6] M. Douglas. D-branes, categories and N = 1 supersymmetry. J. Math. Phys. , Vol. 42, pp.2818–2843, 2001.[7] M. Douglas. Dirichlet branes, homological mirror symmetry, and stability. Proceedings ofthe 1998 ICM , pp. 395–408, 2002. math.AG/0207021.[8] M. Inaba. Toward a definition of moduli of complexes of coherent sheaves on a projectivescheme. J. Math. Kyoto Univ. , Vol. 42-2, pp. 317–329, 2002.[9] D. Joyce. Configurations in abelian categories I. Basic properties and moduli stack. Ad-vances in Math , Vol. 203, pp. 194–255, 2006.[10] D. Joyce. Configurations in abelian categories II. Ringel-Hall algebras. Advances in Math ,Vol. 210, pp. 635–706, 2007.[11] D. Joyce. Configurations in abelian categories III. Stability conditions and identities. Ad-vances in Math , Vol. 215, pp. 153–219, 2007.[12] D. Joyce. Holomorphic generating functions for invariants counting coherent sheaves onCalabi-Yau 3-folds. Geometry and Topology , Vol. 11, pp. 667–725, 2007.[13] D. Joyce. Motivic invariants of Artin stacks and ‘stack functions’. Quarterly Journal ofMathematics , Vol. 58, p. 2007, 2007.[14] D. Joyce. Configurations in abelian categories IV. Invariants and changing stability condi-tions. Advances in Math , Vol. 217, pp. 125–204, 2008.[15] M. Kashiwara. t -structures on the derived categories of holonomic D -modules and cohereht O -modules. Mosc. Math. J. , Vol. 981, pp. 847–868, 2004.[16] G. Laumon and L. Moret-Bailly. Champs alg´ebriques , Vol. 39 of Ergebnisse der Mathematikund ihrer Grenzgebiete . Springer Verlag, Berlin, 2000.[17] M. Lieblich. Moduli of complexes on a proper morphism. J. Algebraic Geom , Vol. 15, pp.175–206, 2006.[18] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov-Witten theory andDonaldson-Thomas theory. I. Compositio. Math , Vol. 142, pp. 1263–1285, 2006.3519] R. Pandharipande and R. P. Thomas. Curve counting via stable pairs in the derivedcategory. preprint . math.AG/0707.2348.[20] R. Pandharipande and R. P. Thomas. The 3-fold vertex via stable pairs. preprint .math.AG/0709.3823.[21] R. Pandharipande and R. P. Thomas. Stable pairs and BPS invariants. preprint .math.AG/0711.3899.[22] A. Rudakov. Stability for an Abelian Category. Journal of Algebra , Vol. 197, pp. 231–235,1997.[23] Y. Toda. Limit stable objects on Calabi-Yau 3-folds. preprint . math.AG/0803.2356.[24] Y. Toda. Birational Calabi-Yau 3-folds and BPS state counting. Communications inNumber Theory and Physics , Vol. 2, pp. 63–112, 2008.Yukinobu TodaInstitute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo,Kashiwano-ha 5-1-5, Kashiwa City, Chiba 277-8582, Japan