Weighted Euler characteristic of the moduli space of higher rank Joyce Song pairs

Transcription

1 European Journal of Mathematics (06) :66 75 DOI 0.007/s RESEARCH ARTICLE Weighted Euler characteristic of the moduli space of higher rank Joyce Song pairs Artan Sheshmani, Received: 8 June 05 / Accepted: 8 March 06 / Published online: May 06 Springer International Publishing AG 06 Abstract The invariants of rank Joyce Song semistable pairs over a Calabi Yau threefold were computed in Sheshmani (Illinois J Math 59():55 8, 06), using the wall-crossing formula of Joyce and Song (A Theory of Generalized Donaldson Thomas Invariants. Memoirs of American Mathematical Society, American Mathematical Society, Providence, 0), and Kontsevich and Soibelman (Stability structures, motivic Donaldson Thomas invariants and cluster transformations, arxiv:08.45, 008). Such wallcrossing computations often depend on the combinatorial properties of certain elements of a Hall-algebra [these are the stack functions defined by Joyce (Adv Math 0():65 706, 007). These combinatorial computations become immediately complicated and hard to carry out, when studying higher rank stable pairs with rank >. The main purpose of this article is to introduce an independent approach to computation of rank stable pair invariants, without applying the wallcrossing formula and rather by stratifying their corresponding moduli space and directly computing the weighted Euler characteristic of the strata. This approach may similarly be used to avoid complex combinatorial wallcrossing calculations in rank > cases. Keywords Generalized Donaldson Thomas invariants Joyce Song stable pairs Stack functions Ringel Hall algebra Weighted Euler characteristic Mathematics Subject Classification 4J 4N5 5D7 4F05 5D45 B Artan Sheshmani artan@mit.edu; artan.sheshmani@gmail.com Department of Mathematics, Massachusetts Institute of Technology (MIT), Building, 77 Massachusetts Avenue, Cambridge, MA , USA Department of Mathematics, Ohio State University, West 8 Avenue, Columbus, OH 40, USA

2 66 A. Sheshmani Introduction The Donaldson Thomas theory (DT in short) of a Calabi Yau threefold X is defined in [,8 via integration against the virtual fundamental class of the moduli space of ideal sheaves. In [4,5, Pandharipande and Thomas introduced objects given by pairs (F, s) where F is a pure sheaf with one dimensional support together with a fixed Hilbert polynomial and s H 0 (X, F) is given as a section of F. The authors computed the invariants of stable pairs, using deformation theory and virtual fundamental classes. Following their work, Joyce and Song defined a similar notion of a (twisted) stable pair, given by a sheaf F and section map s: O( n) F where n 0 was chosen to be a sufficiently large integer so that the cohomology vanishing condition H (F(n)) = 0 is satisfied. These stable pairs were equipped with a stability condition rather different than the one used in [4. Definition. (Joyce Song pair stability) Given a coherent sheaf F, letp F denote the reduced Hilbert polynomial of F with respect to the ample line bundle O X (). A pair φ: O( n) F is called stable if the following conditions are satisfied: p F p F for all proper subsheaves F of F such that F = 0. If φ factors through F (F a proper subsheaf of F), then p F < p F. In this article we refer to this stability as ˆτ-stability. For more on Joyce Song stability look at [, Definition.. One advantage in defining ˆτ-stability, is that it enables one to compute the generalized Donaldson Thomas invariants with respect to the invariants of ˆτ-stable pairs. The generalized Donaldson Thomas invariants could not be calculated using the machinery developed by Thomas in [8, since they were given by invariants of semistable sheaves (not just the stable ones!). After work of Joyce and Song an interesting question was whether one is able to study and compute the invariants of objects composed of a sheaf F and multiple sections given by the morphism s s r : O r ( n) F for r > (we will later denote these by rank r stable pairs for short). In [6, the author introduced the notion of highly frozen triples (same as rank r stable pairs), and used the virtual localization technique introduced by Graber Pandharipande [5 to compute their invariants over local Calabi Yau threefolds (such as local P ). The objects studied in [6 (because of the stability condition chosen) were reminiscent of the higher rank analog of (a twisted version of) the Pandharipande Thomas (PT in short) stable pairs [4. In [7, the author studied the same higher rank objects, but equipped with the ˆτ-stability condition, and computed their invariants using the wallcrossing technique. In this article we would like to introduce a direct method of calculation of such invariants, which involves first stratifying the moduli space of higher rank semistable pairs into disjoint components, where each stratum contains the stable rank pairs and then computing the weighted Euler characteristic of the moduli space of higher rank pairs with respect to the Euler characteristic of the rank strata. In doing so, we need to first define an auxiliary category B p (which was originally introduced by Joyce Song [, Section.). The objects in B p are defined similar to the higher

3 Weighted Euler characteristic of the moduli space rank Joyce Song pairs and they are classified basing on their numerical class (β, r). Here, β denotes the Chern character of F and r denotes the number of sections of F being considered in the construction. The definition of the category B p allows one to define weak stability conditions on B p (look at Definition.6). As we have shown in [7, Theorem 5., the moduli stack of weak semistable objects (we denote this by τ-semistable) in B p is closely related to the parameterizing moduli stack of higher rank ˆτ-semistable pairs, which enables us to obtain the following identity: stp ( ˆτ) = ( )r B ss p (X,β,r, τ). () N β,r The left-hand side of () stands for invariants of ˆτ-semistable pairs and the right-hand side stands for invariants of τ-semistable objects in B p which are, roughly speaking, defined as the weighted Euler characteristic of their corresponding moduli stack. Therefore, using this identity, we aim at calculating the right-hand side of (), using the stratification method mentioned above. We show in this article that the result of our calculation agrees with the results obtained in [7. In particular, we restrict our computations to the rank pairs (r = ), and very explicitly calculate their invariants in some examples. As we will see below, even though the computation of such invariants requires a detailed study of the strata involved in the moduli space, the advantage of the strategy used in here is that it is much more geometric and it avoids complicated combinatorics involved in the method of wallcrossing. Moreover, we suspect that the methods introduced in this article may be used to prove the integrality conjectures for the partition functions of the higher rank Joyce Song invariants in special cases. Toda in [9 has used a similar stratification technique and provided an evidence of such integrality property, proposed by Kontsevich Soibelman [, Conjecture 6. The auxiliary category B p Definition. Let X be a nonsingular projective Calabi Yau threefold equipped with an ample line bundle O X ().Letτ denote the Gieseker stability condition on the abelian category of coherent sheaves on X. Define A p to be the sub-category of coherent sheaves whose objects are zero sheaves and nonzero τ-semistable sheaves with fixed reduced Hilbert polynomial p. Definition. Fix an integer n. Now define the category B p to be the category whose objects are triples (F, V,φ), where F Obj(A p ), V is a finite dimensional C- vector space, and φ: V Hom(O X ( n), F) is a C-linear map. Given (F, V,φ) and (F, V,φ ) in B p, define morphisms (F, V,φ) (F, V,φ ) in B p to be pairs of morphisms ( f, g), where f : F F is a morphism in A p and g: V V is a C-linear map such that the following diagram commutes: Look at [, Definition. for more detail.

4 664 A. Sheshmani V φ Hom(O X ( n), F) g V φ Hom(O X ( n), F ). Our definition of the category B p is compatible with that of [, Definition.. Now we define the numerical class of objects in B p based on [, Section.. Definition. Define the Grothendieck group K (B p ) = K (A p ) Z, where K (A p ) is given by the image of K 0 (A p ) in K (Coh(X)) = K num (Coh(X)).LetC(A p ) denote the positive cone of A p defined as C(A p ) = { E K num (A p ) : 0 = E A p }. Now, given (F, V,φ) B p, we write [(F, V,φ)=([F, dim V ) and define the positive cone of B p by C(B p ) = { (β, d) : β C(A p ), d 0orβ = 0 and d > 0 }. We state the following results by Joyce and Song without proof. Lemma. ([, Lemma.) The category B p is abelian and B p satisfies the condition that if [F =0 K (A p ) then F = 0. Moreover, B p is noetherian and artinian and the moduli stacks M (β,d) B p are of finite type for all (β, d) C(B p ). Remark.4 The category A p embeds as a full and faithful sub-category in B p by F (F, 0, 0). Moreover, it is shown in [, Equation (.) that every object (F, V,φ)sits in a short exact sequence 0 (F, 0, 0) (F, V,φ) (0, V, 0) 0. Next we recall the definition of weak (semi)stability for a general abelian category A. Definition.5 Let A be an abelian category. Let K (A) be the quotient of K 0 (A) by some fixed group. Let C(A) be the positive cone of A. Suppose (T, ) is a totally ordered set and τ: C(A) T is a map. We call (τ, T, ) a stability condition on A if whenever α, β, γ C(A) with β = α + γ then either τ(α) < τ(β) < τ(γ) or τ(α) > τ(β) > τ(γ) or τ(α) = τ(β) = τ(γ). We call (τ, T, ) a weak stability condition on A if whenever α, β, γ C(A) with β = α + γ then either τ(α) τ(β) τ(γ) or τ(α) τ(β) τ(γ). For such (τ, T, ), we say that a nonzero object E in A is τ-semistable if for all S E, where S 0, we have τ([s) τ([e/s), τ-stable if for all S E, where S 0, we have τ([s) <τ([e/s), f

5 Weighted Euler characteristic of the moduli space τ-unstable if it is not τ-semistable. Now we apply the definition of weak stability conditions to the category B p. Definition.6 Define the weak stability condition ( τ, T, ) on B p by T ={0, } with the natural order 0 <, and τ(β,d) = 0ifd = 0 and τ(β,d) = ifd > 0. Definition.6 is compatible with that of [, Definition.5. Moduli stack of objects in B p Before constructing the moduli stack of objects in B p, we would like to provide a different description of objects in B p as complexes in the derived category. This makes it easier to understand the strategy to construct their moduli spaces. By [, Lemma., there exists a natural embedding functor F: B p D(X) which takes (F, V,φ V ) B p to an object in the derived category given by 0 V O X ( n) F 0, where V O X ( n) and F sit in degree and 0. Assume that dim V = r. In that case V O X ( n) = O X ( n) r. Therefore, one may view an object (F, V,φ V ) B p as an object in an abelian sub-category of the derived category, given as [O X ( n) r F. Now let us use this prespective and define the notion of flat families for objects in B p. Definition. Fix a parameterizing scheme of finite type S.Letπ X : X S X and π S : X S S denote the natural projections. Use the natural embedding functor F: B p D(X) in [, Lemma.. Define the S-flat family of objects in B p of type (β, r) as a complex π S M π X O X ( n) ψ S F sitting in degree and 0, such that F is given by an S-flat family of semistable sheaves with fixed reduced Hilbert polynomial p with ch(f) = β and M a vector bundle of rank r over S. A morphism between two such S-flat families is given by a morphism between the complexes π S M π X O X ( n) ψ S F and π S M π X O X ( n) ψ S F, π S M π X O X ( n) ψ S F π S M π X O X ( n) ψ S F. Moreover an isomorphism between two such S-flat families in B p is given by an isomorphism between the associated complexes For more detail on Definition.5 look at [, Definition.5.

6 666 A. Sheshmani π S M π X O X ( n) ψ S F and π S M π X O X ( n) ψ S F, π S M π X O X ( n) ψ S F = π S M π X O X ( n) = ψ S F. From now on, by objects in B p we mean the objects which lie in the image of the natural embedding functor F: B p D(X). Moreover, by the S-flat family of objects in B p and their morphisms (or isomorphisms) we mean their corresponding definitions as stated in Definition.. Now we define the rigidified objects in B p. These are only going to provide the means for construction of moduli stack of objects in B p as a quotient stack.. Rigidified objects and their realization in the derived category As stated in Definition., the objects in B p are defined so that the sheaf sitting in degree is given by a trivial vector bundle of rank r isomorphic to O r X ( n). However we have not fixed any choice of such trivialization. Below we will define closely related objects, which we denote by rigidified objects in B p, by fixing a choice of the trivialization of O r X ( n). These objects are essential for our construction, as their moduli stack forms a GL r (C)-torsor over the (to be defined) moduli stack of objects in B p. Therefore our plan is to essentially construct the moduli stack of objects in B p as the stacky quotient of the moduli stack of rigidified objects in B p, where the group we take the quotient with is GL r (C). Definition. Fix a positive integer r and define the sub-category B R p B p to be the category of rigidified objects in B p of rank r whose objects are defined by tuples (F, C r,ρ) where F is a coherent sheaf with reduced Hilbert polynomial p, ch(f) = β and ρ: C r Hom(O X ( n), F). Given two rigidified objects of fixed type (β, r) as (F, C r,ρ)and (F, C r,ρ ) in B R p, define morphisms (F, C r,ρ) (F, C r,ρ ) to be given by a morphism f : F F in A p such that the following diagram commutes: C r ρ Hom(O X ( n), F) id C r ρ Hom(O X ( n), F ). f Remark. Similar to before, there exists a natural embedding functor F R : B R p D(X) which takes (F, C r,ρ) B R p to an object in the derived category given by 0 C r O X ( n) F 0, where C r O X ( n) sits in

7 Weighted Euler characteristic of the moduli space degree and F sits in degree 0. One may view an object in B R p as a complex φ: O r X ( n) F such that the choice of trivialization of O r X ( n) is fixed. Definition.4 Fix a parametrizing scheme of finite type S. Use the natural embedding functor F R : B R p D(X) in Remark..AnS-flat family of objects of type (β, r) in is given by a complex B R p πs O r S π X O X ( n) ψ S F sitting in degree and 0 such that F is given by an S-flat family of semistable sheaves with fixed reduced Hilbert polynomial p with ch(f s ) = β for all s S. A morphism between two such S-flat families in B R p is given by a morphism between the complexes πs O r S π X O X ( n) ψ S F and π S O r πs O r S π X O X ( n) S ψ S π X O X ( n) ψ S F, F id O X S πs O r S π X O X ( n) ψ S F. Moreover an isomorphism between two such S-flat families in B R p isomorphism between the associated complexes is given by an πs O r S π X O X ( n) ψ S F and π S O r πs O r S π X O X ( n) S ψ S π X O X ( n) ψ S F, F id O X S πs O r S π X O X ( n) = ψ S F. Similar to the way that we treated objects in B p, from now on by objects in B R p we mean the objects which lie in the image of the natural embedding functor F R : B R p D(X) in Remark.. Moreover by the S-flat family of objects in B R p and their morphisms (or isomorphisms) we mean the corresponding definitions as stated in Definition.4. Notation.5 In what follows we define M (β,r) B p (M (β,r) respectively) to be the moduli B R p functors from Sch/C Groupoids which send a C-scheme S to the groupoid of S-flat family of objects of type (β, r) in B p (B R p respectively). We will show that these moduli functors (as groupoid valued functors) are equivalent to algebraic quotient stacks. We will also show that the moduli stack M (β,r) B p is given by a stacky quotient of M (β,r) by GL B R r (C). p

8 668 A. Sheshmani. The underlying parameter scheme According to Definition., an object in the category A p consists of semistable sheaves with fixed reduced Hilbert polynomial p. Note that having fixed a polynomial (in variable t) p(t) as the reduced Hilbert polynomial of F means that the Hilbert polynomial of F can yet be chosen as P F (t) = kp(t)/d! for different values of k where d is the dimension of F. However here we make an assumption that there are only finitely many possible values k = 0,,...,N for which our computation makes sense. We explain the motivation behind this assumption further below. Our analysis inherits this finiteness property directly from applying [, Proposition.7, where the authors show that there are only finitely many nontrivial contributions to their wallcrossing computation which are induced by objects, whose underlying sheaves could only have finitely many fixed Hilbert polynomials. In other words, according to [, Proposition.7, it suffices to consider Hilbert polynomials P F (t) = kp(t)/d! induced by p and only finitely many values of k =,,...,N for some N > 0. Similarly, for us there would only be finitely many such k for which () holds true, which justifies the reason behind our assumption. On the other hand, as discussed in [7, Theorem.7, the family of Giesekersemistable sheaves F on X such that F has a fixed Hilbert polynomial is bounded. Therefore, the family of coherent sheaves with finitely many fixed Hilbert polynomials is also bounded. We will use this boundedness property in our construction of parameterizing moduli stacks. Now fix the Hilbert polynomial P F (t) = P as above and use the fact that, given a bounded family F of coherent sheaves with fixed Hilbert polynomial P (here F denotes a member of the family F), there exists an upper bound for their Castelnuovo Mumford regularity, given by the integer m, such that for each member F of F the twisted sheaf F(m ) is globally generated for all m m. Fix such m and let V be the complex vector space of dimension d = P(m ) given by V = H 0 (F O X (m )). Twisting the sheaf F by the fixed large enough integer m would ensure one to get a surjective morphism of coherent sheaves V O X ( m ) F. One can then construct a scheme parametrizing the flat quotients of V O X ( m ) with fixed given Hilbert polynomial P. This by usual arguments provides us with Grothendieck s Quot-scheme. Here to shorten the notation we use Q to denote Quot P (V O X ( m )). Denote by Q ss Q the sublocus of Gieseker-semistable (τ-semistable for short) sheaves F with fixed Hilbert polynomial P. Definition.6 Let n in Definition. be given so that n m. Define P over Q ss to be a bundle whose fibers parameterize H 0 (F(n)). The fibers of the bundle P r over each point [F ={p}, where p Q ss, parameterize H 0 (F(n)) r. In other words, the fibers of P r parameterize the maps O r X ( n) F (which define the complexes representing the objects in B R p ). There exists a right action of GL(V ) (where V is as above) on the Quot-scheme Q which induces an action on Q ss, after restriction to the open subscheme of τ-semistable sheaves. It is trivially seen that the action of GL(V ) on Q ss induces a right action on P r. However note that, since we have fixed the trivialization of O r X ( n) for the

9 Weighted Euler characteristic of the moduli space objects in B R p, there exists also an extra action of GL r (C) on P r which is described as follows. Let [ O r X ( n) φ F be given as a point in P r.letψ GL r (C) be the map given by ψ: O X ( n) r O X ( n) r. The action of GL r (C) on P r is defined via precomposing the sections of F with ψ as shown in the diagram below: O r X ( n) ψ O r X ( n) φ F. Note that, by the Grothendieck Riemann Roch theorem, fixing a polarization over X and the Chern character of sheaves as ch(f) = β, induces the fixed Hilbert polynomial for such F. Therefore this is the reason why we index our parameterizing moduli stacks by (β, r) instead of (P, r).. Artin stacks M (β,r) B p and M (β,r) B R p By Definitions. and.4, the construction of moduli stack of objects in B p and B R p is done similar to [6, Section 5. Theorem.7 Let P r be as in Definition.6. Then the following statements hold true: (i) Let [ P r GL(V ) be the stack theoretic quotient of P r by GL(V ). Then, there exists an isomorphism of stacks In particular, M (β,r) B R p (ii) The moduli stack M (β,r) B R p M (β,r) B R p is an Artin stack. exists an isomorphism of stacks [ P r = GL(V ). is a GL r (C)-torsor over M (β,r) B p. In particular, there

10 670 A. Sheshmani M (β,r) B p = M(β,r) B R p GL r (C). (iii) It is true that locally in the flat topology M (β,r) B p [ (β,r) Spec(C) = M. B R p GL r (C) This isomorphism does not hold true globally unless r =. Proof The proofs of parts (i), (ii) are essentially the same as [6, Proposition., Remark.4, Theorem.5. Now we prove part (iii) by showing that there exists a forgetful map π: M (β,r) M (β,r) B R p B p which induces a map from [ M (β,r) Spec(C) B R p GL r (C) to M (β,r) B p and show that this map has an inverse locally but not globally unless r =. First we prove the claim for r =. For r =, GL (C) =.ForaC-scheme S, ans-point of [ M (β,) Spec(C) B R p is identified with the data (O X S ( n) F, L S ), where L S is a line bundle over S. Letπ S : X S S be the natural projection onto the second factor. There exists a map that sends this point to an S-point p M (β,) B p which is obtained by tensoring with L S, i.e., O X ( n) L S φ L F π S L S. Note that tensoring O X S ( n) with π S L S does not change the fact that fiber by fiber O X S ( n) s S = OX ( n) L S s S. Moreover, there exists a section map s: M (β,) B p M (β,) Bp R [ Spec(C) Simply take an S-point [O X ( n) L S F ( M (β,) ) B p (S) and send it to an S-point in ( [ ) M (β,) Spec(C) (S) B R p.

11 Weighted Euler characteristic of the moduli space by the map [ OX ( n) L S F ([ O X S ( n) F πs ) L S, LS. Note that since L S is a line bundle over S then it is invertible and hence a section map is always well-defined and M (β,) B p is a -gerbe over M (β,).nowletr >. It is left B R p to show that there exists a map from [ M (β,r) Spec(C) B R p GL r (C) to M (β,r) B p and this map does not have an inverse (section map) globally. To proceed further, we state the following definition. Definition.8 Consider a stack (Y, p Y : Y Sch/C). Given two morphisms of stacks π : X Y and π : X Y, thefibered product of X and X over Y is defined by the category whose objects are defined by triples (x, x,α), where x X and x X respectively and α: π (x) π (x ) is an arrow in Y such that p Y (α) = id. Moreover, the morphisms (x, x,α) (y, y,β)are defined by the tuple (φ: x y,ψ: x y ) such that π (ψ) α = β π (φ): P E (x) P F (y ). Now observe that for r >, there exists a forgetful map π: M (β,r) B R p M (β,r) B p which over S-points takes [O X ( n) O r S F (see Definition.4)to[M O X ( n) F, where M is defined in Definition.. Moreover, there exists a map g : M (β,r) B p BGL r (C) which sends [M O X ( n) F to M by forgetting F. Finally there exists the natural projection i: Spec(C) [ Spec(C) = BGL r (C). GL r (C)

12 67 A. Sheshmani It follows directly from Definition.8 that the diagram M (β,r) B R p g pt = Spec(C) π M (β,r) B p g BGL r (C) = i [ Spec(C) GL r (C) is a fibered diagram and M (β,r) B R p = M (β,r) B p BGLr (C)pt. However, one cannot use the same argument used for the case of r = to conclude that there exists a section map s: M (β,r) B p M (β,r) B R p [ Spec(C), GL r (C) since the S-point of BGL r (C) is a GL r (C) bundle over S and this vector bundle is trivializable locally but not globally. Therefore locally one may think of M (β,r) B p as isomorphic to [ M (β,r) Spec(C) B R p GL r (C) but not globally. Definition.9 Define M (β,r) B p,ss as the substack of M(β,r) B p parameterizing τ-semistable objects in B p. Since M (β,r) B p is of finite type [, Lemma., then M (β,r) B p,ss is of finite type for all (β, r) C(B p ). 4 Stack function identities in the Ringel Hall algebra We review here some basic facts about the stack functions in the Ringel Hall algebras based on constructions in [9. Let M be an Artin C-stack with affine geometric stabilizers. Consider pairs (R,ρ), where R is given by a finite type Artin C-stack with affine geometric stabilizers and ρ = R M is a one-morphism. Now define an equivalence relation for such pairs, where (R,ρ)and (R,ρ ) are called equivalent if there exists a one-morphism ι: R R such that ρ ι and ρ are two-isomorphic onemorphisms R M. Joyce and Song in [, Section. define the space of stack functions SF(M,χ,Q) as the Q-vector space generated by the above equivalence classes of pairs [(R,ρ) such that the following relations are imposed:

13 Weighted Euler characteristic of the moduli space (a) Given a closed substack (G,ρ G ) (R,ρ),wehave [(R,ρ)=[(G,ρ G )+[(R\G,ρ R\G ). () (b) Let R be a C-stack of finite type with affine geometric stabilizers, U denote a quasiprojective C-variety, π R : R U R be the natural projection and ρ: R M be a one-morphism. Then [(R U,ρ π R )=χ([u)[(r,ρ). () (c) Assume R = [X/G, where X is a quasiprojective C-variety and G a very special algebraic C-group acting on X with maximal torus T G, then we have [(R,ρ)= Q Q(G,T G ) F(G, T G, Q) [( [X/Q,ρ ι Q), where the rational coefficients F(G, T G, Q) have a complicated definition explained in [0, Section 6.. Here Q(G, T G ) is the set of closed C-subgroups Q of T G such that Q = T G C G (Q), where C G (Q) ={g G : sg = gs, s Q} and ι Q :[X/Q R = [X/G is the natural projection one-morphism. Similarly, one defines SF(M,χ,Q) by restricting the one-morphisms ρ in parts (a) (c) to be representable. (d) There exist the notions of multiplication, pullback, pushforward of stack functions in SF(M, χ, Q) and SF(M, χ, Q). For further discussions look at [, Definitions.6,.7 and Theorem.9. Joyce and Song in [, Section. define the notion of characteristic stack functions δ ss (β,d) SF(M Bp,χ,Q). Moreover, in the instance where the moduli stack contains strictly semistable objects, the authors define the logarithm of the moduli stack by the stack function ɛ (β,d) given as an element of the Hall-algebra of stack functions supported over virtual indecomposables, we will include these definitions below. Definition 4. Define the stack functions δ (β,r) ss = δ (β,r) M (β,r) Bp,ss in SF al ( M (β,r) B p,ss ) (for definition of SF al look at [, Definition.) for (β, r) C(B p ). Now define elements ɛ (β,r) in SF al ( M (β,r) B p,ss ) ɛ (β,r) = n (β,r ),...,(β n,r n) C(B p) (β,r )+ +(β n,r n)=(β,r) τ(β i,r i )= τ(β,r) i ( ) n n δ (β,r ) ss δ (β,r ) ss δ ss (βn,rn), where is the Ringel Hall multiplication defined in [, Definition.. (4)

14 674 A. Sheshmani Our goal in the remainder of this article is to first evaluate the element of the Hall algebra ɛ (β,r) above, by explicitly computing the right-hand side of (4), and then calculate the invariants. To do this one needs to apply the Joyce Song Lie algebra morphism to ɛ (β,r). In order to clarify the latter, we need further definitions. Definition 4. ([, Definition.) Define the Euler form in B p as χ Bp : K (B p ) K (B p ) Z such that χ Bp ((β, d), (γ, e)) = χ(β,γ) dχ([o X ( n),γ)+ eχ([o X ( n),β), where χ( ) is the Euler form on K (Coh(X)). Definition 4. ([, Definition.) Define S to be the subset of (β, d) in C(B p ) K (B p ) such that P β (t) = kp(t)/d! for k = 0,...,N and d = 0 or or. Then S is a finite set [8, Theorem.7. Define the Lie algebra L(B p ) to be the Q-vector space with the basis of symbols λ (β,d) with (β, d) S with the Lie bracket [ λ (β,d), λ (γ,e) = ( ) χ Bp ((β,d),(γ,e)) χ Bp ((β, d), (γ, e)) λ (β+γ,d+e) (5) for (β + γ,d + e) S and [ λ (β,d), λ (γ,e) =0otherwise. Here it can be seen that χ Bp is antisymmetric and hence (5) satisfies the Jacobi-identity and that makes L(B p ) into a finite-dimensional nilpotent Lie algebra over Q. Now the Joyce Song Lie algebra morphism is defined as the Q-linear map B p: SF ind al M Bp L(B p ) by applying [, Definition 5. to the moduli stack M Bp and L(B p ). Definition 4.4 Define the invariant B ss p (X,β,r, τ)associated to τ-semistable objects of type (β, r) in B p by B p ( ɛ (β,r) ) = B ss p (X,β,r, τ) λ (β,r), where B p is given by the Lie algebra morphism above. From now on, to minimize the unnecessary computational complexity, we restrict our analysis to rank pairs, i.e., r =. First to motivate the intuition behind our computations we study an interesting example. 5 Direct computation of invariants in an example Example 5. Computation of B ss p (X, [P,, τ), where X is given by the total space of O P ( ) P. For further detail look at [, Definition..

15 Weighted Euler characteristic of the moduli space We compute the invariant of τ-semistable objects (F, C,φ C ) of type ([P, ) in B p. In this case F has rank over its support and p F (n) = n + χ(f). Assume that χ(f) = k. In this case by computations in [6, the only semistable sheaf F with ch (F) =[P is given by O P (k ) which is a stable sheaf. First we give the description of M ([P,) B p,ss. By definition, an object of type ([P, ) in B p is identified by a complex O X ( n) ι O P (k ), where ι: P X (from now on we suppress ι in our notation). By the constructions in Sect.., the parameter scheme of τ-semistable objects is obtained by choosing two sections (s, s ) such that s i H 0 (O P (n +k )) for i =,. Moreover since O P (k ) is a stable sheaf, its stabilizer is given by. An important point to note is that, given a τ-semistable object (F, C,φ C ), one is always able to obtain an exact sequence of the form 0 (F, C,φ C ) (F, C,φ C ) (0, C, 0) 0, (6) for every object in the moduli stack and since τ(f, C,φ C ) = τ(0, C, 0) =, one concludes that all objects parametrized by the moduli stack are given by extensions of rank τ-stable objects and hence all objects are τ-strictly-semistable. Moreover, note that giving a τ-semistable object of the form O X ( n) (s,s ) F is equivalent to requiring the condition that (s, s ) = (0, 0), since otherwise, one may be able to obtain an exact sequence 0 (C, 0, 0) (C, F, 0) (0, F, 0) 0 such that τ(c, 0, 0) = > τ(0, F, 0) = 0, hence (C, 0, 0) (weakly) destabilizes (C, F, 0), hence a contradiction. Now use Theorem.7 and find that M ([P,) B p,ss = [ H 0 (O P (n +k )) \{0}/ GL (C) [ P(H = 0 (O P (n +k )) ) GL (C) We need to compute the element of the Hall algebra ɛ ([P,). By applying Definition 4. to M ([P,) B p,ss, we obtain ɛ ([P,) = δ ([P,) ss β k +β l =[P δ (β k,) s. (7) δ (β l,) s. (8) Now we use a stratification strategy in order to decompose M ([P,) B p,ss into a disjoint union of strata as follows. Since the objects in the moduli stack are of type ([P, ),

16 676 A. Sheshmani one would immediately see that the only possible decomposition of a τ-semistable object of type ([P, ) is given by ([P, ) = ([P, ) + (0, ). This means that a strictly τ-semistable object of type ([P, ) is always given by an (split or non-split) extension, involving an object of type ([P, ) and an object of type (0, ). Note that it is allowable to flip the order of an extension, as long as the original strictly semistable object is re-produced. Our stratification then involves a study of the parametrizing moduli stack for the objects, depending on which extension is used to produce them. This will enable us to decompose M ([P,) B p,ss into a disjoint union of split and nonsplit strata (i.e., induced by split or non-split extensions). Definition 5. Define M ([P,) B p,sp,) M([P B p,ss to be the locally closed stratum over which an object of type ([P, ) is given by split extensions involving objects of type ([P, ) and (0, ). Define M ([P,) B p,nsp,) M([P B p,ss to be the locally closed stratum over which an object of type ([P, ) is given by non-split extensions involving objects of type ([P, ) and (0, ). Now we study the structure of each stratum separately. 5. Stacky structure of M ([P,) B p,sp It is easy to see that any τ-semistable objects of type ([P, ) given as [ OX ( n) O P (k ) = [ OX ( n) O P (k ) [O X ( n) 0 has the property that the sections s, s for this object are linearly dependent on one another. Hence the underlying parameter scheme of τ-semistable objects of this given form is given by choosing a nonzero section of O P (n +k ), in other words we obtain H 0 (O P (n +k ))\{0}. Now we need to take the quotient of this space by the stabilizer group of points. We know that the condition required for a τ-semistable object O X ( n) (s,s ) F to be given by split extensions of rank objects is that s and s are linearly dependent on one another. Now pick such an object given by O X ( n) (s,0) F. The automorphisms of this object are given by the group which makes the following diagram commutative:

17 Weighted Euler characteristic of the moduli space O X ( n) (s,0) O P (k ) = O X ( n) (s,0) = O P (k ). Hence it is seen that the left vertical map needs to be given by a subgroup of GL (C) which preserves s, i.e., the Borel ( subgroup of GL (C) whose elements are given by upper triangular matrices k k 0 k ), where k, k and k A.Havingfixed one of the automorphisms via fixing k, k, k, it is seen that, by the commutativity of the square diagram, the right vertical map needs to be given by multiplication by k which is an element of. Note that one needs to take the quotient of the parameter scheme by all isomorphisms between any two objects in the split stratum, not just the automorphisms of one fixed representative. In general for an object to live in the split stratum one requires the sections (s, s ) to be given by (s, a s ). We observed that fixing a representative for a split object of rank (such as fixing (s, s ) = (s, 0) as above) would tell us that its automorphisms are given by G m A. Hence, taking into account all possible representatives implies that the stabilizer group of objects in the spit stratum is given by G m A. Hence we obtain [ M ([P,) H 0 B p,sp = (O P (n +k ))\{0} G = m A [ P(H 0 (O P (n +k ))) G m A. (9) 5. Stacky structure of M ([P,) B p,nsp In this case, the objects in M ([P,) B p,nsp are given by non-split extensions of the form 0 O X ( n) s 0 O P (k ) O X ( n) (s,s ) O X ( n) = F (0) Note that switching the place of O P (k ) and 0 in the bottom row of diagram (0) would produce a split extension and so such extensions cannot lie in the non-split stratum. Now, in order to obtain non-split extensions, one needs to choose two sections s, s such that s and s are linearly independent. The set of all linearly independent choices of s and s spans a two dimensional subspace of H 0 (O P (n + k )) which is given by the Grassmanian G(, n +k). Now, we need to take the quotient of this scheme by the stabilizer group of points in the stratum. We know that the condition required for a τ-semistable object O X ( n) (s,s ) F

18 678 A. Sheshmani to be given by non-split extensions of rank objects is that s and s are linearly independent. Now pick such an object given by O X ( n) (s,s ) F. The automorphisms of this object are given by the group which makes the following diagram commutative: O X ( n) (s,s ) O P (k ) = O X ( n) (s,s ) = O P (k ). Hence it is seen that the left vertical map needs to be given by a subgroup ( of GL (C) whose elements are given by diagonal matrices of the form k 0 0 k ), where k. Having fixed one of the automorphisms via fixing k, it is seen that by the commutativity of the square diagram, the right vertical map needs to be given by multiplication by k which is an element of. Hence we obtain [ M ([P,) G(, n +k) B p,nsp =. () Now we are ready to compute β k +β l =[P δ (β k,) s δ (β l,) s appearing on the right-hand side of (8). We use the fact that δ (β k,) s δ (β l,) s β k +β l =[P () = δ ([P,) s δ s (0,) + δ s (0,) δ ([P,) s and compute each term on the right-hand side of () separately. Remark 5. As we described above, there exists an action of GL (C) on S = P ( H 0 (O P (n +k )) ). This action induces an action of the corresponding Lie algebra on the tangent space of S given by the map O S gl (C) T S, () where gl (C) denotes the Lie algebra associated to the group GL (C). The dimension of the automorphism group of objects representing the elements of S is given by the dimension of the stabilizer (in GL (C)) group of these elements, which itself is given by the dimension of the kernel of the map in (). On the other hand, the dimension

19 Weighted Euler characteristic of the moduli space of the kernel of the map in () is an upper-semicontinious function. Therefore by the usual arguments, we obtain a stratification of S which induces a stratification of [ S GL (C) into locally closed strata, such that over each stratum the dimension of the stabilizer group is constant as we vary over points inside that stratum. Here in Definition 5. we stated without proof that the defined strata are locally closed in M ([P,) B p,ss,thisis discussed in detail and for more general cases in Appendix. 5. Computation of δ ([P,) s δ s (0,) Given δ ([P,) s = δ (0,) s = ([ P(H 0 (O P (n +k ))) ([ ) Spec(C),ρ,ρ ), (ρ,ρ are natural embedding maps to M Bp ), consider the diagram Z Exact Bp π [ S GL (C) [ ( P H 0 ( O P (n + k ) )) [ Spec(C) ρ ρ π π MBp M Bp, where Z is given by the scheme parametrizing the set of commutative diagrams 0 O X ( n) s 0 O P (k ) O X ( n) (s,s ) O X ( n) = F (4) Since that the extensions in (4) have the possibility of being split or non-split, therefore, we consider each case separately and define the multiplication δ ([P,) in each case separately as follows below. s δ s (0,)

20 680 A. Sheshmani Definition 5.4 Let [δ ([P,) s δ s (0,) sp denote the stratum of (π ) Z over which the points are represented by split extensions given by the commutative diagram in (4). Definition 5.5 Let [δ ([P,) s δ s (0,) nsp denote the stratum of (π ) Z over which the points are represented by the non-split extensions given by coomutative diagram in (4). Therefore δ ([P,) s δ (0,) s = [ δ ([P,) s δ (0,) s sp + [ δ ([P,) s δ (0,) s nsp. Now we compute [δ ([P,) s δ s (0,) sp. This amounts to choosing s, s so that s and s are linearly depending on one another. The scheme parametrizing nonzero sections s is given by P(H 0 (O P (n +k ))). Now we take the quotient of this scheme by the stabilizer group of points. Similar to arguments in [7, Lemma., given any point in P(H 0 (O P (n + k ))) represented by the extension in (4), its stabilizer group is given by the semi-direct product G m Hom(E, E ), where each factor of amounts to the stabilizer group of objects given as E = O X ( n) 0 and E = O X ( n) O P (k ) respectively. Note that the extra factor of A will not appear as a part of the stabilizer group since, by the given description of E and E, we know that Hom(E, E ) = 0 for every such E and E. We obtain the following conclusion: [ δ ([P,) s δ s (0,) [ P(H 0 sp = (O P (n +k ))) G. (5) m Now we compute [δ ([P,) s δ s (0,) nsp. This amounts to choosing s, s so that s and s are linearly independent and the extension in diagram (4) becomes nonsplit. Note that for any fixed value of s, one has P worth of choices for s.nowwe need to consider all possible choices of s and in doing so, we require the sections s, s to remain linearly independent. This gives the flag variety F(,, n + k). Hence we obtain [ δ ([P,) s δ s (0,) [ F(,, n + k) nsp =. (6) Note that the factor of in the denominator of (6) is due to the fact that we have used one of the factors in projectivising the bundle of s -choices over the Grassmanian. We finish this section by summarizing our computation. By (5) and (6) one obtains δ ([P,) s δ (0,) s = [ P ( H 0 ( O P (n +k ) )) G m + [ F(,, n + k). (7)

21 Weighted Euler characteristic of the moduli space Computation of δ s (0,) δ ([P,) s Now change the order of δ s (0,) and δ ([P,) s and obtain the diagram Z Exact Bp π [ S GL (C) [ [ ( Spec(C) P H 0 ( O P (n +k ) )) ρ ρ π π MBp M Bp. Here Z is given by the scheme parametrizing the set of commutative diagrams 0 O X ( n) O X ( n) (0,s ) O X ( n) = F O P (k ) 0. 0 (8) Note that the computation in this case is easier since the only possible extensions of the form given in (8) are the split extensions. The computation in this case is similar to computation in (5) except that one needs to take into account that over any point represented by an extension [as in diagram (8) of E = O X ( n) 0 and E = O X ( n) O P (k ) we have Hom(E, E ) = A. Hence by similar discussions we obtain δ ([P,) s δ (0,) s = [ P(H 0 (O P (n +k ))) G m A. (9) 5.5 Computation of ɛ (β,) By (8), (9), (), (), (7) and (9) we obtain [ P(H ɛ (β,) 0 [ (O = P (n +k ))) G(, n + k) G + m A [ P(H 0 (O P (n +k ))) G m [ F(,, n + k) [ P(H 0 (O P (n + k ))) G. m A (0)

22 68 A. Sheshmani Now use the decomposition used by Joyce and Song in [, p. 58 and write [ P(H 0 (O P (n +k ))) G m A = F ( [ P(H G, G 0 m, (O P (n + k ))) ) G m + F ( [ P(H G, G 0 m, G (O P (n +k ))) m), where F(G, G m, ) = and F(G,, ) =. Equation (0) simplifies as follows: [ P(H ɛ (β,) 0 (O = P (n +k [ ))) P(H 0 (O G P (n +k ))) m [ [ G(, n + k) P(H 0 + (O P (n +k ))) G m [ [ F(,, n + k) P(H 0 (O P (n +k ))) G m + [ P(H 0 (O P (n + k ))). Now use Definition 0. and write [ [ G(, n + k) Spec(C) = χ(g(, n +k)) and [ [ F(,, n + k) Spec(C) = χ (F(,, n + k)) [ Spec(C) = χ(p ) χ (G(, n + k)) [ Spec(C) = χ (G(, n + k)), () where the second equality is due to the fact that the topological Euler characteristic of a vector bundle over a base variety is equal to the Euler characteristic of its fibers times the Euler characteristic of the base. By (0) and (), we obtain [ P(H 0 [ (O P (n +k ))) Spec(C) + χ(g(, n +k)) [ Spec(C) χ(g(, n +k))

23 Weighted Euler characteristic of the moduli space = [ P(H 0 (O P (n +k ))) = χ( P(H 0 (O P (n +k ))) ) [ Spec(C) = +k) [ Spec(C) (n. 5.6 Computation of the invariant Now apply the Lie algebra morphism B p in Definition 4. to ɛ ([P,). By definition, B ( p ɛ ([P,) ) ( = χ na k) [ Spec(C) (n +, ( ) ) νm μ i (0,) λ ([P,), Bp where by χ na we mean the naïve Euler characteristic defined by Joyce and Song [, Definition., weighted by the corresponding Behrend function. Note that by (7), M ([P,) B p,ss = [ P(H 0 (O P (n +k )) ) GL (C) and hence [ Spec(C) has relative dimension (n+k 5) = 4 n k over M ([P,) B p,ss. Moreover, [ Spec(C) is given by a single point with Behrend s multiplicity and therefore, (μ i ) ν M (0,) Bp λ ([P,) = ( ) 4 n k ν [ Spec(C) Gm B ( p ɛ ([P,) ) ( = χ na +k) [ Spec(C) (n = ( ) (n + k) λ ([P,). = ν [ Spec(C), Gm,ν [ Spec(C) Gm ) λ ([P,)

24 684 A. Sheshmani Finally by Definition 4.4, we obtain B ss n + k p (X,β,, τ) =. () Note that a simple calculation shows that substituting ([P, ) for (β, ) in [7, Equation (5.) would give the same answer as in (). Hence our result is compatible with wallcrossing calculations. To summarize, in Sect. 5 we introduced our strategy of direct computations over an example where r = and β was given by the irreducible class [P. Our strategy involved computing the weighted Euler characteristic of the right-hand side of (8). After carefully analyzing the stacky structure of the moduli space of objects of type ([P, ) (Sect. 5.), first we computed the second summand on the right-hand side of (8) (Sects. 5. and 5.4). Then, by the stratification of the original moduli stack, we showed that the first summand on the right-hand side of (8) is written as a sum of the characteristic stack functions of the strata in a suitable way, such that essentially the difference of the first and second summands on the right-hand side of (8) turned out to be given as a sum of stack functions supported over virtual indecomposables. This enabled us to apply the Lie algebra morphism defined in Definition 4. to both sides of (8) and obtain the final answer in (). 6 Direct calculations in general cases In this section we extend the computational strategy in Sect. 5 to one level of generalization further, i.e., to the case where r =, however β can have the possibility of being given by a reducible class. Eventually the final generalization, which is to consider semistable pairs with rank >, follows the same strategy as will be described in detail below. Remark 6. As was seen above, a key condition to hold for our strategy to work is that the strata involved in our stratification are locally closed and disjoint from one another. In order to show that such stratification is possible for general cases, we start below by assuming that an object E : O X ( n) F, with F being a semistable sheaf with reducible Chern character, is decomposable into rank objects E = O X ( n) F and E =O X ( n) F, where F, F are stable sheaves. We then show that the moduli space parameterizing E can be decomposed into a disjoint union of locally closed strata, depending on how E is produced by extensions of E, E. In other words, we will show below that the extensions of the form 0 E E E 0 (depending on being split or non-split), or the ones with the order of the extension flipped as in 0 E E E 0 are all locally closed and disjoint from one another. Assumption 6. Throughout this section we assume that a τ-semistable object (F, V,φ V ) B p of type (β, ) has the property that β is either indecomposable or it satisfies the condition that if β = β k + β l (i.e., if β is decomposable) then β k =[F k and β l =[F l such that F k and F l are τ-stable sheaves with fixed reduced Hilbert polynomial p. In other words, β cannot be decomposed into smaller classes whose associated sheaves are not τ-stable.

25 Weighted Euler characteristic of the moduli space Lemma 6. Let S (β k,) s and S (β l,) s (for some β k,β l ) denote the underlying schemes, as in Sect.. given as a bundle P over the τ-stable locus of the Quot-scheme Q s Q, parameterizing maps O X ( n) F such that the Chern character of F is given by β k and β l respectively, satisfying Assumption 6.. Then, given a tuple of objects (E, E ) S (β k,) s and S (β l,) s, the following is true: Hom(E, E ) = A if E = E, Hom(E, E ) = 0 if E E. Proof This is mainly due to the assumption on the stability of the sheaves involved. Fix E =[O X ( n) F S (β k,) s and E =[O X ( n) F S (β l,) s. Consider amapψ: E E. By definition, the morphism between E and E is defined by a morphism ψ F : F F which makes the following diagram commutative: O X ( n) F id O X O X ( n) F. ψ F () By assumption, F and F are given as stable sheaves with fixed reduced Hilbert polynomial p. Hence any nontrivial sheaf homomorphism from F to F is an isomorphism. Moreover by simplicity of stable sheaves any such nontrivial isomorphism is identified with A. Now use the commutativity of the diagram in (). Given a reducible class β, there might be multiple ways to decompose it into underlying smaller classes β k and β l. We will now show that having fixed a class β and moving from one decomposition type, say β = β k + β l, to another given by β = β k + β l, will correspond to moving from one stratum to another in the ambient moduli space of E, such that the corresponding strata are disjoint from one another. Lemma 6. Fix β k and β l such that β k +β l = β and β,β k and β l satisfy the condition in Assumption 6.. Consider an object E given as an element of S (β,) ss (by this notation we mean that an object E may consist of a semistable sheaf with reducible class β) which fits into a non-split extension of objects 0 E E E 0, such that E and E are given by the elements of S (β k,) s and S (β l,) s respectively. Now suppose E and E are objects with classes (β k, ) and (β l, ) respectively such that β k + β l = β, β k = β k, β l = β l and furthermore, β,β k and β l also satisfy the condition in Assumption 6.. Then it is true that E cannot be given as an extension 0 E E E 0.

26 686 A. Sheshmani Proof If E is given by both extensions then we obtain a map between the two short exact sequences 0 E ι E E 0 0 E E = p E 0. Hence we obtain a map p ι: E E. Since by assumption β k = β k (this means β l = β l because β k + β l = β k + β l = β), by Lemma 6., we conclude that p ι is the zero map. Hence p ι factors through the map ι g in the following diagram: 0 E ι E E 0 g 0 E = ι E p E 0. Since E E, by Lemma 6., g is the zero map. By considering the left commutative square in (5) we obtain a contradiction, since the image of E in E cannot always be zero. Now we show that flipping the order of non-split extensions will again induce disjoint parameterizing stratum in the ambient moduli space (Fig. ). Lemma 6. Fix β k and β l such that β k + β l = β as in Assumption 6.. Now consider E S (β,) ss which fits into a non-split extension 0 E E E 0, where E and E are given by the elements of S (β k,) s the object E cannot be given as an extension where [E =(β l, ) and [E =(β k, ). 0 E E E 0, and S (β l,) s respectively. Then Proof We prove by contradiction. Assume E fits in both exact sequences. We obtain a map between the two sequences 0 E ι E E 0 0 E E = p E 0.

27 Weighted Euler characteristic of the moduli space Similar to before, we obtain a map p ι: E E. Since E and E have equal classes, we need to consider two possibilities. First when E = E and second when E E. If E = E, then the image of the map p ι is either multiple of identity over E or the zero map. If the former case happens it means that the exact sequence on the first row is split, contradicting the assumption that E fits in a non-split exact sequence. If the map is given by the zero map, then we can apply the argument in Lemma 6. and obtain a contradiction. If E E then the map p ι is the zero map and the proof similarly reduces to argument in proof of Lemma 6.. 7GL (C)-invariant stratification Now we are ready to introduce a GL (C)-invariant stratification of M (β,) B p,ss for β satisfying the condition in Assumption 6.. Note that, by Theorem.7 and Definition.9, M (β,) ss,b R p [ S (β,) ss = GL(V ), M (β,) B p,ss M(β,) B p,ss = M(β,) ss,b R p, GL (C) which shows us that a suitable stratification of S (β,) ss, after passing to the subsequent quotient stacks and restricting to the τ-semistable locus, induces a GL (C)-invariant stratification of M (β,) B p,ss. The cartoon below explains the intuition behind our strategy: Fig. Stratification of M (β,) B induced by stratification of S(β,) p,ss ss