aa r X i v : . [ m a t h . AG ] A p r WALL-CROSSINGS FOR TWISTED QUIVER BUNDLES
BUMSIG KIM AND HWAYOUNG LEE
Abstract.
Given a double quiver, we study homological alge-bra of twisted quiver sheaves with the moment map relation us-ing the short exact sequence of Crawley-Boevey, Holland, Gothen,and King. Then in a certain one-parameter space of the stabilityconditions, we obtain a wall-crossing formula for the generalizedDonaldson-Thomas invariants of the abelian category of framedtwisted quiver sheaves on a smooth projective curve. To do so, weclosely follow the approach of Chuang, Diaconescu, and Pan in theADHM quiver case, which makes use of the theory of Joyce andSong. The invariants virtually count framed twisted quiver sheaveswith the moment map relation and directly generalize the ADHMinvariants of Diaconescu. Introduction
A Nakajima’s quiver variety is a holomorphic symplectic quotientattached to a double quiver Q , i.e., a quiver whose arrows are paired( a, ¯ a ) such that ¯ a is a reverse arrow of a . This holomorphic symplecticquotient is a GIT quotient of a locus defined by a moment map rela-tion. In [16], the moduli of stable twisted quasimaps to the symplecticquotient from a fixed smooth projective curve X is obtained as an ap-plication of the quasimap construction of [5, 6] and shown to comewith a natural symmetric obstruction theory. This result generalizesDiaconescu’s work [10].The stability of stable twisted quasimaps turns out to be an asymp-totic one in the one dimensional stability parameter space R > of theabelian category A ′ of framed twisted quiver sheaves on X . It is there-fore natural to investigate the wall-crossing phenomena of the modulistack M ssτ ( γ ) of τ -semistable objects with numerical class γ in A ′ as τ varies in R > .For that study of wall-crossings, there are two theories available:the theory of Joyce and Song [15]; and the theory of Kontsevich and Date : October 27, 2012.2000
Mathematics Subject Classification.
Primary 14N35; Secondary 14H60.
Key words and phrases.
Stability conditions, Double quivers, Twisted quiverbundles, generalized Donaldson-Thomas invariants, Wall-crossings.
Soibelman [17]. In this paper, we perform our research according tothe framework of Joyce and Song.First, we study the homological algebra of the category of twistedquiver sheaves with the moment map relation (2.5). This homologicalstudy is a generalization of works Crawley-Boevey and Holland [7],Crawley-Boevey [8, 9], Gothen and King [12], and Diaconescu [10]. Wededuce that a truncated part of the category A ′ behaves like a 3-Calabi-Yau category. For example, a suitably defined antisymmetric bilinearform is numerical (see Proposition 2.7). This property is the first mainresult of this paper and originates from the moment map relation.Next, using the above bilinear form on the numerical K -group of A ′ , we define a Lie algebra L ( A ′ ) and, using the straightforward gen-eralization of the Chern-Simons functional in [10], we construct a Liealgebra homomorphism to L ( A ′ ) from a Ringel-Hall type algebra ofstack functions with algebra stabilizers supported on virtual indecom-posables, as in [15, Theorems 3.16 and 7.13]. This Lie algebra homo-morphism yields the definition of the generalized Donaldson-Thomasinvariant for ( Q, X, γ, τ ) via the log stack function for M ssτ ( γ ).Finally, using the approach of [10, 3, 4], we establish a wall-crossingformula of the invariants (see Theorem 3.8), which is the second mainresult of this paper. In section 3.6, we show that the invariants vanishwhen the framing is zero and, at the same time, the curve X is notrational. The wall-crossing correction could be therefore nontrivial onlywhen X is rational. 2. Homological Algebra
The aim of this section is to prove a suitably defined antisymmetricEuler-like bilinear form of framed twisted quiver sheaves is numeri-cally determined in certain cases (see Proposition 2.7). For this, webegin with finding a partial injective resolution (2.2) of double quiverrepresentations with the moment map relation (2.1).2.1.
Double quivers.
We set up notations for quivers. Let Q be afinite quiver, i.e., a directed graph whose arrow set Q and vertex set Q are finite. The tail map and the head map from Q to Q aredenoted by t and h , respectively. For each arrow a ∈ Q , we denote by¯ a the reverse arrow of a . Define the double quiver Q of Q by adjoininga reverse arrow to each arrow of Q . A path p is an ordered set a ...a m of arrows a i such that ta i = ha i +1 for i = 1 , ..., m −
1. Define the headand the tail of p by hp = ha and tp = ta m , respectively. For eachvertex i ∈ Q , define a trivial path e i . Also set h ( e i ) = t ( e i ) = i . Thelengths of p and e i are by definition m and 0, respectively. Let R be a ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 3 commutative ring with unity 1. The path algebra RQ is the R -algebragenerated by all paths subject to relations by the following rules (seefor instance [12]): p · q = pq if tp = hq , 0 otherwise; p · e tp = p ; and e hp p = p .2.2. Quiver representations.
In this section, we let Q ′ be any nonemptysubset of Q . We fix λ i ∈ R for each i ∈ Q ′ .Following [7], we consider the two-sided ideal ( µ − λ ) generated bythe relation(2.1) X i ∈ Q ′ X a ∈ Q : ha = i ( − | a | aa − λ i e i = 0 , where | a | is 0 if a ∈ Q or 1 otherwise. In fact, (2.1) is the functorialexpression of the moment map equation (see [7, 9]) and hence we call(2.1) the moment map relation for the quiver (or more precisely for( Q, Q ′ , λ )). We will study the homological algebra of the quotientalgebra A := RQ/ ( µ − λ ), i.e., the deformed preprojective algebra whichwas introduced by W. Crawley-Boevey and M. Holland in [7, Section4]. Note that every A -module can be considered as an R -module bythe natural R -module homomorphism R → A , r r P i ∈ Q e i .Let R be the abelian category of (left) A -modules. For V ∈ R , weconstruct a sequence in R , which becomes a partial injective resolutionof V when R is a field. The sequence is defined to be0 → V ǫ → M i ∈ Q Hom R ( e i A, V i )(2.2) g → M a ∈ Q Hom R ( e ta A, V ha ) m → M i ∈ Q ′ Hom R ( e i A, V i ) , where: • we view e i A as an R - A bimodule so that Hom R ( e i A, V j ) is a left A -module by aα ( p ) := α ( pa ) for α ∈ Hom R ( e i A, V j ) , a ∈ A, p ∈ e i A ; • V i := e i V which is a R -submodule of V ; • the A -homomorphisms ε , g , m are defined by: for p i ∈ e i A ⊂ A (1) ǫ ( v )( p i ) = p i v ;(2) g ( α ) a ( p ta ) = α ha ( ap ta ) − a ( α ta ( p ta ));(3) m ( γ )( p i ) = P a ∈ Q : ta = i ( − | a | ( a ( γ a ( p i )) + γ a ( ap i )).The above sequence (2.2) is obtained from exact sequences in [7,Lemma 4.2], [8, Proof of Lemma 1], [9, Proof of Lemma 3.2], [12,(2.1)]. Since A has the ‘moment map relation’ (2.1), g is not surjective BUMSIG KIM AND HWAYOUNG LEE (see [12, (2.1)]). Our moment map relation is slightly different fromthat in [7, 8, 9] in that we only consider the relations in Q ′ , not therelations in Q . Proposition 2.1.
The sequence (2.2) is exact.Proof.
Following the proofs in [7, 8, 9, 12], we record a proof for the sakeof completion. If ǫ ( v )( e i ) = e i v = 0 for all i ∈ Q , then P i e i v = 0.Since P e i = 1, we see that v = 0 . Thus ǫ is injective. It is clearthat Im ǫ ⊂ Ker g . Next we will show that Ker g ⊂ Im ǫ . Consider α ∈ L i ∈ Q Hom R ( e i A, V i ) ⊂ Hom R ( A, V ) such that g ( α ) = 0. Thisimplies that α is A -linear. Therefore Ker g ⊂ Hom A ( A, V ) = ǫ ( V ).Next we can check that Im g ⊂ Ker m since( m ◦ g ( α i )) i ( p i ) = P a ∈ Q : ta = i ( − | a | ( a ( g ( α i ) a ( p i )) + g ( α i ) a ( ap i )) = P a ∈ Q : ta = i ( − | a | ( a ( α ha ( ap i ) − aα i ( p i )) + α i ( aap i ) − a ( α ta ( ap i ))) = 0 . The last equality above follows from the R -linearity of α , the mo-ment map relation, and ta = ha. Finally, let us show the hard partKer m ⊂ Im g. Let ( γ a ) a ∈ Q ∈ Ker m , in other words, for each i ∈ Q ′ , P a ∈ Q : ta = i ( − | a | ( a ( γ a ( p ta )) + γ ¯ a ( ap ta )) = 0 . Let A k be an R -modulegenerated by arrows p whose lengths are less than or equal to k . For ex-ample, A is generated by e i , i ∈ Q and A is generated by a ∈ Q and e i , i ∈ Q . Then consider the filtration A ⊂ A ⊂ A ⊂ · · · . For eachvertex i , there is the corresponding filtration e i A ⊂ e i A ⊂ e i A ⊂ · · · .An R -linear map α will be constructed by induction on the filtration.Define α = ( α i ) ∈ L i ∈ Q Hom R ( e i A, V i ) , α i : e i A → V i as • α i | e i A = 0 and • α i ( ap ta ) = aα ta ( p ta ) + γ a ( p ta ) for a ∈ Q with ha = i .We need to show that α is well-defined. For that, we consider thepath algebra F := RQ without any relation and its correspondingfiltration F i . Note that α is well defined as an element on Hom( F, V )and ˜ g ( α ) = γ if ˜ g : L i ∈ Q Hom R ( e i F, V i ) → L a ∈ Q Hom R ( e ta F, V ha ) isthe homomorphism corresponding to g . Now we claim that α is well-defined on A n . It is clear that the claim is true when n = 0 ,
1. Let F n be the R -submodule of F spanned by length- n elements. Suppose that α is well-defined on A n − . Then note that for p i ∈ e i F n − and i ∈ Q ′ , α i (( X a ∈ Q ,ha = i ( − | a | aa − λ i ) p i )(2.3) = X ( − | a | ( a ( α ta ( ap i )) + γ a ( ap i )) − λ i α i ( p i ) = 0 ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 5 since P ( − | a | γ a ( ap i ) becomes − X ( − | a | aγ a ( p i ) (by m ( γ a ) = 0)= − X ( − | a | a (˜ g ( α i ) a )( p i ) (by γ = ˜ g ( α ))= − X ( − | a | ( a ( α ha ( ap i )) − a ( aα ta ( p i ))) (by the definition of ˜ g ) . Combined with the inductive definition of α , the equation (2.3) impliesthat α = 0 on the two-sided ideal ( µ − λ ) of A n . By the definition of α , it is clear that g ( α ) = γ . (cid:3) Replacing V by W in the sequence (2.2) and taking Hom A ( V, · ) on(2.2), we obtain a sequence0 → Hom A ( V, W ) → M i ∈ Q Hom R ( V i , W i )(2.4) → M a ∈ Q Hom R ( V ta , W ha ) → M i ∈ Q ′ Hom R ( V i , W i )of R -modules. This simplification follows from the adjunction of [12,(2,2)]. Let C ( V, W ) be the complex consisting of the last three termsof the sequence (2.4) with the first term at degree 0. Then we get thefollowing.
Corollary 2.2.
Let R be a field. (1) Hom R ( e i A, V j ) is an injective A -module. (2) For l = 0 , , Ext lA ( V, W ) ∼ = H l ( C ( V, W )) . Proof.
By the above adjunction, we note the equivalence of functors:Hom A ( • , Hom R ( e i A, V j )) ∼ = Hom R ( e i A ⊗ A • , V j ) . The latter is an exact functor since e i A is a projective right A -moduleand V j is an injective R -module. This proves (1). Now (2) follows since(2.2) is a partial injective resolution of V . (cid:3) Quiver sheaves.
We carry out a similar procedure for twistedquiver sheaves in place of quiver representations. To introduce twistedquiver sheaves, let X be a Gorenstein projective variety, let ω X be thedualizing sheaf for X , and let λ i ∈ Γ( X, ω X ). Suppose also that wechoose an invertible sheaf M a on X for each a ∈ Q and an isomorphism f b, ¯ b : M b ⊗ M ¯ b → ω ∨ X for each b ∈ Q . We will set f ¯ b,b = f b, ¯ b for b ∈ Q using the natural isomorphism M ¯ b ⊗ M b ∼ = M b ⊗ M ¯ b . The condition M b ⊗ M ¯ b ≃ ω ∨ X , ∀ b ∈ Q is a direct generalization of the correspondingcondition in [10]. BUMSIG KIM AND HWAYOUNG LEE
Provided with the above data, we define an O X -algebra structure onthe sheaf M all paths p M p by making: • M p := M a ⊗ ... ⊗ M a n if p = a ...a n with a i ∈ Q , and M e i := O X ; • for x p ∈ M p , x q ∈ M q , let x p x q := x p ⊗ x q if tp = hq , and 0 oth-erwise; and in L p M p we have natural identifications M p M e hp = M p ⊗ M e hp = M p = M e tp M p = M e tp ⊗ M p .We denote by M Q this O X -algebra graded by lengths.We want to define an ideal sheaf from the moment map relation.First, for every local section ξ ∈ ω ∨ X , let( µ − λ )( ξ ) := X i ∈ Q ′ X a ∈ Q : ha = i ( − | a | ξ a ⊗ ξ a − h ξ, λ i i e i , where e i stands for the constant 1 in M e i = O X and ξ a ⊗ ξ a ∈ M a ⊗ M a is required to satisfy f a,a ( ξ a ⊗ ξ a ) = ξ . Then since ( µ − λ )( f · ξ ) = f · ( µ − λ )( ξ ) for f ∈ O X , we can define the ideal sheaf ( µ − λ ) of M Q generated by ( µ − λ )( ξ ) for all ξ ∈ ω ∨ X and hence the quotient sheaf B := M Q/ ( µ − λ ).Now we turn into a homological algebra of the abelian category A of B -modules. For an alternative and concrete description of a B -module,we view a B -module as an O X -module as follows. A collection of O X -sheaves E i , i ∈ Q and O X -homomorphisms φ a : M a ⊗ E ta → E ha , a ∈ Q . The collection will be called a M -twisted quiver sheaf on X inour context if the following moment map relation holds:(2.5) X i ∈ Q ′ X a ∈ Q : ha = i ( − | a | φ a ◦ (Id M a ⊗ φ a ) − λ i ⊗ Id E i = 0 . Proposition 2.3.
The category of M -twisted quiver sheaves is equiv-alent to A .Proof. If { E i , φ a } is an M -twisted quiver sheaf, then it is obvious howto give a B -module structure on ⊕ E i . Conversely, for a B -module E ,define E i := e i E = M e i E and φ a : M a ⊗ e ta E → e ha E . Then it is simpleto check the collection { E i , φ a } has an induced M -twisted quiver sheafstructure. (cid:3) ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 7
We remark that a similar path algebra and a similar result were al-ready introduced and used in the original work on quiver representationtheory over a ground field by P. Gabriel [11]; and the O X -algebra M Q was introduced in [2, Section 5] and also used in [12, Section 3]; andProposition 2.3 is a generalization of results of [7] (see Lemma 2.1 andSection 4) in the context of preprojective algebras and Proposition 5.1of [2] in the context of quiver sheaves with no moment map relations.For E ∈ A , there is an exact sequence in A → E ǫ → M i ∈ Q H om O X ( e i B, E i )(2.6) g → M a ∈ Q H om O X ( M a ⊗ O X e ta B, E ha ) m → M i ∈ Q ′ H om O X ( ω ∨ X ⊗ O X e i B, E i ) , where: • e i denotes the constant 1 in M e i = O X ; • we view e i B as an O X - B bimodule so that Hom R ( e i B, E j ) is a left B -module; • E i := e i E which is a O X -submodule of E ; • the B -homomorphisms ε , g , m are defined by: for p i ∈ e i B ⊂ B , ξ = f a,a ( ξ a ⊗ ξ a ),(1) ǫ ( e )( p i ) = p i e ;(2) g ( α ) a ( x a ⊗ p ta ) = α ha ( x a p ta ) − φ a ( x a ⊗ α ta ( p ta )) for x a ∈ M a ;(3) m ( γ )( ξ ⊗ p i ) = P a ∈ Q : ta = i ( − | a | ( φ a ( ξ a ⊗ γ a ( ξ a ⊗ p i ))+ γ a ( ξ a ⊗ ( ξ a p i ))). Proposition 2.4.
The sequence (2.6) is exact.Proof.
The proof is parallel to the proof of Proposition 2.1. (cid:3)
As before, we replace E by F in the sequence (2.2) and take H om B ( E, · )to obtain a sequence0 → H om B ( E, F ) → M i ∈ Q H om O X ( E i , F i )(2.7) → M a ∈ Q H om O X ( M a ⊗ E ta , F ha ) → M i ∈ Q ′ H om O X ( ω ∨ X ⊗ E i , F i )of O X -modules. Complexes (2.4) and (2.7) are direct generalizationsof [10, (3.2)].Let C ( E, F ) be a complex consisting of the last three terms of (2.7).Assume that E is locally free, i.e., by definition, E i are locally free for BUMSIG KIM AND HWAYOUNG LEE all i ∈ Q , In this case, we will observe that C ( E, F ) is quasi-isomorphicto a complex computing Ext iB ( E, F ) for i = 0 ,
1. In what follows, wedenote the i -th hypercohomology group of the complex C ( E, F ) by H i ( X, C ( E, F )).
Corollary 2.5.
Assume that E is locally free. Then, for i = 0 , , Ext iB ( E, F ) ∼ = H i ( X, C ( E, F )) . Proof.
Note that a partial injective resolution J • of F can be obtainedfrom injective resolutions I • i of F i and (2.6) since ⊕ i I ki has an in-duced B -module structure (see [12, Section 3] for detail). Note that H om B ( E, J • ) as an O X -complex is quasi-isomorphic (at 0 ,
1) to0 → M i ∈ Q E ∨ i ⊗ F i → M a ∈ Q M ∨ a ⊗ E ∨ ta ⊗ F ha → M i ∈ Q ′ ω X ⊗ E ∨ i ⊗ F i which is C ( E, F ). (cid:3) We remark that Corollary 2.2 (2) and Corollary 2.5 directly gener-alize Corollary 3.11 of [10].Now we introduce a bilinear form h , i on A following [15]. Definition . Define h E, F i to bedim Ext B ( E, F ) − dim Ext B ( E, F ) + dim Ext B ( F, E ) − dim Ext B ( F, E ) . If E denotes the O X -coherent sheaf L i ∈ Q \ Q ′ E i (when Q = Q ′ ,always E = 0), we finally come to the first main result of this paper. Proposition 2.7.
Suppose that X is a smooth projective curve and E and F are locally free. Assume either E = 0 or F = 0 . (1) For i = 0 , , , ., H i ( X, C ( E, F )) ∼ = H − i ( X, C ( F, E )) ∨ . (2) h , i is numerically determined as follows: h E, F i = X a ∈ Q ( d ( E ta ) r ( F ha ) − d ( F ha ) r ( E ta ) + d ( M a ) r ( E ta ) r ( F ha ) − (1 − g ) r ( E ta ) r ( F ha )) + 2 X i ∈ Q ′ ( − d ( E i ) r ( F i ) + d ( F i ) r ( E i )) , where d ( E i ) and r ( E i ) stand for the degree and the rank of thelocally free sheaf E i , respectively. ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 9
Proof. (1): Note that C ( F, E ) ∨ ⊗ ω X = C ( E, F )[2]. Combined withthe Serre duality, this implies H i ( X, C ( E, F )) ∼ = H i ( X, C ( F, E ) ∨ ⊗ ω X [ − ∼ = H − i ( X, C ( F, E )) ∨ .(2): Note that χ ( X, C ( E, F )) = h E, F i by (1) above. On the otherhand, χ ( X, C ( E, F )) is equal to χ ( X, C ( E, F )) − χ ( X, C ( E, F )) + χ ( X, C ( E, F )), hence the topological expression for h E, F i followsfrom the Riemann-Roch formula. (cid:3) Wall-crossings
From now on, let X be a smooth projective curve, let λ i = 0 for all i ∈ Q ′ , and let Q \ Q ′ = { } . In the abelian category A ′ of twistedquiver sheaves, we will consider stability conditions for τ ∈ R > and, inthe framework of Joyce-Song theory [15], we will define the generalizedDonaldson-Thomas invariants using the moduli space of τ -semistableobjects in A ′ . We will derive a wall-crossing formula following theapproach of Chuang, Diaconescu, and Pan [10, 3, 4].3.1. Chamber structures.
In this subsection, for τ ∈ R > we in-troduce the notion of a τ -stability on twisted quiver sheaves and showthat for each fixed numerical class with a minimal framing the stabilityspace R > has a finite number of critical values. The precise definitionof critical values is not important. The relevant required property willbe only that there are no strictly τ -semistable quiver sheaves for everynoncritical value τ .Let K be a nonzero complex vector space. Denote by A ′ the abeliancategory of M -twisted quiver sheaves E with E = K S ⊗ O X forsome finite set S (depending on E ). In this category A ′ , a mor-phism from ( E i , φ a ) to ( E ′ i , φ ′ a ) is by definition a usual morphism as M -twisted quiver sheaves with the framing condition that the attached O X -homomorphism K S ⊗O X → K S ′ ⊗O X is a block matrix ( c s,s ′ ) ( s,s ′ ) ∈ S × S ′ , c s,s ′ ∈ C . It is straightforward to check that the category A ′ is anabelian category.Let E ∈ A ′ and τ ∈ R > be the stability parameter. For a nonzero M -twisted quiver sheaf E we define the τ -slope of E to be µ τ ( E ) := deg( L i =0 E i )rank( L i =0 E i ) + τ · rank E rank( L i =0 E i )) ∈ ( −∞ , ∞ ] . Definition . A nonzero object E of A ′ is called τ -(semi-)stable if µ τ ( F )( ≤ ) < µ τ ( E ) for any nonzero proper subobject F of E .The definitions of τ -slope of E and τ -(semi-)stability in Definition 3.1are generalizations of a slope function and a (semi-)stability condition introduced in [10], and furthermore, belong to the class of slope stabilityconditions for quiver sheaves introduced in [1, 2] for a larger family ofparameters.Let r = rank( L i =0 E i ), v = rank E , and d = deg( L i =0 E i ). If E is strictly τ -semistable, that is, τ -semistable but not τ -stable, then τ must be of form τ = rd ′ − r ′ dr ′ v or τ = r ′ d − rd ′ ( r − r ′ ) v (3.1)for some r ′ , d ′ ∈ Z with 1 ≤ r ′ ≤ r − C ( A ′ ) = dim K · N × ( N × Z ) Q ′ of numericalclasses of twisted quiver bundles. The class of E has rank E at thefirst entry, rank E i at the middle one for i ∈ Q ′ , and deg E i at thelast one for i ∈ Q ′ . Let γ ∈ C ( A ′ ) and v ( γ ) be the first entry of γ .Suppose that v ( γ ) = dim K . Then by a generalization of [10, Lemma4.7] there is a number N ( γ ) such that there are no strictly τ -semistableobjects with numerical class γ if τ ≥ N ( γ ). Let C ( γ ) be the set of allpossible positive values τ ≤ N ( γ ) in (3.1) so that for τ / ∈ C ( γ ) thereare no strictly τ -semistable objects with γ -class. Note that C ( γ ) hasno accumulation points in R . Hence, C ( γ ) is a finite set. We call anelement of C ( γ ) a critical value .3.2. Chern-Simons functionals.
Let M be the moduli stack param-eterizing all objects E of A ′ . In order to apply the Joyce-Song theory,we need a local description of M as a critical locus of a holomor-phic function on a complex domain (see [15, Theorems 5.4 and 5.5]which makes use of Miyajima’s results in [18]). The theorems beloware straightforward generalizations of [10, Theorems 7.1 and 7.2]. Inparticular, the Chern-Simons functional (3.2) is a direct generalizationof one in [10, (7.7)].Let A ′≤ be a subcategory of A ′ of an object E with ( E ) = K ⊗ O X or 0. Let M si be the coarse moduli space of simple objects in A ′≤ . Theorem 3.2.
For every [ E ] ∈ M si ( C ) , the analytic germ of M si ( C ) at [ E ] is isomorphic to (Crit( f ) , u ) for some holomorphic function f : U → C on a finite dimensional complex manifold U , where u is a pointof U . Let S be an Aut( E )-invariant subscheme of Ext A ′ ( E, E ) parameter-izing a versal family of objects in M ( C ) near E . Theorem 3.3.
For every E ∈ M ( C ) and a maximal compact sub-group G of Aut( E ) , the analytic germ of ( S, is G C -equivariantlyisomorphic to (Crit( f ) , for some G C -invariant holomorphic function ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 11 f : (Ext A ′ ( E, E ) , → ( C , , where G C is the complexification of G in Aut( E ) . The proofs of [10, Theorems 7.1 and 7.2] work for the general case af-ter the replacement of the Chern-Simons functional [10, (7.7)] accordingto the double quiver Q . In what follows, we describe the Chern-Simonsfunctional for the general case.Let E = ( E i , φ a ) i ∈ Q ,a ∈ Q be a framed twisted quiver bundle on X and let ˆ X denote the complex manifold associated to X . Then there isthe gauge-theoretical interpretation ( ˆ E i , ¯ ∂ E i , φ a ) i ∈ Q ′ ,a ∈ Q of E , i.e., ˆ E i is E i regarded as a C ∞ complex vector bundle on ˆ X ,¯ ∂ E i : C ∞ ( ˆ E i ) → C ∞ ( ˆ E i ⊗ Λ , T ∗ ˆ X )is the unique semiconnection on ˆ E i such that local holomorphic sectionsof E i are translated into horizontal sections of ¯ ∂ E i , and φ a ∈ C ∞ ( ˆ M ∨ ta ⊗ ˆ E ∨ ta ⊗ ˆ E ha ) corresponds to φ a . Here the (0 , ∂ E i automatically holds since X is a curve.Now, the Chern-Simons functional CS near E is defined by a gener-alization of [10, (7.7)]: CS ( A i , ϕ a ) = Z X Tr( X a ∈ Q ϕ a ¯ ∂ a ϕ a + X a ∈ Q ,ha ∈ Q ′ ( − | a | A ha ˜ φ a ˜ φ a )(3.2)for( A i , ϕ a ) ∈ Y i ∈ Q ′ ,a ∈ Q C ∞ (End( ˆ E i ) ⊗ Λ , T ∗ ˆ X ) × C ∞ ( ˆ M ∨ ta ⊗ ˆ E ∨ ta ⊗ ˆ E ha ) . Here ¯ ∂ a is the semiconnection on ˆ E ∨ ta ⊗ ˆ E ha , ˜ φ a = φ a + ϕ a , and theproducts in the integrand are naturally given by compositions and cupproducts so that after all they are considered as elements in End( ˆ E ta ) ⊗ Λ , T ∗ ˆ X . The Chern-Simons functional is gauge-invariant and its criticalequations are ¯ ∂ a ϕ a − ˜ φ a A ta + A ha ˜ φ a = 0 , ∀ a ∈ Q ;(3.3) X ha = i ( − | a | ˜ φ a ˜ φ a = 0 , ∀ i ∈ Q ′ . (3.4)We note that (3.3) is the holomorphic condition on ˜ φ a with respect tothe new semiconnection ( ¯ ∂ E ta + A ta ) ∨ ⊗ ( ¯ ∂ E ha + A ha ) and (3.4) is themoment map relation on ˜ φ a . Ringel-Hall type algebras.
Recall that M denotes the modulistack parameterizing all objects E of A ′ . We call a pair ( X , ρ ) a M -valued stack function if X is an Artin stack over C and ρ : X → M is arepresentable 1-morphism. Let SF( M ) be the ‘Grothendieck group’ of M -valued stack functions, i.e., the quotient group of the free abeliangroup generated by stack functions, whose quotient is given by thesubgroup spanned by all elements of form( X , ρ ) − (( Y , ρ | Y ) + ( X \ Y , ρ | X \ Y ))for a closed substack Y of X .There is a multiplication structure on SF( M ) for which the multi-plication ( X , ρ ) ∗ ( X , ρ )is defined to be the fiber product ( X , ρ ) in diagram X −−−→ Exact ( M ) −−−→ π M , y y π × π X × C X −−−−→ ( ρ ,ρ ) M × C M where: • Exact ( M ) is an Artin stack parameterizing short exact sequencesin M and π i is the obvious i -th projection; • the square is the fiber product and ρ is the composition of theupper arrows.The multiplication is associative by [13, Theorem 5.2]. The inducedalgebra SF( M ) is called the Ringel-Hall type algebra .Denote by M ≥ the moduli stack parameterizing all objects E of A ′ with rank E ≥ K . Let SF( M ′≤ ) be the quotient algebra ofSF( M ) factored by the ideal generated by all ρ : X → M which factorthough M ≥ . Finally we consider the subalgebra SF( M ≤ ) of SF( M ′≤ )generated by all ρ : X → M which factor though the moduli stack oflocally free objects.By Proposition 2.7, we may define an antisymmetric bilinear form h , i : (dim K · N × ( N × Z ) Q ′ ) → Z . Let L ( A ′ ) be the Q -vector spacewith basis e γ , γ ∈ { , dim K } × ( N × Z ) Q ′ , equipped with a Lie algebrastructure given by[ e γ , e ˜ γ ] := (cid:26) ( − h γ, ˜ γ i h γ, ˜ γ i e γ +˜ γ if γ + ˜ γ ≤ dim K, ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 13
In the below, we let B C ∗ denote the classifying stack of the multi-plicative group C ∗ . By the local descriptions, Theorems 3.2 and 3.3, ofthe moduli spaces we will have this. Theorem 3.4.
There is a Lie algebra homomorphism
Ψ : SF indalg ( M ≤ ) → L ( A ′ ) satisfying Ψ([ Z × B C ∗ , ρ ]) = − χ ( Z, ρ ∗ ν B M ≤ ) e γ , where: • SF indalg ( M ≤ ) is a certain subalgebra of SF( M ≤ ) , spanned bystack functions with algebra stabilizers supported on virtuallyindecomposable objects; • Z is a variety and ρ ∗ ν B M ≤ is the Z -valued constructible functioninduced from the Behrend function ν B M ≤ for M ≤ ; • χ ( Z, ρ ∗ ν B M ≤ ) is defined to be the weighted topological Euler char-acteristic P n ∈ Z nχ (( ρ ∗ ν B M ≤ ) − ( n )) .Proof. Theorem 3.2 and Theorem 3.3 imply that the Behrend function ν B M ≤ on M ≤ has the same property as one in [15, Theorem 5.11] and[10, Theorem 7.4]. In the analogy with [15, Theorem 5.14], we get theresult (see also [3, Section 2.2 and Theorem 3.2]). (cid:3) Harder-Narasimhan filtrations.
In this section, using Harder-Narasimhan filtrations we express the stack function representing M ssτ − in terms of those representing M ssτ + and unframed moduli spaces. Bya purely algebraic Lemma in [3] this expression induces a wall-crossingformula. From now on, we let τ ∈ C ( r, d ), τ + > τ and τ − < τ suchthat there are no critical values between intervals ( τ , τ + ] and [ τ − , τ ).For E ∈ A ′ and τ ∈ R > , it is easy to see that there is a uniquefiltration 0 = E ⊂ E ⊂ E . . . ⊂ E n = E such that E k /E k − is τ -semistable and µ τ ( E k − /E k − ) > µ τ ( E k /E k − )for k = 1 , . . . n . This so-called Harder-Narasimhan filtration will leadus to the following.
Lemma 3.5.
Let E ∈ A with ( E ) = K ⊗ O X . TFAE. (1) E is τ -semistable (2) E is τ + -semistable or there is a unique subobject E ′ of E satis-fying: E ′ , E/E ′ are τ + -semistable, ( E ′ ) = K ⊗ O X , µ τ + ( E ′ ) >µ τ + ( E/E ′ ) , and µ τ ( E ′ ) = µ τ ( E/E ′ ) . (3) E is τ − -semistable or there is a unique subobject E ′ of E satisfy-ing: E ′ , E/E ′ are τ − -semistable, ( E/E ′ ) = K ⊗O X , µ τ − ( E ′ ) >µ τ − ( E/E ′ ) , and µ τ ( E ′ ) = µ τ ( E/E ′ ) .Proof. (1) ⇒ (2). Let E be τ -semistable. Let 0 = E ⊂ E ⊂ ... ⊂ E n = E be the τ + Harder-Narasimhan filtration of E . We take E ′ := E . If n = 2, ( E ) = 0 for otherwise, µ τ ( E ) = µ τ + ( E ) >µ τ + ( E /E ) ≥ µ τ ( E /E ) which is a contradiction to the τ -semistabilityof E . We prove that n cannot be larger than 2. Suppose that n ≥ i, j such that 1 ≤ i < j ≤ n and(3.5) µ τ + ( E i /E i − ) = µ τ ( E i /E i − ) > µ τ + ( E j /E j − ) = µ τ ( E j /E j − ) . This induces a contradiction as follows.a) When ( E ) = 0, then µ τ ( E ) = µ τ + ( E ) > µ τ + ( E ) ≥ µ τ ( E ).This is a contradiction to the τ -semistability of E .b) When ( E ) = K ⊗ O X , then ( E i ) = K ⊗ O X for all i ≥ µ τ + ( E i ) > µ τ + ( E ) implies that µ τ ( E i ) = µ τ ( E ) for all i . This contradicts (3.5).Now, by the uniqueness of Harder-Narasimhan filtrations, the proofof (1) ⇒ (2) follows.(2) ⇐ (1). The nontrivial case is that E is not τ + -semistable. Let F be a nontrivial subobject of E and let F ′ := Ker( F → E/E ′ ). Then µ τ + ( F ′ ) ≤ µ τ + ( E ′ ) (because E ′ is τ + -semistable) and µ τ + ( F/F ′ ) ≤ µ τ + ( E/E ′ ) (because E/E ′ is τ + -semistable). Now take the limit τ + → τ to the both inequalities in order to conclude that µ τ ( F ) ≤ µ τ ( E )since µ τ ( E ′ ) = µ τ ( E/E ′ ).(1) ⇔ (3). This follows by an argument similar to the proof of (1) ⇔ (2). (cid:3) Let δ τ ( γ ) denote the stack function [ M ssτ ( γ ) , ρ ] ∈ SF( M ≤ ) for thenatural open embedding ρ of the moduli stack M ssτ ( γ ) ⊂ M of τ -semistable objects of A ′ with the numerical class γ ∈ C ( A ′ ). Weuse notation δ ( γ ) for δ τ ( γ ) if v ( γ ) = 0. Then the previous lemma willinduce relationships between δ τ ± ( γ ), δ τ ( γ ), and δ ( γ ) in the Ringel-Halltype algebra as in Lemma 3.6 below. Before describing the lemma, wewill need the following index sets. For l ≥
1, let HN + ( γ, τ , l ) = { ( γ , ..., γ l ) | γ i ∈ C ( A ) , X i γ i = γ, v ( γ ) = v ( γ ) , µ τ ( γ i ) = µ τ ( γ ) ∀ i } ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 15 and HN − ( γ, τ , l ) = { ( γ , ..., γ l ) | γ i ∈ C ( A ) , X i γ i = γ, v ( γ l ) = v ( γ ) , µ τ ( γ i ) = µ τ ( γ ) ∀ i } . Lemma 3.6. In SF( M ≤ ) , the followings hold. (1) δ τ ( γ ) = δ τ + ( γ ) + X ( γ ,γ ) ∈ HN + ( γ,τ , δ τ + ( γ ) ∗ δ ( γ ) .δ τ ( γ ) = δ τ − ( γ ) + X ( γ ,γ ) ∈ HN − ( γ,τ , δ ( γ ) ∗ δ τ − ( γ ) . (2) δ τ + ( γ ) = X l ≥ ( − l − X HN + ( γ,τ ,l ) δ τ ( γ ) ∗ δ ( γ ) ∗ ... ∗ δ ( γ l ) .δ τ − ( γ ) = X l ≥ ( − l − X HN − ( γ,τ ,l ) δ ( γ ) ∗ δ ( γ ) ∗ ... ∗ δ τ ( γ l ) . (3) δ τ − ( γ ) = δ τ + ( γ ) + P l ≥ ( − l − P HN − ( γ,τ ,l ) δ ( γ ) ∗ ... ∗ δ ( γ l − ) ∗ [ δ ( γ l − ) , δ τ + ( γ l )] . Proof.
There are only finite nontrivial terms in each summation of (1),(2), and (3). For example, when l = 2, let us consider an exact sequence0 → E → E → E /E → E , E /E are τ -semistable with µ τ ( E ) = µ τ ( E /E ).It suffices to show that deg( E ) j is bounded above by a number de-pending only on class γ . Since E is τ -semistable, by a generalizationof [10, Lemma 2.4] E is isomorphic to an element in a bounded familyof vector bundles on C . Now using a finite covering π : C → P , wesee that deg π ∗ ( E ) j is bounded above, hence so is deg( E ) j .The statement (1) follows from Lemma 3.5.For (2), we rewrite the first equation of (1) as(3.6) δ τ + ( γ ) = δ τ ( γ ) − X ( γ ,γ ) ∈ HN + ( γ,τ , δ τ + ( γ ) ∗ δ ( γ )and apply (3.6) to δ τ + ( γ ). This iterated procedure must stop sincethere are only finite nontrivial terms in the summation. This proves(2). To prove (3), we start with the second equation of (2) and replace δ τ ( γ l ) by the first equation of (1). (cid:3) Log stack functions.
Following [14, Definition 8.1], we definethe log stack function for γ ∈ C ( A ′ ) as ǫ τ ( γ ) := X l ≥ ( − l − l X P γ i = γ,µ τ ( γ i )= µ τ ( γ ) ∀ i δ τ ( γ ) ∗ · · · ∗ δ τ ( γ l ) . As in the proof of Lemma 3.6, there are only finite nontrivial termsin the sum expression of ǫ τ ( γ ). According to [15, Theorem 3.11], thelog stack function ǫ τ ( γ ) is an element in SF indalg ( M ≤ ). In the below, if v ( γ ) = 0, we let ǫ ( γ ) denote ǫ τ ( γ ). Lemma 3.7. ǫ τ − ( γ ) − ǫ τ + ( γ ) = X l ≥ ( − l − ( l − X HN − ( γ,τ ,l ) [ ǫ ( γ ) , [ ... [ ǫ ( γ l − ) , ǫ τ + ( γ l )] ... ] . Proof.
Lemma 3.6 (3) and the definition of log stack functions enableus to apply (the combinatorial argument of) [3, Lemma 2.4]. (cid:3)
Theorem 3.4 implies that for fixed γ , the τ -invariant J τ ( γ ) ∈ Q canbe defined as Ψ( ǫ τ ( γ )) = − J τ ( γ ) e γ . We call J τ ( γ ) the generalized DT-invariant for ( Q, X, γ, τ ) (see [15,Definition 5.15]). This is a direct generalization of ADHM invariantsby [3, Lemma 3.1, Theorem 3.2]. When the rank v ( γ ) of the frameis 0, then the invariant will be denoted simply by J ( γ ) since it doesnot depend on τ. Now by combining Lemma 3.7 and Theorem 3.4, weconclude the following wall-crossing formula.
Theorem 3.8. ( J τ − ( γ ) − J τ + ( γ )) e γ = P l ≥ l − P HN − ( γ,τ ,l ) [ J ( γ ) e γ , [ ... [ J ( γ l − ) e γ l − , J τ + ( γ l ) e γ l ] ... ] . The action by the Jacobian variety.
Suppose that v ( γ ) = 0and the genus of X is g ≥
1. In this case, by the similar argument forthe proof of [15, Proposition 6.19], the generalized DT invariant J ( γ )vanishes as follows. Let J ( X ) be the Jacobian variety of X and let L be the universal line bundle on X × J ( X ). Then the torus group J ( X )acts on M ssτ ( γ ) by t · E := E ⊗ L t for t ∈ J ( X ). Note that this actionyields a torus fibration on M ssτ ( γ ) and the Behrend function on M ssτ ( γ )is constant on each J ( X )-orbit. Hence we conclude that J ( γ ) = 0 ALL-CROSSINGS FOR TWISTED QUIVER BUNDLES 17 using the expression of J ( γ ) by the weighted Euler characteristics (see[15, Section 5.3]). Acknowledgments.
We would like to express our deep gratitude toKurak Chung for useful discussions and to Emanuel Diaconescu andJae-Hyouk Lee for their invaluable comments. We also thank a refereefor careful comments which improve the presentation of the paper. Thiswork is financially supported by NRF-2007-0093859.
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School of Mathematics, Korea Institute for Advanced Study, 85Hoegiro, Dongdaemun-gu, Seoul, 130-722, Korea
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