The pure cohomology of multiplicative quiver varieties

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Introduction
A quiver Q = (I , ) is a directed graph with vertex set I and edge set . Despite its simple definition, from the datum of a quiver one can build, using various geometric quotient constructions, rich families of symplectic algebraic varieties. The best-known examples of this are [25] Nakajima's quiver varieties. In this paper, however, we will study their cousins, the multiplicative quiver varieties, first introduced by Crawley-Boevey and Shaw [10]. Just as Nakajima's quiver varieties can be understood as (coarse) moduli spaces of semistable representations of a class of algebras known as preprojective algebras their multiplicative analogues can be viewed similarly as moduli spaces of representations of a noncommutative algebra q , the multiplicative preprojective algebra.
The significance of multiplicative quiver varieties is rapidly growing: Crawley-Boevey and Shaw were led to them through their work on the celebrated Deligne-Tom Nevins passed away on February 1, 2020. He was always generous as both a friend and a collaborator. He is sorely missed.
Simpson problem. Subsequently they have been shown to arise as moduli spaces of irregular connections in the work [3,4] of Boalch and Yamakawa (indeed Boalch's work [3] lead him to define an even more general notion of multiplicative quiver variety than that considered here). Bezrukavnikov and Kapranov [2] realise them as moduli of microlocal sheaves on nodal curves (see also the work of Crawley-Boevey [11]), while in symplectic topology the work of Etgü and Lekili [15] shows that the Fukaya categories of certain symplectic four-manifolds, which are built from quiver-type data, are controlled by a derived version of the associated multiplicative preprojective algebra. Moreover, results of Chalykh and Fairon [7] and Braverman-Etingof-Finkelberg [5] reveal exciting new connections between multiplicative quiver varieties and new families of integrable systems which have also been constructed using double affine Hecke algebras.
Recently a number of authors [27,30] have studied the geometry of multiplicative quiver varieties. The present paper is a contribution to the study of their topology, and, as we discuss later, we expect its results will help shed light on questions raised by Hausel and collaborators in [20].

Results
Just as for a Nakajima quiver variety, a multiplicative quiver variety M q θ (α), where α ∈ N I is a dimension vector, is defined as a GIT quotient (at a character χ θ : G → G m ) of the affine algebraic variety Rep( q , α) of (framed) representations of q (α) by the group G = i G L(α i ) / (G m ), a product of general linear groups modulo the diagonal copy of G m ; when it is a free quotient, this endows M to be the "pure part" of the cohomology, where grW denotes the associated graded with respect to the weight filtration, then the image of the pullback map c * on cohomology must inject into the pure part of the cohomology of M q θ (α).
Remark 1.1 Note that for a smooth variety X the weight filtration on H m (X , Q) vanishes below degree m, so that grW m (H m (X , Q)) = W m (H m (X , Q)). Thus for such spaces the pure part of cohomology is a subspace of the ordinary cohomology.
The main result of the present paper is: In light of Theorem 1.2 of [26], Theorem 1.2 is nicely consonant with Hausel's "purity conjecture" (cf. [17] as well as [19,Theorem 1.3.1 and Corollary 1. 3.2], and the discussion around Conjecture 1.1.3 of [20]), which predicts that when M In the special case in which Q is a quiver with a single node and g ≥ 1 loops, the dimension vector is α = n, and q ∈ C × is a primitive nth root of unity, the multiplicative quiver variety M q θ (α) is identified with the G L n -character variety Char( g , G L n , q Id) of a genus g surface with a single puncture with residue q Id, sometimes called a genus g twisted character variety [18]. We obtain: Corollary 1. 3 The pure cohomology P H * Char( g , G L n , q Id) is generated by tautological classes. Corollary 1.3 has already appeared in [29], where it was deduced, via the nonabelian Hodge theorem, from Markman's theorem [24] that the cohomology of the moduli space of G L n -Higgs bundles of degree 1 on a smooth projective genus g curve is generated by tautological classes. A novelty of our result, compared to [29], is that we avoid invoking non-abelian Hodge theory: instead, we deduce Corollary 1.3 (as well as Theorem 1.2) via a more direct and concrete method that invokes only basic facts of ordinary mixed Hodge theory as in [12].
Unlike the situation of quiver varieties in [26], we know of no obvious generalizations of Theorems 1.2 to other even-oriented cohomology theories (such as topological K -theory or elliptic cohomology). However, we do obtain the following analogue of Theorem 1.6 of [26]. (2) There is a finite list of tautological bundles from which every object of D b coh (M) is obtained by finitely many applications of (i) direct sum, (ii) cohomological shift, and (iii) cone.
As for the analogous result in [26], we emphasize that Theorem 1.4 (2) is not simply a formal consequence of Theorem 1.4(1), since we do not include taking direct summands (i.e., retracts) among the operations (i)-(iii). It would be interesting to know generators for D b coh (M) for more general dimension vectors α than in Theorem 1.4. As mentioned above, one source of interest in the cohomology of twisted character varieties (see [18]) and more generally of multiplicative quiver varieties lies in the P = W conjecture and Hausel's purity conjecture. One categorical level higher, Theorems 1.2 and 1.4 may also be expected to have relevance to versions of mirror symmetry for multiplicative quiver varieties (cf. Section 7B of [2] as well as [16]) and the Betti geometric Langlands program [1].

Method of proof
Theorem 1.2 has the following slightly different but equivalent formulation. Choose a subgroup S ⊂ i G L(α i ) whose projection S → G is a finite covering. Then one can form the stack quotient Rep( q , α) θ -s /S, which comes with a morphism π : Rep( q , α) θ -s /S → M q θ (α) s that is a gerbe, in fact a torsor over the commutative group stack B H where H = ker(S → G). We have an isomorphism It is Theorem 1.5 that we prove directly: the tautological bundles Rep( q , α) θ -s × S V that appear naturally and geometrically in our proof do not themselves descend to the multiplicative quiver variety in general, so it is more convenient to work on the Deligne-Mumford stack Rep( q , α) θ -s /S.
Our proof of Theorem 1.5 follows the broad outline of that used in [26] to establish that tautological classes generate the cohomology of Nakajima quiver varieties, however there are considerable additional technical difficulties not present in that setting: A first stage of the proof is devoted to producing a suitable modular compactification of the multiplicative quiver variety (or rather its Deligne-Mumford stack analogue). A major difference from the Nakajima quiver variety case arises already at this stage: one frequently relies on q being an appropriate tuple of primitive roots of unity to deduce that M q θ (α) parameterizes only stable representations, independently of the choice of θ ; whereas in [26], we assumed, without significant loss of generality, that θ was a generic stability condition. We note that such a genericity assumption here would exclude the possibility of applications to the character variety Char( g , G L n , q Id); hence we avoid it. Instead we identify a compactification by a "projective Artin stack" M, a quotient of a quasiprojective scheme by a reductive group whose coarse moduli space is a projective scheme. Known techniques [14,22] allow us to replace the Artin stack compactification by a projective Deligne-Mumford stack at no cost to the validity of our approach.
The second stage is to identify a complex on M q θ (α) × M that, roughly speaking, resolves the graph of the embedding M q θ (α) → M. Again, while this is morally similar to [26], the actual construction and proofs are more complicated and subtle. This is essentially because our compactification of the Nakajima quiver variety relied on a graded 3-Calabi-Yau algebra, whereas the compactification of M q θ (α) uses an algebra, denoted by A in the body of this paper, which may (conjecturally) be in most cases what one might call a "relative 2g-Koszul algebra", but which (as far as we know) is not known to be so. Fortunately it turns out that we can proceed as if the algebra A were known to have various desired properties in order to build a suitable complex, and then carry out some (occasionally delicate) calculations to check by hand that it has the properties we need. We note that the good behaviour of this complex is closely related to the question of whether the multiplicative preprojective algebra is 2-Calabi-Yau-a conjecture known in many important cases thanks to [21]. It can be hoped that our graded algebra A may thus be of some independent interest in relation to this question. Since in the generality in which we work here (and again unlike [26]), we do not know if the complex actually provides a resolution of the structure sheaf of the graph of the embedding, we instead rely on work of Markman [24] to show that an appropriate Chern class of the complex we build is the Poincaré dual of the fundamental class of the graph.
The final step is to deduce the theorem via usual integral transform arguments. In [26], we used Nakajima's result that the (integral) cohomology of a quiver variety is generated by algebraic cycles, hence is naturally isomorphic to a quotient of the cohomology of any compactification. Such an assertion is not true of the multiplicative quiver varieties M q θ (α). Instead, we rely on the beautiful fact that the cohomology of any reasonable smooth compactification-which is always Hodge-theoretically pure-surjects onto the pure part of the cohomology of any open subset. This yields the assertion of the theorem, which in any case would be the best possible result, given that the cohomology H * (BG, Q) is pure. The Hodge-theoretic nature of this result however necessitates working with rational cohomology. It is thus an interesting question to characterize the image of H * (BG, Z) in H * M q θ (α), Z .

Notation
Throughout, k denotes a field of characteristic 0. In Sects. 1 and 6, k = C.

Truncations of graded algebras
We will frequently use certain "truncations" of a Z ≥0 -graded algebra A in what follows. For a Z-graded vector space V and integer n, we write V ≥n = ⊕ m≥n V m , a vector space graded by {n, n + 1, . . . }. We note the vector space injection V ≥n → V that is the identity on the mth graded piece for m ≥ n.

Quivers, doubles, and triples
Let Q = (I , ) be a finite quiver, so that s, t : ⇒ I are the source and target maps: • . The double of Q is a quiver Q dbl = (I , H = ) with the same vertex set I as for Q and the set of arrows H = where is the arrow set of Q and is a set equipped with a bijection to , written a ↔ a * ∈ . We extend this bijection canonically to an involution on H = , still written a → a * , and decree s(a * ) = t(a), t(a * ) = s(a). For each arrow a ∈ H we write Fix an integer N ≥ 1. The graded tripled quiver Q gtr associated to Q (cf. Section 4 of [26]) is a quiver defined as follows. We give Q gtr the vertex set I gtr = I × [0, N ] where I is the vertex set of Q. If is the edge set of Q and H = the associated set of pairs of an edge together with an orientation, we give Q gtr the arrow set s(h, n) = (s(h), n) and t(h, n) = (t(h), n + 1); s(t i,n ) = (i, n) and t(t (i,n) ) = (i, n + 1).

Path algebras
Let S = i ke i be a commutative semisimple algebra over a field k, with orthogonal system of idempotents {e i }. Suppose A is an algebra with homomorphism S → A. We say that x ∈ A has diagonal Peirce decomposition if x ∈ i∈I e i Ae i , or equivalently if it lies in the centralizer Z A (S).
Let Q = (I , H ) be a quiver. The path algebra k Q of Q is defined as follows: Let S denote the finite-dimensional (semisimple commutative) k-algebra S = i∈I ke i with idempotents e i labelled by the vertices i ∈ I . We define an S-bimodule B = B(Q), with k-basis labelled by the arrows, where the S-bimodule structure takes arrows to be directed "left-to-right," so e i ae j = 0 unless i = s(a), j = t(a), and so that e s(a) ae t(a) = a. Then the path algebra k Q is defined to be the tensor algebra T S (B(Q)).
It is natural to grade the path algebra k Q of any quiver Q = (I , H )-for example, k Q dbl -using the normal grading on a tensor algebra, thus the semisimple algebra S lies in degree 0 and the arrows h ∈ H lie in degree 1.
If A is any S-algebra, we write A[t] for the associative S algebra obtained by adjoining a central variable t (thus every element of A commutes with t). The algebra is naturally bi-graded, but we will only use the total grading, with respect to which deg(t) = 1. Using the above grading for the path algebra k Q gtr , we obtain a graded algebra homomorphism where J denotes the two-sided ideal The graded algebra k Q gtr has the property k Q gtr ≥N +1 = 0, so we obtain a homomorphism The above discussion thus shows:

Universal localizations
We briefly review some aspects of universal localizations that may be unfamiliar to the reader, using Chapter 4 of [28] as our reference; see also [9]. Suppose that R is a ring with 1 and is a set of elements of R. Then there is a ring R with a homomorphism R → R that is universal with respect to the property that for every r ∈ , r becomes invertible in R , i.e. r has a two-sided inverse r −1 in R . The ring R is called the universal localization of R at ; an alternative notation that is sometimes preferable is −1 R. The universal localization is constructed as follows: letting −1 denote the set of symbols a −1 for a ∈ , we define This evidently has the universal property claimed, though very little else can be deduced about R from this construction. Another more illuminating construction (which allows one to invert morphisms between arbitrary projective modules) is given in Theorem 4.1 of [28].
We however will only need the following properties, which follow immediately from the universal property.

Proposition 2.3 Suppose R is a ring with 1.
( (3) Given a two-sided ideal I ⊆ R, let denote the image of in R/I and I denote the two-sided ideal in R generated by I . Then (R/I ) ∼ = R /I .

Multiplicative preprojective algebras
We review the multiplicative preprojective algebra of a quiver Q as defined in [10]. Given a quiver Q with double Q dbl = (I , H ), for each arrow a ∈ H of Q dbl , we define g a = 1 + aa * ∈ k Q dbl . Write L Q for the algebra obtained by universal localization of k Q dbl inverting = {g a | a ∈ H }. Identify the tuple q ∈ (k × ) I with the element i∈I q i e i ∈ S. Crawley-Boevey and Shaw choose an ordering of the arrows in H and define ρ CBS = −→ a∈H g (a) a − q (the arrow over the product indicates that it is taken in the chosen order). It is proven in [10, §2] that, up to isomorphism, the quotient algebra L Q /(ρ CBS ) does not depend on the choice of ordering. Thus, in this paper we specifically fix an ordering = {a 1 , . . . , a g } on the arrows in Q, and let (2.3) Definition 2.4 Following [10, Definition 1.2], the associated multiplicative preprojective algebra is defined to be where ρ CBS is defined as in (2.3).

Homogenized multiplicative preprojective algebras
A principal tool in this paper is a certain graded algebra A that "homogenizes" the multiplicative preprojective algebra q . Here we construct the algebra A and collect some basic facts about A and its relation to the multiplicative preprojective algebra q . Thus, fix a quiver Q. Recall that we consider k Q dbl [t] as a nonnegatively graded algebra, with the generators a ∈ H , t all in degree 1, and S = ⊕ i∈I ke i in degree 0. We let

Remark 2.5
Each G a has diagonal Peirce decomposition: more precisely, We note the obvious equalities G a a = aG a * , a * G a = G a * a * . (2.4) Given q ∈ (k × ) I , we identify q with q := i∈I q i e i ∈ k Q dbl , a sum of idempotents in the path algebra (which thus also has diagonal Peirce decomposition). Analogously to [10], we write L t for the universal localization of the algebra k Q dbl [t] in which the elements {G a , a ∈ H }, and t are inverted. The algebra L t is graded and contains invertible elements g a = t −2 ] 0 at the elements g a , a ∈ H . As above, fix an ordering = {a 1 , . . . , a g } on the arrows in Q. Write where (ρ) denotes the two-sided ideal generated by ρ.
(1) A is a graded algebra where a i , a * i and t have degree 1 (and S = i∈I ke i lies in degree 0).
(2) The universal localization of A obtained by inverting all G a , a ∈ H , and t, is a graded algebra, and t ∼ = q (Q)[t ±1 ] where q (Q) =: q denotes the multiplicative preprojective algebra of [10].
Proof This almost all follows from the above discussion. The isomorphism (2.7) of part (2) of Proposition 2.7 follows from Proposition 2.3.

Representations of kQ dbl and kQ gtr
Fixing some N ≥ 2g, where g is the number of arrows in Q, we form the gradedtripled quiver Q gtr associated to Q as above. 1 Given a dimension vector α ∈ Z I ≥0 for the quiver Q dbl , we write α gtr ∈ Z I ×[0,N ] ≥0 for the dimension vector for k Q gtr for which α gtr i,n = α i for all n ∈ [0, N ]. We write Rep(k Q dbl , α) for the space of representations of k Q dbl with on the I -graded vector space V with V i = k α i for all i ∈ I , so that V has dimension vector α and G = i G L(α i ) for the automorphism group; thus Similarly we write Rep(k Q gtr , α gtr ) for the space of representations of k Q gtr with dimension vector α gtr , and G gtr for the automorphism group.
As in the construction of Section 4.3 of [26], there is a natural "induction functor" from the category of representations of k Q dbl with dimension vector α to the category of representations of k Q gtr of dimension vector α gtr . The construction proceeds as follows. To a representation V of k Q dbl we may associate the Z ≥0 -graded vector space V [t], and let arrows h of Q dbl act as multiplication followed by shift-of-grading. This makes -module, and finally apply Lemma 2.2 to get a representation of k Q gtr : in fact, a representation of the quotient k Q gtr /J where J is as in (2.1).
More concretely, the above construction is the following. Suppose we have a representation V = (V i ) i∈I of k Q dbl of dimension vector α. We obtain a representation of k Q dbl [t] on a vector space V •,• of dimension vector α gtr defined by setting (1) the grading is given by V i,n := V i × {n} for all n ∈ [0, N ]; (2) the action of the generators t i,n of k Q gtr for (i, n) ∈ I × [0, N − 1] is given by The construction determines a morphism of algebraic varieties ("induction") G. Then the morphism Ind • is (G, G gtr )-equivariant. We thus get a natural G gtr -equivariant morphism

Representations of A and A [0,N]
Let A -Gr denote the category of graded left A-modules. We also consider the category A

Representations of A and 3 q
We note: In the opposite direction, we have a functor ( t ⊗ A −) 0 : A -Gr ≥0 → q -Mod. We have: Proof The first statement follows from the isomorphism t ∼ = q [t ±1 ] along with the equivalence of categories noted in the above remark. Indeed where the second isomorphism follow from universality. The second statement follows immediately from the definitions.

Representation spaces and group actions
Because the multiplicative preprojective algebra q is the quotient L Q /(ρ CBS ) of the localization L Q of k Q dbl by the ideal generated by ρ CBS , the space Rep( q , α) of left q -modules with dimension vector α is naturally a locally closed subscheme of Its image consists of those representations on which the elements t, G a act invertibly whenever their domain and target lie in the range [0, N ].

Semistability and stability
We next discuss (semi)stability of representations and the corresponding GIT quotients. For any quiver Q = (I , ) with dimension vector α ∈ Z I ≥0 , a GIT stability condition is given by θ ∈ Z I ≥0 satisfying i θ i α i = 0. The vector θ determines a character χ θ : i G L(α i ) → G m , χ(g i ) i∈I = i det(g i ) θ i , and the condition i θ i α i = 0 guarantees that the diagonal copy (G m ) of G m in i G L(α i ) lies in the kernel of χ ; we require this because (G m ) acts trivially on Rep(Q, α). Given dimension vectors β, α, we write β < α if β = α and β i ≤ α i for all i ∈ I .
We now turn to stability conditions for the doubled and tripled quivers Q dbl and Q gtr for a fixed quiver Q. Suppose θ is a stability condition for Q dbl and dimension vector α. We construct a stability condition θ gtr for Q gtr with dimension vector α gtr as follows. For a representation M of k Q gtr of dimension vector α gtr , we write δ i,n (M) := dim(M i,n ); we will write θ gtr as a linear combination of the δ i,n . Also, we note that it suffices to construct a rational linear functional θ gtr , since any positive integer multiple of θ gtr evidently defines the same stable and semistable loci. We fix an ordering on the vertices of Q, identifying I = {1, . . . , r }. and a positive integer T 0. We define: The proof is an easy adaptation of that of Proposition 4.12(4) of [26].
We remark that the above construction does not match [26]: there we chose to construct a stability θ gtr for Q gtr that would be nondegenerate if θ was, whereas here we ignore this possible requirement. While it would be possible to copy the construction of a stability θ gtr from [26] and prove analogues of the statements of [26], there are cases important to multiplicative quiver varieties in which it is not possible to find a stability condition for k Q dbl that is nondegenerate in the sense used in [26]: for example, the case when Q has a single vertex and loops based at that vertex, with dimension vector α = n > 1. However, again for multiplicative quiver varieties, in some interesting cases the choice of the parameter q can guarantee that every semistable representation of q is automatically stable (though not for numerical reasons, as nondegeneracy guarantees). Indeed, we say q = (q i ) i∈I ∈ (k × ) I is a primitive αth root of unity if q α := q α i i = 1 and q β = 1 for all 0 < β < α. We have: (1) Suppose that M is a representation of q with dimension vector α. Then q α = 1.
(2) In particular, if q is a primitive αth root of 1, then every representation of q of dimension vector α is θ -stable for every θ .
For example, if Q = ({ * }, E) where E has g loops at * , α = n, and q is a primitive nth root of 1, then every representation of q of dimension n is stable for every θ ; the corresponding moduli space of representations of q is the character variety Char( g , G L n , q Id) of the introduction.

Moduli stacks and resolutions
The moduli stacks Rep( q , α) θ -ss /G and Rep gr (A [0,N ] , α gtr ) θ gtr -ss /G gtr are never Deligne-Mumford stacks: the diagonal copy of G m in G, respectively G gtr , always acts trivially on Rep( q , α), respectively Rep gr (A [0,N ] , α gtr ) θ gtr -ss . Thus, the moduli stack of stable representations Rep( q , α) θ -s /G is always a G m -gerbe over the moduli space M q θ (α) s of stable representations. However, one can make a choice of subgroup S ⊂ G that ensures that the quotient stack Rep( q , α) θ−s /S is a Deligne-Mumford stack and that Rep( q , α) θ−s /S → M q θ (α) s is a finite gerbe (indeed a principal B H-bundle for a finite abelian group H ). Indeed, for example, we can choose any character ρ : G gtr → G m for which the composite with the diagonal embedding ρ • : G m → G m is nontrivial, hence surjective. Then S gtr := ker(ρ) has the property that G gtr = S gtr · (G m ) and similarly letting S = G ∩ S gtr we have G = S · (G m ). Moreover, since (G m ) is the stabilizer of every point of Rep( q , α) θ -s and H := (G m ) ∩ S is finite, we get:  (A [0,N ] , α gtr )/ / θ gtr G gtr : it is a closed subscheme of Rep(k Q gtr , α gtr )/ / θ gtr G gtr , which (as in [26]) is projective because k Q gtr has no oriented cycles, and hence is itself projective.
We may apply the results of [22] or [14] to Rep(k Q gtr , α gtr ) θ gtr -ss /S gtr and its closed substack M st to obtain a projective Deligne-Mumford stack (i.e., a Deligne-Mumford stack whose coarse space is a projective scheme) M st equipped with a projective morphism M st → M st that is an isomorphism over Rep( q , α) θ -s /S. The stack M st is itself, by construction, a global quotient of a quasiprojective variety by S, and thus we may apply equivariant resolution to resolve the singularities of M st , to obtain:

Bimodule of derivations
Recall that we have fixed an ordering = {a 1 , . . . , a g } on the arrows in Q. For j = 1, . . . , g we write [10, p. 190], the bimodule that is the target of the universal S-linear bimodule derivation of k Q dbl [t] satisfies

Let B denote the sub-(S[t], S[t])-bimodule of k Q dbl [t] spanned by the arrows, so that k Q dbl [t] is identified with the tensor algebra T S[t] (B). As in
is identified with a → 1 ⊗ a ⊗ 1. As in [10, p. 190], for the universal localization L t we also get L t with the obvious identification of the universal derivation δ L t /S [t] . We write: A. (4.1) The module P 1 is evidently projective as a bimodule. Via the above description, we obtain a collection of bimodule basis elements

An exact sequence
We write Write η i = e i ⊗ 1 = 1 ⊗ e i , i ∈ I , for the obvious bimodule generators of P 0 . Define graded bimodule maps by β(η a ) = aη s(a) − η t(a) a for arrows a of Q gtr , and where a = δ(G a ) (where δ denotes the universal derivation). It is then immediate that α(η i ) = e i · δ(ρ); in particular, letting θ : P 0 → (ρ)/(ρ 2 ) denote the map defined by θ( p ⊗ q) = pρq and writing φ for the isomorphism defined by (4.1), we have: Imitating the proof of Lemma 3.1 of [10] gives: where γ (p ⊗ q) = pq, is an exact sequence of Z-graded bimodules.
Proof As in [28,Theorem 10.3], one gets an exact sequence As in [10], splicing this sequence and the defining sequence for S [t] A and applying (4.4) gives a commutative diagram The vertical arrows φ, ψ are isomorphisms and θ is surjective, yielding the assertion.

Dual of the map P 0 (−2g)− → P 1 (−1)
Recall that the enveloping algebra of A over k [t] is We consider A e as a left A e -module where a ⊗ a ∈ A e acts by We remark that A e naturally also has a right A e -module structure commuting with the left A e -action, where a ⊗ a ∈ A e acts on the right by Given a finitely generated left A e -module, we form P ∨ = Hom A e (P, A e ), the dual over the enveloping algebra; by the above discussion, this module has a right A e -module structure, which we can identify with a left A e -module structure via the isomorphism We now want to calculate the dual α ∨ of the map α of (4.2) using the formula (4.3). Note that (G a ) = aδ(a * ) + δ(a)a * = aη a * + η a a * .
We thus find from Formula (4.3) that the η a -component of α is given by and zero otherwise. Let {η ∨ a } denote the basis of P ∨ 1 dual to the basis {η a } of P 1 ; we note that η ∨ a ∈ e t(a) P ∨ 1 e s(a) . (4.6) The above formulas then imply: (4.7)

Lemma 4.2 For all a ∈ , we have
Proof These formulas follow by direct calculation using 4.7.

Lemma 4.3 If a ∈ H, s(a) = i, then G a Dη
Proof The element D is a product of elements of diagonal Peirce type, hence itself is of diagonal Peirce type. Thus, using e s(a) η ∨ i = 0 = η ∨ i e s(a) , we get This completes the proof.
Suppose now that M is a graded right t -module; then M = M ≥0 is a graded right A-submodule of M. For example, we could take M = t itself, as in (2.7). We consider the map Proof (1) We first prove that m G a j Dη ∨ i − Dη ∨ i G a j ∈ Im(1 M ⊗ α ∨ ) by (strong) induction on j. Base case j = 1. By Lemma 4.3, the assertion is true for i = s(a 1 ). From Lemma 4.2, we have This completes the base case.

Induction step Assume m G a k Dη
where the last equality applies the inductive hypothesis for each k < j (and various m ∈ M, see Remark 4.4). This completes the induction step, thus proving the assertion for the elements G a j Dη ∨ i − Dη ∨ i G a j . The proof for G a * j Dη ∨ i − Dη ∨ i G a * j follows the analogous descending induction on j.
(2) Taking note of Remark 4.4, from (4.7) we have Applying part (1) of the proposition to the right-hand side of this formula gives where the last equality uses (2.4); in particular this gives the first assertion of Part (2) of the proposition. The second assertion follows similarly.

The complex (4.5) and the Hom-functor
Let M, N be graded left A-modules such that M is finitely generated and projective as a k[t]-module. To the exact sequence we apply the functor Hom A (−, N ) to obtain an exact sequence We continue the sequence (5.1) using Thus, we would like to compute the cokernel of the map (5.2).
We note the following identities, which are immediate from adjunction: N ).

The Ext-complex
Fix N ≥ 2g. Let V be a finite-dimensional representation of q of dimension vector α, and let V = V [t] be the corresponding graded A-module as in Sect. 3.2, and specifically as in Lemma 3.3. Suppose W is a Z ≥0 -graded A [0,N ] = A/A ≥N +1 -module, identified with a representation of Q gtr that has dimension vector α gtr . Thus τ [0,N ] V is also identified with a representation of Q gtr that has dimension vector α gtr . Let P • denote the complex of (4.2). We consider the complex Hom A (P • ⊗ A V , W ). Since the sources and target of the Homs in this complex are graded A-modules, each Hom-space can be regarded as a graded vector space; we write for its degree 0 graded piece. As in [26], using Lemma 5.3 we may identify Ext with: where ∂ 0 = β ∨ 0 and ∂ 1 = α ∨ 0 .

Cohomology of varieties and stacks
In the remainder of the paper, the base field k is assumed to be C.
Here as throughout the paper, we use H * (X ) to denote cohomology with Qcoefficients, and H BM * (X ) to denote Borel-Moore homology with Q-coefficients; if X is a smooth Deligne-Mumford stack, there is a canonical isomorphism H * (X ) ∼ = H BM * (X ).

Mixed hodge structure on the cohomology of an algebraic stack
Suppose that X is an algebraic stack of finite type over C. The Gysin map satisfies the projection formula: for classes c ∈ H * (X ), c ∈ H * (Y ), we have We now consider the general case. By the assumptions, X and X • are (separated) quasi-projective smooth Deligne-Mumford stacks that are global quotients. By Theorem 1 of [23], there exist a smooth quasi-projective scheme W and a finite flat LCI morphism W → X; the fiber product X • × X W → X • is then also finite, flat, and LCI. Using the commutative square and base change, we find: are surjective (indeed, q * q * and q • * (q • ) * are multiplication by the degree of q). (2) Since the Gysin maps q • * , q * are morphisms of mixed Hodge structures by Proposition 6.2, , by the conclusion of the previous paragraph.
The assertion is now immediate.

Markman's formula for Chern classes of complexes
Suppose that M is a smooth Deligne-Mumford stack and is a complex of locally free sheaves on M of ranks r −1 , r 0 , r 1 respectively. Proposition 6.5 (Lemma 4 of [24]) Suppose that ⊂ M is a smooth closed substack of pure codimension m, and that the complex C of (6.2) satisfies: (1) H −1 (C) = 0, Remark 6. 6 Markman's Lemma 4 is ostensibly stated for smooth varieties M, but Section 3 of op. cit. generalizes the assertion to smooth Deligne-Mumford stacks.

Proofs of Theorems 1.5 and 1.2
Fix a quiver Q, stability condition θ for Q dbl and the corresponding stability condition θ gtr for Q gtr as in Sect. 3.5. Choosing a subgroup S ⊂ G as in Sect. 3.6, we obtain a "graph immersion" in a product of Deligne-Mumford stacks Rep( q , α) θ -s /S ι − → Rep( q , α) θ -s /S × Rep gr (A [0,N ] , α gtr ) θ gtr -ss /S gtr . (6.3) We write ι for the immersion and = Im(ι) for its image, a smooth closed substack. We remark that ι is not a closed immersion unless H is trivial; however, the morphism ι identifies ∼ = Rep( q , α) θ -s /S × B H. , α gtr ) θ gtr -ss /S gtr come equipped with universal representations V , W respectively. The complex Ext defined in Sect. 5.2 descends to the product Rep( q , α) θ -s /S × Rep gr (A [0,N ] , α gtr ) θ gtr -ss /S gtr . We recall from Proposition 3.13 the compactification Rep( q , α) θ -s /S of Rep( q , α) θ -s /S. This carries a natural map to Rep gr (A [0,N ] , α gtr ) θ gtr -ss /S gtr which induces an isomorphism on the open substack Rep( q , α) θ -s /S. Pulling the complex Ext back to the product Rep( q , α) θ -s /S × Rep( q , α) θ -s /S, we get a complex that we will denote C.
Direct calculation shows that the rank of C is m − 2 = codim( ) − 2 (we note that its rank depends only on Q and α: only the differentials distinguish between the ordinary and multiplicative preprojective algebras). It follows from Proposition 5.4 that C has the following properties: (1) H −1 (C) = 0, (2) H 1 (C) and H 1 (C ∨ ) are set-theoretically supported on , and their schemetheoretic restrictions to are line bundles.
Thus, in order to show that satisfies the hypotheses of Proposition 6.5, it suffices to show that is the scheme-theoretic support of both H 1 (C) and H 1 (C ∨ ). We do this by considering a morphism proof of Proposition 2.4(ii) of [26]), this completes the proof of Theorem 1.5, hence also of Theorem 1.2.

Proof of Theorem 1.4
The proof of Theorem 1.4 is essentially identical to that of Theorem 1.6 of [26] (and we note that Theorem 1.4 holds whenever k is any field of characteristic zero and q ∈ k × ). Indeed, the assumption that there is a vertex i 0 ∈ I for which α i 0 = 1 guarantees the following. First, we may take S = i =i 0 G L(α i ), which acts freely on the stable locus: thus, M q θ (α) s is a fine moduli space for stable representations of q . Second, exactly as in the proof of Theorem 1.6 of [26], in the complex (5.3), there are direct sum decompositions L(V 0 , W 0 ) = Hom(V 0,i 0 , W 0,i 0 ) ⊕ ⊕ i =i 0 Hom(V 0,i , W 0,i ) and so that the complex obtained by modifying (5.3) given by has no cohomology at the ends, and in the middle has cohomology H that is a rank m = codim( ) vector bundle. Moreover, the remaining map k = Hom(V 0,i 0 , W 0,i 0 ) → E(V 0 , W 1 ) defines a section s of H whose scheme-theoretic zero locus is Z (s) = .
The remainder of the proof now copies that of Theorem 1.6 of [26].