The integrality conjecture and the cohomology of preprojective stacks

We study the Borel-Moore homology of stacks of representations of preprojective algebras $\Pi_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion $\Pi_Q[x]$. Via a result on the supports of the BPS sheaves for $\Pi_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of $\Pi_Q[x]$-modules, and define BPS sheaves for stacks of $\Pi_Q$-modules. These are mixed Hodge modules on the coarse moduli space of $\Pi_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure, and thus that the Borel-Moore homology of stacks of $\Pi_Q$-modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of $\Pi_Q$-modules. Among these and other applications, we use our results to prove positivity of a number of"restricted"Kac polynomials, determine the critical cohomology of $\mathrm{Hilb}_n(\mathbb{A}^3)$, and the Borel-Moore homology of genus one character stacks, as well as various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of preprojective algebras, including the PBW theorem, and torsion-freeness.

1. Introduction 1.1. Background. This paper concerns the Borel-Moore homology of stacks of representations of preprojective algebras Π Q , which play a prominent role in many branches of mathematics, and which we study through the prism of cohomological Donaldson-Thomas theory and BPS cohomology. The Borel-Moore homology of stacks of preprojective algebras occurs as the underlying vector space of the cohomological Hall algebra containing all raising operators for the cohomology of Nakajima quiver varieties [33,34], which themselves can be presented as certain stacks of semistable representations of preprojective algebras. More generally, stacks of representations of preprojective algebras model the local geometry of complex 2-Calabi-Yau categories possessing good moduli spaces [9], for example coherent sheaves on K3 and Abelian surfaces, Higgs bundles on smooth projective curves, local systems on Riemann surfaces, and moduli of semistable objects in Kuznetsov components.
Via dimensional reduction we study the Borel-Moore homology of the stack of Π Q -modules by relating it to the BPS sheaves for the stack of objects in the 3-Calabi-Yau completion C ΠQ (as defined by Keller in [25]) of the category of Π Qmodules. This paper is devoted to understanding the BPS sheaves (as defined in [10]) of the 3CY categories C ΠQ formed this way. By studying these BPS sheaves and the associated BPS cohomology, we prove a number of theorems regarding the Borel-Moore homology of stacks of Π Q -representations, Nakajima quiver varieties, stacks of coherent sheaves on surfaces, as well as vanishing cycle cohomology of Hilb n (A 3 ), and vanishing cycle cohomology of stacks of objects in C ΠQ .
1.2. The role of purity. In Donaldson-Thomas theory, as well as many of the other subjects this paper touches on, we are typically interested in motivic invariants. See e.g. [22,27] and references therein for extensive background on motivic DT theory. This just means that we are interested in invariantsχ of objects in a triangulated category D that factor through the Grothendieck group of D, i.e. if V ′ → V → V ′′ is a distinguished triangle in D then we requirẽ Alternatively, by "motivic", people mean invariants of varieties X such that if U ⊂ X is open, with complement Z, thenχ(X) =χ(U ) +χ(Z) (the "cut-and-paste relation"). The link between the two meanings is provided by the distinguished triangle H c (U, Q) → H c (X, Q) → H c (Z, Q), so that a motivic invariant in the first sense induces one in the second sense. A very basic example of a motivic invariant is the Euler characteristic of a complex of vector spaces χ(V ) = i∈Z (−1) i dim(V i ). A basic example of a nonmotivic invariant is the Poincaré polynomial P (V, q) = i∈Z dim(V i )q i ; since the connecting morphisms in a long exact sequence of vector spaces may be nonzero, the Poincaré polynomial may not satisfy (1), for example P (H c (A 1 , Q), q) = q 2 = P (H c (C * , Q), q) + P (H c (pt, Q), q).
1.2.1. In refined Donaldson-Thomas theory the invariantχ that we consider is a motivic invariant that is defined via Hodge theory. Recall that a Hodge structure on a rational vector space V is the data of an ascending weight filtration W • V , along with a descending Hodge filtration F • V C of the complexification, such that the Hodge filtration induces a weight n Hodge structure on the nth piece Gr W n (V ) of the associated graded object with respect to the weight filtration.
Since both the E series and weight series involve an alternating sum over cohomological degrees, it is easy to see that they are motivic invariants. Enumerative theories like DT theory, and point-counting over finite fields, provide powerful tools for determining motivic invariants. However, from a naive point of view they may produce the "wrong kind" of invariants, since we are often most interested in the Poincaré polynomial of cohomologically graded vector spaces, or might even hope to describe the cohomology itself (rather than its class in some Grothendieck group). We say that a cohomologically graded mixed Hodge structure L is pure if its ath cohomologically graded piece is pure of weight a, i.e. if Gr W b H a (L) = 0 for b = a. Our interest in pure mixed Hodge structures comes from the fact that if L is pure then P (L, q) = χ wt (L, q). Moreover, when the Borel-Moore homology of a stack is pure we have a much better chance of being able to actually calculate it, as we will demonstrate in this paper.
1.3. The purity theorem. Let Q be a quiver with vertices Q 0 and arrows Q 1 . The quiver Q, which is the double of Q, is obtained by adding an arrow a * for every arrow of a, with the reverse orientation. Then the preprojective algebra is defined as the quotient of the free path algebra of Q: We define N := Z ≥0 . Let d ∈ N Q0 be a dimension vector for Q. Define is pure, of Tate type.
We prove a more general version of Theorem A, concerning Borel-Moore homology of stacks of semistable Π Q -modules: see §6, and Theorem 6. 4.
In Theorem A the symbol ∨ denotes the dual in the category of cohomologically graded mixed Hodge structures. Purity means that Deligne's mixed Hodge structure on each cohomologically graded piece H n c (M(Π Q ) d , Q) is pure of weight n, and the statement that a cohomologically graded mixed Hodge structure L is of Tate type is the statement that we can write given the usual weight 2 pure Hodge structure, concentrated in cohomological degree 2. Purity is the further statement that a m,n = 0 for n = 0.
1.3.2. Theorem A concerns compactly supported cohomology. Since µ −1 Q,d (0) is a cone, and hence homotopic to a point, there is an isomorphism in usual singular cohomology, and it is known that the right hand side of (3) is pure [11]. On the other hand, compactly supported cohomology is not preserved by homotopy equivalence, and the highly singular nature of µ −1 Q,d (0)/ GL d means that its compactly supported cohomology is a great deal more complicated than its cohomology. In fact, purity requires an essentially new type of argument, requiring the full force of cohomological Donaldson-Thomas theory. In particular, outside of finite type Q, there is no way known (to date) of proving this purity statement without invoking the cohomological integrality theorem for the Donaldson-Thomas theory of quivers with potential, along with dimensional reduction.

1.3.3.
Categorification. If L is a cohomologically graded mixed Hodge structure that is pure, of Tate type, then its isomorphism class is determined by its Poincaré/Hodge/weight/E series. The weight series of the mixed Hodge structures (2) are described in terms of the Kac polynomials [23] of Q, via the results of [21] and [32], and so Theorem A enables us to calculate the compactly supported cohomology H c (M(Π Q ) d , Q) itself. Purity enables us to go beyond motivic invariants and actually determine the compactly supported cohomology of the stack M(Π Q ) d , not just its class in some Grothendieck group.

1.3.4.
Okounkov's conjecture. Theorem A is a singular stack-theoretic cousin of the result that the cohomology of Nakajima quiver varieties is pure, with Hodge polynomial expressible as a polynomial in xyz 2 (this can be obtained by combining the proof of [18,Thm.1] with [20, Thm.6.1.2(3)]). In fact we recover this result (Corollary 6.8). The purity of Nakajima quiver varieties provides one of the main motivations for the purity statement in Theorem A.
In a little more detail, it is conjectured that the cohomological Hall algebra A ΠQ obtained by taking the direct sum of H BM (M(Π Q ) d , Q) across all dimension vectors d is isomorphic to the positive half of the Yangian Y MO,Q constructed by Maulik and Okounkov in [29]. This in turn would imply that the graded dimensions of g MO,Q are given by Kac polynomials, as conjectured by Okounkov. Since the algebra Y MO,Q is defined as a subalgebra of the endomorphism algebra of the cohomology of Nakajima quiver varieties, the purity of Y MO,Q follows from purity for these quiver varieties. Our purity theorem provides evidence towards the conjecture that A ΠQ ∼ = Y + MO,Q .
1.4. From Donaldson-Thomas theory to symplectic geometry. Consider the following general setup, of which our situation with X(Q) d being acted on by GL d is a special case. Let X be a complex symplectic manifold, with the affine algebraic group G acting on it via a Hamiltonian action, with (G-equivariant) moment map µ : X → g * . Then define the function This function is G-invariant, and so defines a function on the stack-theoretic quotient g : (X × g)/G → C.
Via dimensional reduction [5,Thm.A.1] there is a natural isomorphism in compactly supported cohomology where φ g Q is the mixed Hodge module complex of vanishing cycles for g. This explains the appearance of vanishing cycles in what follows.
Note that φ g Q is supported on the critical locus of g. A guiding principle for Donaldson-Thomas theory (e.g. as expressed in [45]) is that a given moduli stack N of coherent sheaves on a Calabi-Yau 3-fold can be locally expressed as the critical locus of a function g on some smooth ambient stack M. Donaldson-Thomas invariants are then defined by taking invariants, factoring through the Grothendieck group of mixed Hodge structures, of H c (M, φ g Q) = H c (crit(g), φ g Q) = H c (N, φ g Q).
The link between Donaldson-Thomas theory and symplectic geometry is completed by the observation of [14,Sec.4.2] (see also [32]) that associated to any quiver Q there is a tripled quiver with potential (Q,W ) such that (X(Q) d × gl d )/ GL d is identified with the smooth stack of d-dimensional representations of CQ, and the critical locus of the function Tr(W ) (which is the function g from (4) Cohomological DT theory enables us to prove powerful theorems regarding the right hand side of (6), which we use to deduce results regarding the left hand side.
1.5. BPS sheaves and their supports. We prove Theorem A via an analysis of BPS sheaves. These were introduced in [10], in the course of the proof of the relative cohomological integrality/PBW theorem for the critical cohomological Hall algebras introduced by Konstevich and Soibelman [28]. This theorem states that for a symmetric quiver Q ′ with potential W ′ , and stability condition ζ the direct image of the mixed Hodge module of vanishing cycles for the function Tr(W ′ ) along the morphism JH from the moduli stack of ζ-semistable CQ ′ -modules to the coarse moduli space is obtained by taking the free symmetric algebra generated by an explicitly defined mixed Hodge module BPS ζ Q ′ ,W ′ , called the BPS sheaf, tensored with a half Tate twist of H(BC * , Q). The BPS cohomology BPS ζ Q ′ ,W ′ is defined to be the hypercohomology of this sheaf.
Although the direct image of the mixed Hodge module of vanishing cycles along JH is concentrated in infinitely many cohomological degrees, this BPS sheaf is a genuine mixed Hodge module, i.e. its underlying complex of constructible sheaves is a perverse sheaf. This is what we mean by "integrality". As a consequence, for any dimension vector d ∈ N Q0 , the associated BPS cohomology BPS ζ Q ′ ,W ′ ,d lives in bounded degrees. 1.5.1. Unless the pair Q ′ , W ′ is quite special, it is difficult to actually determine BPS ζ Q ′ ,W ′ . In particular, the hypercohomology of this sheaf can fail to be pure. However, in this paper, we show that for the quiverQ with potentialW appearing in the previous section, the situation is much more promising. A key role is played by a support lemma, Lemma 4.1, which imposes strong restrictions on the support of the BPS sheaf in the case (Q ′ , W ′ ) = (Q,W ) for Q any quiver. This is a crucial lemma on the way to proving purity of BPS cohomology for Jac(Q,W ). In combination with this purity result, the lemma also enables us to provide some of the first nontrivial calculations of BPS sheaves; see in particular §5, lifting the work of Behrend, Bryan and Szendrői on motivic degree zero invariants to the level of BPS sheaves. This lemma is also one of the crucial ingredients in proving the purity of the BPS sheaves BPS ζQ ,W themselves, and the definition of the "less perverse filtration": see [8,9] for developments in this direction.
1.6. 2d BPS sheaves. Aside from purity of BPS cohomology, one of the main applications of the support lemma is that it enables us to define 2d BPS sheaves: be the morphism extending a Π Q -module to a CQ-module by letting each of the extra loops ω i act via scalar multiplication by z ∈ A 1 . Then there is a Verdier self-dual mixed Hodge module , which we call the 2d BPS sheaf, such that The pure intersection complex IC A 1 (Q) is defined in §3.1. The 2d BPS sheaves enjoy a number of properties: (1) They categorify the Kac polynomials; we elaborate upon this in §8.
(2) They are Verdier self-dual (see §4.2), which we expect to have a role in producing geometric doubles of BPS Lie algebras. (3) Their hypercohomology carries a Lie algebra structure, the so-called BPS Lie algebra g ΠQ . (4) They are pure as mixed Hodge modules, enabling us to relate generators of g ΠQ to intersection cohomology. These last two properties are explained and explored in the paper [8], which is devoted to the further study of 2d BPS sheaves. 1.7. Serre subcategories. Working with the BPS sheaf BPS ΠQ , as opposed to its hypercohomology, enables us to calculate the compactly supported cohomology of substacks of the stack M(Π Q ) corresponding to Serre subcategories, even when these stacks are not pure, leading to e.g. applications for genus one character stacks.
A Serre subcategory S ⊂ CQ -mod is a full subcategory such that for every short exact sequence 0 which is defined to be the substack of CQ-modules belonging to S, is the same as restricting to the preimage of a particular subspace under the semisimplification map from the stack of CQ-modules to the coarse moduli space M(Q).
Because many of our results can be stated in the category of mixed Hodge modules 2 on M(Q), we can prove results on the Borel-Moore homology of M(Π Q ) S via restriction functors and base change.
For example consider the quiver Q Jor , with one vertex and two loops X, Y , and set S to be the category of representations for which the two loops X and Y are sent to invertible morphisms. The resulting stack is the character stack for the genus one Riemann surface, and we use the above ideas to calculate its compactly supported cohomology, even though it is not pure; see §5.2 for details.
1.8. Structural results. We prove two general structural results regarding the compactly supported cohomology of stacks M(Π Q ) S for arbitrary finite quiver Q and Serre subcategory S, stated below as Theorems C and D. The first is a kind of cohomological wall-crossing isomorphism: Theorem C. Let Q be a quiver, let S ⊂ CQ -mod be a Serre subcategory, let ζ ∈ H Q0 + be a stability condition, and let ̺ be the slope function defined with respect to ζ. Then there is an isomorphism of N Q0 -graded mixed Hodge structures and M(Π Q ) S,ζ -ss d is the stack of d-dimensional ζ-semistable Π Q -modules in S.
1.8.1. Taking the Hodge series of both sides of (8), there is an equality of generating series The compactly supported cohomology of M(Π Q ) S,ζ -ss d can fail to be pure, and fail to be of Tate type, but we show that the isomorphism (8) exists nonetheless, and hence, taking the Hodge series of both sides, equation (9) holds. We explain how a specialisation of a special case of equation (9) yields Hausel's formula for the Betti polynomials of Nakajima quiver varieties [18] in Section 7.3.
2 In cohomological Donaldson-Thomas theory, this is what is meant by the "relative" in the relative integrality conjecture.
1.8.2. PBW/integrality isomorphism. Our next result is an analogue of the PBW isomorphism from Donaldson-Thomas theory. We fix a quiver Q, a stability condition ζ ∈ H Q0 + , a slope θ ∈ (−∞, ∞), and a Serre subcategory S of the category of CQ-modules. We write This graded mixed Hodge module carries a Hall algebra structure, see §9.1 for a generalisation of the construction.
Theorem D. Let ̺ be the slope function defined with respect to a stability condition ζ ∈ H Q0 + , let θ ∈ (−∞, ∞) be a slope. Define the 2d BPS sheaf BPS ζ ΠQ,θ as in Theorem B and the BPS cohomology to be the mixed Hodge structure Then there is an isomorphism Moreover, there is a PBW isomorphism Since BPS ζ ΠQ,d is Verdier self-dual by Theorem B, BPS S,ζ ΠQ,d is the hypercohomology of the !-restriction of the BPS sheaf on the coarse moduli space of ζ-semistable d-dimensional CQ-modules to the subspace of points representing modules in S.
1.9. Positivity of restricted Kac polynomials. For an arbitrary quiver Q, it was proven by Kac in [23] that for each dimension vector d ∈ N Q0 there is a polynomial a Q,d (q) ∈ Z[q] which is equal to the number of absolutely indecomposable d-dimensional representations of Q over the field of order q, whenever q is equal to a prime power.
In the case of the degenerate stability condition, for which all modules are semistable of the same slope, and so the superscript ζ and the subscript θ can be dropped, (11) gives Taking weight series of both sides of (12) yields where is the "S-restricted Kac polynomial". The right hand side of (13) is defined in terms of the plethystic exponential: where the + subscript means that b i,0 = 0 for all i ∈ Z.
1.9.1. Calculating the right hand side of (14) looks daunting, but the mere existence of isomorphism (12) can tell us something highly nontrivial about a S Q,d (q 1/2 ) without knowing how to do this calculation (or even knowing what the definition of BPS ΠQ,d or BPSQ ,W ,d is!). Namely, if the left hand side of (12) is pure, then the BPS cohomology BPS S ΠQ,d must also be pure, and so a S d (q 1/2 ) has positive coefficients (expressed as a polynomial in −q 1/2 ).
In particular, for the case S = CQ -mod, the S-restricted Kac polynomial is the same as Kac's original polynomial, and our purity theorem (Theorem A) implies Kac's positivity conjecture, originally proved by Hausel, Letellier and Rodriguez-Villegas [19].
1.9.2. A rich source of proper Serre subcategories of CQ is provided by demanding nilpotence of certain paths in CQ. Via this construction Bozec, Schiffmann and Vasserot [3,43] define the subcategory of nilpotent, *-semi-nilpotent and *-stronglysemi-nilpotent CQ-representations; see §7.1 for definitions. By the above method, in §8 we prove positivity of all of the resulting polynomials: Theorem E (Theorem 8.2, Remark 8.4). Let Q be an arbitrary finite quiver, and let d ∈ N Q0 be a dimension vector. Setting S to be any out of the the full subcategory of nilpotent, *-semi-nilpotent, or *-strongly-semi-nilpotent CQ-representations, the S-restricted Kac polynomial a S Q,d (q) has positive coefficients. 1.10. Structure of the paper. In §2 we give definitions and notation concerning quivers. In Section 3 we collect together all required definitions and background theorems from noncommutative/cohomological Donaldson-Thomas theory. Then in §4 we prove Theorem A, regarding purity of the compactly supported cohomology of the stack M(Π Q ). Along the way we prove the support lemma (Lemma 1.1 above) and Theorem B.
In §5 we focus on the Jordan quiver, corresponding to "degree zero" DT theory. The associated Jacobi algebra Jac( Q Jor ,W ) is isomorphic to the commutative algebra C[x, y, z], so that our work makes contact with classical algebraic geometry. We determine the BPS sheaves for the pair Q Jor ,W , and thus determine the compactly supported/vanishing cycle cohomology of various stacks in geometry and nonabelian Hodge theory. In particular, in this section we calculate the vanishing cycle cohomology of Hilb n (A 3 ), via a proof that it is pure, categorifying the results of Behrend, Bryan and Szendrői [1].
In §6 we turn back to the geometry of representations of Π Q for a general Q. Thanks to a second support lemma (Lemma 6.3), essentially all moduli spaces and stacks of representations of Π Q -representations have a (categorically) 3-dimensional analogue, by which we mean that their compactly supported cohomologies fit into isomorphisms like (6), and so are recovered from the DT theory of Jac(Q,W ). This enables us to prove a generalisation of Theorem A incorporating stability conditions.
In §7 we use this second support lemma to prove Theorems C and D. Then in §8 we use the PBW isomorphism in order to relate restricted Kac polynomials to hypercohomology of restrictions of BPS sheaves, and prove Theorem E.
In Sections §9 and 10 we explore the implications of purity for the structure of the cohomological Hall algebra A ΠQ built out of H c (M(Π Q ), Q). Firstly, in §9 we use equivariant formality (a consequence of purity) to relate this Hall algebra to Hall algebras defined with extra equivariant parameters, introduced by letting a torus T scale the arrows ofQ. Then in §10 we prove (a generalisation of) Conjecture 4.4 of [42], stating that the cohomological Hall algebra A τ,ΠQ for an arbitrary preprojective algebra Π Q , deformed by incorporating extra equivariant parameters, is naturally a subalgebra of an elementary shuffle algebra. With these results it becomes possible to perform concrete calculations in A ΠQ , and we finish with some example applications. In particular, we prove that the cohomological Hall algebra A Coh(A 2 ) of finite length coherent sheaves on A 2 , without any extra equivariant parameters, is not commutative, and that the cohomological Hall algebra A τ,ΠQ can fail to be torsion free if T is chosen to be one of certain bad 1-dimensional tori.

Conventions.
• For G a complex algebraic group, we set H G := H(BG, Q).
• All functors are assumed to be derived unless explicitly stated otherwise.
• All quivers are assumed to be finite.
• For X a complex variety, or global quotient stack, we continue to denote • We continue to write N = Z ≥0 .
• Wherever an object appears with a subscript that is a bold Roman letter, that letter refers to a dimension vector, and • d is the subobject corresponding to that dimension vector. If any such object appears with a Greek letter such as θ as a subscript, then θ will refer to a slope, and • θ will refer to the subobject corresponding to dimension vectors in Λ ζ θ . Finally, if an expected subscript is missing altogether, then the entire object is intended.
• We generally use capital Roman letters to refer to spaces of representations before taking any kind of quotient, calligraphic letters to refer to GIT moduli spaces, and fraktur letters to refer to moduli stacks. • For D a triangulated category equipped with a t structure, we define the total cohomology functor H( Convention 1.2. Where a space or object is defined with respect to a stability condition ζ, that stability condition will appear as a superscript. In the event that the superscript is missing, we assume that ζ is the degenerate King stability condition (i, . . . , i) ∈ H Q0 + . With respect to this stability condition all representations have the same slope and are semistable, semisimple representations are the polystable representations, and the stable representations are exactly the simple ones.
1.12. Acknowledgements. During the writing of this paper, I was a postdoctoral researcher at EPFL, supported by the Advanced Grant "Arithmetic and physics of Higgs moduli spaces" No. 320593 of the European Research Council. During redrafting of the paper, I was supported by the starter grant "Categorified Donaldson-Thomas theory" No. 759967 of the European Research Council. I was also supported by a Royal Society university research fellowship.
I would like to thank Sasha Minets, Tristan Bozec, Olivier Schiffmann, Eric Vasserot, Davesh Maulik and Victor Ginzburg for illuminating conversations that contributed greatly to the paper. In particular, the idea for the proof of the crucial "support lemma" (Lemma 4.1) came from seeing Victor Ginzburg talk about the results of [13] at the Warwick EPSRC symposium "Derived Algebraic Geometry, with a focus on derived symplectic techniques", and the final section of the paper benefitted greatly from Sasha Minets' careful reading of an earlier draft.

Quiver representations
2.1. Quivers and potentials. Throughout the paper, Q will be used to denote a finite quiver, i.e. a pair of finite sets Q 0 and Q 1 (the vertices and arrows, respectively), and a pair of maps s, t : Q 1 → Q 0 (taking an arrow to its source and target, respectively). We denote by CQ the path algebra of Q, i.e. the algebra over C having as a basis the paths in Q, with structure constants for the multiplication given by concatenation of paths. For each vertex i ∈ Q 0 , there is a "lazy" path of length 0 starting and ending at i, and we denote by e i the resulting element of CQ.
A potential on a quiver Q is an element W ∈ CQ/[CQ, CQ] vect . A potential is given by a linear combination of cyclic words in Q, where two cyclic words are considered to be the same if one can be cyclically permuted to the other. If W is a single cyclic word, and a ∈ Q 1 , we define and we extend this definition linearly to general W . We define Jac(Q, W ) := CQ/ ∂W/∂a|a ∈ Q 1 , the Jacobi algebra associated to the quiver with potential (Q, W ). We will often abbreviate "quiver with potential" to just "QP".
Given a quiver Q, we denote by Q the quiver obtained by doubling Q. This is defined by setting Q 0 := Q 0 and Q 1 = {a, a * |a ∈ Q 1 }, and extending s and t to maps Q 1 → Q 0 by setting s(a * ) =t(a) t(a * ) =s(a).
We denote by Π Q the preprojectve algebra of Q, defined by We denote byQ the quiver obtained from Q by setting where each ω i is an arrow satisfying s(ω i ) = t(ω i ) = i. If a quiver Q is fixed, we define the potentialW as in [14,Sec.4.2] and [32] by setting If A is an algebra, we denote by A -mod the category of finite-dimensional Amodules.
Proposition 2.1. Define C ΠQ to be the category whose objects are pairs (M, f ), where M is a finite-dimensional Π Q -module, and f ∈ End ΠQ -mod (M ), and define Hom CΠ Q (M, f ), (M ′ , f ′ ) to be the subspace of morphisms g ∈ Hom ΠQ -mod (M, M ′ ) such that the diagram Then there is an isomorphism of categories Proof. From the relations ∂W /∂ω i , for i ∈ Q 0 , we deduce that the natural inclusion CQ ⊂ CQ induces an inclusion Π Q ⊂ Jac(Q,W ). So a Jac(Q,W )-module is given by a Π Q -module M , along with linear maps M (ω i ) ∈ End C (e i ·M ) satisfying These are precisely the conditions for the elements {M (ω i )} i∈Q0 to define an endomorphism of M , considered as a Π Q -module. and We define gl d = i∈Q0 gl di (C), and define Then as substacks of M(Q) d , there is an equality µ −1 Q,d (0)/ GL d = M(Π Q ) d . As in the introduction, we define the function We define M(Q) ω -nilp d ⊂ M(Q) d to be the reduced stack defined by the vanishing of the functions The geometric points of M(Q) ω -nilp d over a field extension K ⊃ C correspond to d-dimensional KQ representations ρ such that for each i ∈ Q 0 , the endomorphism ρ(ω i ) is a nilpotent K-linear endomorphism.
A stability condition for Q is defined to be an element of H Q0 Definition 2.2. For a fixed stability condition ζ ∈ H Q0 + , we define the central charge We define the slope of a dimension vector d ∈ N Q0 \ {0} by setting If ρ is a representation of Q, we define ̺(ρ) := ̺(dim(ρ)). A representation ρ is called ζ-semistable if for all proper subrepresentations ρ ′ ⊂ ρ we have ̺(ρ ′ ) ≤ ̺(ρ), and ρ is called ζ-stable if the inequality is strict.
We will always assume that our stability conditions are King stability conditions, meaning that for each 1 i ∈ N Q0 in the natural generating set, ℑm(Z(1 i )) = 1 and If ζ is a King stability condition, then for each d ∈ N Q0 there is a geometric invariant theory (GIT) coarse moduli space of ζ-semistable Q-representations of dimension d, constructed in [26], which we denote M(Q) ζ -ss is the open subscheme whose geometric points correspond to ζ-semistable Q-representations.
We denote by the morphism from the stack to the coarse moduli space. At the level of points, this map takes a semistable representation ρ to the direct sum of the subquotients appearing in the Jordan-Hölder filtration of ρ, considered as an object in the category of ζ-semistable representations of slope ̺(d). If there is no ambiguity, we omit the subscript Q from the definition of JH.
We denote by q ζ Q,d : M(Q) ζ -ss d → M(Q) d the morphism from the GIT quotient to the affinization. This morphism is proper, as can be seen from the construction of the domain via GIT. At the level of points, q ζ Q,d takes a ζ-semistable module to its semisimplification.
We define two pairings on Z Q0 Again, we will drop the subscript Q when the choice of quiver is obvious from the context. For θ ∈ (−∞, ∞) a slope, we denote by the submonoid of dimension vectors d such that d = 0 or ̺(d) = θ.
Definition 2.3. A stability condition ζ ∈ H Q0 + is θ-generic if for all d, e ∈ Λ ζ θ , d, e = 0, and we say that ζ is generic if it is θ-generic for all θ.  If Q is symmetric, then all stability conditions ζ ∈ H Q0 + are generic. The degenerate stability condition is generic if and only if Q is symmetric. In particular, for all quivers Q, the degenerate stability condition is generic for Q andQ.
Definition 2.6. We define by dim ζ : M(Q) ζ -ss → N Q0 the map taking a polystable quiver representation to its dimension vector, and define where JH ζ Q is as in (17).
If S is a Serre subcategory of the category of CQ -mod, we denote by . If X is a quasiprojective complex variety, and so there is a closed embedding X ⊂ Y inside a smooth complex variety, and f extends to a function f on Y , we which takes a complex of mixed Hodge modules F to its underlying complex of perverse sheaves, and commutes with f * , f ! , f * , f ! , D X and tensor product. In addition, the functors φ p f and ψ p f lift to exact functors for the category of mixed Hodge modules. We denote by φ f the lift of φ p f .
Remark 3.1. If f is a regular function on the smooth variety X, then supp(φ p f Q X ) = supp(φ f Q X ) = crit(f ).
3.1.1. In the context of Donaldson-Thomas theory of general Jacobi algebras it is necessary to work in a larger category than MHM(X), called the category of monodromic mixed Hodge modules on X, denoted MMHM(X). This category is equivalent to the full subcategory of mixed Hodge modules on X × C * such that along each fibre {x} × C * the total cohomology of the restriction is an admissible variation of mixed Hodge structure. See [28, Sec.7] or [10,Sec.2] for an introduction to this category, along with its slightly subtle monoidal product. Shifted pullback along the inclusion X × {1} → X × C * gives a faithful functor MMHM(X) → MHM(X) -one should think of this functor as "forgetting monodromy." There is an embedding of monoidal categories τ : MHM(X) → MMHM(X) defined by One should think of this functor as turning a mixed Hodge module into a monodromic mixed Hodge module by stipulating that the monodromy is trivial, and of the essential image of this functor as being "monodromy-free" monodromic mixed Hodge modules. This is a symmetric monoidal functor.
The functor φ f : MHM(X) → MHM(X) lifts to a functor Here u is the coordinate on C * .
For g a regular function on a complex variety Y , set to be the canonical natural transformation of functors of mixed Hodge modules. Let f be a regular function on X. Then the natural transformation provides a natural transformation ν mon where τ is as in (19). The natural transformation ν mon f is a lift of the natural transformation ν f to the category MMHM(X).
The reason for introducing monodromic mixed Hodge modules is that for a general pair (Q, W ), if one restates the cohomological integrality theorem (Theorem 3.9) purely in terms of the ordinary tensor category of mixed Hodge modules, with φ instead of φ mon , it is not true -the subtlety here regards tensor products of monodromic mixed Hodge modules. For our purposes though, this headache will not occur -see Remark 3.8. Definition 3.3. We define D b (MMHM(X)) to be the bounded derived category of monodromic mixed Hodge modules on X. If X is connected, we define D ≥ (MMHM(X)) to be the inverse limit of the diagram of categories Explicitly, an object of D ≥ (MMHM(X)) is given by a Z-tuple of objects F n in D ≥ (MMHM(X)), along with isomorphisms τ ≤n−1 F n ∼ = F n−1 . For F an object in D ≥ (MMHM(X)) we write τ ≤n F = F n and H n (F ) = H n (F n ). For an object F of D ≥ (MMHM(X)), the cohomological amplitude of the objects F n are universally bounded below.
Similarly, we define D ≤ (MMHM(X)) to be the inverse limit of the diagram For general X, we define and D ≥ (MMHM(X)) similarly.
A monodromic mixed Hodge module F comes with a filtration the weight filtration, which is equal to the usual weight filtration if F is a genuine mixed Hodge module.
We define L := H c (A 1 , Q), considered as a cohomologically graded mixed Hodge structure, i.e. as a pure cohomologically graded mixed Hodge structure concentrated in cohomological degree two. Via the embedding (19) we may consider this object alternatively as a cohomologically graded monodromic mixed Hodge structure, or a cohomologically graded monodromic mixed Hodge module on a point. Working in the category MMHM(pt), we define , to obtain a tensor square root of L. In other words we have L 1/2 ⊗ L 1/2 ∼ = L.
Warning 3.5. Using the Thom-Sebastiani isomorphism and Theorem 3.7 below, one can show that there are two equally natural choices for this isomorphism, depending on which "dimensional reduction" of x 2 1 + x 2 2 = (x 1 + ix 2 )(x 1 − ix 2 ) we consider. These isomorphisms are not the same! This issue can be largely ignored in this paper, but is the reason for the signs appearing in (93). Convention 3.6. Let X be a complex variety, such that each connected component contains a connected dense smooth locus. In this paper we will shift the definition of the intersection complex mixed Hodge module for X so that it is pure of weight zero, while its underlying element in D b (Perv(X)) is a perverse sheaf. This we achieve by setting Since the smooth stack BC * has complex dimension -1, we extend this notation in the natural way by setting

Pushforwards from stacks.
Assume that X is a smooth complex variety, carrying the action of the algebraic group G, and let f be a G-invariant regular function on X, and let p : X/G → Y be a morphism from the global quotient stack to a scheme Y . Then we recall (following [10, Sec.2]) how to define ). We recall the definition for the case in which X is connected -the general definition is obtained by taking the direct sum over connected components. The definition is a minor modification of Totaro's well-known construction [46].
Firstly, let V 0 ⊂ V 1 ⊂ . . . be an ascending chain of G-representations, and let be an ascending sequence of closed inclusions of G-equivariant varieties, with each U i ⊂ X × V i an open dense subvariety. We assume furthermore that that G acts freely on U i for all i, and that the principal bundle U i → U i /G exists in the category of complex varieties. Then we define . For fixed n and sufficiently large i, the maps ) are isomorphisms (see e.g. [6, Sec.3.4]), stabilising to a monodromic mixed Hodge module that is independent of our choice of . . .
= 0 for m < dim(X) and n > dim(X). So for fixed n and sufficiently large i the morphisms This can be seen as a special case of the previous definition, with f = 0.
Let Z ⊂ X be a subvariety, preserved by the G-action, and denote by ι : Z/G ֒→ X/G the inclusion of stacks. We obtain inclusions and we define the restricted pushforward of vanishing cycle cohomology As a particular case, setting Y to be a point, we obtain

3.3.
Dimensional reduction. Assume that we are given a decomposition X = X ′ × A n of varieties, and that C * acts on X via the product of the trivial action on X ′ , and the scaling action on A n . Assume that f has weight one. Denote by π : X → X ′ the natural projection. Then we can write where f i are functions on X ′ , and x i are coordinates for A n . Define to be the shared vanishing locus of all the functions f 1 , . . . , f n , and denote Note that Z ⊂ X 0 := f −1 (0), and so we can postcompose the canonical natural transformation ν mon to obtain a natural transformation This is a cohomological analogue of the dimensional reduction theorem of [1]. It implies (see [5,Cor.A.7]) that if X is the total space of a G-equivariant affine fibration π : X → X ′ for G an algebraic group, and S ⊂ X ′ is a G-invariant subspace of the base, there is a natural isomorphism in compactly supported cohomology Remark 3.8. The natural transformation π ! υπ * is considered as a natural transformation between two functors D b (MHM(X ′ )) → D b (MMHM(X ′ )) (see Remark 3.2). However, the target functor is defined as such a functor via the embedding D b (MHM(X ′ )) → D b (MMHM(X ′ )). So Theorem 3.7 implies that under suitable equivariance conditions, the monodromy on π ! φ mon f π * is trivial, and we can replace π ! φ mon f π * with the more standard functor π ! φ f π * .
3.4. Integrality and PBW isomorphisms. Let Q be a finite quiver. We consider N Q0 -graded monodromic mixed Hodge structures as monodromic mixed Hodge modules on the space N Q0 in the obvious way: a monodromic mixed Hodge module on a point is just a monodromic mixed Hodge structure, and N Q0 is a union of points d ∈ N Q0 , and so a monodromic mixed Hodge module on N Q0 is given by a formal direct sum ֒→ M(Q) ζ -ss , which at the level of complex points, corresponds to the inclusion of the zero module. The morphism dim ζ : M(Q) ζ -ss → N Q0 , taking a representation to its dimension vector, is a morphism of monoids, where the morphism If X is a commutative monoid in the category of locally finite type complex schemes, with finite type monoid morphism then by [30, Thm.1.9] the categories D ≥ (MMHM(X)), and D ≤ (MMHM(X)) of Definition 3.3 carry symmetric monoidal structures defined by In particular, the categories D ≥ MMHM(M(Q) ζ -ss ) and D ≤ MMHM(M(Q) ζ -ss ) carry symmetric monoidal structures defined by The following theorem allows for the definition of BPS sheaves and BPS cohomology. It is a cohomological lift of the property known in DT theory as integrality. (18), define the monodromic mixed Hodge module and define BPS ζ Q,W,θ : Since Verdier duality naturally commutes with the vanishing cycles functor, and Remark 3.10. Technically, the result quoted from [10] is stated for H(•) of the LHS of (21) and (22), and not the LHS as stated above. That there is an isomorphism H(LHS) ∼ = LHS is a consequence of approximation by projective morphisms and the decomposition theorem, see Corollary 3.15.
3.4.1. (3d) BPS cohomology. Now let S be a Serre subcategory of the category of CQ-modules. Recall that we denote by ι ′ : M(Q) S,ζ -ss ֒→ M(Q) ζ -ss the inclusion of objects in S. We define the BPS cohomology: where the isomorphism follows from Verdier self-duality of the BPS sheaf.

The cohomologically graded mixed Hodge structure
carries a Hall algebra multiplication, defined in [28,5], via pullback and pushforward of vanishing cycle sheaves; see §9.1 for a generalisation of the construction. Applying the natural transformation τ ≤1 → id to (21) we obtain the morphism Applying H ι ′ * ι ′! to (24), we obtain the embedding Since H C * acts on the target, this extends to a morphism Let Q be a quiver. For the moment we do not assume that Q is symmetric. Let d, f ∈ N Q0 be a pair of dimension vectors. Following [10,Sec.3.3] we define Q f to be the quiver obtained from Q by setting Given a King stability condition ζ for Q, and a slope θ ∈ ( − ∞, ∞), we extend ζ to a stability condition ζ (θ) for Q f by fixing the slope -stable, and this holds if and only if the underlying Q-representation of ρ is ζ-semistable, and for all proper Q fsubrepresentations ρ ′ ⊂ ρ, if dim(ρ ′ ) ∞ = 1 then the underlying Q-representation of ρ ′ has slope strictly less than θ.
. Then V f ,d carries a GL d -action, given by the product of the GL di (C)-actions on C di . Furthermore, there is an obvious decomposition If L, L ′ are vector spaces, we define Hom surj (L, L ′ ) ⊂ Hom(L, L ′ ) to be the subvariety of surjective homomorphisms. Then the subspace The group GL d acts freely on X(Q f ) ζ (θ) -ss (1,d) . In the notation of the start of the section, we may set U i = X(Q i·(1,...,1) ) ζ (θ) -ss (1,d) to obtain our promised chain of GL dequivariant varieties.
the induced map from the quotient.
Proposition 3.13. The map π f ,d above is proper.
Proof. This is standard, and follows from the valuative criterion of properness and the fact that in the following diagram over the common affinization of the domain the unmarked arrows are GIT maps, and hence proper.
as per the definition in Section 3.2. Equation (26) states that the cohomology of is obtained as a limit of direct images of related vanishing cycle complexes along projective morphisms from smooth complex varieties. It is in this sense that JH ζ is "approximated by proper maps", and the outcome is that many theorems regarding projective morphisms are true of JH ζ . For instance, it follows from the W = 0 case of equation (26) and the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber, that Lemma 3.14. Let Q be quiver, let ζ be a stability condition on Q, let W ∈ CQ/[CQ, CQ] be a potential, and let d, f ∈ N Q0 be a pair of dimension vectors. As above, we let π ζ f ,d : be the forgetful map taking a stable framed ζ-semistable representation to its underlying ζ-polystable representation. Then there is an isomorphism i.e. the left hand side of (27) is isomorphic to its total cohomology.
Proof. This follows from the existence of the chain of isomorphisms commutativity of vanishing cycles with proper maps.
Lemma 3.14 can be thought of as saying that "one half" of the BBDG decomposition theorem is true, even with the introduction of the vanishing cycles functor (which may destroy purity, i.e. the other half of the theorem). Corollary 3.15. Let Q be a quiver, let ζ be a stability condition on Q, let W ∈ CQ/[CQ, CQ] be a potential, and let d ∈ N Q0 be a dimension vector. Then there are isomorphisms in Proposition 3.16. Let ζ be a θ-generic stability condition, and assume that crit(Tr(W )) ⊂ Proof. By [31,Prop.4.3] there is an equality in the Grothendieck group of mixed Hodge modules on M(Q) ζ -ss On the other hand, the terms in square brackets in (29) and (30) The proposition follows from applying φ mon T r(W ) ζ θ to both sides of (31), and using the fact that the vanishing cycle functor commutes with taking direct image along proper maps [39, Thm.2.14], as well as commuting with the monoidal structure ⊠ ⊕ on D ≤ (MMHM(M(Q) ζ -ss θ )) by Saito's version of the Thom-Sebastiani theorem [37], as well as the enhancement of this monoidal structure to a symmetric monoidal structure, by [10, Prop.3.8].

Purity and supports
In this section we prove Theorem A. A crucial role in the proof is played by the support lemma (Lemma 4.1), which also enables us to prove Theorem B.

4.1.
Proof of Theorem A. Fix a quiver Q. We define (Q,W ) as in Section 2.1. Define BPS ω -nilp,∨ the compactly supported cohomology, and the restricted compactly supported cohomology, respectively, of the BPS sheaf from Theorem 3.9. As in Definition 2.6, dim : M(Q) → N Q0 is the map taking a semisimple representation to its dimension vector. Note that no stability condition appears in (32) (see Convention 1.2). As explained at the beginning of Section 3.4, we consider a N Q0 -graded mixed Hodge structure as essentially the same thing as a mixed Hodge module on N Q0 , and so we consider both of the above objects equivalently as mixed Hodge module complexes on the discrete space N Q0 , or N Q0 -graded mixed Hodge structures. We break the proof of Theorem A into several steps. We use the following three lemmas. and By Theorem A the target is pure, of Tate type, and so BPSQ ,W ,d ⊗L 1/2 is also pure, of Tate type. It follows that the Tate twist BPSQ ,W ,d is pure, of Tate type.
The proof of both Lemmas 4.2 and 4.3 will use the dimensional reduction theorem, recalled as Theorem 3.7. Let Q + be obtained fromQ by deleting all of the arrows a * , and let Q op be obtained fromQ by deleting all the arrows a and all the loops ω i . We decompose If we let C * act on X(Q) d via the trivial action on X(Q + ) d and the weight one action on X(Q op ) d , then Tr(W ) d is C * -equivariant in the manner required to apply Theorem 3.7. In the notation of Theorem 3.7, we have that Z ′ ⊂ X(Q + ) d is determined by the vanishing of the matrix valued functions, for a ∈ Q 1 Concretely, the stack Z ′ / GL d is isomorphic to the stack of pairs (ρ, f ), where ρ is a d-dimensional Q-representation, and f : ρ → ρ is an endomorphism in the category of Q-representations.
We fix X(Q + ) ω -nilp d ⊂ X(Q + ) d to be the subspace of representations such that each ρ(ω i ) is nilpotent. We deduce from Theorem 3.7 that there is a natural isomorphism in compactly supported cohomology (37) Lemma 4.3 is proved by analyzing the right hand side of (37). Note that there is no overall Tate twist in (37) -the Tate twist in the definition of the left hand side is cancelled by the Tate twist appearing in Theorem 3.7.
The first isomorphism in Lemma 4.2 is obtained in similar fashion. Let L ⊂Q be the quiver obtained by deleting all of the arrows a and a * , for a ∈ Q 1 . Then we can decompose and let C * act on X(Q) d via the trivial action on X(Q) d and the scaling action on X(L) d . This time the role of Z ′ in Theorem 3.7 is played by µ −1 Q,d (0) ⊂ X(Q) d , and we deduce that Proof of Lemma 4.3. This is [7, Thm.3.4]; we recall a sketch of the proof and refer the reader to [7] for more details. The space (36) and the condition that ω i acts nilpotently, for every i. It follows, as in the proof of Proposition 2.1, that the stack (Z ′ ∩ X(Q + ) ω -nilp d )/ GL d is isomorphic to the stack for which the C-points are pairs (ρ, f ), where ρ is a d-dimensional CQ-module, and is a nilpotent endomorphism of ρ. This stack decomposes into finitely many disjoint locally closed strata indexed by multipartitions π of d (i.e. Q 0 -tuples (π (1) , . . . , π (n) ) of partitions, such that |π (i) | = d i for i ∈ Q 0 ), where a multipartition determines the Jordan normal form of each ρ(ω i ) in the obvious way. We label these strata M π .
Given a partition π of a number d, containing each number i a total of π i -times, we define the C[x]-module Hom(N π (s(a)) , N π (t(a)) ) GL π = i∈Q0 Aut(N π (i) ). Then Each of the stacks M π can thus be described as a stack-theoretic quotient of an affine space by a unipotent extension of a product of general linear groups, from which it follows that each H c (M π , Q) is pure, of Tate type. It follows that the connecting maps in the long exact sequences of compactly supported cohomology associated to the stratification indexed by multipartitions are zero, and the resulting short exact sequences are split. It follows by induction that Proof of Lemma 4.2. Since the map dim : M(Q) → N Q0 is a morphism of commutative monoids, with proper monoid maps ⊕ and + respectively, by [30,Sec.1.12] there is a natural equivalence of functors We denote by ι ′ d : M(Q) ω -nilp d ֒→ M(Q) d the inclusion. Taking the direct sum over all d ∈ N Q0 , applying base change, and using the relative cohomological integrality theorem (Theorem 3.9): giving the isomorphism (35).
Taking the direct sum of the isomorphisms (38) over d ∈ N Q0 gives the isomorphism (33). Applying dim ! to (22) we have the isomorphisms To prove the existence of the isomorphism (34), then, it is sufficient to prove that BPSQ ,W ∼ = BPS ω -nilp Q,W ⊗L.
We complete the proof of Theorem A by proving the support lemma.
Proof of Lemma 4.1. To ease the notation we prove the lemma under the assumption that ζ is the degenerate stability condition: the proof for the general case is unchanged.
Since the support of BPSQ ,W is the same as the support of the underlying perverse sheaf, and all complexes that we encounter in the following proof are quasi-isomorphic to their total cohomology, throughout the proof we work in the category of cohomologically graded perverse sheaves. Let x ∈ M(Q) d be a point corresponding to a semisimple CQ-module ρ, and assume that there are at least two distinct eigenvalues ǫ 1 , ǫ 2 for the set of operators {ρ(ω i )|i ∈ Q 0 }. Assume, for a contradiction, that x ∈ supp(BPSQ ,W ), so that in particular x ∈ supp JH * φ p Tr(W ) IC M(Q) (Q) and so by (15) and Remark 3.1, there exists a Jac(Q,W ) module with semisimplification given by ρ, and so ρ is a semisimple Jac(Q,W )-module.
Under our assumptions, there are disjoint (analytic) open sets U 1 , U 2 ⊂ C with ǫ 1 ∈ U 1 and ǫ 2 ∈ U 2 , and with all of the generalised eigenvalues of ρ contained in 3 Note that this is not true of a general point in M U 1 ∪U 2 (Q) -the crucial fact is that the operation i∈Q 0 ρ(ω i )· defines a module homomorphism for a Jac(Q,W )-module ρ, since i∈Q 0 ω i is central in Jac(Q,W ).
We claim that there is an isomorphism This follows from Lemma 4.6 below. Assuming the claim, applying (21) to the RHS of (40) we have isomorphisms On the other hand, restricting the isomorphism of (21) to the LHS of (40) yields Comparing (41) and (42), we deduce that We deduce that and so since the restriction of BPSQ ,W to x is zero, which is the required contradiction.
For the final statement of the lemma, it suffices to prove that if ρ is a simple Jac(Q,W )-module, then i∈Q0 ρ(ω i ) acts via scalar multiplication. In the decomposition of ρ into generalised eigenspaces for the action of the operator i∈Q0 ρ(ω i )· we have already shown that there is only one generalised eigenvalue, which we denote λ. Then ρ is filtered by the nilpotence degree of the nilpotent operator Ψ := i∈Q0 ρ(ω i ) · −λ Id ρ , and so since ρ is simple, Ψ = 0 and we are done.  We have seen in the proof of Lemma 4.2 that BPS ζQ ,W ,d is A 1 -equivariant, where the A 1 -action on the subspace A 1 × M(Π Q ) ζ -ss d is via translation in the first factor. It follows that we can write Finally, BPS ζQ ,W ,d = φ mon Tr(W ) IC M(Q) ζ -ss d is Verdier self-dual [10], as is IC A 1 (Q). So from (43) we deduce and Verdier self-duality of BPS ζ ΠQ,d follows. 4.2.1. Although we will not use it, we remark that via the same proof as Lemma 4.1 we deduce the following This isomorphism does not follow directly from the Thom-Sebastiani isomorphism, since we need to compare the vanishing cycle sheaf of the function Tr(W ) ⊞ Tr(W ) on M U1 (Q) d ′ × M U2 (Q) d ′′ with the vanishing cycle sheaf for the function Tr(W ) on M U1∪U2 (Q) d , and these ambient smooth stacks are different.
Proof of Lemma 4.6. Again, it is sufficient to prove the lemma for the degenerate stability condition (the general case then follows by restriction to the ζ-semistable locus). Writing Note that rank(V + ) = rank(V − ). Denote by z : B → Y the inclusion of the zero section. Writing f, g for the functions on Y, B induced by Tr(W ), it is sufficient to show that there is an isomorphism of GL d ′ ×d ′′ -equivariant perverse sheaves . Let C * act on V + and V − with weights 1 and −1 respectively, then f is C *invariant. It follows that g| Tot(V + ) = f • π + , where π + : Tot(V + ) → B is the projection. So there is a natural isomorphism We claim that the natural morphism is an isomorphism, where we denote by i + : Tot(V + ) → Y the inclusion. This can be checked locally on the base B. Pick b ∈ B, and let x 1 , . . . , x α , y 1 , . . . , y β , z 1 , . . . , z β be a set of elements of the local ring O X(Q),b , providing a basis for m b /m 2 b , where x i all have weight zero, y i have weight 1, and z i have weight −1 for the C * -action. The weight −1 partial derivatives of f are provided by ∂g/∂y i , and so since the critical locus of g (restricted to a neighbourhood of b) lies on the zero section B, it follows that we can change coordinates and pick z i = ∂g/∂y i . Then we have By the Thom-Sebastiani theorem, after restricting to a neighbourhood U ∋ b φ p g Q U×V ∼ = φ p h Q U ⊠ φ p k Q V and the claim reduces to the claim that is an isomorphism, which is a simple calculation, or a trivial application of the dimensional reduction theorem.
Combining (44) and (45) yields the isomorphism φ p we obtain the required isomorphism by applying z * to this isomorphism and shifting cohomological degree by dim B.

4.3.
Calculating H c (M(Π Q ) d , Q). We use Theorem A and existing results on the E series of M(Π Q ) d to determine the compactly supported cohomology of M(Π Q ) d , along with its mixed Hodge structure. The E series (see Subsection 1.2) of H c (M(Π Q ) d , Q) was calculated in [32]. Before we recall this series, we recall some definitions.
Recall the plethystic exponential defined in §1.9. For our purposes, it is profitable to think of the plethystic exponential as the decategorification of the endofunctor of tensor categories taking an object to the underlying object of the free symmetric algebra generated by that object. We define the ring Z((X 1 , . . . , X m ))[[Y 1 , . . . , Y n ]] of formal Laurent power series g(X 1 , . . . , X m , Y 1 , . . . , Y n ) such that for each (a 1 , . . . , a n ) ∈ N n the Y a1 1 · · · Y an n coefficient of g(X 1 , . . . , X m , Y 1 , . . . , Y n )X c1 1 · · · X cm m is in Z[[X 1 , . . . , X m ]] for sufficiently large c 1 , . . . , c m . This is isomorphic to the Grothendieck ring of the category D ⋄ (Vect Z m ⊕Z n ), which we define to be the subcategory of the unbounded derived category of Z m ⊕ Z n -graded vector spaces V such that (1) For each (e, d) ∈ Z m ⊕Z n the total cohomology H(V ) e,d is finite-dimensional We define D ⋄ (Vect + Z m ⊕Z n ) ⊂ D ⋄ (Vect Z m ⊕Z n ) to be the full subcategory satisfying the extra condition that the total cohomology H(V ) (e,0) is zero for all e ∈ Z m . Then χ induces an isomorphism where m is the maximal ideal generated by Y 1 , . . . , Y n . We may define plethystic exponentiation via the formula for V ∈ D ⋄ (Vect + Z m ⊕Z n ). Then the E series for H c (M(Π Q ) d , Q) is given by [32] Taking Hodge series of (48) yields the following refinement of (47):

Degree zero BPS sheaves.
For n ∈ N we define Q (n) to be a quiver with one vertex, which we denote 0, and n loops. We will be particularly interested in the quiver Q Jor := Q (1) : the Jordan quiver. We identify We denote by x, y, z the three arrows of Q (3) . ThenW = x[y, z]. The ideas in the proof of Theorem A allow us to prove rather more for the QP ( Q Jor ,W ), essentially because this QP is invariant (up to sign) under permutation of the loops, so that we can apply the support lemma (Lemma 4.1) three times.
Let d ∈ N with d ≥ 1. The support of JH QJor,! φ Tr(W ) d IC M( QJor) d (Q) is given by the coarse moduli space of d-dimensional representations of the Jacobi algebra C[x, y, z], i.e. the space of semisimple representations of C[x, y, z]. This space is in turn isomorphic to Sym d (A 3 ), since any simple representation ρ of C[x, y, z] is one-dimensional, and determined up to isomorphism by the three complex numbers ρ(x), ρ(y), ρ(z).
for all d.
Proof. By the same argument as for Lemma 4.1, the support of BPS QJor,W ,d is contained in the image of the morphism By the argument in the proof of Lemma 4.2, BPS QJor,W ,d is constant on its support, so and so from (49) there is an isomorphism and we finally deduce that L d ∼ = Q, with the standard pure weight zero mixed Hodge structure, as required.

We set Coh
Then define the N-graded, cohomologically graded mixed Hodge structure Combining Theorems 3.9, 3.11 and 5.1 gives the following and a PBW isomorphism of N-graded mixed Hodge structures Proof. We just construct isomorphism (51) as a special case of (22); via the same argument we then realise (52) as a special case of Theorem 3.11. In fact it is sufficient to construct the isomorphism in the case U = C 3 , since then the general case is given by restriction to ι U (Sym(U )). In this case, since supp(JH QJor,! φ mon Q) vir , which follows from (22) and Theorem 5.1.

5.2.
Applications to surfaces and character stacks. Let j : V ֒→ A 2 be the inclusion of a constructible subset, and write We consider the commutative diagram where the horizontal morphisms are the forgetful morphisms. We denote by C d = C d / GL d (C) the stack of commuting pairs of matrices, and set C = d∈N C d . We define the inclusions as in the previous section. We denote by ∪ : Sym(V )×Sym(V ) → Sym(V ) the morphism taking a pair of multisets of points to their union (so that ι V is a morphism of monoids in the category of schemes). We define We denote by i : C ֒→ M(Q Jor ) the inclusion. By Theorem 3.7, there is an isomorphism of complexes of mixed Hodge modules We denote by Coh V (A 2 ) the reduced substack of coherent sheaves on A 2 settheoretically supported on V with zero-dimensional support, and by p : Coh V (A 2 ) → Sym(V ) the morphism taking such a sheaf to its support, counted with multiplicity, so that p restricts to a morphism Proof. We denote by ι V : Sym(V ) → M(Q Jor ) the inclusion. Then we compose the ismorphisms where (54) comes from (53) and the isomorphism (55) comes from Corollary (5.2). This establishes the first isomorphism, the PBW isomorphism follows by the same argument, and (52).
In nonabelian Hodge theory, an interesting special case of Corollary 5.2 comes from setting V = (C * ) 2 . Set A = C x ±1 , y ±1 . Then there is a natural identification of substacks of M(Q Jor ) where the final stack is the stack of finite-dimensional representations of the fundamental group of a genus 1 closed Riemann surface. As a special case of Corollary 5.3 we deduce The CoHA structure on the left hand side of (56) is introduced and studied in [4].
From Corollary 5.4 and (57) we deduce the g = 1 part of the following; the genus zero case follows from [28, Sec.1].
6. Generalisations of the purity theorem 6.1. The wall crossing isomorphism in DT theory. The wall crossing isomorphism in cohomological DT theory (e.g. [10, Thm.B]) provides a powerful way to deduce purity of Borel-Moore homology of moduli spaces of semistable quiver representations, for some stability condition ζ, from purity of Borel-Moore homology for some other stability condition ζ ′ (see e.g. [6] for an application of this principle for quantum cluster algebras). We will use this idea to prove a generalisation of Theorem 2 incorporating stability conditions. Fix a quiver Q, and a stability condition ζ ∈ H Q0 + . Let ρ be a finite-dimensional CQ-module, then ρ admits a unique Harder-Narasimhan filtration 0 = ρ 0 ⊂ . . . ⊂ ρ s = ρ such that each ρ t /ρ t−1 is ζ-semistable, and the slopes ̺(ρ 1 /ρ 0 ), . . . , ̺(ρ s /ρ s−1 ) are strictly descending. Given a dimension vector d ∈ N Q0 , we denote by the set of Harder-Narasimhan types for CQ-modules of dimension d. For α = (d 1 , . . . , d s ) ∈ HN d , we denote d j by α j , and write s(α) = s. For each α ∈ HN d , there is a locally closed quasiprojective subvariety for which the closed points correspond exactly to those CQ-modules ρ of Harder-Narasimhan type α. For α ∈ HN d , define by the subspace of linear maps preserving the Q 0 -graded flag and such that each subquotient is ζ-semistable, and denote by P α ⊂ GL d the subgroup preserving this same flag. Then the natural map the locally closed inclusion of substacks. By [35,Prop.3.4] there is a decomposition into locally closed substacks The following are the relative and absolute versions of the cohomological wallcrossing isomorphism, respectively [10, Thm.B]. Since we state them in the general case, which may involve nontrivial monodromy, we first state them in terms of the functor φ mon Tr(W ) of Section 3. When we come to use the theorem, we will be back in the trivial monodromy situation, and we will be able to revert to using the functor φ Tr(W ) , as explained in Remark 3.8.
Theorem 6.1. For Q a quiver, W ∈ CQ/[CQ, CQ] a potential, and stability condition ζ, there is an isomorphism in D ≤ (MMHM(M(Q))): Taking the direct image to N Q0 , there is an isomorphism in D ≤ (MMHM(N Q0 )): If Q is symmetric, the function f in the above proposition is identically zero.
Corollary 6.2. For any stability condition ζ ∈ H Q0 + , the cohomologically graded mixed Hodge structure is pure of Tate type.
Proof. Firstly, strictly speaking, the left hand side of (60), as well as both sides of (59), should be considered as monodromic mixed Hodge structures. By Lemma 4.2, for the case in which our QP is of the form (Q,W ) for some quiver Q, the left By Theorem 3.7 there is an isomorphism There are equalities Thus the natural map is an isomorphism. Combining (66) and (67) with Corollary 6.2 we deduce the result.
6.3. Framed quivers. For Q ′ a quiver, f , d ∈ Q ′ 0 a pair of dimension vectors, and ζ ∈ H Q ′ 0 + a stability condition for Q ′ , recall from Section 3.5 the construction of the We consider this construction in the case where Q ′ =Q, the tripled quiver associated to a quiver Q. As in Equation (25)  Theorem 6.5. Fix a finite quiver Q, a dimension vector f ∈ N Q0 , a King stability condition ζ ∈ H Q0 + , and d ∈ N Q0 . Then the N Q0 -graded mixed Hodge structure on the total vanishing cycle cohomology on the fine moduli space of ζ-semistable f -framed CQ-modules is pure, of Tate type.
Proof. Applying dim ζ θ,! to the isomorphism (28) we obtain the isomorphism On the other hand, from Corollary 4.4 each of the complexes BPS ζQ ,W ,d are pure. The purity of the right hand side of (68) follows, and so does the theorem. 6.4. Critical cohomology of Hilb(A 3 ). We consider again the special case in which Q = Q Jor , and soQ is a quiver with one vertex and three loops, which we label x, y, z, andW = x[y, z]. Setting f = 1, there is a natural isomorphism of schemes (see [1]) where the right hand side of (69) is the usual Hilbert scheme parameterising codimension n ideals I ⊂ C[x, y, z]. The following is then a corollary of Theorem 6.5: Corollary 6.6. The mixed Hodge structure (1−(xyz 2 ) 1−k t n ) −1 .
Indeed, we can determine the critical cohomology of Hilb n (A 3 ) itself, without passing to any motivic invariants: Corollary 6.7. There is an isomorphism of N-graded, cohomologically graded mixed Hodge structures: Proof. By Corollary 6.6 the left hand side of (70) is pure of Tate type, as is the right hand side (by definition). A cohomologically graded mixed Hodge structure that is pure, of Tate type, is determined by its weight polynomial. The required equality of weight polynomials follows from the main result of [1].
6.5. Nakajima quiver varieties. In Section 6.3 we considered the mixed Hodge structures on the vanishing cycle cohomology of framed representations of the quiver Q, where the framing results in a quiver that is not symmetric, i.e. we perform the operation of framing the quiver after the operation Q →Q. By reversing the order of these operations, we derive our results on Nakajima quiver varieties.
Let Q be an arbitrary quiver, and let ζ ∈ H Q0 + be a stability condition. Let f ∈ N Q0 be a framing vector. Throughout this section we assume that f = 0. Consider the quiver Q f , where the tilde covers the f as well as the Q; this is the quiver obtained by doubling the framed quiver Q f and then adding a loop ω i at every vertex (including the vertex ∞).
For each of the vertices i ∈ Q 0 , the condition µ (1,d) (ρ) = 0 imposes the conditions which are the usual Nakajima quiver variety relations [33] [34], while at the vertex ∞, the relation imposed is By cyclic invariance of the trace, we have i∈(Q f )0 Tr(T i ) = 0 and so T ∞ = Tr(T ∞ ) = 0 follows already from the relations (71), and (72) is redundant. It follows that / GL d is the usual Nakajima quiver variety, which we will denote M ζ (d, f ). There is an isomorphism  The standard construction for P is as follows. For a quiver Q, let C(Q) denote the set of equivalence classes of cycles in Q, i.e. the set of cyclic paths, where if ll ′ and l ′ l are both cyclic paths, they are considered to be equivalent. For every cycle c ∈ C(Q), we pick a constructible subset U c ⊂ C, and we say that a CQ-module ρ has property P if and only if the generalised eigenvalues of ρ(c) belong to U c , for each c a representative of c ∈ C(U ).
we obtain the condition for the Lusztig nilpotent variety if Q has no loops. In general, the Serre subcategory S ⊂ CQ -mod determined by this choice of U c is the subcategory of modules M for which there exists a filtration by Q 0 -graded vector spaces 0 ⊂ L 1 ⊂ . . . ⊂ L n of the underlying Q 0 -graded vector space of M , such that a · L s ⊂ L s for all s, and a * · L s ⊂ L s−1 . This second property is obviously of Serre type.
The equivalence of these two Serre properties is demonstrated as follows. Say M is a CQ-module in the Serre subcategory determined by the above choices of U c . Then every p ∈ CQ ≥1 \ CQ ≥1 acts on M via a nilpotent operator. By Engel's theorem there is a filtration of vector spaces 0 = M 0 ⊂ M 1 ⊂ . . . ⊂ M n such that for each such p there is an inclusion p · M s ⊂ M s−1 . Now set L s = CQ · M s to obtain the required filtration by CQ-modules, observing that This condition is introduced under the name of *-semi-nilpotency in [3].
Example 7.3. For a final example we turn to [2]. Set

7.2.
Proof of Theorem C and D. Applying the functor dim ! ι * to the isomorphism constructed in the next theorem yields Theorem C.
Theorem 7.4. Pick a stability condition ζ ∈ H Q0 + . There is an isomorphism in D ≤ (MHM(M(Q))) Proof of Theorem 7.4. We consider the commutative diagram With V defined as in (65). By Theorem 3.7 there are isomorphisms is contained in the image of the natural inclusion j, and so from (74) and the above commutative diagram we obtain the isomorphism Thus, applying τ ′ Q,! to the isomorphism (58) applied to the QP (Q,W ) yields the required isomorphism.
We again use Lemma 6.3 to deduce Theorem D from the analogous result for general quivers with potential: be the morphism extending a Π Q -module to a Jac(Q,W )-module by letting all of the loops ω i act by multiplication by a fixed scalar in A 1 . Then by Theorem B there is an isomorphism Arguing as in the proof of Theorem 7.4, there are isomorphisms The construction of the PBW isomorphism is similar; via dimensional reduction and Lemma 6.3 there is an isomorphism Q,W ,θ , whereS is the Serre subcategory of CQ-modules for which the underlying CQmodule is in S, defining a Hall algebra structure on A S,ζ ΠQ,θ . Then the required PBW isomorphism is constructed from Theorem 3.11 and the isomorphisms BPSS

7.3.
Applications for Nakajima quiver varieties. We explain the special case of Theorem C which gives rise to Hausel's original formula for the Poincaré polynomials of Nakajima quiver varieties. In brief, we choose Π Q f to be the preprojective algebra for a framed quiver Q f , pick ζ to be the usual stability condition defining the Nakajima quiver variety, set S = CQ f -mod and specialise the Hodge series to the Poincaré series, to derive Hausel's result. For this set of choices, an analogue of equation (9) was demonstrated by Dimitri Wyss [47], working in the naive Grothendieck ring of exponential motives. We describe in a little more detail how our derivation runs.
Firstly, let Q be a quiver, and let S ⊂ CQ -mod be a Serre subcategory. Let f ∈ N Q0 be a framing vector, assumed nonzero, and let S f ⊂ Π Q f -mod be the Serre subcategory consisting of those modules for which the underlying CQ-module is in S. We let ζ = (i, . . . , i) be the degenerate stability condition on Q, and define ζ (0) as in Section 3.5. If X is an Artin stack, we define its Poincaré series via P(X, q) = h(H c (X, Q), 1, 1, q).
Equating coefficients in (9) for which d ∞ = 1, and specialising, we obtain where we have used the isomorphism (73) for the final equality, and M(f , d) S is the subvariety of the Nakajima quiver variety for the dimension vector d and framing vector f corresponding to those points for which the underlying Q-representation is in S. Putting S = CQ -mod (or, equivalently, removing S from the above formulae) and using Hua's formula [21] to rewrite (76) and the first term of (77) as rational functions in q defined in terms of Kac polynomials, we recover Theorem 5 of [17]. The advance that Theorem C gives us is an upgrade from an equality of generating series to an isomorphism in cohomology, i.e. it tells us that (76) is induced by a graded isomorphism of (pure) Hodge structures by taking Poincaré series of the two sides of the isomorphism.

Restricted Kac polynomials
8.1. Definitions. In this section we explain how Theorem D enables one to define and categorify the Kac polynomial a S Q,d (q 1/2 ) associated to a quiver Q, a Serre subcategory S ⊂ CQ, and a dimension vector d. Furthermore, we explain a general mechanism for deducing positivity of such Kac polynomials from purity, and in particular prove Theorem E.
Defining BPS S ΠQ , as per our conventions, as BPS S,ζ ΠQ , for the degenerate stability condition ζ = (i, . . . , i) (equivalently, without any stability condition), the dual of isomorphism (11) yields Isomorphism (78) can be restated as saying that BPS S,∨ ΠQ categorifies the restricted Kac polynomials a S Q,d (q 1/2 ), defined by the plethystic logarithm (the inverse to the plethystic exponential) Isomorphism (78) and (46) imply . This is indeed a polynomial: despite its high-tech definition, BPS S ΠQ,d is, after all, the hypercohomology of a bounded complex of mixed Hodge modules on an algebraic variety.

Positivity of Kac polynomials. A corollary of the existence of the isomorphism (78) is that if d∈NQ
is pure, then so is BPS S ΠQ , and as a result, a S Q,d (q 1/2 ) has only positive coefficients, when expressed as a polynomial in −q 1/2 . This brings us to the special case of Theorem D that, along with Theorem A, implies the Kac positivity conjecture, first proved by Hausel, Letellier and Villegas in [19] via arithmetic Fourier analysis for smooth Nakajima quiver varieties. Namely, we set S = CQ -mod, and we set ζ = (i, . . . , i) to be the degenerate stability condition. Then Theorem D states that there is an isomorphism while Theorem 3.7 states that there is an isomorphism On the other hand by [32,Thm.5.1] there is an equality where a Q,d (q) is Kac's original polynomial, from which we deduce that On the other hand, from Corollary 4.4, we deduce that each BPS ΠQ,d ∼ = BPSQ ,W ,d ⊗L 1/2 is pure, and so χ wt (BPS ΠQ,d , q 1/2 ) is a polynomial in −q 1/2 with positive coefficients. In particular, since a Q,d (q) is a polynomial in q, we have reproved the following theorem:  N , SN , SSN ⊂ CQ -mod to be the full subcategory of nilpotent, *-semi-nilpotent and *-strongly-semi-nilpotent CQ-modules, respectively, we define In this way we obtain a new description of the nilpotent, semi-nilpotent and strongly-seminilpotent Kac polynomials of [3]. Via the results of [3] the polynomials a SN Q,d (q) and a SSN Q,d (q) have an enumerative definition when q is a prime power: the former counts absolutely indecomposable d-dimensional F q Q-modules such that each loop acts via a nilpotent operator, while the latter counts absolutely indecomposable d-dimensional nilpotent F q Q-modules.
Note that the above proof of the Kac positivity conjecture (Theorem 8.1) used only the purity of H c (Π Q , Q). We deduce the following variant of the positivity theorem for Kac polynomials, conjectured in [3].   [41] from the case of a quiver without loops. In the case of quivers without loops, this Kac polynomial identity is explained [41] by Poincaré duality for smooth Nakajima quiver varieties. In general it may be explained by self-Verdier duality of 2d BPS sheaves.  9. Deformations of Hall algebras 9.1. Kontsevich-Soibelman CoHAs. In [28], a method was given for associating a cohomological Hall algebra (CoHA for short) to the data of an arbitrary QP (Q, W ). The construction provides a mathematically rigorous approach to defining algebras of BPS states -see [16] for the physical motivation. We will work with a slight generalisation of the original definition, denoted A τ,Q,W , incorporating extra parameters depending on a weight function τ .
Definition 9.1. If (Q, W ) is a QP, a W -invariant grading for Q is a function τ : Q 1 → Z s such that every cyclic word appearing in W is homogeneous of weight zero.
Example 9.2. For s = 0, the function τ = 0 : Q 1 → Z 0 gives a W -invariant grading for any potential W , and we will recover below the original definition of Kontsevich and Soibelman, by considering this grading.
From now on we will only consider the case in which our quiver with potential is (Q,W ) for some quiver Q. SinceQ is symmetric we avoid some troublesome Tate twists, and since the potentialW is linear in the ω i direction we also avoid the notion of monodromic mixed Hodge modules; in this section we deal only with monodromic mixed Hodge modules for which Theorem 3.7 applies, and so we use the usual mixed Hodge module complex φ Tr(W ) Q, as opposed to the monodromic mixed Hodge module complex φ mon Tr(W ) Q (see Remark 3.8).
9.1.1. Given a grading τ : Q 1 → Z s , define T τ := Hom(Z s , C * ). Given a dimension vector d ∈ N Q0 we form the extended gauge group . For fixed υ ∈ T τ , the action of υ on the category of CQ-modules is functorial, and preserves dimension vectors. It follows that if ζ ∈ H Q0 + is a stability condition, the spaces X(Q) ζ -ss For the rest of the section we will only consider the degenerate stability condition ζ = (i, . . . , i) and so we drop ζ from our notation, as per Convention 1. 9.1.2. We endow A τ,Q,W with the structure of an algebra object in the category of complexes of mixed Hodge modules on N Q0 . In order to achieve this, as in §3.2, a little care has to be taken to approximate morphisms of stacks by morphisms of varieties, so that we can apply Saito's theory of mixed Hodge modules to these morphisms. We spell this out in detail. We define We let GL τ d act on V τ,d,N via the product of the natural action of GL d on the first component, and the action of T τ on t τ given by the embedding (C * ) s ⊂ C s = t τ , and componentwise multiplication. We define U τ,d,N ⊂ V τ,d,N to be the subset consisting of those ({g i } i∈Q0 , f ) ∈ V τ,d,N such that each g i is surjective, and f is too. Then GL τ d acts freely on U τ,d,N . We break the multiplication into two parts. Fix a pair of dimension vectors d ′ , d ′′ and set d = d ′ + d ′′ . We write We embed GL d ′ ×d ′′ and GL d ′ ,d ′′ into GL d as a Q 0 -indexed product of Levi or parabolic subgroups, respectively. We define GL τ d , GL τ d ′ ,d ′′ and GL τ d ′ ×d ′′ , as the product of T τ with GL d , GL d ′ ,d ′′ and GL d ′ ×d ′′ , respectively.
For G an algebraic group with a fixed embedding G ⊂ GL τ d , we define the functor on G-equivariant varieties X If f : X → Y is a G-invariant morphism, we denote by f N : A N (X) → Y the induced morphism. For ι : Y ֒→ X a G-invariant subvariety, then as discussed in Section 3.2, for fixed i the mixed Hodge for N ≫ 0 depending on i. Consider the commutative diagram where q 1 and q 2 are the natural affine fibrations, inducing isomorphisms

Consider the composition of proper maps
Since r N and s N are proper, there is a natural morphism Applying Dim τ N, * φ Tr(W ) and letting N → ∞, the morphism (86) induces the morphism Defining where TS is the Thom-Sebastiani isomorphism, gives the multiplication We write A S Q,W for the special case in which T τ is the zero-dimensional torus (as in Example 9.2). In this case, the above multiplication is exactly the multiplication defined by Kontsevich and Soibelman in [28]. The proof that for general T τ the multiplication is associative is standard, and is in particular unchanged from the proof given in [28, Sec.7], to which we refer for fuller details.
9.2. The degeneration result. The extra equivariant parameters arising from the torus action on M(Q) are not considered in the original paper [28], but were introduced, for the particular cohomological Hall algebras we are considering, in [36] and [48,49]. In general, such extra parameters are of most interest when they provide a geometric deformation of the original algebra, i.e. when they provide a flat family of algebras over Spec(H T ), such that the specialization at the central fibre is our original algebra, which in this case is AQ ,W . For T τ the torus associated to aW -invariant grading ofQ this is precisely the result we prove in this section.
Let τ :Q 1 → Z s be aW -invariant grading, with associated torus T . Let υ : Z s → Z s ′ be a surjective morphism of groups, inducing the inclusion of tori T ′ ֒→ T , where T ′ is the torus associated to theW -invariant grading τ ′ = υ • τ . Write Then picking a splitting of υ, i.e. an extension of υ to an isomorphism Z s → Z s ′ ⊕ Z s ′′ , induces an isomorphism where χ :Q 1 → Z s ′′ is induced by the splitting. The splitting of υ induces a splitting We define Y τ,d,N := X(Q) d × GL τ d U τ,d,N . We consider the natural maps v d,N : Y τ,d,N → Hom surj (C N , t χ )/T χ =: S χ,N defined by the morphism
Proof. By choosing a splitting t χ = C ⊕s ′′ and considering the entries of a morphism f ∈ S χ,N one by one, we obtain a sequence of morphisms where l e is a (A N \ A e )/C * -fibration, with C * acting on A N via scaling. All the claims follow from this description.
Each of the maps v d,N is a fibre bundle with fibre Y τ ′ ,d,N . Picking d,N is the projection, and so we deduce that the mixed Hodge modules are locally trivial in the analytic topology, with fibre given by H q (Y τ ′ ,d,N , φ Tr(W ) τ ′ ,d,N Q Y τ ′ ,d,N ), and are furthermore globally trivial by the rigidity theorem [44,Thm.4.20], since the base of v d,N is simply connected.
As a result, the Leray spectral sequence Set In similar fashion, we obtain spectral sequences respectively. As in the construction of A τ,Q,W we obtain a commutative diagram of morphisms of spectral sequences, with vertical morphisms provided by restriction morphisms in cohomology The argument for all three statements is the same: fixing p and q, the limit E p,q υ,d,N,∞ depends only on a finite portion of E p,q υ,d,N,s , which therefore stabilises for sufficiently large N = N p,q . The (p, q)-term of both the second and third expression of (90) are then given by E p,q υ,d,Np,q,∞ . We may define the cohomological Hall algebra multiplication on A τ,Q,W via the commutative diagram obtained from (84) or as the morphism induced in the double limit by the composition of the horizontal morphisms in (89). Via the morphism in cohomology to a morphism in the category of algebra objects in the category of complexes of mixed Hodge structures.
Proof. First we consider the special case s ′ = 0, τ ′ = 0. Then by Theorem A and Lemma 9.5, the right hand side of (88) is a pure Hodge structure, and so the spectral sequence E •,• υ,d,∞,• degenerates at the second sheet, and the existence of the isomorphism (92) follows, along with the fact that (91) is an isomorphism.
As a consequence, A τ,Q,W is pure for all τ . So it follows that for general υ, the right hand side of (88) is pure, and the general case follows via the same argument as the special case.
is an isomorphism of algebras. Since Ψ τ,Q,d is a morphism of H T -modules, we deduce the following corollary of Theorem 9.6.
Corollary 9.7. Let m be the maximal homogeneous ideal in H T . Then A τ,ΠQ is free as a H T -module, and the natural morphism of algebras is an isomorphism.
Proof. While the vertical dimension reduction isomorphisms in the following commutative diagram are not morphisms of algebras (due to sign issues), the horizontal morphisms are, and the top one is an isomorphism since the bottom one is by Theorem 9.6: Set k := H T . Since X(Q) d is equivariantly contractible, there is an isomorphism in cohomology Here S d = i∈Q0 S di is the product of symmetric groups, with S di acting by permuting the variables x i,1 , . . . , x i,di . For d ′ + d ′′ = d, we define Sh d ′ ,d ′′ ⊂ S d to be the subset of permutations (σ i ) i∈Q0 such that for each i ∈ Q 0 we have inequalities σ i (1) < σ i (2) < . . . < σ i (d ′ i ) and σ i (d ′ i + 1) < . . . < σ i (d i ).
We fix generators t 1 , . . . , t s of H T , with t i corresponding to the generator of the equivariant cohomology of Hom(Z i , C * ), where Z i is the ith copy of Z inside Z s .
For a ∈Q 1 define E a (z) = z + i≤s τ (a) i t i .
Consider the stratification of the space N d by Jordan types, introduced in the proof of Lemma 4.3: if π = (π (i) ) i∈Q0 is a tuple of partitions, with each π (i) a partition of d i , the stratum N π ⊂ N d is the space for which the Jordan normal form of the operator assigned to ω i has blocks with sizes given by π (i) . Then it is easy to check that each H BM (N π , Q) has no I 1 -torsion, and the claim that H BM (N d , Q) has no I 1 -torsion follows from the long exact sequences in compactly supported cohomology induced by the stratification of N d . Remark 10.5. It is possible for the strata N π to have S 2 -torsion, so we cannot substitute I 2 for I 1 in the above proof, and merely insist on the inclusion C * 2 ⊂ T . Indeed we show in §10.4 that this (stronger) version of the statement of Theorem 10.2 with (weaker) assumptions is false.
Remark 10.6. For example, let Q be the Jordan quiver, and consider the partition π = (2) of the dimension vector 2 ∈ N Q0 . We set T = C * 1 × C * 2 , and write H T = Q[t 1 , t 2 ]. The space N π is smooth, and we claim that H T ×GL2(C) (N π , Q) has S 2 -torsion. There is an equivariant homotopy equivalence N π ≃ N ′ π , where N ′ π ⊂ Mat 2×2 (C) is the subspace of nonzero nilpotent matrices. We denote by T ′ ⊂ GL 2 (C) the usual maximal torus, and write There is precisely one GL 2 (C)-orbit inside N ′ π . So there is an isomorphism H T ×GL2(C) (N ′ π , Q) ∼ = H A where A is the stabiliser subgroup of the nilpotent matrix M = 0 1 0 0 .
We calculate H A = Q[z 1 , z 2 , t 1 , t 2 ]/(z 1 − z 2 − t 2 ) with the morphism H T ×GL2(C) → H A given by the natural surjection defined by the composition where S 2 fixes the variables t 1 , t 2 . The above factorization corresponds to the composition of inclusions where T ′′ ⊂ T × T ′ is the 3-torus stabilising M , and we are using that T ′′ is homotopic to A. In particular, the action of γ = t 2 2 − (z 1 − z 2 ) 2 on H A is not injective, although γ ∈ S 2 . On the other hand, clearly H A has no S 1 -torsion. 10.3. Noncommutativity. Theorem 10.2 enables explicit calculations inside A τ,ΠQ . Furthermore, although (as we have seen in Remarks 10.5 and 10.6, and will see further, with Proposition 10.11) it is important that we work equivariantly with respect to a sufficiently large torus T in Theorem 10.2, we will demonstrate in this section how Theorem 10.2 enables us to perform concrete calculations for trivial T , i.e. in the undeformed preprojective CoHA A ΠQ .
1 ] does not vanish. In particular, the algebra A ΠQ Jor is noncommutative.
Proof. We set Q = Q Jor . Pick τ as in Example 9.3, with associated torus T ∼ = C * 1 × C * 2 in the notation of the proof of Theorem 10.2. By Theorem 10.2 the morphism ι : A τ,ΠQ → A τ,Q is an inclusion of algebras. Write A ′ ⊂ A τ,Q for the image of this inclusion. Then by Corollary 9.7 there is an isomorphism of algebras We write A τ,ΠQ,1 ∼ = A ΠQ,1 ⊗ H T , and defineα First we calculate the commutator in A τ,Q : This element has cohomological degree −2. We claim that the unique nonzero element of cohomological degree less than −2 in ι(A ′ ) is x 0 1 ⋆ x 0 1 (up to scalar). Firstly, x 0 1 ⋆ x 0 1 has cohomological degree −4, since x 0 1 has cohomological degree −2. Secondly, it is indeed nonzero, as we calculate below. Finally, it follows from e.g. Corollary 5.3 that the stack C 2 ∼ = Coh d (A 2 ) of pairs of commuting 2 × 2 matrices has a unique irreducible component of (complex) dimension greater than 1, and that component has dimension 2. Now we calculate =2(x 1 − x 2 ) 2 − 2(t 2 1 + t 1 t 2 + t 2 2 ).
So we find s = 1 as required. we have seen that [1, u] = 0 (where the commutator is taken in A τ,ΠQ and lands in A τ,ΠQ,2 ). For T = C * l with l = 1, 2, 3, the shuffle algebra A τ, Q is commutative; e.g. for the C * 2 case, by inspection of (94) and the equations E a * (z) =z for a the unique arrow in Q it follows that A τ, Q is commutative. So Φ 2 (Im([·, ·])) = 0 and Φ 2 is not injective, proving both parts of the proposition. Now let ♯ = ∅, and E = 0. is nonzero, and the proof continues as in the ♯ = ∅ case.
The calculation of the commutator [x 1 , x 0 1 ] in the proof of Lemma 10.7 offers hope that C * 1 , C * 2 , C * 3 are the only three "bad" 1-dimensional tori, i.e. those for which Proposition 10.11 holds.