Torus-Equivariant Chow Rings of Quiver Moduli

We compute equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the equivariant Chow ring of the fixed point locus, and we compare the two descriptions.

the one-dimensional orbits. Our description of the image of the localization map then follows from a result of Brion [3].
Acknowledgements. I would like to thank Markus Reineke and Silvia Sabatini for inspiring discussions on this subject. At the time this research was conducted I was supported by the DFG SFB/TR 191 "Symplectic structures in geometry, algebra, and dynamics".

Generalities on Quiver Moduli
Fix an algebraically closed field k. Let Q be a quiver. We denote its set of vertices with Q 0 and its set of arrows with Q 1 . With s(α) and t(α) we denote the source and target of an arrow α. We assume throughout that Q is connected, which means that in the underlying unoriented graph, all vertices are connected by a path. We assume that the reader is familiar with the basics of representation theory of quivers. For an introduction to the subject we refer to [2,II.1].
For d ∈ Z ≥0 we fix k-vector spaces V i of dimension d i and we consider the vector space Hom(V s(α) , V t(α) ).
Its elements are representations of Q of dimension vector d. On R(Q, d) we have an action of the group G d = i∈Q 0 GL d i (k) by g · M = (g t(α) M α g −1 s(α) ) α∈Q 1 . Two elements of R(Q, d), considered as representations of Q, are isomorphic if and only if they lie in the same G d -orbit. Note that the image ∆ ⊆ G d of the diagonal embedding of G m acts trivially on R(Q, d) so the action of G d descends to an action of P G d = G d /∆.
A Z-linear map θ : Z Q 0 → Z is called a stability condition. For a vector d ∈ Z Q 0 ≥0 − {0} we define the slope of d as King shows in [6,Prop. 3.1] that the set of semi-stable points of R(Q, d) agrees with the set of semistable points with respect to the G d -linearization of the (trivial) line bundle on R(Q, d) which is given by the character Note that χ θ is trivial on ∆ whence it descends to a character of P G d . The twist in the exponent in the definition of the character χ θ was introduced in [10, 3.4] to get rid of the requirement θ(d) = 0. The set R(Q, d) θ−st is the set of properly stable points (in the sense of Mumford [8,Def. 1.8]) with respect to the aforementioned linearized line bundle. This is for the isotropy group of a stable representation is ∆ by Schur's lemma.
We define M θ−sst (Q, d) = R(Q, d) θ−sst / /P G d and M θ−st (Q, d) = R(Q, d) θ−st /P G d . The quotient map R(Q, d) θ−st → M θ−st (Q, d) is a fiber bundle in the étale topology with a smooth total space, so we conclude that M θ−st (Q, d) is smooth. The induced morphism M θ−sst (Q, d) → M 0−sst (Q, d) = R(Q, d)/ /P G d is projective. A result of Le Bruyn and Procesi [7,Thm. 1] states that the ring of invariants is generated by traces along oriented cycles in Q. If Q has no oriented cycles, in which case we call Q acyclic, then M θ−sst (Q, d) is projective. If R(Q, d) θ−sst = R(Q, d) θ−st we call θ generic for d and we write R(Q, d) θ for the (semi-)stable locus and M θ (Q, d) for the quotient.
Let V i be the (trivial) vector bundle on R(Q, d) with fiber V i equipped with the G d -linearization given by g(M, v) = (g · M, g i v). Note that ∆ does not act trivially on the fibers and hence V i does not descend along the geometric

Torus Actions on Quiver Moduli
Fix a quiver Q and a dimension vector d.
As it commutes with the P G d -action on R(Q, d), the action of T descends to an action on the geometric quotient M θ−st (Q, d).
The locus of fixed points M θ−st (Q, d) T can be described in terms of the universal abelian covering quiver Q which is given by The character x α ∈ X(T ) in the right-hand expression above is defined as x α (t) = t α . We say that a dimension vector β of Q covers d if χ β i,χ = d i for every vertex i. There is an action of Z Q 1 on the lattice of dimension vectors of Q given by ξ · β = (β i,χ+ξ ) (i,χ) . Two dimension vectors in the same orbit will be called translates of one another. Translates clearly cover the same dimension vector of Q. We define a stability condition θ for Q by θ (i,χ) = θ i . The following result is due to Weist: as a disjoint union of irreducible components X β , where β ranges over all dimension vectors of Q up to translation which cover d and each X β is isomorphic to

Torus Equivariant Tautological Relations
Again let Q be a quiver, let d be a dimension vector for Q and let θ be a stability condition.
. This is not far away from the G d × T -equivariant Chow ring. So we would first like to compute the rings We will be able to derive a presentation after passing to rational coefficients.
Choose a basis for each of the V i . Let T d be the maximal torus of G d of diagonal matrices and let ξ i,r ∈ X(T d ) be the character that selects in the matrix corresponding to the vertex i the r th diagonal entry. Recall that x α ∈ X(T ) is the character that selects the entry which corresponds to α. For a character In the above equation x i,r is the r th elementary symmetric function in the variables ξ i,1 , . . . , be the open embeddings. The pull-backs of j 1 and j 2 induce surjective homomorphisms of graded rings The push-forwards of i 1 and i 2 give surjections onto the kernels of j * 1 resp. j * 2 . The same arguments as in [4, Thm. 8.1, Thm. 9.1] show that the images of i * 1 and i * 2 can -after base change to Q -be re-written in terms of the T -equivariant CoHA multiplication. More precisely, we consider the correspondence diagram The left hand map is the projection and the right-hand map is the inclusion as a linear subspace. These maps are equivariant with respect to Passing to equivariant Chow groups with rational coefficients we obtain a map which sends f ⊗ g to f * g which is given by ) .
In the above formula π = (π i ) i∈Q 0 ranges over all elements of i∈Q 0 S d i for which each π i is a (d ′ i , d ′′ i )-shuffle permutation and The contribution of x α in the last summand can be explained as follows: inside is the zero locus of a section of a T d × T -equivariant vector bundle. The bundle is isomorphic to

Theorem 2. The kernels of the surjections
Now to the T -equivariant Chow ring of M θ−st (Q, d). The maximal torus T d of G d contains ∆. The quotient P T d := T d /∆ is a maximal torus of P G d . Its character lattice is given by be the pure tensor of unit vectors e r ⊗ e s embedded into the (i, j) th direct summand of Z d . On Z d there is an action of W d in the obvious way. The map Z d → X(P T d ) which sends ζ i,j r,t to ξ i,r − ξ j,t is well-defined, W d -equivariant, and surjective. It hence gives rise to a surjective homomorphim Remark 3. Suppose that k = C. The same arguments as in [4,Thm. 5.1] show that the G d × Tequivariant and also the P G d × T -equivariant cohomology of R(Q, d) θ−sst vanishes in odd degrees and that the cycle maps However, if semi-stability and stability do not agree then the equivariant cohomology groups of R(Q, d) θ−st will in general not vanish in odd degrees and the cycle maps will not be isomorphisms.

Localization at Torus Fixed Points
By Weist's result the locus of torus fixed points M θ (Q, d) decomposes into components which are isomorphic to M θ ( Q, β). The closed immersion To compute the pull-back ofι β in equivariant intersection theory we choose a basis e i,χ,1 , . . . , e i,χ,β i,χ of V i,χ and a bijection For convenience, we are going to neglect the dependency on β in the notation whenever possible. Let e i,r := e i,χ i,r ,s i,r . Then e i,1 , . . . , e i,d i is a basis. Consider the maximal torus T d of G d of diagonal matrices with respect to that basis; it is contained in the Levi subgroup G β . Its character lattice is In principle the image of x i,r underι * β can also be expressed in terms of the but the intermediate terms are more complicated. That is very explicit but it is not the action on R( Q, β) that we want to consider: If we look at the θ-unstable locus, it is a zero locus of a section of a G β × T -equivariant bundle where T does not act trivially. But we want to instead consider the G β × T -action where G β acts as usual and T acts trivially. Fortunately this is possible on R( Q, β) as we can rescale the action of t α using elements of G β . More concretely we consider the automorphism of the group G β × T given by T acts once as described above and once via the usual base change action with the G β -component and trivially with the T -component. We summarize the situation in the following diagram:

and the induced morphism in equivariant intersection theory is given by
This shows: Lemma 4. Let β be a dimension vector which covers d. What we actually want to determine is the image of the localization map ι * : (ξ i,χ i,rν ,s i,rν − χ i,rν − ξ j,χ j,tµ ,s j,tµ + χ j,tµ ).

(1) The pull-back map in equivariant intersection theory of the
Note that ξ i,χ,s − ξ j,η,t are the (non-equivariant) Chern roots of the bundle V i,χ ⊗ V ∨ j,η on M θ−st ( Q, β). Note also that f β (z i,j k,l ) is independent of the choice of a representative the class of translates of β. This shows: Theorem 5. The image of the pull-back map of the inclusion of the fixed point locus is the subring which is generated by the elements (f β (z i,j k,l )) β where i = j are vertices of Q and k = 1, . . . , d i , l = 1, . . . , d j and by elements of the form (1, . . . , 1)⊗x α with α ∈ Q 1 .
The description of the image in Theorem 5 is hard to handle in general but the case of an action with isolated fixed points is more manageable. Assume that each of the covering dimension vectors β of d is a real root of Q. In this case, each of the fixed point components M θ−st ( Q, β) is a single point. As there are only finitely many covering dimension vectors up to translation for d, this means that T acts with finitely many fixed points. In this case of the inclusion of the fixed point locus is the subring which is generated by the elements In the right-hand picture α 1 , . . . , α 4 ∈ {a, b, c} such that α 1 = α 2 = α 3 = α 4 (up to S 2 -symmetry, there are 12 of those combinations). It can be shown that in this case there are infinitely many one-dimensional T -orbits. Let u 1 = ξ i,1 , u 2 = ξ i,2 be the equivariant Chern roots of the G d × T -equivariant bundle V i and let v s = ξ j,s (with s = 1, 2, 3) be the equivariant Chern roots of V j . In S(Z d ) W d Q , we write z k,l := z i,j k,l , as we have only two vertices. We just need to consider the elements z 1,1 , z 2,1 , z 1,2 , and z 1,3 as their images under f : (j, c) (abusing notation, we write a for the character x a , and so on) and dimension vector β = (2, 1, 1, 1). After an appropriate choice of bases, the characters χ (β) i,r are given as χ (2) We consider a fixed point [M ] of the second kind. Up to translation the support of the covering dimension vector is In this case the characters χ i,r are is the entry in the t th row and r th column of the following table.
Using these, we compute the elements f β (z k,l ) as We can now determine the image of the map ι * : 13 . By Corollary 6, it is the subring generated by (a, . . . , a) T , (b, . . . , b) T , (c, . . . , c) T and the vectors 

Thin Quiver Moduli
We consider the special case of an acyclic quiver Q and the dimension vector d = 1 := (1, . . . , 1) (formally d i = 1 for every i ∈ Q 0 ). In this case the group G 1 = (G m ) Q 0 is a torus. A representation M ∈ R(Q, 1) consists of M α ∈ k and g ∈ G 1 acts via g · M = (g t(α) g −1 s(α) M α ) α . Again, the action descends to an action of P G 1 = G 1 /∆. Let T = (G m ) Q 1 which acts, like in the general case, by scaling. The action of P G 1 can be recovered from the T -action by embedding P G 1 as a subtorus via the map ) α whose kernel is precisely ∆; note that we assume Q to be connected. Let T 0 be the cokernel of this map. That means we have an exact sequence of tori Let θ be a stability condition. Without loss of generality we may assume θ(1) = 0. As the image of P G 1 inside T acts trivially on M θ−st (Q, 1) we obtain an action of T 0 on the moduli space. The torus T 0 acts with a dense orbit so the moduli space is toric [1]. By virtue of the exact sequence of tori above -which splits -we obtain an isomorphism of stacks which are induced by the open embeddings j 1 : R(Q, 1) θ−sst → R(Q, 1) and j 2 : R(Q, 1) θ−st → R(Q, 1) are given as follows: for a subset I ⊆ Q 0 let Then ker(j * 1 ) is the ideal generated by all x I with θ(1 I ) > 0 and ker(j * 2 ) is generated by all expressions x I for which θ(1 I ) ≥ 0.
In the above statement 1 I ∈ Z Q 0 denotes the characteristic function on the subset I ⊆ Q 0 . It is easy to read off a Q-linear basis from this characterizations as the ideal that we are dividing out is generated by monomials. For a tuple γ = (γ α ) α∈Q 1 ∈ Z Q 1 ≥0 write x γ := α∈Q 1 x γα α . For a subset I ⊆ Q 0 put J(I) := {α ∈ Q 1 | s(α) ∈ I, t(α) / ∈ I}. Then x I = x 1 I . A basis of A * T (R(Q, 1) θ−sst ) Q is given by all monomials x γ where supp(γ) contains no subset J(I) for which θ(1 I ) > 0; the monomials x γ for which J(I) supp(γ) for all I ⊆ Q 0 with θ(1 I ) ≥ 0 form a basis of A * T (R(Q, 1) θ−st ) Q . A basis of A * T (R(Q, 1) θ−sst ) Q can be obtained in a similar way. As a next step we would like to determine the pull-back of the embedding of the fixed point locus M θ−st (Q, 1) → M θ−st (Q, 1) T 0 . We introduce the following notion: a subset H ⊆ Q 1 is called a spanning tree if the underlying graph of (Q 0 , H) is a tree, which means it is connected and cycle-free. Introduce the formal symbol α −1 for every arrow α ∈ Q 1 and formally define s(α −1 ) = t(α) and t(α −1 ) = s(α). An unoriented path is a sequence p = α εr r . . . α ε 1 1 such that s(α ε ν+1 ν+1 ) = t(α εν ν ) for ν = 1, . . . , r − 1. We define s(p) = s(α ε 1 1 ) and t(p) = t(α εr r ). By the spanning tree property there exists for every i, j ∈ Q 0 an unoriented path p = α εr r . . . α ε 1 1 in H such that s(p) = i and t(p) = j. Let H ⊆ Q 1 be any subset. Define the representation M H ∈ R(Q, 1) by Note that for a representation M of Q with support H := supp(M ) the representation M H lies in the same T -orbit as M because T acts by scaling along the arrows. Now assume that H is a spanning tree. We say H is θ-stable if the representation M H is θ-stable. Denote by T H c the subtorus of all t = (t α ) α∈Q 1 ∈ T for which t α = 1 whenever α ∈ H. We use H c as a short-hand for Q 1 − H. (1) This can easily be deduced from the description of the fixed point locus in Theorem 1.
(2) Note that the number of arrows in a spanning tree of Q is ♯Q 0 − 1, which implies that the ranks of T H c and T 0 agree. It therefore suffices to show that the map T H c → T 0 is injective as a surjective map between two free abelian groups of the same finite rank is an isomorphism. Let t ∈ T H c be contained in the image of the map G 1 → T . Then there exists g ∈ G 1 such that For α ∈ H c let p be the unique unoriented path in H such that s(p) = s(α) and t(p) = t(α). Then t α = g t(α) g −1 s(α) = (g t(αr ) g −1 s(αr ) ) εr . . . (g t(α 1 ) g −1 s(α 1 ) ) ε 1 = 1. This proves the second assertion of the lemma.
We hence may identify for a θ-stable spanning tree H the equivariant Chow ring The pull-back of the inclusion i H of the fixed point then corresponds to the map We We need to require Q to be acyclic and θ to be generic for 1 in order to ensure the moduli space is smooth and projective. The action of T 0 on M θ (Q, 1) possesses just finitely many one-dimensional orbits. To describe them we introduce the notion of a spanning almost tree. A subset Ω ⊆ Q 1 is called a spanning almost tree if it is not a spanning tree but there exists an arrow α ∈ Ω such that Ω − {α} is a spanning tree. Note that this forces (Q 0 , Ω) to be connected. Given a spanning almost tree we again define a representation M Ω ∈ R(Q, 1) by assigning M α = 1 for α ∈ Ω and M α = 0 otherwise. We say Ω is θ-stable if M Ω is θ-stable. Proof. Let M be a θ-stable representation with a one-dimensional orbit. Put Ω := supp(M ). Note that stability of M implies connectedness of (Q 0 , Ω). Assume that Ω is not a θ-stable spanning almost tree. As M Ω is contained in the T -orbit of M we may assume M = M Ω . Let B = {α 1 , . . . , α r } be a maximal subset of Ω such that (Q 0 , Ω − B) is connected. Then r ≥ 2 as Ω is neither a spanning almost tree nor a spanning tree. Moreover, H := Ω − B is a spanning tree (which is not necessarily θ-stable). Let (t 1 , . . . , t r ) ∈ (k × ) r . Define the representation M t by Clearly M t is in the T -orbit of M . Moreover two such representations M t and M s are not isomorphic as a g ∈ G 1 with g · M t = M s would need to satisfy g · M H = M H which implies g = 1 by Lemma 9 (stability is not necessary for this argument to hold). This shows that the orbit of [M ] would be (at least) r-dimensional which is a contradiction. The converse direction is obvious.
The closure of such a one-dimensional orbit is isomorphic to P 1 and hence contains precisely two fixed points. This means that from a θ-stable spanning almost tree there are precisely two ways to remove an arrow and obtain a θ-stable spanning tree. This does not seem to be obvious from the combinatorics of stable spanning almost trees. Let these arrows be α 0 and α ∞ . Let Ω 0 = Ω − {α 0 } and Ω ∞ = Ω − {α ∞ }. These are the two θ-stable spanning trees which represent the two fixed points in the closure of the orbit of which send r Ω,0 (x α 0 ) = 0 and r Ω,∞ (x α∞ ) = 0 and act as the identity on the other variables. Let us write f | Ω c for the image of a function f ∈ A * under the maps r Ω,0 or r Ω,∞ , respectively.
Theorem 12. The pull-back of the embedding of the fixed point locus i : It is injective and its image consists precisely of those tuples (f H ) H for which f Ω 0 | Ω c = f Ω∞ | Ω c for every θ-stable spanning almost tree Ω.
The direct sum in the theorem ranges over all θ-stable spanning trees H of Q.
Proof. This is an application of Theorem 10. Let us show how the statement of our theorem follows from it. Let Ω be a θ-stable spanning almost tree. Let C be the closure of the We identify the T Ω c 0 -orbit of [M Ω ] with G m by the entry which corresponds to the arrow α 0 . Then T Ω c 0 acts on G m by the character x α 0 . Take the resulting identification of C with P 1 . The limit for t α 0 → 0 is [M Ω 0 ] and the limit for t α 0 → ∞ must hence be [M Ω∞ ]. Consider the composition of the isomorphisms of tori T Ω c 0 → T 0 → T Ω c ∞ from Lemma 9. Call it ϕ. To determine ϕ, let t ∈ T Ω c 0 and find the unique s ∈ T Ω c ∞ for which there exists a g ∈ G 1 such that g t(α) t α g −1 s(α) = s α for all α ∈ Q 1 . The element g is uniquely determined up to scaling. We get four equations: As Ω ∞ is a spanning tree, (Q 0 , Ω − {α 0 , α ∞ }) is not connected. Let C 1 , C 2 ⊆ Q 0 be its connected components. The vertices i 0 := s(α 0 ) and j 0 := t(α 0 ) lie in different components, so we may assume i 0 ∈ C 1 and j 0 ∈ C 2 . As Ω 0 is connected, i ∞ := s(α ∞ ) and j ∞ := t(α ∞ ) cannot be contained in the same component. We first consider the case where i ∞ ∈ C 1 and j ∞ ∈ C 2 . We get g i is constant on C 1 and on C 2 . Let its value on C 1 be g 1 and its value on C 2 be g 2 . Now we obtain in Ω c we again have to distinguish three cases. If i and j are in the same component then s α = t α . If i ∈ C 1 and j ∈ C 2 then we get s α = g 2 t α g −1 Finally if i ∈ C 2 and j ∈ C 1 then we obtain s α = g 1 t α g −1 2 = t α 0 t α . We define We get We now deal with the second case where i ∞ ∈ C 2 and j ∞ ∈ C 1 . Then ϕ looks the same except for ϕ(t) α∞ = t α 0 . The torus T Ω c ∞ acts on C by the character x α∞ . But as lim tα 0 →∞ ϕ(t).[M Ω ] = lim tα 0 →∞ t.[M Ω ] = [M Ω∞ ] we see that ϕ(t) α∞ = t −1 α 0 . So the second case will not occur. The induced map by ϕ on the character lattices is ϕ * : X(T Ω c ∞ ) → X(T Ω c 0 ) which is given by ϕ * (x α∞ ) = −x α 0 ϕ * (x α ) = x α + δ α x α 0 for α ∈ Ω c . Brion's theorem now tells us that the image of i * consists precisely of those tuples (f H ) H for which modulo x α 0 . This is the same as to say f Ω 0 | Ω c = f Ω∞ | Ω c .
Remark 13. It would be interesting to determine the canonical classes from [9] in the toric case and compare them to our result.
We end by illustrating this result in an example.
A Q-vector space basis of this algebra is given by all monomials x γ where γ = (γ ji ) ∈ M 3×2 (Z ≥0 ) is a matrix with at most one non-zero entry in each row and each column. Now to the fixed points and the one-dimensional orbits of the action of the rank 2 torus T 0 on M θ (Q, 1). We have to determine the stable spanning trees and the stable spanning almost trees. We describe them in the following picture: The matrices at the vertices of the above graph are the representations M H which correspond to the 6 stable spanning trees H of Q and the edges are the representations M Ω assigned to the 6 stable spanning almost trees Ω of Q. For a stable spanning almost tree Ω, the spanning trees attached to the adjacent vertices correspond to the fixed points which lie in the closure of the one-dimensional orbit associated with Ω. By Theorem 12 the pull-back i * of the embedding of the fixed point locus is the map which is induced by where the components are defined by f 1 1