Torus-Equivariant Chow Rings of Quiver Moduli
We compute rational 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 Chow ring of the fixed point locus, and we compare the two descriptions.
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| description | We compute rational 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 Chow ring of the fixed point locus, and we compare the two descriptions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 096, 22 pages
Torus-Equivariant Chow Rings of Quiver Moduli
Hans FRANZEN
Fakultät für Mathematik, Ruhr-Universität Bochum,
Universitätsstraße 150, 44780 Bochum, Germany
E-mail: hans.franzen@rub.de
URL: www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Algebra/franzen.html.en
Received March 14, 2020, in final form September 16, 2020; Published online September 30, 2020
https://doi.org/10.3842/SIGMA.2020.096
Abstract. We compute rational 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 Chow ring of the fixed point locus, and we
compare the two descriptions.
Key words: torus actions; equivariant Chow rings; torus localization; quiver moduli
2020 Mathematics Subject Classification: 14C15; 16G20
1 Introduction
In this paper we study actions of tori on moduli spaces of quiver representations and determine
their rational equivariant Chow rings as introduced by Edidin and Graham in [5]. More precisely,
given a quiver Q, a dimension vector d, and a stability condition θ, we look at the action of
T = GQ1
m which acts on M θ−st(Q, d) by scaling along the arrows of Q. This action was introduced
by Weist in [19] and has since been used successfully, for instance in [16, 17, 18]. The main objec-
tive of this paper is to determine the pull-back i∗ : A∗T
(
M θ−st(Q, d)
)
Q → A∗T
(
M θ−st(Q, d)T
)
Q of
the embedding i : M θ−st(Q, d)T → M θ−st(Q, d) of the fixed point locus explicitly and hence
to find the T -equivariant Chow ring of M θ−st(Q, d) as a subring of A∗
(
M θ−st(Q, d)T
)
Q ⊗
A∗T (pt)Q. This is called torus localization and it is an idea that goes back to Goresky–Kottwitz–
MacPherson’s paper [8] and even further to Chang–Skjelbred [4], where equivariant cohomology
rings of equivariantly formal spaces with respect to a compact torus are studied. Brion developed
in [3] the algebraic counterpart of this theory. He gives descriptions of the equivariant Chow ring
for an action of an algebraic torus on a variety. Both articles provide results of various generali-
ties depending on the nature of the action of the torus. The most explicit description is possible
if there are only finitely many fixed points and finitely many one-dimensional orbits. For quiver
moduli however, this is hardly ever fulfilled. The fixed points of the aforementioned torus action
are usually not isolated and even if they are, there might be infinitely many one-dimensional
orbits. Still there is one particularly easy class of quiver moduli in which this condition holds.
If we assume that the entries of d are all 1 then the moduli space is toric. This toric variety is
well understood, see [1]. We study this case separately.
One of the main results of [19] asserts that each connected component of the locus of T -fixed
points of M θ−st(Q, d) is isomorphic to a stable moduli space of the universal abelian covering
quiver Q̂ of Q. This result is the key ingredient to determine the localization map. We use
that M θ−st(Q, d) is an algebraic quotient R(Q, d)θ−st/PGd by a reductive group to identify the
T -equivariant Chow ring of the quotient with the PGd × T -equivariant Chow ring of the total
space. We can adapt a method of [6] to obtain a presentation of this ring in terms of Chern roots
of certain PGd×T -equivariant vector bundles. This is Theorem 2. We then exhibit in Theorem 5
the images of these generators under the localization map i∗ using Weist’s characterization of
mailto:hans.franzen@rub.de
www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Algebra/franzen.html.en
https://doi.org/10.3842/SIGMA.2020.096
2 H. Franzen
the fixed points. This description is most useful if the torus acts with finitely many fixed points.
We illustrate this result in an example where there are finitely many fixed points but infinitely
many one-dimensional orbits. After that we turn to the case where d = 1, i.e., d consists of ones
entirely. In this case, PG1 is itself a torus which embeds into T . We study the action of the
cokernel T0 on the moduli space. We determine the fixed points and the one-dimensional orbits.
Our description of the image of the localization map then follows from a result of Brion [3].
2 Generalities on quiver moduli
Fix an algebraically closed field k. Let Q be a quiver. We denote its set of vertices with Q0
and its set of arrows with Q1. 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. Furthermore, we suppose that for every two vertices
i, j ∈ Q0 there are just finitely many α ∈ Q1 such that s(α) = i and t(α) = j. If Q0 (and hence
also Q1) is finite, then we say that Q is a finite quiver.
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, Section II.1].
Let Λ(Q) be the set of vectors d ∈ ZQ0 such that di = 0 for all but finitely many i ∈ Q0.
For d ∈ Λ+(Q) := Λ(Q) ∩ ZQ0
≥0 we fix k-vector spaces Vi of dimension di and we consider the
finite-dimensional vector space
R(Q, d) =
⊕
α∈Q1
Hom(Vs(α), Vt(α)).
Its elements are representations of Q of dimension vector d. On R(Q, d) we have an action of
the group Gd =
∏
i∈Q0
GLdi(k) by
g ·M =
(
gt(α)Mαg
−1
s(α)
)
α∈Q1
.
Two elements of R(Q, d), considered as representations of Q, are isomorphic if and only if they
lie in the same Gd-orbit. Note that the image ∆ ⊆ Gd of the diagonal embedding of Gm acts
trivially on R(Q, d) so the action of Gd descends to an action of PGd = Gd/∆.
A Z-linear map θ : Λ(Q) → Z is called a stability condition. For a vector d ∈ Λ+(Q) − {0}
we define the slope of d as
µ(d) =
θ(d)∑
i di
.
Define µ(M) := µ(dimM). A representation M of Q is called θ-semi-stable (θ-stable) if µ(M ′) ≤
µ(M) (resp. µ(M ′) < µ(M)) for every non-zero proper subrepresentation M ′ of M .
We regard the vector space R(Q, d) as a variety and Gd as a linear algebraic group. The sets
R(Q, d)θ−sst and R(Q, d)θ−st of semi-stable resp. stable points of R(Q, d) are Zariski open (but
they might be empty). Obviously
R(Q, d)θ−st ⊆ R(Q, d)θ−sst ⊆ R(Q, d).
King shows in [9, Proposition 3.1] that the set of semi-stable points of R(Q, d) agrees with the set
of semi-stable points with respect to the Gd-linearization of the (trivial) line bundle on R(Q, d)
which is given by the character
χθ(g) =
∏
i∈Q0
det(gi)
θ(d)−θi
∑
j dj .
Torus-Equivariant Chow Rings of Quiver Moduli 3
Note that χθ is trivial on ∆ whence it descends to a character of PGd. The twist in the
exponent in the definition of the character χθ was introduced in [15, Section 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 [13, Defenition 1.8]) with respect to the aforementioned linearized line bundle. This
holds as the isotropy group of a stable representation is ∆ by Schur’s lemma.
We define M θ−sst(Q, d) = R(Q, d)θ−sst//PGd and M θ−st(Q, d) = R(Q, d)θ−st/PGd. The
quotient map R(Q, d)θ−st →M θ−st(Q, d) is a PGd-torsor in the étale topology (see [13, Propo-
sition 0.9]) with a smooth total space, so we conclude that M θ−st(Q, d) is smooth. The induced
morphism M θ−sst(Q, d)→M0−sst(Q, d) = R(Q, d)//PGd is projective. A result of Le Bruyn and
Procesi [11, Theorem 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 Vi be the (trivial) vector bundle on R(Q, d) with fiber Vi equipped with the Gd-lineari-
zation given by
g(M,v) = (g ·M, giv).
Note that ∆ does not act trivially on the fibers and hence Vi does not descend along the geometric
quotient R(Q, d)θ−st →M θ−st(Q, d). However, the bundles V∨i ⊗ Vj do.
3 Torus actions on quiver moduli
Fix a finite quiver Q and a dimension vector d. Let T = GQ1
m act on R(Q, d) as follows: an
element t = (tα)α ∈ T acts on M = (Mα)α ∈ R(Q, d) by t.M = (tαMα)α. As it commutes with
the PGd-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 Q̂. The (infinite) quiver Q̂ is given by Q̂0 = Q0 × ZQ1 , Q̂1 = Q1 × ZQ1 and for
α ∈ Q1 and χ ∈ ZQ1 the source and target of the arrow (α, χ) of Q̂ are
s(α, χ) = (s(α), χ), t(α, χ) = (t(α), χ+ xα).
The character xα ∈ X(T ) in the right-hand expression above is defined as xα(t) = tα.
We say that a dimension vector β ∈ Λ+
(
Q̂
)
covers d if
∑
χβi,χ = di for every vertex i. There
is an action of ZQ1 on Λ+
(
Q̂
)
given by ξ · β = (βi,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 θ̂ for Q̂ by θ̂(i,χ) = θi. The following result is due to Weist:
Theorem 1 ([19, Theorem 3.8]). The fixed point locus M θ−st(Q, d)T decomposes as
M θ−st(Q, d)T =
⊔
β
Xβ,
a disjoint union of irreducible components Xβ, where the union ranges over all β ∈ Λ+
(
Q̂
)
up
to translation which cover d. Each Xβ is isomorphic to
Xβ
∼= M θ̂−st
(
Q̂, β
)
.
4 H. Franzen
4 Torus-equivariant tautological relations
Again let Q be a finite quiver, let d be a dimension vector for Q and let θ be a stability condition.
We compute a presentation for the rational equivariant Chow ring A∗T
(
M θ−st(Q, d)
)
Q. For the
definition and general results on equivariant Chow rings we refer to [5]. We will work with Chow
rings with rational coefficients throughout.
The T -equivariant Chow ring A∗T
(
M θ−st(Q, d)
)
Q coincides with the PGd × T -equivariant
Chow ring A∗PGd×T (R(Q, d)θ−st)Q by [12, Lemma 2.1]. This is not far away from the Gd × T -
equivariant Chow ring. So we would first like to compute the rings
A∗Gd×T
(
R(Q, d)θ−st
)
Q, A∗Gd×T
(
R(Q, d)θ−sst
)
Q.
We will derive a presentation of these rings.
Choose a basis for each of the Vi. Let Td be the maximal torus of Gd of diagonal matrices.
Let us recap the definitions of the groups that we are using:
Gd =
∏
i∈Q0
GL(Vi) ∼=
∏
i∈Q0
GLdi(k),
Td ⊆ Gd maximal torus of invertible diagonal matrices,
∆ = {(z idVi)i | z ∈ k×} ⊆ Gd,
PGd = Gd/∆,
PTd = Td/∆,
T = GQ1
m .
Let ξi,r ∈ X(Td) be the character that selects in the matrix corresponding to the vertex i
the rth diagonal entry. Recall that xα ∈ X(T ) is the character that selects the entry which
corresponds to α. For a character χ ∈ X(Td × T ) = X(Td) ⊕X(T ) let L(χ) be the trivial line
bundle on R(Q, d) equipped with the Td × T -linearization induced by χ. So cTd×T1 (L(χ)) = χ.
As Gd × T acts linearly on the vector space R(Q, d), we obtain
A∗Gd×T (R(Q, d))Q ∼= A∗Gd×T (pt)Q ∼=
(⊗
i∈Q0
Q[ξi,1, . . . , ξi,di ]
Sdi
)
⊗Q Q[xα]α∈Q1
=
(⊗
i∈Q0
Q[xi,1, . . . , xi,di ]
)
⊗Q Q[xα]α∈Q1 .
In the above equation xi,r is the rth elementary symmetric function in the variables ξi,1, . . . , ξi,di .
Note that xi,r = cGd×Tr (Vi). Let j1 : R(Q, d)θ−sst → R(Q, d) and j2 : R(Q, d)θ−st → R(Q, d) be
the open embeddings. The pull-backs of j1 and j2 induce surjective homomorphisms of graded
rings
A∗Gd×T (R(Q, d))Q A∗Gd×T
(
R(Q, d)θ−sst
)
Q
A∗Gd×T
(
R(Q, d)θ−st
)
Q.
j∗1
j∗2
Let i1 and i2 be the closed embeddings of R(Q, d) − R(Q, d)θ−sst and R(Q, d) − R(Q, d)θ−st,
respectively. The push-forwards of i1 and i2 give surjections onto the kernels of j∗1 resp. j∗2 .
The same arguments as in [6, Theorems 8.1 and 9.1] show that the images of i∗1 and i∗2 can be
Torus-Equivariant Chow Rings of Quiver Moduli 5
re-written in terms of the T -equivariant CoHA multiplication. More precisely, we consider the
correspondence diagram
R(Q, d′)×R(Q, d′′)←
(
R(Q, d′) ∗
0 R(Q, d′′)
)
→ R(Q, d),
where d′ and d′′ are dimension vectors of Q such that d′ + d′′ = d. 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
Gd′ ×Gd′′ × T ←
(
Gd′ ∗
0 Gd′′
)
× T → Gd × T.
Passing to equivariant Chow groups we obtain a map
A∗Gd′×T (R(Q, d′))Q ⊗Q A
∗
Gd′′×T (R(Q, d′′))Q → A∗Gd×T (R(Q, d))Q,
which sends f ⊗ g to f ∗ g which is given by
f ∗ g =
∑
π
f(ξi,πi(r), xα)r=1,...,d′i
· g(ξi,πi(r), xα)r=d′i+1,...,di ·
∆1(ξi,πi(r), xα)
∆0(ξi,πi(r))
.
In the above formula π = (πi)i∈Q0 ranges over all elements of
∏
i∈Q0
Sdi for which each πi is
a (d′i, d
′′
i )-shuffle permutation and
∆0 =
∏
i∈Q0
d′i∏
r=1
di∏
s=d′i+1
(ξi,s − ξi,r),
∆1 =
∏
α:i→j
d′i∏
r=1
dj∏
s=d′j+1
(ξj,s − ξi,r + xα).
Recall that being a (d′i, d
′′
i )-shuffle permutation means πi(1) < · · · < πi(d′i) and πi(d′i + 1) <
· · · < πi(di). The contribution of xα in the expression for ∆1 can be explained as follows:
inside
(R(Q,d′) ∗
0 R(Q,d′′)
)
, the subset R(Q, d′)×R(Q, d′′) is the zero locus of a section of a Td×T -
equivariant vector bundle. The bundle is isomorphic to
⊕
α : i→j
d′i⊕
r=1
dj⊕
s=d′j+1
L(ξi,r)
∨ ⊗ L(ξj,s)⊗ L(xα).
Theorem 2. The kernels of the surjections(⊗
i∈Q0
Q[ξi,1, . . . , ξi,di ]
Sdi
)
⊗QQ[xα]α∈Q1 = A∗Gd×T (R(Q, d))Q A∗Gd×T
(
R(Q, d)θ−sst
)
Q
A∗Gd×T
(
R(Q, d)θ−st
)
Q
j∗1
j∗2
are the Q-linear subspaces
ker(j∗1) =
∑
d′,d′′∈ZQ0
≥0
d′+d′′=d
µ(d′)>µ(d′′)
{
f ∗ g | f ∈ A∗Gd′×T (R(Q, d′))Q, g ∈ A∗Gd′′×T (R(Q, d′′))Q
}
,
ker(j∗2) =
∑
d′,d′′∈ZQ0
≥0
d′+d′′=d
µ(d′)≥µ(d′′)
{
f ∗ g | f ∈ A∗Gd′×T (R(Q, d′))Q, g ∈ A∗Gd′′×T (R(Q, d′′))Q
}
.
6 H. Franzen
Proof. The statement of the theorem follows from a suitable adaption of the arguments in [6,
Theorems 8.1 and 9.1]. �
Now to the T -equivariant Chow ring of M θ−st(Q, d). The maximal torus Td of Gd contains ∆.
The quotient PTd := Td/∆ is a maximal torus of PGd. Its character lattice is given by
X(PTd) =
η =
∑
i∈Q0
di∑
r=1
bi,rξi,r |
∑
i∈Q0
di∑
r=1
bi,r = 0
⊆ X(Td).
The Weyl group of PTd inside PGd is also Wd :=
∏
i∈Q0
Sdi . Therefore
A∗PGd×T (R(Q, d))Q ∼= S(X(PTd))
Wd
Q ⊗Q Q[xα]α∈Q1
by [5, Proposition 6]. The ring homomorphism A∗PGd×T (R(Q, d))Q → A∗Gd×T (R(Q, d))Q which
comes from Gd → PGd corresponds to the homomorphism that is induced by the inclusion
X(PTd) ↪→ X(Td). We can give generators for S(X(PTd))
Wd
Q . Fix an order on Q0, say Q0 =
{1, . . . , n}. Then consider the lattice Zd =
⊕
1≤i<j≤nZdi ⊗ Zdj . Let ζi,jr,s be the pure tensor
of unit vectors er ⊗ es embedded into the (i, j)th direct summand of Zd. On Zd there is an
action of Wd in the obvious way. The map Zd → X(PTd) which sends ζi,jr,t to ξi,r − ξj,t is
well-defined, Wd-equivariant, and surjective. It hence gives rise to a surjective homomorphim
f : S(Zd)
Wd
Q → S(X(PTd))
Wd
Q . The ring S(Zd)
Wd
Q is generated by the algebraically independent
elements
zi,jk,l :=
∑
1≤r1<···<rk≤di
1≤t1<···<tl≤dj
k∏
ν=1
l∏
µ=1
ζrν ,tµ .
Remark 3. Suppose that k = C. The same arguments as in [6, Theorem 5.1] show that
the Gd × T -equivariant and also the PGd × T -equivariant cohomology of R(Q, d)θ−sst with ra-
tional coefficients vanishes in odd degrees and that the cycle maps AiGd×T
(
R(Q, d)θ−sst
)
Q →
H2i
Gd×T
(
R(Q, d)θ−sst;Q
)
and AiPGd×T
(
R(Q, d)θ−sst
)
Q → H2i
PGd×T
(
R(Q, d)θ−sst;Q
)
are isomor-
phisms. 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.
5 Localization at torus fixed points
In this section we compute the pull-back ι∗ : A∗T
(
M θ−st(Q, d)
)
Q → A∗T
(
M θ−st(Q, d)T
)
Q along
the inclusion ι : M θ−st(Q, d)T → M θ−st(Q, d). By Weist’s result the locus of torus fixed points
M θ(Q, d) decomposes into components which are isomorphic to M θ̂
(
Q̂, β
)
. It therefore suffices
to compute the pull-back
ι∗β : A∗T
(
M θ−st(Q, d)
)
Q → A∗T
(
M θ̂−st
(
Q̂, β
))
Q
along the regular closed immersion
ιβ : M θ̂
(
Q̂, β
)
→M θ(Q, d).
Fix for every i ∈ Q0 a decomposition Vi =
⊕
χ∈X(T ) Vi,χ into subspaces of dimension dimVi,χ =
βi,χ. This amounts to embedding Gβ as a Levi subgroup of Gd. The immersion ιβ is provided
by the map
ι̃β : R
(
Q̂, β
)
→ R(Q, d)
Torus-Equivariant Chow Rings of Quiver Moduli 7
which sends a representation N ∈ R
(
Q̂, β
)
to M = ι̃β(N) ∈ R(Q, d) where Mα : Vi → Vj is
defined by
Mα
(∑
χ
vχ
)
=
∑
χ
Nα,χ(vχ)
for vi,χ ∈ Vi,χ. The map ι̃β is Gβ × T -equivariant with respect to the Gβ × T -action a1 on
R(Q̂, β) which is defined by
a1 : (Gβ × T )×R
(
Q̂, β
)
→ R
(
Q̂, β
)
, ((g, t), N) 7→
(
tαgj,χ+xαNα,χg
−1
i,χ
)
α,χ
.
With respect to this action, R
(
Q̂, β
)θ̂−st
is invariant and the action descends to an action of
PGβ × T . To compute the pull-back of ι̃β in equivariant intersection theory we choose a basis
ei,χ,1, . . . , ei,χ,βi,χ of Vi,χ and a bijection
ϕ
(β)
i : {1, . . . , di} → {(χ, s) |χ ∈ X(T ), s ∈ {1, . . . , βi,χ}},
r 7→
(
χ
(β)
i,r , s
(β)
i,r
)
.
For convenience, we are going to neglect the dependency on β in the notation whenever possible.
Let ei,r := ei,χi,r,si,r . Then ei,1, . . . , ei,di is a basis of Vi. Consider the maximal torus Td of Gd
of diagonal matrices with respect to that basis; it is contained in the Levi subgroup Gβ. Its
character lattice is
⊕
i∈Q0
di⊕
r=1
Zξi,r =
⊕
i∈Q0
⊕
χ∈X(T )
βi,χ⊕
s=1
Zξi,χ,s,
where ξi,r := ξi,χi,r,si,r . Now
A∗Gd×T (R(Q, d))Q =
(⊗
i∈Q0
Q[ξi,1, . . . , ξi,di ]
Sdi
)
⊗Q Q[xα]α∈Q1
=
(⊗
i∈Q0
Q[xi,1, . . . , xi,di ]
)
⊗Q Q[xα]α∈Q1 ,
A∗Gβ×T (R(Q̂, β))Q =
(⊗
i∈Q0
⊗
χ∈X(T )
Q[ξi,χ,1, . . . , ξi,χ,βi,χ ]
Sβi,χ
)
⊗Q Q[xα]α∈Q1
=
(⊗
i∈Q0
⊗
χ∈X(T )
Q[xi,χ,1, . . . , xi,χ,βi,χ ]
)
⊗Q Q[xα]α∈Q1 ,
where xi,r = er(ξi,1, . . . , ξi,di) and xi,χ,s = es(ξi,χ,1, . . . , ξi,χ,βi,χ). The map ι̃∗β is given by ι̃∗β(xα) =
xα and by ι̃∗β(ξi,r) = ξi,χi,r,si,r . This implies
ι̃∗β(xi,r) = er(ξi,χ1,s1 , . . . , ξi,χdi ,sdi ).
In principle the image of xi,r under ι̃∗β can also be expressed in terms of the xi,χ,s’s, like ι̃∗β(xi,1) =∑
χ xi,χ,1 and ι̃∗β(xi,di) =
∏
χ xi,χ,βi,χ , but the intermediate terms are more complicated.
8 H. Franzen
We obtain the following commutative diagram:
A∗T
(
M θ−st(Q, d)
)
Q A∗T
(
M θ̂−st(Q̂, β)
)
Q
A∗PGd×T
(
R(Q, d)θ−st
)
Q A∗PGβ×T
(
R(Q̂, β)θ̂−st
)
Q
A∗Gd×T
(
R(Q, d)θ−st
)
Q A∗Gβ×T
(
R(Q̂, β)θ̂−st
)
Q
A∗PGd×T (R(Q, d))Q A∗PGβ×T (R(Q̂, β))Q
A∗Gd×T (R(Q, d))Q A∗Gβ×T (R(Q̂, β))Q
ι∗β
∼= ∼=
ι̃∗β
j∗β
ι̃∗β
j∗β
where jβ : R
(
Q̂, β
)θ̂−st → R
(
Q̂, β
)
is the open embedding of the stable locus. In the above
diagram the action of Gβ × T and PGβ × T is by a1. As T acts trivially on M θ̂−st
(
Q̂, β
)
, we
may identify
A∗T
(
M θ̂−st(Q̂, β)
)
Q
∼= A∗
(
M θ̂−st
(
Q̂, β
))
Q ⊗Q S(X(T ))Q
∼= A∗PGβ
(
R
(
Q̂, β
)θ̂−st)
Q ⊗Q S(X(T ))Q.
We introduce another action a2 of Gβ × T on R
(
Q̂, β
)
. Define
a2 : (Gβ × T )×R
(
Q̂, β
)
→ R
(
Q̂, β
)
, ((g, t), N) 7→
(
gj,χ+xαNα,χg
−1
i,χ
)
α,χ
.
The action a2 leaves the θ̂-stable locus invariant and descends to an action of PGβ × T . With
respect to the action a2 we obtain
A∗PGβ
(
R
(
Q̂, β
)θ̂−st)
Q ⊗Q S(X(T ))Q ∼= A∗PGβ×T
(
R
(
Q̂, β
)θ̂−st)
Q.
We are going to provide an isomorphism between the equivariant Chow rings with respect to
the actions a1 and a2. To this end we consider the automorphism of the group Gβ × T given by
Φ: Gβ × T → Gβ × T, (g, t) 7→ ((χ(t)gi,χ)i,χ, t).
Then we get a commutative diagram
(Gβ × T )×R
(
Q̂, β
)
(Gβ × T )×R
(
Q̂, β
)
R
(
Q̂, β
)
R
(
Q̂, β
)
.
Φ×id
a1 a2
id
This yields an isomorphism of quotient stacks ϕ :
[
R
(
Q̂, β
)
/(Gβ × T )
]
via a1
→
[
R
(
Q̂, β
)
/(Gβ ×
T )
]
via a2
. We regard the equivariant Chow ring as the Chow ring of the corresponding quotient
stack. This is justified by arguments by Edidin and Graham [5, Section 5.3] and by Kresch [10].
Torus-Equivariant Chow Rings of Quiver Moduli 9
The pull-back ϕ∗ in intersection theory sends ϕ∗(xα) = xα and ϕ∗(ξi,χ,s) = ξi,χ,s + χ. We get
another commutative diagram:
A∗T
(
M θ̂−st
(
Q̂, β
))
Q A∗
(
M θ̂−st
(
Q̂, β
))
Q ⊗Q S
A∗PGβ×T
(
R
(
Q̂, β
)θ̂−st)
Q A∗PGβ×T
(
R
(
Q̂, β
)θ̂−st)
Q A∗PGβ
(
R
(
Q̂, β
)θ̂−st)
Q ⊗Q S
A∗Gβ×T
(
R
(
Q̂, β
)θ̂−st)
Q A∗Gβ×T
(
R
(
Q̂, β
)θ̂−st)
Q A∗Gβ
(
R
(
Q̂, β
)θ̂−st)
Q ⊗Q S
A∗PGβ×T
(
R
(
Q̂, β
))
Q A∗PGβ×T
(
R
(
Q̂, β
))
Q A∗PGβ
(
R
(
Q̂, β
))
Q ⊗Q S
A∗Gβ×T
(
R
(
Q̂, β
))
Q A∗Gβ×T
(
R
(
Q̂, β
))
Q A∗Gβ
(
R
(
Q̂, β
))
Q ⊗Q S
∼=
∼= ∼=
∼=
j∗β
∼=
j∗β
j∗β⊗id
∼= ∼=
(ϕ−1)∗
∼=
∼=
(ϕ−1)∗
∼=
j∗β
j∗β
∼=
j∗β⊗id
Here we have used the short-hand S := S(X(T ))Q = Q[xα]α∈Q1 . In the left-hand side wall of
the diagram, the action is via a1 while in the middle plane the action is via a2. In the right-hand
side wall of the diagram, all morphisms are of the form r⊗ id for some homomorphism r between
the respective Chow rings.
Let iβ := ι̃β ◦ ϕ−1. It is a morphism of quotient stacks
iβ :
[
R
(
Q̂, β
)
/(Gβ × T )
]
via a2
→ [R(Q, d)/(Gd × T )].
The pull-back along this morphism exists and is given as follows:
Lemma 4. Let β be a dimension vector which covers d.
1. The pull-back map in equivariant intersection theory of iβ is the map
i∗β :
(⊗
i∈Q0
Q[xi,1, . . . , xi,di ]
)
⊗Q Q[xα]α∈Q1
→
(⊗
i∈Q0
⊗
χ∈X(T )
Q[xi,χ,1, . . . , xi,χ,βi,χ ]
)
⊗Q Q[xα]α∈Q1
defined by i∗β(xi,r) = er(ξi,χ1,s1 −χ1, . . . , ξi,χdi ,sdi −χdi) and by i∗β(xα) = xα where α ∈ Q1.
2. The pull-backs of the restrictions
[
R
(
Q̂, β
)θ̂−sst
/(Gβ × T )
]
via a2
→
[
R(Q, d)θ−sst/(Gd × T )
]
,[
R
(
Q̂, β
)θ̂−st
/(Gβ × T )
]
via a2
→
[
R(Q, d)θ−st/(Gd × T )
]
of iβ are the induced maps by i∗β on the quotients by the ideals described in Theorem 2.
Next we determine the image of the localization map
ι∗ : A∗T (M θ−st(Q, d))Q → A∗(M θ−st(Q, d)T )Q ⊗Q S(X(T ))Q.
10 H. Franzen
Recall that S(Zd)
Wd
Q surjects onto A∗PGd(R(Q, d))Q. Consider the commutative diagram
S(Zd)Q A∗PTd×T (R(Q, d))Q A∗PTd×T
(
R(Q, d)θ−st
)
Q A∗PTβ×T
(
R
(
Q̂, β
)θ̂−st)
Q
S(Zd)
Wd
Q A∗PTd×T (R(Q, d))
Wd
Q A∗PTd×T
(
R(Q, d)θ−st
)Wd
Q A∗PTβ×T
(
R
(
Q̂, β
)θ̂−st)Wβ
Q
A∗PGd×T (R(Q, d))Q A∗PGd×T
(
R(Q, d)θ−st
)
Q A∗PGβ
(
R
(
Q̂, β
)θ̂−st)
Q ⊗Q[xα]α∈Q1
A∗T
(
Mθ−st(Q, d)
)
Q A∗
(
M θ̂−st
(
Q̂, β
))
Q ⊗Q Q[xα]α∈Q1
∼= ∼=
i∗β
∼=
ι∗β
∼= ∼=
In the right-most column, PGβ × T acts via a2. Let fβ : S(Zd)
Wd
Q → A∗
(
M θ̂−st
(
Q̂, β
))
Q ⊗Q
Q[xα]α∈Q1 be the composition of two diagonal arrows with the arrow ι∗β. It maps
fβ(zi,jk,l) =
∑
1≤r1<···<rk≤di
1≤t1<···<tl≤dj
k∏
ν=1
l∏
µ=1
(ξi,χi,rν ,si,rν − χi,rν − ξj,χj,tµ ,sj,tµ + χj,tµ).
Note that ξi,χ,s − ξj,η,t are the (non-equivariant) Chern roots of the bundle Vi,χ ⊗ V∨j,η on
M θ̂−st
(
Q̂, β
)
. Note also that fβ
(
zi,jk,l
)
is independent of the choice of a representative from
the class of translates of β. This shows:
Theorem 5. The image of the pull-back map
ι∗ : A∗T
(
M θ−st(Q, d)
)
Q → A∗
(
M θ−st(Q, d)T
)
Q ⊗Q S(X(T ))Q
=
(⊕
β
A∗
(
M θ̂−st
(
Q̂, β
)))
⊗Q Q[xα]α∈Q1
of the inclusion of the fixed point locus is the subring which is generated by the elements(
fβ
(
zi,jk,l
))
β
where i 6= j are vertices of Q and k = 1, . . . , di, l = 1, . . . , dj and by elements
of the form (1, . . . , 1)⊗ xα with α ∈ Q1.
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 roots β
of d is a real root of Q̂. In this case, each of the fixed point components M θ̂−st
(
Q̂, β
)
is a single
point. As there are only finitely many covering dimension vectors with connected support up to
translation for d, this means that T acts with finitely many fixed points. In this case
fβ
(
zi,jk,l
)
=
∑
1≤r1<···<rk≤di
1≤t1<···<tl≤dj
k∏
ν=1
l∏
µ=1
(
χ
(β)
j,tµ
− χ(β)
i,rν
)
.
Corollary 6. Suppose that every covering root of d is a real root of Q̂. Let B = {β1, . . . , βN}
be a set of representatives of translation classes of covering roots of d for which there exists
a θ̂-stable representation. Then the image of the pull-back map
ι∗ : A∗T
(
M θ−st(Q, d)
)
Q → Q[xα]⊕Nα∈Q1
of the inclusion of the fixed point locus is the subring which is generated by the elements( ∑
1≤r1<···<rk≤di
1≤t1<···<tl≤dj
k∏
ν=1
l∏
µ=1
(
χ
(β1)
j,tµ
− χ(β1)
i,rν
)
, . . . ,
∑
1≤r1<···<rk≤di
1≤t1<···<tl≤dj
k∏
ν=1
l∏
µ=1
(
χ
(βN )
j,tµ
− χ(βN )
i,rν
))
,
Torus-Equivariant Chow Rings of Quiver Moduli 11
where i 6= j are vertices of Q and k = 1, . . . , di, l = 1, . . . , dj and by the elements (xα, . . . , xα)
for α ∈ Q1.
Example 7. Let us confirm the above formula in a well-known example. Let Q be the n + 1-
Kronecker quiver. That is the quiver with vertices i and j and arrows a0, . . . , an : i → j.
Let d = (1, 1) and let θ = (1,−1). A representation of Q of dimension vector d is a tuple
(p0, . . . , pn) ∈ kn. It is (semi-)stable with respect to θ if and only if (p0, . . . , pn) 6= (0, . . . , 0). An
isomorphism M θ(Q, d)→ Pn is provided by (p0, . . . , pn) 7→ [p0 : · · · : pn].
The torus T = Gn+1
m acts on Pn by
t.p = [t0p0 : · · · : tnpn].
Let xi be the character of T given by xi(t) = ti. We equip the line bundle O(−1) with a
T -linearization as follows: for v = (v0, . . . , vn) in the fiber of O(−1) over a point p let t.v =
(t0v0, . . . , tnvn). Then define
h := cT1 (O(1)).
Let s0, . . . , sn ∈ H0
(
Pn,O(1)
)
be the usual sections. Then sν is of weight −xν with respect
to the induced T -action on global sections. This implies that sν is a T -invariant section of the
bundle O(1)⊗L(xν), where L(xν) is the trivial bundle on Pn equipped with the T -linearization
given by xν . The bundle O(1)⊗(L(x0)⊕· · ·⊕L(xn)) hence has a no-where vanishing T -invariant
global section which shows (x0 + h) · · · (xn + h) = 0 in A∗T
(
Pn
)
Q. It can be shown that this is
the only relation, so
A∗T (Pn)Q = Q[x0, . . . , xn, h]/(x0 + h) · · · (xn + h).
The locus of T -fixed points of Pn is {e0, . . . , en}, where eν = [0 : · · · : 1 : · · · : 0] is the unit vector
with an entry in the νth position. Let ιν be the inclusion of {eν} into Pn. Then ι∗νO(1) = L(−xν),
so ι∗ν(h) = −xν . The pull-back of the inclusion ι : (Pn)T → Pn is therefore the map
ι∗ : Q[x0, . . . , xn, h]/(x0 + h) · · · (xn + h)→ Q[x0, . . . , xn]⊕n+1
given by ι∗(xν) = (xν , . . . , xν) and ι∗(h) = −(x0, . . . , xn).
Let us try and match this with the formula from the previous corollary. The n + 1 fixed
points of the T -action on M θ(Q, d) are given by dimension vectors β0, . . . , βn; the entries of
these dimension vectors are 0 or 1 and the support of βν is the sub-quiver
(i, 0)
aν−→ (j, xν)
of Q̂. Corollary 6 tells us that the image of ι∗ is the subring of Q[x0, . . . , xn]⊕n+1 which is
generated by the elements (xν , . . . , xν) and by zi,j1,1 = (x0, . . . , xn). So the two descriptions of
the image of ι∗ do indeed agree.
Example 8. Let Q be the 3-Kronecker quiver. That is the quiver with two vertices i and j
and 3 arrows a, b, c : i → j. Consider the stability condition θ = (3,−2) and the dimension
vector d = (2, 3) (so θ(d) = 0). The dimension of the moduli space M θ(Q, d) is 1− 〈d, d〉Q = 6.
The torus T = G3
m acts by scaling the linear maps Ma, Mb, and Mc individually. There are 13
T -fixed points (from which we see that the Euler characteristic of M θ is 13) which correspond
12 H. Franzen
to the following two types of covering quivers:
1
2 1
1
a
b
c
1
1
1
1
1
α1
α2
α3
α4
In the right-hand picture α1, . . . , α4 ∈ {a, b, c} such that α1 6= α2 6= α3 6= α4 (up to S2-symmetry,
there are 12 of those combinations).
We show that in this case there are infinitely many one-dimensional T -orbits. For a pair
(x, y) ∈ k × k let M(x, y) be the representation
M(x, y) =
0 0
0 1
x y
,
1 0
0 0
0 0
,
0 0
1 0
0 1
.
It can be seen that every representation M(x, y) is θ-stable and that the Gd-orbits of M(x, y) and
M(x′, y′) are disjoint, provided that (x, y) 6= (x′, y′). Let t = (u, v, w) ∈ T . Then t.M(x, y) is
contained in the Gd-orbit of M((u2/w2)x, (u/w)y); indeed g · t.M(x, y) = M((u2/w2)x, (u/w)y)
for
g =
(w 0
0 u
)
,
w/v 0 0
0 1 0
0 0 u/w
.
For x, y ∈ k×, the T -orbit of the isomorphism class [M(x, y)], regarded as a point of M θ(Q, d),
is therefore T [M(x, y)] =
{[
M
(
z2x, zy
)]
| z ∈ k×
}
; in particular, it is one-dimensional. We see
that the value x/y2 is a well-defined invariant of the T -orbit T [M(x, y)] and it separates all these
T -orbits. We have thus found a family of one-dimensional T -orbits indexed by k×.
Let u1 = ξi,1, u2 = ξi,2 be the equivariant Chern roots of the Gd × T -equivariant bundle Vi
and let vs = ξj,s (with s = 1, 2, 3) be the equivariant Chern roots of Vj . In S(Zd)
Wd
Q , we write
zk,l := zi,jk,l, as we have only two vertices. We just need to consider the elements z1,1, z2,1, z1,2,
and z1,3 as their images under f : S(Zd)
Wd
Q → S(X(PTd))
Wd
Q span S(X(PTd))
Wd
Q as a ring.
1. We look at the fixed point [M ] which corresponds to the covering quiver
(j, a)
(i, 0) (j, b)
(j, c)
(abusing notation, we write a for the character xa, 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
(β)
i,1 = 0 = χ
(β)
i,2 ,
χ
(β)
j,1 = a, χ
(β)
j,2 = b, and χ
(β)
j,3 = c. We now compute the values fβ(zk,l) as
fβ(z1,1) = 2(a+ b+ c), fβ(z2,1) = a2 + b2 + c2,
fβ(z1,2) = 2e2(a, b, c), fβ(z1,3) = 2abc.
Torus-Equivariant Chow Rings of Quiver Moduli 13
2. We consider a fixed point [M ] of the second kind. Up to translation the support of the
covering dimension vector is
(2, 0)
(1,−α1)
(2,−α1 + α2)
(1,−α1 + α2 − α3)
(2,−α1 + α2 − α3 + α4)
In this case the characters χ
(β)
i,r are χ
(β)
i,1 = −α1, χ
(β)
i,2 = −α1 +α2−α3, χ
(β)
j,1 = 0, χ
(β)
j,2 = −α1 +α2,
and χ
(β)
j,3 = −α1 + α2 − α3 + α4. The difference χ
(β)
j,t − χ
(β)
i,r is the entry in the tth row and rth
column of the following table:
1 2
1 α1 α1 − α2 + α3
2 α2 α3
3 α2 − α3 + α4 α4
Using these, we compute the elements fβ(zk,l) as
fβ(z1,1) = 2α1 + α2 + α3 + 2α4,
fβ(z2,1) = α2
1 − α1α2 + α1α3 + α2α3 + α2α4 − α3α4 + α2
4,
fβ(z1,2) = 2α1α2 + 2α1α4 + α2
2 − 2α2α3 + α2
3 + 2α3α4,
fβ(z1,3) = α1α
2
2 − α1α2α3 + α1α2α4 + α1α3α4 − α2α3α4 + α2
3α4.
Let C =
{
(α1, . . . , α4) ∈ {a, b, c}4 |α1 6= α2 6= α3 6= α4
}
/S2 where S2 reverses the order of
the tuple. Denote the element of C represented by (α1, . . . , α4) by α1 · · ·α4. Then
C = {abab, abac, abca, abcb, acab, acac, acbc, babc, bacb, bcac, bcbc, cabc}.
We can now determine the image of the map ι∗ : A∗T
(
M θ(Q, d)
)
Q →
(
Q⊕QC
)
⊗Q S(X(T ))Q ∼=
Q[a, b, c]13. By Corollary 6, it is the subring generated by (a, . . . , a)T , (b, . . . , b)T , (c, . . . , c)T
and the vectors
2(a+ b+ c)
3a+ 3b
3a+ b+ 2c
4a+ b+ c
2a+ 3b+ c
3a+ 2b+ c
3a+ 3c
2a+ b+ 3c
a+ 3b+ 2c
a+ 4b+ c
a+ 2b+ 3c
3b+ 3c
a+ b+ 4c
,
a2 + b2 + c2
2a2 − ab+ 2b2
2a2 − ac+ bc+ c2
2a2 + bc
a2 − ab+ 2b2 + ac
2a2 − ab+ b2 + bc
2a2 − ac+ 2c2
a2 + ab− ac+ 2c2
2b2 + ac− bc+ c2
2b2 + ac
ab+ b2 − bc+ 2c2
2b2 − bc+ 2c2
ab+ 2c2
,
14 H. Franzen
2ab+ 2ac+ 2bc
a2 + 4ab+ b2
a2 + b2 + 4ac
2a2 + 2ab+ b2 + 2ac− 2bc+ c2
4ab+ b2 + c2
a2 + 4ab+ c2
a2 + 4ac+ c2
b2 + 4ac+ c2
a2 + b2 + 4bc
a2 + 2ab+ 2b2 − 2ac+ 2bc+ c2
a2 + 4bc+ c2
b2 + 4bc+ c2
a2 − 2ab+ b2 + 2ac+ 2bc+ 2c2
,
2abc
a2b+ ab2
−a2b+ ab2 + 2a2c
a2b+ ab2 + a2c− 2abc+ ac2
2ab2 − b2c+ bc2
2a2b− a2c+ ac2
a2c+ ac2
b2c+ 2ac2 − bc2
a2b− ab2 + 2b2c
a2b+ ab2 − 2abc+ b2c+ bc2
a2c− ac2 + 2bc2
b2c+ bc2
a2c− 2abc+ b2c+ ac2 + bc2
.
6 Thin quiver moduli
We consider the special case of an acyclic finite quiver Q and the dimension vector d = 1 :=
(1, . . . , 1) (formally di = 1 for every i ∈ Q0). In this case the group G1 = (Gm)Q0 is a torus.
A representation M ∈ R(Q,1) consists of Mα ∈ k; an element g ∈ G1 acts via g · M =(
gt(α)g
−1
s(α)Mα
)
α
. Again, the action descends to an action of PG1 = G1/∆. Let T = (Gm)Q1
which acts, like in the general case, by scaling. The action of PG1 can be recovered from the
T -action by embedding PG1 as a subtorus via the map
σQ : G1 → T, g 7→
(
gt(α)g
−1
s(α)
)
α
,
whose kernel is precisely ∆; note that we assume Q to be connected. Let T0 be the cokernel of
this map. That means we have an exact sequence of tori
1→ PG1 → T → T0 → 1.
Let θ be a stability condition. Without loss of generality we may assume θ(1) = 0. As the image
of PG1 inside T acts trivially on M θ−st(Q,1) we obtain an action of T0 on the moduli space.
The torus T0 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
[
M θ−st(Q,1)/T0
] ∼=[
R(Q,1)θ−st/T
]
. Hence, to determine the T0-equivariant Chow ring of M θ−st(Q,1) it suffices to
compute the T -equivariant Chow ring of R(Q,1)θ−st. Using similar arguments as in the third
section, we obtain the following characterizations of A∗T
(
R(Q,1)θ−sst
)
and A∗T
(
R(Q,1)θ−st
)
in
terms of generators and relations:
Proposition 9. The kernels of the surjections
Q[xα]α∈Q1 = A∗T (R(Q,1))Q A∗T
(
R(Q,1)θ−sst
)
Q
A∗T
(
R(Q,1)θ−st
)
Q,
j∗1
j∗2
which are induced by the open embeddings j1 : R(Q,1)θ−sst → R(Q,1) and j2 : R(Q,1)θ−st →
R(Q,1) are given as follows: for a subset I ⊆ Q0 let
xI :=
∏
α∈Q1
s(α)∈I, t(α)/∈I
xα.
Torus-Equivariant Chow Rings of Quiver Moduli 15
Then ker(j∗1) is the ideal generated by all xI with θ(1I) > 0 and ker(j∗2) is generated by all
expressions xI for which θ(1I) ≥ 0.
In the above statement 1I ∈ ZQ0 denotes the characteristic function on the subset I ⊆ Q0.
It is easy to read off a Q-linear basis from this characterization as the ideal that we are dividing
out is generated by monomials. For a tuple γ = (γα)α∈Q1 ∈ ZQ1
≥0 write xγ :=
∏
α∈Q1
xγαα . For
a subset I ⊆ Q0 put J(I) := {α ∈ Q1 | s(α) ∈ I, t(α) /∈ I}. Then xI = x1I . A basis of
A∗T
(
R(Q,1)θ−sst
)
Q is given by all monomials xγ where supp(γ) contains no subset J(I) for
which θ(1I) > 0; the monomials xγ for which J(I) * supp(γ) for all I ⊆ Q0 with θ(1I) ≥ 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)T0 . We introduce the following notion: a subset H ⊆ Q1 is
called a spanning tree if the underlying graph of (Q0, H) is a tree, which means it is connected
and cycle-free. Introduce the formal symbol α−1 for every arrow α ∈ Q1 and formally define
s
(
α−1
)
= t(α) and t
(
α−1
)
= s(α). An unoriented path is a sequence p = αεrr · · ·α
ε1
1 such that
s
(
α
εν+1
ν+1
)
= t
(
αενν
)
for ν = 1, . . . , r − 1. We define s(p) = s
(
αε11
)
and t(p) = t
(
αεrr
)
. By the
spanning tree property there exists for every i, j ∈ Q0 an unoriented path p = αεrr · · ·α
ε1
1 in H
such that s(p) = i and t(p) = j.
Let H ⊆ Q1 be any subset. Define the representation MH ∈ R(Q,1) by
Mα =
{
1, α ∈ H,
0, α /∈ H.
Note that for a representation M of Q with support H := supp(M) the representation MH 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 MH is
θ-stable. Denote by THc the subtorus of all t = (tα)α∈Q1 ∈ T for which tα = 1 whenever α ∈ H.
We use Hc as a short-hand for Q1 −H.
Lemma 10. Let M ∈ R(Q,1)θ−st.
1. The isomorphism class [M ] is a fixed point of the T0-action on M θ−st(Q,1) if and only if
supp(M) is a θ-stable spanning tree. In this case M and MH are isomorphic.
2. Let H be a spanning tree of Q. Then the composition THc → T → T0 is an isomorphism.
Proof. This can easily be deduced from the description of the fixed point locus in Theorem 1.
Denote ϕ : THc → T → T0. For α in Q1 let p = αεrr · · ·α
ε1
1 be the unique unoriented path
in H such that s(p) = s(α) and t(p) = t(α); note that if α ∈ H then p = α. Define a morphism
ψ̃ : T → THc by ψ̃(t) = s, where
sα := tα ·
(
tεrαr · · · t
ε1
α1
)−1
for all α ∈ Q1. We argue that the morphism ψ̃ descends to a morphism ψ : T0 → HHc . Let
g ∈ G1. Define t ∈ T by tα = gt(α)g
−1
s(α) (all α ∈ Q1). We obtain ψ̃(t) = s, where
sα = gt(α)g
−1
s(α)
(
gεrt(αr)g
−εr
s(αr)
. . . gε1t(α1)g
−ε1
s(α1)︸ ︷︷ ︸
=gt(p)g
−1
s(p)
)−1
= 1
for every α ∈ Q1. We show that ϕ and ψ are mutually inverse. It is easy to see that ψϕ, which
is the restriction ψ̃|THc , is the identity. To show that ϕψ = idT0 , it is sufficient to prove that for
every t ∈ T , the product r := tψ̃(t)−1 lies in the image of σQ : G1 → T . We compute
rα = tεrαr · · · t
ε1
α1
16 H. Franzen
for all α ∈ Q1. So rα = tα for every α ∈ H. By the following lemma there exists g ∈ G1 such
that tα = gt(α)g
−1
s(α) for every α ∈ H. For an arbitrary α ∈ Q1, we then obtain
rα = gt(p)g
−1
s(p) = gt(α)g
−1
s(α).
Therefore, r does lie in imσQ. This proves the second assertion of the lemma. �
Lemma 11. Let H be a quiver whose underlying unoriented graph is a tree. Then the morphism
σH : G1 → TH = GH1
m is surjective.
Proof. We argue by induction on the number n of vertices of H. The base n = 1 is trivial. For
n > 1 choose a leaf l ∈ H0 of H, i.e., a vertex which is adjacent to precisely one other vertex.
A leaf exists because the underlying graph of H is a tree. Without loss of generality we assume
that l is a sink; the case of a source is completely analogous. Let γ : k → l be the unique arrow
adjacent to l. Consider the subquiver H ′ defined by H ′0 = H0 − {l} and H ′1 = H1 − {γ}. Let
G′ := G1H′ . Then G1 = G′ × Gm. By induction assumption there exists g′ ∈ G′ such that
g′t(α)g
′−1
s(α) for all α ∈ H ′1. Define gl := tγg
′
k. Then g := (g′, gl) is an inverse image of t under
σH : G1 → TH . �
We hence may identify for a θ-stable spanning treeH the equivariant Chow ring A∗T0
({[MH ]})Q
of the (isolated) fixed point [MH ] ∈M θ−st(Q,1) with
A∗THc (pt)Q ∼= A∗T
(
R(H,1)θ−st
)
Q.
The pull-back of the inclusion iH of the fixed point then corresponds to the map
i∗H : A∗T
(
R(Q,1)θ−st
)
Q → A∗THc (pt)Q = Q[xα]α∈Hc ,
which is defined by
i∗H(xα) =
{
xα, α ∈ Hc,
0, α ∈ H.
We now want to describe the image of the pull-back of the inclusion of the fixed point locus.
We use the following result of Brion. It is the algebraic analog of [8, Theorem 1.2.2].
Theorem 12 ([3, Theorem 3.4]). Let X be a smooth projective variety on which a torus T acts
with finitely many fixed points x1, . . . , xN and with finitely many one-dimensional orbits. Then
the image of the localization map
i∗ : A∗T (X)Q → S(X(T ))⊕NQ
consists of all tuples (f1, . . . , fN ) such that fi ≡ fj modulo χ whenever there exists a one-
dimensional T -orbit on which T acts with the character χ and whose closure contains xi and xj.
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 T0 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 ⊆ Q1 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 (Q0,Ω) to be
connected. Given a spanning almost tree we again define a representation MΩ ∈ R(Q,1) by
assigning Mα = 1 for α ∈ Ω and Mα = 0 otherwise. We say Ω is θ-stable if MΩ is θ-stable.
Lemma 13. Let M ∈ R(Q,1)θ. The orbit of [M ] under T0 is one-dimensional if and only if
supp(M) is a θ-stable spanning almost tree.
Torus-Equivariant Chow Rings of Quiver Moduli 17
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 (Q0,Ω). Assume that Ω is not a θ-stable
spanning almost tree. As MΩ is contained in the T -orbit of M we may assume M = MΩ. Let
B = {α1, . . . , αr} be a maximal subset of Ω such that (Q0,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 (t1, . . . , tr) ∈ (k×)r. Define the representation Mt by
(Mt)α =
tk, α = αk,
1, α ∈ H,
0, otherwise.
Clearly Mt is in the T -orbit of M . Moreover two such representations Mt and Ms are not iso-
morphic as a g ∈ G1 with g ·Mt = Ms would need to satisfy g ·MH = MH which implies g = 1 by
Lemma 10 (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 P1 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 α0 and α∞. Let
Ω0 = Ω−{α0} 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 [MΩ]. The inclusions Ωc
0 ⊇ Ωc ⊆ Ωc
∞ give rise
to the maps
A∗TΩc0
(pt)
rΩ,0−−→ A∗TΩc
(pt)
rΩ,∞←−−− A∗TΩc∞
(pt),
which send rΩ,0(xα0) = 0 and rΩ,∞(xα∞) = 0 and act as the identity on the other variables.
Let us write f |Ωc for the image of a function f ∈ A∗TΩc0
(pt)Q = Q[xα]α∈Ωc0
or f ∈ A∗TΩc∞
(pt)Q =
Q[xα]α∈Ωc∞ under the maps rΩ,0 or rΩ,∞, respectively.
Theorem 14. The pull-back of the embedding of the fixed point locus i : M θ(Q,1)T0 ↪→M θ(Q,1)
is the map
i∗ : A∗T
(
R(Q,1)θ
)
Q →
⊕
H
Q[xα]α∈Hc ,
which sends f to (i∗H(f))H . It is injective and its image consists precisely of those tuples (fH)H
for which fΩ0 |Ωc = fΩ∞ |Ωc 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 12. 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 T0-orbit
of [MΩ] in M θ(Q,1). Let t ∈ TΩc0
. We obtain
(t.M)α =
tα0 , α = α0,
1, α ∈ Ω0,
0, α ∈ Ωc.
We identify the TΩc0
-orbit of [MΩ] with Gm by the entry which corresponds to the arrow α0.
Then TΩc0
acts on Gm by the character xα0 . Take the resulting identification of C with P1.
The limit for tα0 → 0 is [MΩ0 ] and the limit for tα0 → ∞ must hence be [MΩ∞ ]. Consider
18 H. Franzen
the composition of the isomorphisms of tori TΩc0
→ T0 → TΩc∞ from Lemma 10. Call it ϕ. To
determine ϕ, let t ∈ TΩc0
and find the unique s ∈ TΩc∞ for which there exists a g ∈ G1 such that
gt(α)tαg
−1
s(α) = sα for all α ∈ Q1. The element g is uniquely determined up to scaling. We get
four equations:
gt(α)g
−1
s(α) = 1, α ∈ Ω− {α0, α∞},
gt(α)tαg
−1
s(α) = sα α ∈ Ωc,
gt(α0)tα0g
−1
s(α0) = 1,
gt(α∞)g
−1
s(α∞) = sα∞ .
As Ω∞ is a spanning tree, (Q0,Ω−{α0, α∞}) is not connected. Let C1, C2 ⊆ Q0 be its connected
components. The vertices i0 := s(α0) and j0 := t(α0) lie in different components, so we may
assume i0 ∈ C1 and j0 ∈ C2. As Ω0 is connected, i∞ := s(α∞) and j∞ := t(α∞) cannot be
contained in the same component. We first consider the case where i∞ ∈ C1 and j∞ ∈ C2. We
get gi is constant on C1 and on C2. Let its value on C1 be g1 and its value on C2 be g2. Now
we obtain
g2tα0g
−1
1 = 1, g2g
−1
1 = sα∞ .
This implies sα∞ = t−1
α0
and g2g
−1
1 = t−1
α0
. For α : i→ j in Ωc we again have to distinguish three
cases. If i and j are in the same component then sα = tα. If i ∈ C1 and j ∈ C2 then we get
sα = g2tαg
−1
1 = t−1
α0
tα. Finally if i ∈ C2 and j ∈ C1 then we obtain sα = g1tαg
−1
2 = tα0tα. We
define
δα :=
0, i, j ∈ C1 or i, j ∈ C2,
−1, i ∈ C1, j ∈ C2,
+1, i ∈ C2, j ∈ C1.
We get
ϕ(t)α =
t−1
α0
, α = α∞,
tδαα0
tα, α ∈ Ωc,
1, α ∈ Ω∞.
We now deal with the second case where i∞ ∈ C2 and j∞ ∈ C1. Then ϕ looks the same except
for ϕ(t)α∞ = tα0 . The torus TΩc∞ acts on C by the character xα∞ . But as lim
tα0→∞
ϕ(t).[MΩ] =
lim
tα0→∞
t.[MΩ] = [MΩ∞ ] we see that ϕ(t)α∞ = t−1
α0
. So the second case will not occur.
The induced map by ϕ on the character lattices is ϕ∗ : X(TΩc∞)→ X(TΩc0
) which is given by
ϕ∗(xα∞) = −xα0 , ϕ∗(xα) = xα + δαxα0
for α ∈ Ωc. Brion’s theorem now tells us that the image of i∗ consists precisely of those tuples
(fH)H for which
fΩ0(xα0 , xα)α∈Ωc ≡ fΩ∞(−xα0 , xα + δαxα0)α∈Ωc
modulo xα0 . This is the same as to say fΩ0 |Ωc = fΩ∞ |Ωc . �
Remark 15. In [14] Pabiniak and Sabatini find a basis for the T -equivariant cohomology
ring H∗T (M) regarded as a module over S(X(T )) for a compact symplectic toric manifold M
Torus-Equivariant Chow Rings of Quiver Moduli 19
with torus T . Let us briefly recap their result. They fix a moment map and a generic compo-
nent µ of it. A result of Kirwan states that for every T -fixed point p ∈MT there exists a class
νp ∈ H2λp(M) such that its restriction νp|p ∈ H2λp({p}) agrees with the equivariant Euler class
of the negative normal bundle N−p of µ at p, while νp|q = 0 for every q ∈MT with µ(q) < µ(p).
Such a class νp is called a Kirwan class. Any collection of Kirwan classes {νp}p∈MT is a basis
of H∗T (M) over S(X(T )). Kirwan classes, however, are not unique. In [14, Definition 4.5] a spe-
cial type of Kirwan class is defined, the so-called i-canonical class. This is a Kirwan class τp such
that the local index of τp at p is 1, whereas the local index at every other fixed point vanishes.
In [14, Theorem 1.2] it is shown that a set {τp}p∈MT of i-canonical classes exists and is unique.
It would be interesting to determine the i-canonical classes in the toric case and compare them
to our result.
We end by illustrating this result in two examples.
Example 16. Let Q be the n + 1-Kronecker quiver, d = 1 = (1, 1) and θ = (1,−1) as
in Example 7. We have already identified M θ(Q,1) with Pn. The torus T0 is the cokernel
(Gm)n+1/Gm of the diagonal embedding Gm → Gn+1
m . So its character group is X(T0) ={∑n
ν=0 bνxν |
∑n
ν=0 bν = 0
}
. The ring S := S(X(T0))Q is the subring of Q[x0, . . . , xn] generated
by the differences xν − xµ. The one-dimensional T0-orbits on Pn are the orbits of the points
pµ,ν = [0 : · · · : 0 : 1 : 0 : · · · : 0 : 1 : 0 : · · · : 0] with an entry in the µth and in the νth position.
On that orbit, T0 acts with the character ±(xν − xµ). Brion’s description of the image of the
localization map (i.e., Theorem 12) tells us that a tuple f = (f0, . . . , fn) ∈ S⊕n+1 belongs to
im i∗ if and only if fµ ≡ fν modulo xν − xµ for all 0 ≤ µ < ν ≤ n.
To match this with the description of Theorem 14, we consider the isomorphism⊕
k 6=ν
Zxk → X(T0),
which sends xk to xk−xν . This provides us with an isomorphism of rings ψ : R :=
⊕n
ν=0Q[x0, . . . ,
x̂ν , . . . , xn]→ S⊕n+1. Let g = (g0, . . . , gn) ∈ R. We analyze when ψ(g) = (f0, . . . , fn) fulfills the
condition of Brion’s theorem. For µ < ν define g̃µ ∈ Q[x0, . . . , x̂ν , . . . , xn] as
g̃µ := gµ(x1 − xµ, . . . , xµ−1 − xµ, xµ+1 − xµ, . . . , xν−1 − xµ,−xµ, xν+1 − xµ, . . . , xn − xµ).
Then fµ ≡ fν modulo (xν − xµ) if and only if g̃µ ≡ gν modulo xµ. But the reduction of g̃µ
modulo xµ in Q[x0, . . . , x̂ν , . . . , xn]/(xµ) ∼= Q[x0, . . . , x̂µ, . . . , x̂ν , . . . , xn] agrees with the reduc-
tion of gµ modulo xν . This shows that the image of ι∗ consists of those g = (g0, . . . , gn) ∈ R
such that for all µ < ν the images of gµ and gν under the maps
Q[x0, . . . , x̂µ, . . . , xn]→ Q[x0, . . . , x̂µ, . . . , x̂ν , . . . , xn]← Q[x0, . . . , x̂ν , . . . , xn]
agree. This is precisely the description in Theorem 14.
Example 17. Let Q = K(2, 3) be the full bipartite quiver with 2 sources and three sinks.
A representation of Q of dimension vector 1 is a (3× 2)-matrix. The structure group G1 is the
torus G2
m ×G3
m which acts via
(g1, g2, h1, h2, h3) ·
a11 a12
a21 a22
a31 a32
=
h1g
−1
1 a11 h1g
−1
2 a12
h2g
−1
1 a21 h2g
−1
2 a22
h3g
−1
1 a31 h3g
−1
2 a32
.
We consider the stability condition θ = (3, 3,−2,−2,−2). This stability condition is generic
for 1. The resulting moduli space M θ(Q,1) is isomorphic to the blow-up Bl3
(
P2
)
of P2 in three
20 H. Franzen
points not on a line. This can be seen, for instance, using the results of [7]: as θ agrees with the
canonical stability condition and ample stability is fulfilled, the moduli space is a Fano variety.
Its dimension is two and its Picard rank is 4, so it has to be isomorphic to Bl3
(
P2
)
.
By Proposition 9 we obtain the following description of the T -equivariant Chow ring of
R(Q, 1)θ in terms of generators and relations:
A∗T
(
R(Q,1)θ
)
Q
∼=
Q
[
x11 x12
x21 x22
x31 x32
]
(xj1ixj2i | i = 1, 2, 1 ≤ j1 < j2 ≤ 3) + (xj1xj2 | j = 1, 2, 3)
.
A Q-vector space basis of this algebra is given by all monomials xγ where γ = (γji) ∈M3×2(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 T0
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:
(
1 1
1 0
0 1
) (
1 0
1 1
0 1
)
(
0 1
1 0
1 1
) (
1 0
0 1
1 1
)
(
0 1
1 1
1 0
) (
1 1
0 1
1 0
)
(
1 1
1 1
0 1
)
(
1 1
1 0
1 1
) (
1 0
1 1
1 1
)
(
0 1
1 1
1 1
) (
1 1
0 1
1 1
)
(
1 1
1 1
1 0
)
The matrices at the vertices of the above graph are the representations MH which correspond
to the 6 stable spanning trees H of Q and the edges are the representations MΩ 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 14 the pull-back i∗ of the embedding of
the fixed point locus is the map which is induced by
Q
[
x11 x12
x21 x22
x31 x32
]
→ Q
[
x22
x31
]
⊕Q
[
x11
x22
]
⊕Q
[
x11
x32
]
⊕Q
[
x21
x32
]
⊕Q
[
x12
x21
]
⊕Q
[
x12
x31
]
,
f →
(
f( 1 1
1 0
0 1
), f( 0 1
1 0
1 1
), f( 0 1
1 1
1 0
), f( 1 1
0 1
1 0
), f( 1 0
0 1
1 1
), f( 1 0
1 1
0 1
)),
where the components are defined by
f( 1 1
1 0
0 1
) = f
(
0 0
0 x22
x31 0
)
, f( 0 1
1 0
1 1
) = f
(
x11 0
0 x22
0 0
)
, f( 0 1
1 1
1 0
) = f
(
x11 0
0 0
0 x32
)
,
f( 1 1
0 1
1 0
) = f
(
0 0
x21 0
0 x32
)
, f( 1 0
0 1
1 1
) = f
(
0 x12
x21 0
0 0
)
, f( 1 0
1 1
0 1
) = f
(
0 x12
0 0
x31 0
)
.
The image of i∗ consist of all tuples
p =
(
p( 1 1
1 0
0 1
), p( 0 1
1 0
1 1
), p( 0 1
1 1
1 0
), p( 1 1
0 1
1 0
), p( 1 0
0 1
1 1
), p( 1 0
1 1
0 1
)
)
,
Torus-Equivariant Chow Rings of Quiver Moduli 21
such that the following six conditions hold:
p( 1 1
1 0
0 1
)( x22
0
)
= p( 0 1
1 0
1 1
) ( 0
x22
)
, p( 0 1
1 0
1 1
) ( x11
0
)
= p( 0 1
1 1
1 0
)( x11
0
)
,
p( 0 1
1 1
1 0
)( 0
x32
)
= p( 1 1
0 1
1 0
)( 0
x32
)
, p( 1 1
0 1
1 0
)( x21
0
)
= p( 1 0
0 1
1 1
)( 0
x21
)
,
p( 1 0
0 1
1 1
)( x12
0
)
= p( 1 0
1 1
0 1
)( x12
0
)
, p( 1 0
1 1
0 1
)( 0
x31
)
= p( 1 1
1 0
0 1
)( 0
x31
)
.
A basis of the image is given by the following elements:
i∗(1) = (1, 1, 1, 1, 1, 1),
i∗
(
t
(
0 0
0 m
0 0
))
=
(
xm22, x
m
22, 0, 0, 0, 0
)
, i∗
(
t
(
m 0
0 0
0 0
))
=
(
0, xm11, x
m
11, 0, 0, 0
)
,
i∗
(
t
(
0 0
0 0
0 m
))
=
(
0, 0, xm32, x
m
32, 0, 0
)
, i∗
(
t
(
0 0
m 0
0 0
))
=
(
0, 0, 0, xm21, x
m
21, 0
)
,
i∗
(
t
(
0 m
0 0
0 0
))
=
(
0, 0, 0, 0, xm12, x
m
12
)
, i∗
(
t
(
0 0
0 0
m 0
))
=
(
xm31, 0, 0, 0, 0, x
m
31
)
,
i∗
(
t
(
0 0
0 m
n 0
))
=
(
xm22x
n
31, 0, 0, 0, 0, 0
)
, i∗
(
t
(
n 0
0 m
0 0
))
=
(
0, xm22x
n
11, 0, 0, 0, 0
)
,
i∗
(
t
(
n 0
0 0
0 m
))
=
(
0, 0, xm32x
n
11, 0, 0, 0
)
, i∗
(
t
(
0 0
n 0
0 m
))
=
(
0, 0, 0, xm32x
n
21, 0, 0
)
,
i∗
(
t
(
0 m
n 0
0 0
))
=
(
0, 0, 0, 0, xm12x
n
21, 0
)
, i∗
(
t
(
0 m
0 0
n 0
))
=
(
0, 0, 0, 0, 0, xm12x
n
31
)
.
Acknowledgements
I would like to thank Markus Reineke and Silvia Sabatini for inspiring discussions on this subject.
I also want to thank the two referees for their valuable comments. At the time this research
was conducted I was supported by the DFG SFB/TR 191 “Symplectic structures in geometry,
algebra, and dynamics”.
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1 Introduction
2 Generalities on quiver moduli
3 Torus actions on quiver moduli
4 Torus-equivariant tautological relations
5 Localization at torus fixed points
6 Thin quiver moduli
References
|
| id | nasplib_isofts_kiev_ua-123456789-211024 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T05:52:01Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Franzen, Hans 2025-12-22T09:31:41Z 2020 Torus-Equivariant Chow Rings of Quiver Moduli. Hans Franzen. SIGMA 16 (2020), 096, 22 pages 1815-0659 2020 Mathematics Subject Classification: 14C15; 16G20 arXiv:1911.03288 https://nasplib.isofts.kiev.ua/handle/123456789/211024 https://doi.org/10.3842/SIGMA.2020.096 We compute rational 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 Chow ring of the fixed point locus, and we compare the two descriptions. I would like to thank Markus Reineke and Silvia Sabatini for inspiring discussions on this subject. I also want to thank the two referees for their valuable comments. At the time this research was conducted, I was supported by the DFG SFB/TR 191 "Symplectic structures in geometry, algebra, and dynamics". en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Torus-Equivariant Chow Rings of Quiver Moduli Article published earlier |
| spellingShingle | Torus-Equivariant Chow Rings of Quiver Moduli Franzen, Hans |
| title | Torus-Equivariant Chow Rings of Quiver Moduli |
| title_full | Torus-Equivariant Chow Rings of Quiver Moduli |
| title_fullStr | Torus-Equivariant Chow Rings of Quiver Moduli |
| title_full_unstemmed | Torus-Equivariant Chow Rings of Quiver Moduli |
| title_short | Torus-Equivariant Chow Rings of Quiver Moduli |
| title_sort | torus-equivariant chow rings of quiver moduli |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211024 |
| work_keys_str_mv | AT franzenhans torusequivariantchowringsofquivermoduli |