Orbifold cohomology of abelian symplectic reductions and the case of
  weighted projective spaces
Orbifold cohomology of abelian symplectic reductions and
the case of weighted projective spaces
Tara S. Holm
Abstract. These notes accompany a lecture about the topology of symplectic
(and other) quotients. The aim is two-fold: first to advertise the ease of
computation in the symplectic category; and second to give an account of
some new computations for weighted projective spaces. We start with a brief
exposition of how orbifolds arise in the symplectic category, and discuss the
techniques used to understand their topology. We then show how these results
can be used to compute the Chen-Ruan orbifold cohomology ring of abelian
symplectic reductions. We conclude by comparing the several rings associated
to a weighted projective space. We make these computations directly, avoiding
any mention of a stacky fan or of a labeled moment polytope.
Contents
1. Symplectic manifolds and quotients 2
2. Orbifolds and their cohomology 5
3. Why the symplectic category is convenient 9
4. The case of weighted projective spaces 12
References 18
The notion of an orbifold has been present in topology since the 1950’s [S1,
S2]. More recently, orbifolds have played an important role in differential and alge-
braic geometry, and in mathematical physics. A fundamental theme is to compute
topological invariants associated to an orbifold, with one ostensible goal to under-
stand Gromov-Witten invariants for these spaces. The aim of the present article is
modest: to expound how techniques from symplectic geometry may be used to un-
derstand the degree-zero genus-zero Gromov-Witten invariants with three marked
points, the so-called Chen-Ruan orbifold cohomology ring; and to make ex-
plicit the details of these techniques in the case of weighted projective spaces.
In the symplectic category, orbifolds arise as symplectic quotients. We recount
the techniques from symplectic geometry that may be used to compute topolog-
ical invariants of a symplectic quotient. This is based on Kirwan’s seminal work
1991 Mathematics Subject Classification. Primary 53D20; Secondary 14N35, 53D45, 57R91.
Key words and phrases. Symplectic quotient, orbifold, cohomology.
TSH is grateful for the support of the NSF through the grant DMS-0604807.
http://arxiv.org/abs/0704.0257v1
2 TARA S. HOLM
[Ki]; and for orbifold invariants, the author’s joint work with Goldin and Knutson
[GHK]. The quotients we consider are by a compact connected abelian group. We
employ techniques coming from algebraic topology, most notably using equivariant
cohomology. For those used to working with finite groups, it is important to note
that, whereas for finite groups the invariant part of a cohomology ring is identical
to the equivariant cohomology, this is not the case for connected groups.
The main example in this article is a weighted projective space CPn
. Its
definition depends on a sequence (b) = (b0, . . . , bn) of positive integers. Kawasaki
showed that the ordinary cohomology groups, with integer coefficients, of the un-
derlying topological space of a weighted projective space are identical to the co-
homology groups of a smooth projective space [Ka], but there is a twisted ring
structure. We review the details of his work. Then in Theorem 4.2, we compute
the cohomology of the orbifold [CPn
], proving that
(0.1) H∗([CPn(b)];Z) = H
(S2n+1;Z) ∼=
〈b0 · · · bnun+1〉
Whereas Kawasaki finds a twist in the the ring structure, we find torsion in high
degrees of the ring (0.1). There is a natural map from Kawasaki’s ring to this one,
and we describe the map explicitly. Finally in Theorem 4.3, we compute the Chen-
Ruan cohomology ring of this orbifold. We make this computation using integer
coefficients, generalizing results in [J, Ma1, Ma2]. Moreover, we give explicit
generators and relations, and avoid mentioning a stacky fan [BCS] or a labeled
polytope [LT, GHK].
The definitions in this article make sense for arbitrary coefficient rings. Indeed,
all computations in the final section use integer coefficients. Moore and Witten
have suggested that the torsion in K-theory has more physical significance than
torsion in cohomology [MoWi]. The author together with Goldin, Harada and
Kimura, is investigating a K-theoretic version of [GHK] and of the computations
herein, building on the work of Harada and Landweber [HL].
The remainder of the paper is organized as follows. In Section 1 we give a quick
exposition of how orbifolds arise in the symplectic category. We then introduce
several cohomology rings associated to an orbifold in Section 2. We advertise the
ease of computation for these rings in Section 3. The novel results in this article
are the computations in Section 4. We include detailed proofs that avoid much of
the symplectic machinery used in [GHK].
Acknowledgments. Many thanks are due to Tony Bahri, Matthias Franz, Re-
becca Goldin, Megumi Harada, Ralph Kaufmann, Takashi Kimura, Allen Knutson,
Eugene Lerman, Reyer Sjamaar, and Alan Weinstein for many helpful conversa-
tions; and to Yoshiaki Maeda and the organizers and sponsors of Poisson 2006 in
Tokyo, Japan, where this work was presented.
1. Symplectic manifolds and quotients
We begin with a very brief introduction to the symplectic category; a more
detailed account of the subject can be found in [CdS]. A symplectic form on a
manifold M is a closed non-degenerate two-form ω ∈ Ω2(M). Thus, for any tangent
vectors X ,Y ∈ TpM , ωp(X ,Y) ∈ R. The key examples include the following.
Example 1.1: M = S2 = CP 1 with ωp(X ,Y) equal to the signed area of the
parallelogram spanned by X and Y. This is the Fubini-Study form on CP 1.
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 3
Example 1.2: M any orientable Riemann surface with ω as in Example 1.1. Note
that orientability is a necessary condition on a symplectic manifold M , because the
top exterior power of the symplectic form is a volume form.
Example 1.3: M = R2d with ω =
dxi ∧ dyi.
Example 1.4: M = Oλ a coadjoint orbit of a compact connected semisimple Lie
group, equipped with ω the Kostant-Kirillov-Soriau form.
Example 1.3 gains particular importance because of
Darboux’s Theorem 1.5. Let M be a symplectic 2d-manifold with symplectic
form ω. Then for every point p ∈ M , there exists a coordinate chart U about p with
coordinates x1, . . . , xd, y1, . . . , yd so that on this chart,
(1.1) ω =
dxi ∧ dyi.
Thus, whereas Riemannian geometry uses local invariants such as curvature to
distinguish metrics, symplectic forms are locally indistinguishable.
The symmetries of a symplectic manifold may be encoded as a group action.
Here we restrict ourselves to a compact connected abelian group T = (S1)n. An
action of T on M is symplectic if it preserves ω; that is, ρ∗gω = ω, for each g ∈ T ,
where ρg is the diffeomorphism corresponding to the group element g. The action
is Hamiltonian if in addition, for every ξ ∈ t, the vector field
(1.2) Xξ =
[exp(tξ)]|t=0
is a Hamiltonian vector field. That is, we require that ω(Xξ, ·) = dφ
ξ is an exact
one-form. Each φξ is a smooth function on M , determined up to a constant. Taking
them together, we may define a moment map
(1.3)
Φ : M −→ t∗
p 7−→
Φ(p) : t → R
ξ 7→ φξ(p)
Returning to our examples, we have Hamiltonian actions in all but the second
example.
Example 1.6: The circle S1 acts on M = S2 = CP 1 by rotations. If we use
angle and height coordinates on S2, then the vector field this action generates is
tangent to the latitude lines, so in coordinates, X ξ = ∂
, and since ω = dθ ∧ dh,
ω(Xξ, ·) = dh, so a moment map is the height function on S
2, as shown in Figure 1.1
below.
Example 1.7: If M is a two-torus M = T 2 = S1 × S1, then S1 × S1 acts on itself
by multiplication. This action is symplectic, but is not Hamiltonian. In fact, no
Riemann surface with non-zero genus has a nontrivial Hamiltonian torus action.
Example 1.8: The torus T d = (S1)d ⊂ Cd acts by coordinate-wise multiplication
on M = R2d = Cd. This action rotates each copy of C = R2 (at unit speed), and is
Hamiltonian. Identifying t∗ ∼= Rd, a moment map is
(1.4) Φ(z1, . . . , zn) = (|z1|
2, . . . , |zd|
up to a constant multiple.
4 TARA S. HOLM
PSfrag replacements
Figure 1.1. The vector field and moment map for S1 acting by
rotations on S2.
Example 1.9: Each coadjoint orbit M = Oλ ⊆ g
∗ may be identified as a homoge-
nous space G/L, where L is a Levi subgroup of the Lie group G. Thus G and its
maximal torus T act on M by left multiplication. A G-moment map is inclusion
(1.5) ΦG : Oλ →֒ g
and a T -moment map is the G-moment map composed with the natural projection
g∗ → t∗ that is dual to the inclusion t →֒ g.
In each of these examples, the image of the (torus) moment map is a convex
subset of Rn. This is true more generally.
Convexity Theorem 1.10 ([A],[GuSt]). If M is a compact Hamiltonian T -space,
then Φ(M) is a convex polytope. It is the convex hull of Φ(MT ), the images of the
T -fixed points.
The convexity theorem is an example of a localization phenomenon: a global
feature (the image of the moment map) that is determined by local features of the
fixed points (their images under the moment map). The convexity property is a
recurring theme in symplectic geometry; its many guises are illustrated in [GuSj].
The moment map is a T -invariant map: it maps entire T -orbits to the same
point in t∗. Thus when α is a regular value, the level set Φ−1(α) is a T -invariant
submanifold of M . Moreover, the action of T on a regular level set is locally free:
it has only finite stabilizers. This follows directly from the moment map condition:
at a regular value, dφξ is never zero, implying that Xξ is not zero, so there is no
1-parameter subgroup fixing points in the level set. Thus, at a regular value the
symplectic reduction M//T (α) = Φ−1(α) is an orbifold. In fact, Marsden and
Weinstein proved
Theorem 1.11 ([MaWe]). If M is a Hamiltonian T -space and α is a regular value
of the moment map Φ, then the symplectic reduction M//T (α) is a symplectic
orbifold.
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 5
More generally, the symplectic reduction M//T (α) at a critical value is a symplectic
stratified space [SL].
Symplectic reduction is an important technique for constructing new symplec-
tic manifolds from old. From our examples, we may construct several classes of
symplectic manifolds.
Example 1.12: For the action of S1 on M = S2 = CP 1 by rotation, the level
set of a regular value is a latitude line, which the circle rotates. The quotient is
a point. Note that if S1 acts by rotation S2 at twice the usual speed, then the
quotient, as an orbifold, is [pt/Z2]. Thus, every orbifold [pt/Zk] is a quotient of S
by a rotation action.
Example 1.13: T d acts on M = R2d = Cn by rotation of each copy of C = R2.
The level set of a regular value is a copy of T d, and so again the quotient is a point
(or potentially an orbipoint, for different actions of T d). However, we may also
restrict our attention to subtori K ⊆ T n. The action of K is still Hamiltonian, and
for certain choices of K, Cn//K(α) is a symplectic toric orbifold. Lerman and
Tolman show that every effective symplectic toric orbifold may be constructed in
this way [LT].
Example 1.14: For the T -action on a coadjoint orbit M = Oλ, the symplectic
reduction M//T (α) is known as a weight variety. One may determine the possi-
ble orbifold singularities by analyzing the combinatorics of G and its Weyl group.
See, for instance, [Kn] or [GHK]. This reduced space plays an important role in
representation theory.
2. Orbifolds and their cohomology
We now turn to orbifolds in the topological category. In terms of local models,
an orbifold is a topological space where each point has a neighborhood homeo-
morphic (or diffeomorphic) to the quotient of a (fixed dimensional) vector space
by a finite group. Satake introduced this notion in the 1950’s [S1, S2], originally
calling the spaces V -manifolds. Thurston coined the term orbifold when he redis-
covered them in the 1970’s (see [Th]) in his study of 3-manifolds. This local model,
however, makes it difficult to define very basic pieces in the theory of orbifolds:
overlap conditions on orbifold charts, suborbifolds, and maps between orbifolds.
As is evident already in the work of Haefliger [Hæ], the proper way to think of an
orbifold is as a Morita equivalence class of groupoids, one of which is a proper
étale groupoid (see [Moe]); or equivalently as a smooth Deligne-Mumford stack
(see [DM]). For example, using this structure, a map of orbifolds should simply be
a morphism of the appropriate objects.
While groupoids or stacks provide the correct mathematical framework, the
technology is a bit beyond the scope of this article. Indeed, for us it is sufficient to
work with the local models, largely because we restrict our attention to orbifolds
that arise as global quotients. Nevertheless, we will need to distinguish between an
orbifold X or [X ] and its underlying topological space (or coarse moduli space)
X . In particular, when X is presented as a global quotient of a manifold M by a
group G, we will use square brackets [M/G] to denote the orbifold, and M/G to
denote the underlying coarse moduli.
For an orbifold X, at each point x ∈ X, we have a local isotropy group Γx at x.
We will be interested in almost complex orbifolds, that is orbifolds that have
6 TARA S. HOLM
local models isomorphic to Cd/Γx, with Γx ⊆ U(d) a finite group acting unitarily
on Cd. Our main example in this section is the orbisphere shown in the figure
below. This is a symplectic toric orbifold in the sense of Tolman and Weitsman
Figure 2.1. An orbisphere with two orbifold points, a Zp singu-
larity at the north pole and a Zq singularity at the south pole. All
other points in the space have local isotropy group the one-element
group. When p and q are relatively prime, this is a weighted pro-
jective space CP 1p,q.
[LT]. In that context, it corresponds to the labeled polytope that is an edge, with
the vertices labeled p and q. This orbifold cannot (always) be presented as a global
quotient by a finite group, although it can be presented as a symplectic reduction.
When p and q are relatively prime, it is the reduction of C2 by the Hamiltonian
S1-action
(2.1) e2πiθ · (z1, z2) = (e
p·2πiθ · z1, e
q·2πiθ · z2).
Thus, it is a global quotient of the level set Φ−1(α) ≈ S3 by a locally free S1 action.
In this case, it is also the global quotient of CP 2 by a Zp × Zq action
1. When p
and q are not relatively prime, the orbisphere is not a global quotient by a finite
group, but it is still a symplectic quotient of C2 by a Hamiltonian S1 × Zg action,
where g = gcd(p, q) (following [LT]). Note also that an orbisphere is isomorphic
to a weighted projective space CP 2p,q exactly when p and q are relatively prime. If
p and q are not relatively prime, the weighted projective space is not reduced: it
has a global stabilizer. On the other hand, the orbisphere described above is always
reduced.
Next we turn to algebraic invariants that we may attach to an orbifold X. The
hope is that these invariants are computable and at the same time retain some
information of the orbifold structure of X. All the invariants are isomorphic to
singular cohomology if X = X is in fact a manifold.
Definition 2.1. The ordinary cohomology ring of an orbifold X is the singular
cohomology of the underlying topological space X,
(2.2) H∗(X ;R),
with coefficients in a commutative ring R.
1With an apology to number theorists, we take the topologist’s notation: Zp denotes the
integers modulo p.
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 7
This ring is computable using standard techniques from algebraic topology, but it
does not distinguish between the orbisphere in Figure 2.1 and a smooth sphere.
For the second invariant, we restrict our attention to orbifolds X presented as
the quotient of a manifold M by the locally free action of a Lie group G. It is
conjectured that every orbifold can be expressed as such a global quotient, and it
is known to be true for effective or reduced orbifolds, those that do not have a
global finite stabilizer (see, for example, [EHKV, Theorem 2.18]). A presentation
as a global quotient is desirable because then the topology or geometry of the
quotient X is simply the G-equivariant topology or geometry of the manifold M .
This principle motivates the following definition.
Definition 2.2. Given a presentation of an orbifold X = [M/G] as a global quo-
tient, the cohomology ring of the orbifold X is the equivariant cohomology
(2.3) H∗(X;R) := H∗G(M ;R),
with coefficients in a commutative ring R.
Recall that equivariant cohomology is a generalized cohomology theory in the
equivariant category. Using the Borel model, we define
(2.4) H∗G(M ;R) := H
∗((M × EG)/G;R),
where EG is a contractible (though infinite dimensional) space with a free G action,
andG acts diagonally onM×EG. There are well-developed methods for computing
equivariant cohomology, hence this invariant is still computable.
Whenever X = [M/G] is a global quotient, the associated quotient map
(2.5) q : M → X
is a G-invariant map. This induces a continuous map
(2.6) q : (M × EG)/G → X.
When X = X is a manifold (i.e. G acts on M locally freely), this map is a fibration
with fiber BG. The map q induces a map in cohomology,
(2.7) q∗ : H∗(X ;R) −→ H∗((M × EG)/G;R) = H∗G(M ;R).
This induced map is an isomorphism when
1. G acts freely on M ;
2. G is a finite group and R is a ring in which |G| is invertible; and
3. G acts locally freely on M and R is a field of characteristic 0.
This last item implies that the cohomology of the orbifold differs from the coho-
mology of its coarse moduli only in its torsion. Notably, when R = Z, this ring
does in fact distinguish between the orbisphere in Figure 2.1 and a smooth sphere.
The third invariant was introduced by Chen and Ruan [CR] to explain mathe-
matically the stringy Betti numbers and stringy Hodge numbers that physi-
cists have attached to orbifolds. To define this third invariant, we need to introduce
the first inertia orbifold
(2.8) I1(X) :=
(x, (g)Γx)
x ∈ X and (g)Γx is a conjugacy class in Γx
This is again an orbifold, and X is the suborbifold called the identity sector
whose pairs consist of a point in X together with the identity element coset. The
other connected components of I1(X) are called the twisted sectors. For a global
8 TARA S. HOLM
quotient X = [M/G] with G abelian, we may identify I1(X) =
g∈G[M
g/G]. On
the other hand, when X = [M/G] is global quotient with G finite, we may identify
I1(X) =
g∈T [M
g/C(g)], where the union is over T a set of representatives of
conjugacy classes in G. For the orbisphere example, the inertia orbifold is shown
in Figure 2.2.
PSfrag replacements
· · ·
· · ·
Figure 2.2. The inertia orbifold for the the orbisphere with two
orbifold points as in Figure 2.1. Each of the p − 1 points to the
right of the north pole represents a [pt/Zp] and each of the q − 1
to the right of the south pole a [pt/Zq].
Definition 2.3 ([CR]). Given a presentation of an orbifold X = [M/G] as a global
quotient, the Chen-Ruan orbifold cohomology of X, as a vector space, is defined
to be the cohomology of the first inertia orbifold,
(2.9) HCR(X;R) := H(I
1(X);R),
with coefficients in a commutative ring R.
The Chen-Ruan ring is endowed with a Q grading, different from the grading
coming from singular cohomology. For a connected component Z of I1(X) that lies
in the (g) piece of the first inertia, the grading of H(Z;R) is shifted by a rational
number which is twice the age of Z. The age is determined by the weights of
the action of the group element g on the normal bundle to Z inside of X. This
is precisely where we need X to be (stably) almost complex. These rational shifts
ensure that the ranks of the Chen-Ruan cohomology groups agree with the stringy
invariants of an orbifold. The dependence of the rational shifts on the normal
bundles ν(Z ⊂ X) means that the Chen-Ruan ring is not in general functorial for
arbitrary morphisms of orbifolds.
To define a product on the Chen-Ruan cohomology, we must define higher
inertia. The nth inertia orbifold consists of tuples, a point in the orbifold and an
n-tuple of conjugacy classes. Restricting to the 2nd inertia, there are natural maps
(2.10) e1, e2, e3 : I
2(X) −→ I1(X)
defined by
e1(p, (g)Γp , (h)Γp) = (p, (g)Γp),(2.11)
e2(p, (g)Γp , (h)Γp) = (p, (h)Γp), and(2.12)
e3(p, (g)Γp , (h)Γp) = (p, ((gh)
−1)Γp)(2.13)
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 9
Chen and Ruan define the product of two classes α, β ∈ HCR(X;R) to be be
(2.14) α ⌣ β := (e3)∗(e
1α ∪ e
2β ∪ ε),
where e∗1 and e
2 are pull-back maps, (e3)∗ is the push-forward, ∪ is the usual
cup product, and ε is the Euler class of the obstruction bundle. This Euler class
should be viewed as a quantum correction term. It ensures that the product respects
the Q-grading and that the product is associative. Neither of these properties is
immediately obvious, and proving the latter requires a rather substantial argument.
Since we can avoid mention of the obstruction bundle in our computations, we
suppress further details here, and refer the curious reader to [ℵGV, CR, GHK]
for additional information.
The Chen-Ruan cohomology ring is the degree 0 part of the (small) quantum
cohomology ring. Hence, it is generally the most difficult of these three invariants to
compute. It has been computed for orbifolds that are global quotients of a manifold
by a finite group in [FG]. The definition was extended to the algebraic category in
[ℵGV], and in [BCS], the ring is computed for toric Deligne-Mumford stacks with
Q, R and C coefficients. In the next section, we review how to compute this ring
for abelian symplectic quotients, as demonstrated in [GHK]. In the last section,
we will compute each of these invariants explicitly for weighted projective spaces.
We conclude this section with a table of these rings for an orbisphere that has
a Z2 singularity at the north pole and is otherwise smooth. This is an example of
a weighted projective space, and is denoted CP 11,2.
(2.15)
X = [CP 11,2] Ring Grading
H∗(X ;Z) Z[x]/〈x2〉 deg(x) = 2
H∗(X;Z) Z[x]/〈2x2〉 deg(x) = 2
H ∗CR(X;Z) Z[x, u]/〈2x
2, 2xu, u2 − x〉 deg(x) = 2, deg(u) = 1
3. Why the symplectic category is convenient
In the past thirty years, tremendous progress has been made in understanding
the equivariant topology of Hamiltonian T -spaces and its relationship to the ordi-
nary topology of their quotients. For a compact torus T = (S1)n, the classifying
bundle ET is an n-fold product of infinite dimensional spheres, and the classifying
space BT is an n-fold product of copies of CP∞. Thus,
(3.1) H∗T (pt;Z) = H
∗(BT ;Z) = Z[x1, . . . , xn],
where deg(xi) = 2. The key ingredient to understanding topology of Hamiltonian
T -spaces is the moment map. Frankel [F] proved that for a Hamiltonian T -action
on a Kähler manifold, each component φξ of the moment map is a Morse-Bott
function on M , and generically the critical set is the fixed point set MT . In his
paper [A] on the Convexity Theorem 1.10, Atiyah generalized this work to the
purely symplectic setting.
Building on the work of Frankel and Atiyah, Kirwan developed techniques to
prove two fundamental theorems that allow us to understand the cohomology of
Hamiltonian T -spaces and their quotients. The first is a version of localization:
it allows us to make global computations by understanding fixed point data. While
10 TARA S. HOLM
this theorem is not explicitly stated in her book [Ki], it does follow immediately
from her work in Chapter 5.
Injectivity Theorem 3.1 ([Ki]). Let M be a compact Hamiltonian T -space. The
inclusion map MT →֒ M induces
(3.2) i∗ : H∗T (M ;Q) −→ H
T ;Q)
an injection in equivariant cohomology.
The compactness hypothesis is stronger than strictly necessary. We may re-
place it with a properness condition on the moment map. The proof relies on the
fact that a generic component of the moment map is an equivariantly perfect
Morse-Bott function on M . The image of this injection has been computed in
many examples, including toric varieties, coadjoint orbits of compact connected
semisimple Lie groups, and coadjoint orbits of Kac-Moody groups. These compu-
tations initially appeared in [CS, GKM] and further generalizations are described
in [GoH, GuH, GuZ, HHH].
The second theorem relates the equivariant topology of a Hamiltonian T -space
to the ordinary topology of its reduction.
Surjectivity Theorem 3.2 ([Ki]). Let M be a compact Hamiltonian T -space,
and α a regular value of the moment map. The inclusion Φ−1(α) →֒ M induces
(3.3) κ : H∗T (M ;Q) −→ H
−1(α);Q)
a surjection in equivariant cohomology.
Again for surjectivity, compactness is more than is necessary. Most importantly,
this result does apply to linear actions of a torus on Cd with a proper moment
map. The key idea in the proof is to use the function ||Φ − α||2 as a Morse-like
function, now known as a Morse-Kirwan function. The critical sets are not
non-degenerate, but one may still explicitly understand them via a local normal
form. It is then possible to prove that ||Φ−α||2 is an equivariantly perfect function
on M .
The kernel of the map κ can be computed using methods in [Go, JK, TW].
Using the fact that at a regular value, H∗T (Φ
−1(α);Q) ∼= H∗(M//T (α);Q), we have
a diagram
(3.4)
0 // ker(κ)
// H∗T (M ;Q))
κ // //
H∗(M//T (α);Q) // 0
H∗T (M
T ;Q)
Thus, by computing im(i∗) and ker(κ), we may derive an explicit presentation of
the cohomology H∗([M//T (α)];Q) of an orbifold arising as a symplectic quotient.
We now turn to a generalization of Theorem 3.2 in the context of orbifolds and
the Chen-Ruan ring.
CR Surjectivity Theorem 3.3 ([GHK]). Let M be a compact Hamiltonian T -
space, and α a regular value of the moment map. The inclusion Φ−1(α) →֒ M
induces
(3.5) K :
H∗T (M
g;Q) −→
H∗T (Φ
−1(α)g ;Q)
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 11
a surjection. Moreover, these are R×T -graded rings, K is a map of graded rings,
and there is an isomorphism of graded rings
(3.6)
H∗T (Φ
−1(α)g ;Q) ∼= H
CR(M//T (α);Q).
The surjectivity (3.5) is a direct consequence of the Surjectivity Theorem 3.2
applied to each space Mg (and again the compactness is not strictly necessary).
The hard work is defining the grading and ring structure, and proving that K is
a map of graded rings. Goldin, Knutson, and the author define a ring structure
⌣ that generalizes a definition (for G finite) of Fantechi and Göttsche [FG]. Us-
ing this definition, we may deduce (3.6); on the other hand, associativity of this
product is not at all obvious. Making use of the injection i∗ on each piece, there
is an alternative product ⋆ on the ring
g∈T H
g;Q) that is much simpler to
compute, and clearly associative. This alternative product has the advantage that
it avoids all mention of the obstruction bundle; instead it relies only on fixed point
data (i.e. the topology of the fixed point set and isotropy data for the action of the
torus on the normal bundles to the fixed point components).
Another key point is that although the ring on the left of (3.5) is quite large,
there is a finite subgroup Γ of T , generated by all finite-order elements that stabilize
some regular point in M , so that the Γ-subring
(3.7)
H∗T (M
still surjects onto the Chen-Ruan cohomology of the reduction. Thus, while it
appears that we have made the computation much more complicated, it turns that
there is still an effective algorithm to complete it. For full details, please refer to
[GHK]. We now return to our examples.
Example 3.4: For a symplectic toric orbifold Cn//K(α), we may use the com-
binatorics of its labeled polytope to establish an explicit presentation of the Chen-
Ruan cohomology of these orbifolds [GHK, § 9]. In the cases where the symplectic
picture is identical to the algebraic, the [GHK] results replicate those of [BCS].
Example 3.5: For the T -action on a coadjoint orbit M = Oλ, the symplectic
reduction M//T (α) is a weight variety. The equivariant cohomology of M may
be read directly from its moment polytope, as may the orbifold singularities of the
reduction. In this case, the Theorem 3.3 yields an explicit combinatorial description
of the Chen-Ruan cohomology of the weight variety.
We conclude this section with a brief remark on coefficients. In both Theo-
rems 3.1 and 3.2, the rational coefficients are necessary. For the Injectivity The-
orem 3.1, we may prove the result over Z with the additional hypothesis that
H∗(MT ;Z) contains no torsion. The Surjectivity Theorem 3.2 over Z requires much
stronger hypotheses. We will see in the next section that we may compute inte-
grally for weighted projective spaces, but that a simple product of two weighted
projective spaces yields a counter-example to the general theorem. Tolman and
Weitsman verify surjectivity over Z for a rather restrictive class of torus actions
[TW]. This topic is being more closely examined for a larger collection of actions
by Susan Tolman and the author [HT].
12 TARA S. HOLM
4. The case of weighted projective spaces
Let b = (b0, . . . , bn) be an (n+1)-tuple of positive integers. Consider the circle
action on Cn+1 given by
(4.1) t · (z0, . . . , zn) = (t
b0z0, . . . , t
bnzn)
for each t ∈ S1. This action preserves the unit sphere S2n+1, and the weighted
projective space CPn
is the quotient of S2n+1 by this locally free circle action.
This is a symplectic reduction because S2n+1 is (up to equivariant homeomorphism)
a regular level set for a moment map Φ for the weighted S1 action on Cn+1. Thus,
(4.2) CPn(b)
n+1//S1.
Nonetheless, we may continue our analysis without invoking the full symplectic
machinery: the arguments simplify greatly in this special case.
When the bi are relatively prime (that is, gcd(b0, . . . , bn) = 1), this fits into
the framework of symplectic toric orbifolds discussed in Example 3.4. When the bi
are not relatively prime, we let g = gcd(b0, . . . , bn), and note that there is a global
Zg stabilizer. In this case, the orbifold [CP
] is not reduced. Its coarse moduli
space is the same as the coarse moduli space of CPn
(b/g)
, where (b/g) denotes the
sequence of integers (b0/g, . . . , bn/g); and as a non-reduced orbifold, it corresponds
to a gerbe
[CPn(b)] → [CP
(b/g)].
It is important to include the case when the bi are not relatively prime, because
such non-reduced weighted projective spaces may well show up as suborbifolds of
a reduced weighted projective space.
The cohomology of the topological space CPn
. Kawasaki studied the sin-
gular cohomology ring of (the coarse moduli space of) weighted projective spaces
[Ka]. To present the product structure, we will need the integers
(4.3) ℓk = ℓ
k := lcm
bi0 · · · bik
gcd(bi0 , . . . , bik)
0 ≤ i0 < · · · < ik ≤ n
for each 1 ≤ k ≤ n.
Theorem 4.1 ([Ka]). The integral cohomology of CPn
(4.4) Hi(CPn(b);Z)
Z if i = 2k, 0 ≤ k ≤ n,
0 otherwise.
Moreover, letting γi denote the generator of H
2i(CPn
;Z), we have
(4.5) γk ∪ γm =
ℓk · ℓm
γm+k.
Outline of the Proof. Let Gk denote the group of k
th roots of unity. Then
as a topological space, CPn
is homeomorphic to a quotient of ordinary projective
space CPn by the finite group
(4.6) G(b) = Gb0 × · · · ×Gbn .
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 13
Explicitly, using standard homogeneous coordinates, the map
pb : CP
n −→ CPn(b)(4.7)
[z0 : · · · : zn] 7−→ [z
0 : · · · : z
n ](4.8)
induces the homeomorphism CPn/G(b) ∼= CP
. This then induces an isomor-
phism in singular cohomology with rational coefficients, since over Q we have the
isomorphisms
(4.9) H∗(CPn;Q) ∼= H
∗(CPn;Q)G(b) ∼= H
∗(CPn/G(b);Q).
Over the integers, the computation is a bit more subtle. Using twisted lens
spaces, Kawasaki verifies that just as for CPn, the cohomology ring H∗(CPn
is torsion-free with a copy of Z in each even degree between 0 and 2n; however the
product structure is twisted by the weights bi. Moreover, there are cases when we
need all n generators γ1, . . . , γn to present this ring.
Kawasaki showed that the map
(4.10) p∗b : H
2k(CPn(b);Z) −→ H
2k(CPn;Z)
is multiplication by ℓk for all 1 ≤ k ≤ n. From this, we may deduce that
(4.11) γ1 ∪ γk =
ℓ1 · ℓk+1
γk+1, and hence γk ∪ γm =
ℓk · ℓm
γm+k.
The result now follows. �
It is important to note that pb is not an isomorphism of orbifolds. Indeed, the
isotropy group at any point in CPn
is the stabilizer group of any lift of the point
in S2n+1. Thus, all isotropy groups for CPn
are cyclic. On the other hand, the
orbifold CPn/G(b) has points with isotropy group G(b), which may not be cyclic.
Nevertheless we will make use of the map pb to understand the structure of the
cohomology of the orbifold [CPn
] = [S2n+1/S1
]. Here, the subscript (b) on S1
indicates that the circle action is weighted by the integers (b) = (b0, . . . , bn).
The cohomology of the orbifold [CPn(b)]. Since the weighted projective space
] is a symplectic reduction, it is possible invoke the results from Section 3 to
determine the cohomology of the orbifold H∗([CPn
];Q), and to apply results from
[GHK, §9] to obtain a presentation over Z. We give a direct argument here that
is similar in spirit, but that avoids much of this big machinery; we then compare
this to Kawasaki’s Theorem 4.1.
Theorem 4.2. The cohomology of the orbifold [CPn
(4.12) H∗([CPn(b)];Z) = H
(S2n+1;Z) ∼=
〈b0 · · · bnun+1〉
Moreover, the natural map
(4.13) q∗(b) : H
∗(S2n+1/S1(b);Z) −→ H
(S2n+1;Z)
is completely determined by q∗
(γ1) = ℓ1 · u = lcm(b0, . . . , bn) · u.
Proof. Consider the (weighted) circle action of S1 on Cn+1 given by
(4.14) t · (z0, · · · , zn) = (t
b0 · z0, · · · , t
bn · zn).
14 TARA S. HOLM
The unit sphere S2n+1 is invariant under this action, so we get a long exact sequence
in S1-equivariant cohomology for the pair (Cn+1, S2n+1),
(4.15) · · · → HiS1
(Cn+1, S2n+1;Z)
→ HiS1
(Cn+1;Z)
→ HiS1
(S2n+1;Z) → · · · .
Thinking of (Cn+1, S2n+1) as a disk and sphere bundle over a point, we may use
the Thom isomorphism to identify Hi
(Cn+1, S2n+1;Z) ∼= H
i−2(n+1)
(Cn+1;Z).
Under this identification, the map α is the cup product with the equivariant Euler
class
(4.16) eS1
(Cn+1) = bo · · · bnu
Thus, the map α is injective, so the long exact sequence splits into short exact
sequences
(4.17) 0 → HiS1
(Cn+1, S2n+1;Z)
→ HiS1
(Cn+1;Z)
→ HiS1
(S2n+1;Z) → 0.
Thus, we have a surjection
(4.18) β : Z[u] ∼= H
(Cn+1;Z) −→ H∗S1
(S2n+1;Z).
Moreover, the exactness of (4.17) means that the kernel of β is equal to the image
of α, namely all multiples of the equivariant Euler class. This establishes (4.12).
Turning to (4.13), the map
(4.19) q∗(b) : H
∗(S2n+1/S1(b);Z) −→ H
(S2n+1;Z),
is exactly the one defined in (2.7). We know that this is an isomorphism over Q,
so q∗
must map γ1 to a multiple of u. Moreover, because b0 · · · bnu
n+1 is zero in
(S2n+1;Z), we must have that
(4.20) (q∗(b)(γ1))
n+1 ∈
b0 · · · bnu
To determine the image of the class γ1, we return to the map pb : CP
n → CPn(b).
This map lifts to maps on S2n+1 and Cn+1 given by
(4.21)
Πb //
S2n+1
πb //?
S2n+1
0 6= (z0, · · · , zn) 7−→
|zbii |
zb00 , · · · , z
(4.22)
(0, . . . , 0) 7−→ (0, . . . , 0)(4.23)
The maps in this diagram are all equivariant with respect to the standard circle
action on the left-hand spaces and the (b)-weighted circle action on the right-hand
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 15
spaces. Thus, we have a diagram of maps
(4.24)
(Cn+1 × S1)/S1
Πb // (Cn+1 × S1
(S2n+1 × S1)/S1
πb //
(S2n+1 × S1(b))/S
CPn pb
// CPn
Applying singular cohomology H∗( ;Z) and identifying equivariant cohomology,
we have a commutative diagram
(4.25)
Z[x] H∗
(Cn+1;Z) oo
(Cn+1;Z) Z[u]
〈xn+1〉 H
(S2n+1;Z) oo
(S2n+1;Z)
〈b0···bnun+1〉
〈xn+1〉 H
∗(CPn;Z) oo
∗(CPn(b);Z) Z⊕ Zγ1 ⊕ · · · ⊕ Zγn
Because Cn+1 equivariantly deformation retracts to a point, the map Π∗b maps the
generator u to x. The commutativity of the top square then implies that π∗b (u) = x.
Thus, we know that
π∗b (q
(b)(γ1) = q
∗(p∗b(γ1))(4.26)
= q∗(ℓ1 · x), by Kawasaki’s result,(4.27)
= ℓ1 · x, since q
∗ is an equality,(4.28)
= π∗b (ℓ1 · u).(4.29)
In low degree, π∗b is injective, so we may conclude that q
(b)(γ1) = ℓ1 · u. Noting
that ℓ1 = lcm(b0, . . . , bn) completes the proof. �
Over the integers, this invariant does distinguish a weighted projective space
from the standard one; however, it may not differentiate between two weighted
projective spaces. For example, the cohomology rings of the orbifolds [CP 12,2] and
[CP 14,1] are identical. They are both
(4.30)
〈4u2〉
We note that these surjectivity techniques do not generally work over the inte-
gers. To see this, we note that for any abelian reduction of affine space, the domain
of the Kirwan map H∗T (C
N ;Z) has terms only in even degrees. If we consider the
simple product [CP 11,2 × CP
1,2], we may compute the cohomology of this orbifold
using the above result and the Künneth formula. Since [CP 11,2] has 2-torsion in high
degrees, the Tor term from the Künneth formula plays a role, yielding 2-torsion in
16 TARA S. HOLM
high odd degrees in the cohomology of the orbifold [CP 11,2 × CP
1,2]. Thus, surjec-
tivity must fail over the integers in this example. We note that any failure over Z
must be due to problems with torsion, because surjectivity does hold over Q.
The Chen-Ruan orbifold cohomology of [CPn
]. When computing the Chen-
Ruan ring, it is important to recall that a weighted projective space is a circle reduc-
tion. Thus, the finite group Γ for which the Γ-piece surjects onto H ∗CR([CP
(b)];Z)
is a cyclic group. For any vector v that is non-zero is a single coordinate, say the
ith coordinate, the stabilizer of v is Zbi . Thus, the group Γ generated by all finite
stabilizers is the ℓth roots of unity Zℓ ⊂ S
1, where ℓ = lcm(b0, . . . , bn). Hence, we
have a surjection
(4.31) Z[u, α0, α1, . . . , αℓ−1] // // H
CR([CP
(b)];Z).
In this case, thinking of e
ℓ = ζk ∈ Zℓ ⊂ S
1, αk denotes a generator for
(4.32) H∗S1
((Cn+1)ζk ;Z).
To complete the computation, we must determine the orbifold product
(4.33) αi ⌣ αj = αi ⋆ αj
and the kernel of the orbifold Kirwan map (3.5). For any integer m ∈ Z, we let
[m] denote the smallest non-negative integer congruent to m modulo ℓ. For any
rational number q ∈ Q, 〈q〉f denotes its fractional part. Finally, we let
(4.34) ak(m) :=
[bk ·m]
bk ·m
This is the rational number such that ζm acts on the k
th coordinate by e2πiak(m).
Theorem 4.3. The Chen-Ruan orbifold cohomology of CPn
(4.35) H ∗CR([CP
(b)];Z)
Z[u, α0, α1, . . . , αℓ−1]
I + J
where u is a class in degree 2,
(4.36) deg(αj) = 2
ak(j).
Here, I is the ideal
(4.37) I =
αiαj −
(bku)
ak(i)+ak(j)−ak(i+j)
α[i+j]
generated by the ⋆ product structure, and J is
(4.38) J =
ak(j)=0
the kernel of the surjection K of the orbifold Kirwan map.
Remark 4.4: The generator u is the generator of S1-equivariant cohomology and
hence has degree 2. The generator αk is a placeholder for the cohomology of the
ζk-sector.
COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 17
Remark 4.5: Note that the generator α0 is the placeholder for the identity sector.
Indeed, we always have
(4.39) α0 ⋆ α0 = α0
as a consequence of the relation in (4.37) where i = j = 0, hence we may think of
α0 as 1.
Remark 4.6: The reader may use this theorem to check that H ∗CR( ;Z) does
distinguish [CP 12,2] from [CP
4,1].
Proof. We use the ⋆ product given by Equation (2.1) in [GHK]. In the case of
a weighted circle action on Cn+1, there is exactly one fixed point (the origin), any
generator αi restricted to that fixed point is 1, and the equivariant Euler class for
the kth coordinate is precisely bku, whence
(4.40) αi ⋆ αj =
(bku)
ak(i)+ak(j)−ak(i+j)
α[i+j].
Turning to the kernel computation, each (Cn+1)ζj has a weighted S1 action,
and so we apply Theorem 4.2 to this subspace. Thus, for the ζj-sector, the kernel
contribution is the equivariant Euler class of (Cn+1)ζj times the placeholder αj . We
note that (Cn+1)ζj contains the kth coordinate subspace precisely when ak(j) = 0.
Hence,
(4.41) eS1
((Cn+1)ζj ) =
ak(j)=0
and the theorem follows. �
This theorem is an immediate consequence of Theorem 4.2 and [GHK]. The
importance of this description is its ease in computation, since it avoids any com-
putation of a labeled moment polytope (á la [LT]) or of a stacky fan (á la [BCS]).
We demonstrate this computational facility in the following concluding example.
Example 4.7: Consider the weighted projective space [CP 51,2,2,3,3,3]. This is a
symplectic reduction of C6, the group Γ is Z/6Z, and so the Chen-Ruan orbifold
cohomology of [CP 51,2,2,3,3,3] is a quotient of
(4.42) Z[u, α0, α1, α2, α3, α4, α5].
18 TARA S. HOLM
The following chart contains the data needed to compute the ideals I and J .
(4.43)
g ζ0 ζ1 ζ2 ζ3 ζ4 ζ5
(C6)g C6 {0} 3C(3) 2C(2) 3C(3) {0}
aC(1)(g) 0
aC(2)(g) 0
aC(3)(g) 0
2 · age(g) 0 14
generator of
((C6)g;Z)
α0 α1 α2 α3 α4 α5
((C6)g) 108u6 1 27u3 4u2 27u3 1
Note that because of the multiplicities,
(4.44) 2 · age(g) = 2 ·
a1(g) + 2a2(g) + 3a3(g)
Since α0 = 1, and α1 and α5 are in the kernel ideal J , we only need to compute
the products among α2, α3 and α4. For example, we may compute
(4.45) α2 ⋆ α2 = (u)
(3u)0+0−0
α4 = 4u
All of the products contributing to I, then, are summarized in the following table.
(4.46)
⋆ α2 α3 α4
α2 4u
2α4 α5 = 0 4u
α3 27u
4 uα1 = 0
α4 uα2
Thus, as a ring,
(4.47) H ∗CR([CP
1,2,2,3,3,3];Z)
Z[u, α0, α1, α2, α3, α4, α5]
I + 〈108u6, α1, 27u3α2, 4u2α3, 27u3α4, α5〉
This generalizes Jiang’s computation [J] to a computation over Z.
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	1. Symplectic manifolds and quotients
	2. Orbifolds and their cohomology
	3. Why the symplectic category is convenient
	4. The case of weighted projective spaces
	References
