Toric symplectic singular spaces I: isolated singularities

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by  D. Burns

TORIC SYMPLECTIC SINGULAR SPACES I: ISOLATED SINGULARITIES
D.BURNS,V.GUILLEMIN,ANDE.LERMAN
ABSTRACT. WegeneralizeatheoremofDelzantclassifyingcompactconnectedsymplectic
manifolds with completely integrable torus actions to certain singular symplectic spaces.
Theassumptiononsingularitiesisthatiftheyarenotfinitequotientthentheyareisolated.
5
0
0
2

n 1. INTRODUCTION
a
J
Itisanoldquestionastowhatthesymplecticgeometryanalogueofasingularalgebraic
0
variety should be. For example, it is an exercise in Gromov’s book [Gr]. One would like
2
to have a definition of a singular object in symplectic geometry (not to be confused with

]
G symplectic varieties in algebraic geometry) that includes, at the very least, symplectic
quotientsbyproperLiegroup actionsand,perhaps,affinesubvarietiesofcomplexvector
S
. spacesCn. There are anumberof reasons for wantingsuch adefinition.
h
at (1) Symplectic analoguesof algebro-geometric objects shed newlight on questions of
m symplectic and algebraic geometry, providing new understanding and new tech-
[ niques. For instance Alexeev and Brion [AB] recently proved the existence of de-

1 generations of spherical varieties to toric varieties. The proof is not geometric;
v it uses commutative algebra. It would be interesting to find a purely symplectic
0
proof. And forthat one needstounderstand whatsymplectic singular spacesare.
1
3 (2) An intrinsic definition of singular symplectic spaces will play a useful role in un-
1
derstandingthe dynamicsofsymmetric Hamiltonian systems, in particularin un-
0
5 derstanding the stability and bifurcation of relative equilibria and the equivariant
0 analogue of the Liapunov - Weinstein - Moser center theorem (see for example
/
h [GL]and referencestherein).
t
a (3) In the same spirit, a good definition of a singular symplectic space should allow
m
one to develop an analogue of Floer homology and to start proving the existence
: ofrelativeperiodicorbitsinsymmetricHamiltoniansystemswithoutresorttoper-
v
i turbative methods, thatis, awayfrom relative equilibria.
X
(4) Such adefinition should also shednewlight on quantization ofsingular systems.
r
a
What should a symplectic singular space be? Fourteen years ago Sjamaar and Lerman
proved [SL] that symplectic quotients by actions of compact Lie groups are stratified
spaces (the result is also true for proper actions of non-compact groups, see [BL] and
[GL]). They also proved that symplectic quotients have a number of interesting proper-
ties, some of which were abstracted in a definition of a symplectic stratified space [SL,
Definition 1.12]:
SupportedinpartbyNSFgrantsDMS-0104047(DB),DMS-0104116(VG)andDMS-0204448(EL).
1
Definition1.1. AsymplecticstratifiedspaceisastratifiedspaceXtogetherwithasubalgebra
C (X) (“the algebra of smooth functions”) of the algebra of continuous functions C0(X)
∞
such that
(1) each stratum Sof Xisa symplectic manifold,
(2) C (X) isa Poisson algebra, and
∞
(3) the embeddingS ֒ XisPoisson for each stratum S.
→
The last line means that for any function f C (X) its restriction f| to any stratum S
S
∈ ∞
is a smooth function in the ordinary sense, and for any two functions f,g C (X) we
∈ ∞
have{f| ,g| } = {f,g}| ,whereontheleft{ , } denotesthePoissonbracketdefinedbythe
S S S S S
· ·
symplectic form on the stratum S. This tells us that a singular symplectic space should
bea stratifiedspace. Recallthatastratified space isa topological space with alocally finite
decomposition into manifolds, which are called strata. Moreover, stratified spaces are
definedrecursively. Namely, a neighborhood of a point ishomeomorphic to a product of
a ball with a cone on a compact stratified space, called the link of the singularity at the
point. Points in the samestratum haveisomorphic links.
The results of [SL] also seem to suggest that a singular symplectic space should be
some sort of a differential space in the sense of Aronszajn [Ar], Spallek [Sp] and Sikorski
[Si] (see [NS] for the state of the art on differential spaces and a large bibliography). We
feel,however,thatthisismaynotbethebestwaytoapproachsymplecticsingularspaces.
For instance, consider a complex algebraic subvariety Xof Cn. The most natural space of
smooth functionsonXisthespaceofrestrictions C (Cn)| ,thespaceofWhitneysmooth
X
∞
functions. ButthereisnoobviousPoissonbracketinsight. Hereisanotherexample. Take
X = R2with the standard symplectic form and stratify it asX = (R2r{0}) {0}. Then for
∪
any n, the space of Z/n-invariant functions on R2 is a Poisson algebra, and embeddings
of strata are Poisson. In this example there are infinitely many choices of C (X) and no
∞
preferred one.
In this paper we set ourselves a more modest goal. We investigate what a toric sym-
plectic singular space (with isolated singularities) should be. The paradigm here is a
theorem of Delzant [D]: compact symplectic manifolds with completely integrable torus
actionsaresmooth projective toric varieties. Thelesson hereisthattoricsymmetriesturn
a floppysymplectic manifold into arigid algebraicvariety.
In the case of toric symplectic singular spaces the links of singularities are toric contact
spaces. As the classification of contact toric manifolds shows [L2], in order for the mo-
ment map image to be locally convex, these contact toric spaces have to be of Reeb type.
Recallthatthe notion ofReebtype wasintroduced byBoyerand Galicki in theirstudyof
Sasakian toric manifolds[BG].
Since symplectic orbifolds can be treated on the same footing as symplectic manifolds
[LT] and since the category of orbifolds is stable under symplectic cuts, we will assume
that ourspacesare stratified byorbifolds, not manifolds.
The main result of the paper is Theorem 1. We prove that compact symplectic toric
spaces, such that their non-orbifold singularities are isolated, are classified by convex
rationalpolytopesthataresimpleawayfromthevertices. Consequentlytheyareisomor-
phictosymplecticquotientsofCNbyactionsofcompactabelianLiegroups(i.e.,products
oftori Rn/Znandof finite abeliangroups). Itthen follows thatall such spacesare projec-
tive toric varieties.
2
2. CLASSIFICATION OF TORIC SYMPLECTIC SPACES WITH ISOLATED SINGULARITIES
In this section we define symplectic toric stratified spaces with the property that their
non-orbifoldsingularitiesareisolated1 andprovethattheyareclassifiedbyrationalpoly-
topesthataresimpleawayfromvertices. Westartwithafewpreliminarydefinitionsand
remarks.
One of the techniques of the proof involves blowing up singularities. Nowin the sym-
plectic category blow-ups are symplectic cuts [L1]. Symplectic cuts are symplectic quo-
tients. So orbifold singularities are generic and thus unavoidable. On the other hand,
the symplectic cuts of orbifolds are again, generically, orbifolds. So it is more convenient
to work with orbifolds throughout. Therefore we will consider stratified spaces strati-
fied by orbifolds.2 Thus, for us, a singular space with isolated singularities is a Hausdorff
topological space Mtogether with a discrete setof points {x } Msothat
α
⊂
M := Mr{x }
reg α
is a smooth (meaning C ) orbifold. We also want the neighborhoods of singularities x
α
∞
to have bounded topology. Therefore we require that for each index α there is a com-
pact orbifold L (the link of the singularity at x ) so that a neighorhood of x in M is
α α α
homemorphic to the open cone
c˚(L ) := ([0, ) L )/({0} L ).
α α α
× ×
∞
In fact it will be more convenient for us to define the topological cone c˚(L) on a space L
by
c˚(L) := ([− , ) L)/({− } L),
× ×
∞ ∞ ∞
and to refer to the image of ({− } L) in c˚(L) as the vertex of the cone. Here we think
of [− , ) as the topolog∗ical spa∞ce ×which is homeomorphic to [0, ) and contains R =
(− ,∞ )∞asa denseopen subset. ∞
B∞ya∞torus wemeanacompactconnected abelianLiegroup. Wedenotethe Liealgebra
of a torus G by g. We let g∗ = Hom(g,R), the dual of the Lie algebra, and Z = ker{exp :
G
g G}, the integral lattice of G. A vector v Z is primitive if it is not a positive integer
G
m→ultiple of another vector in the lattice Z .∈A codimension one face of a polytope is a
G
facet. A polytope in g∗ is rational if the normals of all of its facets lie in the integral lattice
Z . Ingeneralthenormalstothefacetsofapolytopeing∗ liein(g∗)∗ = g. Ourconvention
G
is that all the normals to the facets are primitive and outward pointing. A polytope in g∗
issimpleifallthefacetsareingeneralposition. Inparticularthereareexactlydimg∗ edges
coming out of every vertex. Thus a cube is simple while an octahedron is not. However,
an octahedron issimple awayfrom the vertices.
1Theirorbifoldsingularitiesarearbitrary.
2There is a fair amount of confusion in the literature regarding the notions of orbifold and orbifold
morphisms. Inthispaperwewillonlydealwithreduced(effective)orbifolds, thatis,orbifoldsforwhich
thestructuregroupsallacteffectively,asdefined,forexample,byMoerdijkandPronk[MP]. Itseemstous
thatthecorrectnotionofmorphismofeffectiveorbifoldsisthatofastrongmapofMoerdijkandProng(op.
cit.).Inmanyinstancesinsymplecticgeometrythenotionofareducedorbifoldistoorestrictive.Themain
reasonis thatslice representationsfor locallyfreeproperactionsof Lie groupsneednot be effective. The
general correct notion of orbifolds and their morphisms (“good maps”) is probably the one due to Chen
and Ruan [CR]. Itis a folklore wisdom that Chenand Ruan’s definitions make orbifolds into stacksover
thecategoryoforbifolds.
3
A symplectic toric G-orbifold is a triple (M,ω,Φ : M g∗) where M is a connected
orbifold,ωisasymplecticformonMandΦisamoment→mapforaneffectivecompletely
integrable action of a torus G on (M,ω). We don’t assume that our toric orbifolds are
compact, nordo weassume that the momentmapsare proper.
Remark2.1. AclassificationtheoremofLermanandTolman[LT]assertsthatcompactcon-
nectedeffectivesymplectictoricG-orbifoldsareclassifiedbysimplerationalpolytopesin
g∗ together with positive integers labels attached to each facet. The need for the integer
labels can be seen in the following example. Consider the teardrop orbifold X , that is,
n
consider an orbifold whose underlying topological space is homeomorphic to the two
sphere and whose only singularity is modelled on C/(Z/n). The orbifold carries a sym-
plecticformwhichisinvariantunderaHamiltonianactionofthecircleS1. Theassociated
polytope isaclosedintervaling∗ R. Theclassification theorem inquestion tellsusthat
≃
the image of the singularity is one of the end points of the interval and that this facet of
the interval should be labelled n. More generally it turns out that for symplectic toric
orbifolds,theorbifoldstructure groupsofpointsthatthemomentmapsendstofacetsare
cyclic. The integerlabelsare simplythe orders of these groups.
Forusasymplecticconeisasymplecticorbifold(N,ω)togetherwithafreeproperaction
ρ of the reals such that ρ∗ω = etω for all t R. Just as for manifolds there is a one-
t t ∈
to-one correspondence between symplectic cones and (cooriented) contact orbifolds. If
(N,ω,ρ ) is a symplectic cone, then L = N/R is a contact orbifold. Conversely, if L is a
t
contact cooriented orbifold then its symplectization is a symplectic cone. These facts are
standard for manifolds. Their proofs in the orbifold case carries over mutatis mutandis
from the manifold case: replace everyoccurance of the word “manifold”by “orbifold.”
We will say that a symplectic cone (N,ω,ρ ) is G-toric if there is an effective action of
t
a torus G on N preserving ω and commuting with dilations ρ and having the property
t
that 2dimG = dimN. Now, whenever a group action on a symplectic cone preserves a
symplectic form and commutes with dilations ρ , it is Hamiltonian since it preserves the
t
contraction of ω with the vector field ζ generating the dilations ρ . Note that the associ-
t
atedmomentmapΦdefinedbytheprimitive ι(ζ)ωishomogeneous: Φ(ρ (x)) = etΦ(x).
t
Hence lim Φ(ρ (x)) = 0, and Φ extends to a continous map on the topological cone
t − t
c˚(N/R) = {→}∞N. Consequently any moment map on a toric symplectic cone N extends
to acontinu∗ou∪s map on the cone c˚(N/R)
Asymplectictoriccone(N,ω,ρ ,Φ : N g∗)isofReebtype ifthereisavectorX gso
t
∈
that Φ(x),X > 0 for all x N. Here a→gain Φ denotes a homogeneous moment map.
We whill see bielow (Lemma 2∈.8) that if additionally the quotient N/R is compact and
connected then the moment map image Φ(N) {0} g∗ is a cone on a simple rational
∪ ⊂
polytope. If N is a manifold, the converse is true as well. This follows from the classifi-
cation of contact toric manifolds [L2]. Note that in general, even if N/R is compact, the
cone Φ(N) {0} need not be a convex cone. This happens, for example, when N is the
∪
symplectization of anovertwisted 3-sphere [L2].
We are now in a position to define the main object of this section: toric symplectic
singular spaceswith isolated non-orbifold singularities.
4
Definition 2.2. A toric symplectic singular G-space with isolated singularities is a Hausdorff
topological space Mwith an effective action of a torus G, which has a number of proper-
ties. Namely, weassume that
(1) Thereisadiscretesetofpoints{x } MsuchthatM := Mr{x }isasymplectic
α reg α
⊂
toric G-orbifold.
(2) Topologically the neighborhoods of the singular points x are cones on compact
α
orbifolds, which are not quotients ofodd dimensional spheresS2n−1by finite sub-
groups ofTn.
(3) Symplectically andequivariantly thepunctured neighborhoods of singularpoints
x are neighborhoods of − in toric symplectic cones. Hence by the preceding
α
remarksthemomentmapΦ∞onM extendstoacontinuousfunctiononMwhich
reg
we will continue todenote by Φ.
(4) For each singular point x there is a vector Y g so that Φ−Φ(x ),Y > 0 on
α α α α
U rx ,whereU isaG-invariantneighborhoo∈dofx . Thahtis,theneighbiorhoods
α α α α
of singularities are symplectic conesof Reebtype.
The condition (2) on the links insures that the singularties are genuine non-orbifold
singularities.
Theorem 1. Let (M,ω,Φ : M g∗) be a compactconnected toricsymplecticsingular G-space
with isolatedsingularities. Then→
(1) Themomentimage∆ := Φ(M) g∗isarationalpolytope,whichissimpleexceptpossibly
⊂
atvertices.
(2) Thefibersof the momentmap Φ : M g∗ are G-orbits.
(3) The moment polytope ∆ plus positive→integers attached to the facets of ∆ uniquely deter-
mine the topological space M, the singularities {x } and the symplectic toric G-orbifold
α
M = Mr{x }.
reg α
Remark2.3. Thereisaneasyconverse toTheorem1. Namely,givenanyrational polytope
∆ g∗ which is simple away from vertices and a set of positive integers attached to
⊂
facets, there exists a singular symplectic G-variety with moment image ∆. The proof is
essentially the same as in Delzant’s original paper [D] modified as in [LT] to take care of
the positive integer labelsattached tofacets. Wewill saymore about itlaterin the paper.
We now proceed with the proof of Theorem 1. We start by recalling what symplectic
cuts are [L1]. Let(M,ω)be a symplectic orbifold with aneffective Hamiltonian action of
a circle T and an associated moment map µ : M R. Let a be a regular value of µ. Then
the action ofT on the levelset µ−1(a)is locallyfr→ee and
M// T := µ−1(a)/T
a
is again naturally a symplectic orbifold by the reduction theorem of Marsden, Weinstein
and Meyer. Consider nowan orbifold with boundary
{µ a} := {x M | µ(x) a}.
≥ ∈ ≥
Define an equivalence relation ∼ on {µ a} by identifying points on the boundary that
≥
lie on the same orbit of T. The following theorem is one of the main observations of [L1].
Wewill not describe its proof here.
5
Theorem 2.4. Inthe above notation, the topologicalspace
M := {µ a}/ ∼
≥a
≥
canbenaturallygiventhestructureofasymplecticorbifoldsothattheinclusionM// T ֒ M
a ≥a
isasymplecticembeddingandthedifferenceM rM// T issymplectomorphictotheope→nsubset
≥a a
{µ > a}of (M,ω). Moreovertherestrictionµ| descendstoamomentmapµ¯ : M Rfor
{µ>a} ≥a
the actionof S1on thecutspaceM . →
≥a
Remark2.5. Mutatis mutandis the statementholds alsofor
M := {µ a}/ ∼ .
≤a
≤
Remark 2.6. Theorem 2.4 is local: If Σ (M,ω) is a hypersuface which separates M into
⊂
two manifolds with boundary M+ and M− and if the null foliation of ω| consists of
Σ
closed leaves, then we may collapse the leaves of the null foliation in the boundaries of
M+and M−and obtain symplectic orbifolds without boundary.
Remark2.7. TheresultsofTheorem2.4holdequivariantly. Inparticular,supposeatorusG
acts on (M,ω)with moment mapΦ : M g∗ and T Gisa closed subgroup generated
⊂
by X g so that µ = Φ,X . Then the cut→space M inherits a Hamiltonian action of G.
≥a
∈ h i
Moreover the restriction Φ| descendstoa G-moment map Φ¯ on the cut spaceand
{µ>a}
Φ¯(M ) = Φ({µ > a}) = Φ(M) {η g∗ | η,X a}.
≥a
∩ ∈ h i ≥
Theorem 2.4wasmotivated by symplectic blowups. Namelylet
(M,ω) = (Cn,√−1Xdzj∧z¯j)
and let µ(z) = |z|2. Then M is the symplectic blow-up of Cn at the origin and M is
≥a ≤a
CPn.
Nextwe sort out neighborhoods ofsingular points.
Lemma 2.8. Let (N,ω,ρ ) be a symplectic cone with an action of a torus K which preserves
t
the symplectic form and commutes with dilations ρ . Let µ : N k∗ denote the associated
t
homogeneous momentmap. Suppose thereisa vectorX k suchthat→
∈
(2.1) µ(x),X > 0 for allx N,
h i ∈
that is, suppose (N,ω,ρ ,µ : N k∗) is of Reeb type. If the base B = N/R is compact and
t
connected then µ is proper as a ma→p into k∗ r {0}, the fibers of µ are connected and the moment
cone µ(N) {0} isacone ona convexrational polytope.
∪
Proof. Since the base B of the cone is compact, (2.1) is an open condition. Therefore we
may assume that T = {exptX | t R} is a circle. Then f = µ,X is the moment map for
∈ h i
the action ofT on (N,ω). Recallthat µisdefinedby
µ,Y = ω(ζ,Y )
N
h i
forallY k,whereζistheisthevectorfieldgeneratingρ andY denotesthevectorfield
t N
on N ind∈uced by Y. Consequently µ(ρ (x)) = etµ(x) for all t R, x M. Note also that
t
∈ ∈
since f = ω(ζ,X ) > 0 by assumption, X = 0 anywhere on N; hence the action of T on
N N
6
N islocallyfree.
6
Wenowarguethatf−1(1)isdiffeomorphictoB,henceiscompactandconnected. Since
f(ρ (x)) = etf(x)
t
the ρ -orbit map π : f−1(1) B is injective. Moreover, it is onto: pick a section s : B N
t
ofπ : N B(anyprincipal→R-bundle istrivial sosuch a global section doesexist). L→et
→
τ(x) = −log(f(s(x))).
Then
f(ρ (s(x))) = eτ(x)f(s(x)) = 1.
τ(x)
Note that f−1(1) ֒ N isa K-invariantsection for π : N B. Hence NisK-equivariantly
diffeomorphic to B→ R. Moreover under this identificati→on of N with B R, f is simply
× ×
the map f(m,t) = et. Therefore f is proper as a map into (0, ). Hence µ : N {η k∗ |
∈
η,X > 0}isproperaswell. By[LMTW, Theorem4.3],µ(N)∞isconvexandthe→fibersofµ
h i
are connected. But more can be said.
The restriction µ| descends to a moment map µ for the action of K on the quo-
f−1(1)
tient f−1(1)/T, which is a compact symplectic orbifold. By the orbifold version of the
Atiyah-Guillemin-Sternberg convexity theorem [LT] µ(f−1(1)/T) is a rational polytope.
Hence µ(f−1(1)) = µ(f−1(1)/T) is a rational polytope as well. Therefore µ(N) {0} =
R>0µ(f−1(1)) {0} isacone on a rational polytope. ∪ (cid:3)
∪
Proof of Theorem1. SinceMiscompact,thesetofsingularities{x }isfinite. Byassumption
α
aneighborhoodU ofeachsingularpointx istheneighboroohof− inatoricsymplectic
i i
coneofReebtype. InparticularthereisavectorY intheintegralla∞tticeZ ofGsuchthat
i G
Φ − Φ(x ),Y > 0 on U . By Lemma 2.8 the restriction of the function f = Φ,Y to
i i i i i
h i h i
U is proper as a map onto its image. By the proof of the lemma the action of the circle
i
G := {exptY | t R}is locallyfree on U r{x }.
i i i i
∈
Nowfor everysufficiently small ε > 0consider
M(ε) := Mr[{m Ui | fi(m) < fi(xi)+ε};
∈
i
it is a compact symplectic orbifold with boundary. The connected components of the
boundary are the contact toric orbifolds {m U | f (m) = ε + f (x )}. Moreover by
i i i i
∈
Theorem2.4andsubsequentremarks,ifwedivideeachcomponentbythecorresponding
group G , the various components ofthe boundarydisappearandthe result isa compact
i
symplectic toric orbifold M(ε). Furthermore, Φ| descends to a moment map Φ¯ on
M(ε)
M(ε) for the induced action of G. By construction Φ(M(ε)) = Φ¯(M(ε)) and the fibers
of Φ| are G-orbits if and only if the fibers of Φ¯ are. Since M(ε) is a symplectic toric
M(ε)
orbifold,Φ¯(M(ε))isasimplerationalpolytope[LT]. Moreover,thefacetsofΦ¯(M(ε))that
come from cuts have labels 1 attached to them (see Remark 2.1). Therefore Φ(M(ε)) is a
simple rational polytope and the fibers ofΦ| are G-orbits. Note that
M(ε)
[ M(ε) = Mreg.
{ε>0}
WearenowinapositiontoarguethatΦ(M)isconvexandthatthefibersofΦ : M g∗
areG-orbits. Supposea,b M aretwopoints. Pickεsmallenoughsothata,b M→(ε).
reg
∈ ∈
7
Since Φ(M(ε)) Φ(M ) is convex, [Φ(a)Φ(b)] Φ(M(ε)). Therefore Φ(M ) is con-
reg reg
⊂ ⊂
vex. Also, if a,b M and Φ(a) = Φ(b) then we can choose ε so that a,b M(ε).
reg
∈ ∈
Hencea G b. Therefore the fibersofΦ : M g∗ areG-orbits. Wewillneedthisfactin
∈ ·
the proof of uniquenessbelow. →
SinceM isdenseinM,Φ(M)iscontainedintheclosureΦ(M ). Ontheotherhand,
reg reg
since M is compact, Φ : M g∗ is proper. Hence Φ(M) = Φ(M ). In particular Φ(M)
reg
isconvex. →
Wenow argue that Φ(M)is a polytope, that the points Φ(x ) are among the vertices of
i
thispolytope andthatΦ−1(Φ(x )) = {x }foralli. Wecutofftheconicalneighborhoods of
i i
singularities. Thatis, for sufficiently small εwe have
M = M(ε) G{m Ui | fi(m) fi(xi)+ε}
∪ ∈ ≤
i
Weknow that both Φ(M(ε)) and
∆ (ε) := Φ({f f (x )+ε})
i i i i
≤
are convex polytopes which have disjoint interiors (the latter is true because the fibers
of Φ : M g∗ are G-orbits). Also, for each i, ∆ (ε) intersects Φ(M(ε)) in a facet
reg i
that doesn’t co→ntain the vertex Φ(x ). Hence Φ(M) is a polytope, Φ(x ) are vertices (not
i i
necessarily all the vertices) and, since Φ| hasthe property that the fiber
{m∈Ui|fi(m)≤fi(xi)+ε}
above Φ(x ) is the point x , we have that the preimages of vertices of Φ(M) are single
i i
points. We leave it to the reader to check that Φ(M) is simple except possibly at the
vertices.
Uniqueness. Suppose (M ,ω ,Φ : M g∗,{x(i),...,x(i)}), i = 1,2 are two toric sym-
i i i i 1 N
plectic singular G-spaces. Suppose Φ (M→) = Φ (M ) and the associated integer labels
1 1 2 2
on the facets agree. Since by assumption the links of singularities are not finite quotients
of odd dimensional spheres, a vertex of the polytope Φ (M ) is simple if and only if it is
i i
the image ofa point in (M ) . Therefore, after some renumbering, Φ (x(1)) = Φ (x(2)).
i reg 1 i 2 i
The symplectic toric orbifolds ((M ) ,ω ,Φ ), i = 1,2, are locally isomorphic in the
i reg i i
sense of [LT, p. 4222]. That is, every point p Φ ((M ) ) = Φ ((M ) ) has an open
1 1 reg 2 2 reg
∈
neighborhood U ∆ andanequivariantdiffeomorphismφ : Φ−1(U) Φ−1(U)with
⊂ reg U 1 2
φ∗ω = ω andφ∗Φ = Φ on Φ−1(U). Hence to prove uniquenessitis e→nough to argue
U 2 1 2 1 1
that the group H2(Φ ((M ) ,Z ) is trivial [LT, Section 7]. But Φ ((M ) ) is a polytope
i i reg G i i reg
with some vertices possibly deleted, hence is contractible. Thus the uniqueness follows
and we get an equivariant diffeomorphism φ : (M ) (M ) with φ∗ω = ω and
1 reg 2 reg 2 1
φ∗Φ = Φ . Itremainsto argue that φ extendsto a contin→uous map φ : M M sothat
2 1 1 2
φ∗Φ = Φ still holds. →
2 1
Fixi. Take anysequence{y } in (M ) converging to x(1). Then
n 1 reg i
lim Φ (φ(y )) = lim(Φ (y )) = Φ (x(1)) = Φ (x(2)).
2 n 1 n 1 i 2 i
n n
→∞ →∞
Since M iscompact, we may assume that {φ(y )} converges. Since Φ (lim φ(y )) =
2 n 2 n n
Φ (x(2)), and since Φ−1(Φ (x(2))) = {x(2)}, lim φ(y ) = x(2). Since the seq→u∞ence {y } is
2 i 2 2 i i n n i n
arbitrary, φ doesextend to a continuous map φ→∞: M M . (cid:3)
1 2
8 →
Theorem 1 has an easy converse which is a slight generalization of [LT, Theorem 8.2].
Namely,
Theorem 2.9. Let G be a torus. Let g denote its Lie algebra, and let Z g denote its integral
G
⊂
lattice. Given a rational polytope ∆ g∗, which is simple except possibly at the vertices, and
⊂
a positive integer m attached to each open facet F˚ of ∆, there exists a compact toric symplectic
F˚
singular G-space (M,ω,Φ : M g∗,{x }) such that Φ(M) = ∆ and the orbifold structure
α
group of apointinMwhichmaps→toa openfacetF˚ isZ/m Z.
F˚
Moreover, (M,ω,Φ : M g∗,{x }) is a symplectic quotient of CN by a closed abelian sub-
α
group of SU(N),whereNis t→henumber of facetsof ∆minusdimG.
The proof is a straightforward modification of the proof of [LT, Theorem 8.2]. We omit
it.
Remark 2.10. Combining the above theorem with the uniqueness part of Theorem 1 we
see that every toric symplectic singular space is a symplectic quotient of CN. Therefore,
by [Sj, Theorem 2.8], every toric symplectic singular space is a complex analytic space.
Furthermore by [Sj, Lemma 2.16] it is a Ka¨hler space in the sense of Grauert. The Ka¨hler
structures on these spacesisdescribed elsewhere [BGL].
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M.I.T.,CAMBRIDGE, MA02139ANDUNIVERSITY OF MICHIGAN, ANNARBOR, MI48109
E-mailaddress: [email protected]
M.I.T.,CAMBRIDGE, MA02139
E-mailaddress: [email protected]
UNIVERSITY OF ILLINOIS, URBANA,IL 61801AND
AUSTRALIANNATIONAL UNIVERSITY, CANBERRA,ACT 0200,AUSTRALIA
E-mailaddress: [email protected], [email protected]
10