The K-theory of abelian symplectic quotients

0.23 MB·English

by  Megumi Harada

THE K-THEORY OF ABELIAN SYMPLECTIC QUOTIENTS
MEGUMIHARADAANDGREGORYD.LANDWEBER
ABSTRACT. LetT beacompacttorusand(M,ω)aHamiltonianT-space. Inapreviouspaper,the
8 authorsshowedthattheT-equivariantK-theoryofthemanifoldM surjectsontotheordinaryinte-
0
gralK-theoryofthesymplecticquotientM//T, undercertaintechnical conditionsonthe moment
0
map.Inthispaper,weuseequivariantMorsetheorytogiveamethodforcomputingtheK-theoryof
2
M//T byobtaininganexplicitdescriptionofthekernelofthesurjectionκ : KT∗(M) ։ K∗(M//T).
n
OurresultsareK-theoreticanaloguesoftheworkofTolmanandWeitsmanforBorelequivariantco-
a
J homology. Further,weprovethatundersuitabletechnicalconditionsontheT-orbitstratificationof
2 M,thereisanexplicitGoresky-Kottwitz-MacPherson(“GKM”)typecombinatorialdescriptionofthe
K-theoryofaHamiltonianT-spaceintermsoffixedpointdata.Finally,weillustrateourmethodsby

] computingtheordinaryK-theoryofcompactsymplectictoricmanifolds,whichariseassymplectic
G
quotientsofanaffinespaceCN byalineartorusaction.
S
.
h
t
a
m CONTENTS
[
1. Introduction 1

3
2. InjectivityintotheK-theoryofthefixedpoints 4
v
0 3. ThekerneloftheKirwanmap 6
6 4. GKMtheoryinK-theory 8
6
5. Example: symplectictoricmanifolds 11
2
1 References 14
6
0
/
h
at 1. INTRODUCTION
m
The main result of this manuscript is an explicit description of the ordinary K-theory of an
:
v abelian symplectic quotient M//T in terms of the equivariant K-theory K∗(M) of the original
T
i
X Hamiltonian T-space M, where T is a compact torus. Symplectic quotients by Lie groups arise
r naturally in many different fields; well-known examples are toric varieties and moduli spaces of
a
bundles over Riemann surfaces. Further, symplectic quotients can often be identified with Geo-
metric Invariant Theory (“GIT”) quotients in complex algebraic geometry. Hence, many moduli
spacesthatariseasGITquotientsalsohavesymplecticrealizations,andthetopologicalinvariants
ofsuchmodulispacesgiveusefulconstraintsonmoduliproblems. Inaddition,thetheoryofgeo-
metric quantization provides a fundamental link between the topology of symplectic quotients
and representation theory (see e.g. [22, Section 7]). Our methods, as developed in this manu-
script, give a general procedure for computing the K-theory of such spaces when the Lie group
is acompact torus. ByK-theory,wemeantopological, integralK-theory,takingK0(X) tobe the
isomorphism classes of virtual complex vector bundles over X when X is compact, or for more
Date:February2,2008.
2000MathematicsSubjectClassification. Primary:53D20;Secondary:19L47.
Keywordsandphrases. symplecticquotient,K-theory,equivariantK-theory,Kirwanmap.
1
2 MEGUMIHARADAANDGREGORYD.LANDWEBER
general X, taking [X,Fred(H)] for a complex separable Hilbert space H. In the equivariant case,
by K (X) we mean Atiyah-Segal T-equivariant K-theory [26], built from T-equivariant vector
T
bundles if X is a compact T-space, and T-equivariant maps [X,Fred(H )] if X is noncompact
T T
(hereH containseveryirreduciblerepresentationofT withinfinitemultiplicity,seee.g. [4]).
T
In the setting of rational Borel equivariant cohomology, a fundamental symplecto-geometric
resultofKirwan[21]statesthatthereisanaturalsurjectiveringhomomorphism
(1.1) κ : H∗(M;Q) ։ H∗(M//T;Q),
H T
where H∗(M;Q) is the T-equivariant cohomology ring of the original Hamiltonian T-space M
T
fromwhichM//T isconstructed. Hence,inordertocomputeH∗(M//T;Q),itsufficestocompute
two objects: the equivariant cohomology ring H∗(M;Q) and the kernel of κ . For both of these
T H
computations,onecan useequivariant techniquesthatare unavailable onthequotient. Following
Kirwan’s original theorem [21], this “Kirwan method” has been well developed to yield explicit
methods to compute both H∗(M;Q) and ker(κ ). In this paper, we generalize to K-theory the
T H
explicit computation of the kernel of κ given by Tolman and Weitsman [28]. Further, we also
H
take to the K-theory setting the GKM-type combinatorial description of the T-equivariant coho-
mology of Hamiltonian T-spaces, which is motivated by the original work of Goresky, Kottwitz,
andMacPherson[11].
We now explain the setting of our results. Suppose that M is a symplectic manifold with a
Hamiltonian T-action, i.e., there exists a moment map µ : M → t∗. Assuming that T acts freely
on the level set µ−1(0), the symplectic quotient is then defined as M//T := µ−1(0)/T. We wish
to compute the ordinary K-theory K∗(M//T). The first step in this direction is the K-theoretic
analogue of the Kirwan surjectivity result (1.1) above, proven in [16]: just as in the cohomology
case, there is a natural ring homomorphism κ induced by the natural inclusion µ−1(0) ֒→ M as
follows:
(1.2) K∗(M) // K∗ µ−1(0)
T NNNNNκNNNNNNT&& (cid:0) (cid:15)(cid:15) ∼= (cid:1)
K∗(M//T),
andκissurjective. Themainresultofthispaperistogiveanexplicitcomputationofthekernelof
theK-theoreticKirwan map κabove. This,togetherwith thesurjectivityofκ, givesusamethod
toexplicitlydescribetheK-theoryofabelian symplecticquotients.
A keyelement in our arguments is the K-theoreticAtiyah-Bott lemma, which is a K-theoretic
analogue ofa fact originally proven in [3] in the settingof Borel equivariant cohomology. In this
manuscript, weusetheformulation given in [16, Lemma2.1]; a versionin thealgebraic category
isgiven in [29,Lemma4.2]. Aswenotedin [16], theAtiyah-Bottlemmais acrucial stepin many
Morse-theoretic proofs in symplectic geometry using the moment map µ, and once we have a
K-theoreticAtiyah-Bottlemma,itmaybeexpectedthatmanysymplectic-geometricresultsinthe
settingofrationalBorelequivariantcohomologycarryovertothatof(integral)K-theory. Indeed,
theresultsofthismanuscriptcanbeviewedasillustrationsofthisprinciple.
We now briefly outline the contents of this manuscript. In Section 2 we prove an injectivity
result, Theorem 2.5, which states that the T-equivariant K-theory of the Hamiltonian T-space,
under some technical conditions on the moment map, injects into the T-equivariant K-theory of
the fixed point set MT via the natural restriction map ı∗ : K∗(M) → K∗(MT). Similar results
T T
were obtained in the compact Hamiltonian setting already in [13], and in the algebraic category
THEK-THEORYOFABELIANSYMPLECTICQUOTIENTS 3
(in algebraic K-theory, for actions of diagonalizable group schemes on smooth proper schemes
over perfect fields) in [29]. Our contribution is to extend the result in the Hamiltonian setting to
a situation where M may not be compact; the proofusesstandard Morse-theoretictechniques in
symplecticgeometryusingagenericcomponentofthemomentmapandtheK-theoreticAtiyah-
Bottlemmamentionedabove. Thisinjectivityresultisanimportantpreliminarystepforourmain
result because the T-action on MT is trivial, and the ring isomorphism K∗(MT) ∼= K∗(MT) ⊗
T
K∗(pt)makestheequivariantK-theoryofthefixedpointsetstraightforwardtocompute.
T
Withthesetoolsinhand,weprovethefollowinginSection3:
Theorem1.1. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M → t∗.
Suppose there exists a component of the moment map which is proper and bounded below, and further
supposethatMT hasonlyfinitelymanyconnectedcomponents. Let
(1.3) Z := µ(C) C aconnectedcomponentof Crit(kµk2) ⊆ M ⊆ t∗ ∼= t
be the set of images un(cid:8)der µ of(cid:12)connected components of the critical set of the(cid:9)norm-square of µ. Suppose
(cid:12)
thatT actsfreelyonµ−1(0)andletM//T := µ−1(0)/T bethesymplectic quotient. Forξ ∈ t,define
M := x ∈ M hµ(x),ξi ≤ 0 ,
ξ
Kξ := (cid:8)α ∈ KT∗(cid:12)(cid:12)(M) α|Mξ =(cid:9)0 , and
K := (cid:8) K . (cid:12) (cid:9)
ξ (cid:12)
ξ∈Z⊆t
X
Thenthereisashortexactsequence
0 // K // K∗(M) κ // K∗(M//T) // 0,
T
whereκ :K∗(M) → K∗(M//T)istheK-theoretic Kirwanmap.
T
ThistheoremisaK-theoreticanalogueandaslightrefinementofthecomputationofthekernel
of the Kirwan map for rational Borel equivariant cohomology given by Tolman and Weitsman
[28]. In their paper, Tolman and Weitsman give an expression for the kernel ker(κ ) which is a
H
sumofidealsK (definedsimilarly tothosegivenabove)overallξ ∈ t,and it isnotimmediately
ξ
evident that such an infinite sum, in the case when M is not compact, yields a finite algorithm
for the computation ofthe kernel. In our version of the computation, we refine the statementby
explicitly exhibiting K as a sum over a finite set Z. For a discussion of a different simplification
ofthekernelcomputationforrationalBorelequivariantcohomologyinthecompactHamiltonian
case,see[10].
InSection4weproveaK-theoreticversionofaGoresky-Kottwitz-MacPherson(“GKM”)com-
binatorial description of the T-equivariant K-theory of a Hamiltonian T-space satisfying certain
conditions on the orbit type stratification. In this setting, we give an explicit combinatorial de-
scriptionoftheimageofı∗ inK∗(MT) ∼= K∗(MT)⊗K∗(pt).Thisresultshouldbeinterpretedas
T T
a part of a large body of work motivated by, and in many ways generalizing, the original work
ofGoresky,Kottwitz,andMacPherson[11], whoconsideredalgebraic torusactionsonprojective
algebraic varieties and Borel equivariant cohomology with C coefficients. For instance, it is now
knownthatsimilarGKMresultsholdinthesettingofBorelequivariantcohomologyforHamilton-
ianT-spaces(seee.g. [27,15]). SimilarGKM-typeresultsforHamiltonianT-spacesinequivariant
K-theorywith C coefficients are discussedin [23, 13], for otherequivariant cohomologytheories
(undersuitablehypothesesonthecohomologytheory)andformoregeneralT-spacesin[14],and
4 MEGUMIHARADAANDGREGORYD.LANDWEBER
for equivariant K-theory in the algebraic category in [29]. Our contribution in this section is to
prove,usingresultsin[14],thatsuchaGKMtheoremalsoholdsinequivariantK-theory(overZ)
forHamiltonianT-spaceswhichsatisfycertainconditionsontheT-orbitstratification.
Finally, in Section 5 we use our methods to give a computation of the ordinary K-theory of
smooth compact projective toric varieties, which can be obtained by a symplectic quotient of an
affine space CN. This rederives, using symplectic Morse-theoretic techniques, a description of
the K-theory of these toric varieties analogous to the Stanley-Reisner presentation (see e.g. [29,
Section6.2],[5]).
The work in this manuscript opensmany avenues for future research, of which we now men-
tionafewexamples. First,itwouldbeofinteresttogiveexplicitcomputationsoftheK-theoryof
moreexamplesofabeliansymplecticquotients,suchaspolygonspaces[18,19]or,moregenerally,
weightvarieties[9]. Furthermore,althoughwerestrictourattentioninthismanuscripttothecase
where the symplectic quotient M//G is a manifold, we expect that an orbifold version of our re-
sultswillstillholdinthesituationwhereGactsonlylocallyfreelyonµ−1(0),makingthequotient
X = µ−1(0)/G an orbifold. Here we use the definition of the “full orbifold K-theory” K (·) of
orb
an orbifold given in [20], where it is also shown that there is an orbifold Chern character map
from the full orbifold K-theory K (X) to the Chen-Ruan orbifold cohomology H∗ (X) of the
orb CR
orbifold X. Methods for computing the Chen-Ruan orbifold cohomology of orbifold symplectic
quotients were given in [8] by using Kirwan surjectivity methods in addition to explicit compu-
tationsofthekerneloftheKirwan map in themanifold case. Weexpectthat,using theresultsof
thismanuscriptandanapproachsimilarto[8],wecanalsocomputethefullorbifoldK-theoryof
orbifold symplectic quotients. Finally, it would be of interest to prove a K-theoretic analogue of
thesimplificationofthecomputationofker(κ)givenbyGoldinin[10]inthecaseofrationalBorel
equivariant cohomology. Goldin restrictedher considerationstothecase ofcompact Hamiltonian
spacesin[10],butweexpectthatasimilarstatementshouldstillhold. Weintendtoexplorethese
andrelatedtopicsinfuturework.
Acknowledgements. We thank Jonathan Weitsman for helpful discussions. The second author
thanks the University of Toronto and the Fields Institute for their hospitality and support while
conductinga portionofthis research. Bothauthorsthank theAmerican InstituteofMathematics
andtheBanffInternationalResearchStationfortheirhospitality.
2. INJECTIVITY INTO THE K-THEORY OF THE FIXED POINTS
We begin with a K-theoreticversion of the injectivity theoremof Kirwan, which is a keytech-
nical tool that we will need in the later sections. Suppose given M a Hamiltonian T-space with
moment map µ : M → t∗, with a component which is proper and bounded below. In this sec-
tion only, we will for notationalconvenience use Σ todenotethe T-fixedpoint setMT ofM. We
additionally assume that Σ has only finitely many connected components. In this situation, it is
well-knownthattheinclusionı :Σ ֒→ M inducesamapinrationalBorelequivariantcohomology
(2.1) ı∗ :H∗(M;Q) // H∗(Σ;Q),
T T
whichisaninjection. WewillproveaK-theoreticversionofthisinjectivity(2.2)byMorsetheory
using a generic component of the T-moment map µ. Our proof will follow that given in [15,
Theorem2.6].
Recall that the componentsµξ := hµ,ξi for ξ ∈ t of the moment map are Morse-Bottfunctions
onM. Wecallacomponentgeneric ifthecritical setofµξ ispreciselythefixedpointsetΣ. Thisis
THEK-THEORYOFABELIANSYMPLECTICQUOTIENTS 5
trueofanopendensesetofdirectionsξ ∈ t.Ourassumptionabovethatthereexistsacomponent
whichisproperandboundedbelowguaranteesthatourMorse-theoreticargumentswillwork.
InthecourseoftheproofwewilluseaspecialcaseoftheK-theoreticAtiyah-Bottlemma,which
werestatehereforreference.
Lemma 2.1. ([29, Lemma 4.2], [16, Lemma 2.3]) Let a compact connected Lie group G act fiberwise
linearly on a complex vector bundle π : E → X over a compact connected G-manifold X. Assume that
a circle subgroup S1 ⊆ G acts on E so that the fixed point set is precisely the zero section X. Choose an
invariant metric on E and let D(E) and S(E) denote the disc and sphere bundles, respectively. Then the
longexactsequenceforthepair(D(E),S(E))inequivariantK-theorysplitsintoshortexactsequences
0 // K∗ D(E),S(E) // K∗ D(E) // K∗ S(E) // 0.
G G G
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Remark2.2. Recallthatthesplittingofthelongexactsequenceisequivalent,bytheThomisomor-
phism, to the statement that the K-theoretic equivariant Euler class of E is not a zero divisor in
K∗(X). Wewillmainlyusethislatterperspective.
G
Inwhatfollows,weneedonlyconsidertheAtiyah-BottlemmaforthecaseG = T.Moreover,in
ourapplicationsofLemma2.1,theT-actiononthebasemanifoldX willbetrivial(sincethebase
manifold isacomponentofaT-fixedset). Hence,inordertousethelemma, weneedonlycheck
thatthereexistsasubgroupS1 ⊆ T actingonE fixingpreciselythezerosection.
Beforeproceedingtothemainargument,weprovethefollowingsimpletechnicallemma:
Lemma2.3. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ :M → t∗.
Suppose there exists a componentofµ whichis proper andbounded below, andfurther suppose that Σ has
only finitely many connected components. Then there exists a generic component f := µξ of µ which is
proper, boundedbelow,andwithrespecttowhichallcomponentsofΣhavedifferent criticalvalues.
Proof. Ifthereisacomponentµη ofµwhich isproperandboundedbelow,thenµitselfisproper,
since a component is given by a linear projection. A result of Lerman, Meinrenken, Tolman, and
Woodward [25, Theorem 4.2] states that in this situation the image µ(M) of M is convex and
locally polyhedral. Thus, by taking a small enough perturbation ξ of η, we may arrange that all
components of the fixed point set MT have different critical values with respect to µξ and still
haveµξ properandboundedbelow. (cid:3)
Withthislemmainhand,wemaynowstatethepropositionwhichisourmaintechnicaltoolto
showinjectivity. ThisistheK-theoreticanalogueof[15,Proposition2.4].
Proposition2.4. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M →
t∗. Suppose there exists a component of the moment map which is proper and bounded below, and further
suppose that Σ has onlyfinitely manyconnected components. Let f := µξ beageneric componentof µas
in Lemma 2.3. Let c be a critical value of f, and pick ε > 0 such that c is the only critical value of f in
(c−ε,c+ε). Let Σ be the component of Σ with f(Σ ) = c,and let M± := f−1(−∞,c±ε). Then the
c c c
longexactsequenceofthepair(M+,M−)splitsintoshortexactsequences
c c
0 // K∗ M+,M− // K∗ M+ k∗ // K∗ M− // 0,
T c c T c T c
where k : M− ֒→ M+ is the i(cid:0)nclusion. (cid:1)Moreover, t(cid:0)he res(cid:1)triction ma(cid:0)p K∗(cid:1)(M+) → K∗(Σ ), given by
c c T c T c
the inclusion Σ ֒→ M+, induces an isomorphism from the kernel of k∗ to the classes of K∗(Σ ) that are
c c T c
multiplesoftheequivariantEulerclasse(Σ )ofthenegativenormalbundletoΣ withrespect tof.
c c
6 MEGUMIHARADAANDGREGORYD.LANDWEBER
Proof. ThisisaK-theoreticversionoftheproofgivenin[27]. LetD(ν )andS(ν )denotethedisc
c c
and spherebundlesofthenegativenormalbundleν tothefixedsetΣ withrespecttof. Bythe
c c
assumptiononε,thecomponentΣ istheonlycomponentofCrit(f)inf−1(c−ε,c+ε). Similarly
c
sincef isproperandT-invariant,thenegativegradientflowwithrespecttof usingaT-invariant
metricgivesaT-equivariantretractionofthepair (M+,M−)tothepair(D(ν ),S(ν )). Usingthe
c c c c
T-equivariantThomisomorphism,weconcludethatthelongexactsequenceofthepair
··· → K∗(M+,M−) → K∗(M+) → K∗(M−) → ···
T c c T c T c
splitsintoshortexactsequencesifandonlyiftheequivariantEulerclasse(ν )isnotazerodivisor.
c
(Wealsonotethat,unliketherationalcohomologycase,sincef isacomponentofamomentmap,
the Morse index of f is even. Since K∗−λ(D(ν )) ∼= K∗(D(ν )) by Bott periodicity, there is no
T c T c
degreeshifthere.) Moreover,thenegativenormalbundleE iscomplexbyaresultofKirwan[21].
The group T fixes exactly the zero section Σ because Σ is defined to be a component of the T-
c c
fixed point set. We may apply Lemma 2.1 taking G = T,X = Σ , and E = ν , together with a
c c
suitablechoiceofS1 ⊂ T,toconcludethate(Σ )isnotazerodivisor. Thepropositionfollows. (cid:3)
c
Wenowcometothemaintheoremofthissection. WithLemmas2.1and2.3andProposition2.4
in hand,theproofnowexactlyfollowsthatgivenfor[15, Theorem2.5], sowewill notreproduce
ithere.
Theorem2.5. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M → t∗.
Suppose there exists a component of the moment map which is proper and bounded below, and further
suppose that Σ has onlyfinitely manyconnected components. Let f := µξ beageneric componentof µas
inLemma2.3. Letı : Σ֒→ M betheinclusionofthefixedpointsetintoM. Thentherestriction map
(2.2) ı∗ : K∗(M) → K∗(Σ)
T T
isinjective.
Remark 2.6. Since the Atiyah-Bott lemma and Proposition 2.4 tell us that the Euler classes e(Σ )
c
of the negative normal bundles to the fixed point components are not zero divisors, in order to
obtaintheinjectivity(2.2),wecouldalsotaketheMorsestratificationM = S ,wheretheS are
c c c
the flow-down manifolds associated to Σ , and apply the injectivity theorem [14, Theorem 2.3],
c
F
which holds for more generalequivariant cohomologytheories. We presentthis straightforward
Morse-theoreticargumentheresinceweexpectittobemorefamiliar tosomereaders.
3. THE KERNEL OF THE KIRWAN MAP
Wehavealreadyobservedintheintroductionthatinordertomakeexplicitcomputationsofthe
K-theory rings of abelian symplectic quotients, it is necessary to explicitly identify the kernel of
theKirwanmapκ :K∗(M) → K∗(M//T).InthesettingofrationalBorelequivariantcohomology,
T
Tolman and Weitsman showed in [28] that the kernel of κ can be identified as a sum of certain
ideals in H∗(M;Q). In this sectionwe stateand provethe (integral)K-theoreticanalogue ofthis
T
result. ThiswillallowustomaketheexplicitcomputationinSection5.
Beforeproceedingtothemaintheorem,wetakeamomenttodiscusstherelationshipsbetween
thevarioustechnicalMorse-theoretichypothesesonamomentmapµwhichareusedintheliter-
ature. Theseare:
(1) thereexistsacomponentµξ ofthemomentmapwhichisproperandboundedbelow,
(2) µisproper,and
THEK-THEORYOFABELIANSYMPLECTICQUOTIENTS 7
(3) kµk2 isproper.
It is straightforward to see that (1) implies (2), and that (2) and (3) are equivalent. When the
Hamiltonian space M is compact, then any of these conditions automatically holds; the point is
that when M is not compact, these conditions ensure that the Morse-theoretic arguments given
belowusingµstillwork. FortheproofoftheK-theoreticKirwansurjectivitytheoremin[16],we
needonlythesecond(orequivalentlythird)condition,namely,thatµisproper. However,forour
explicit kernelcomputationswewillbeusingTheorem2.5which requiresthehypothesis(1)and
additionally the condition that MT has finitely many components, so we continue making these
strongerassumptions. Inpracticethisisnotveryrestrictive.1 Moreover,ourhypothesesguarantee
thatthealgorithmwepresentforcomputingthekernelofκisfinite,aswillbeseeninLemma3.2.
We first give the statement of the main theorem of this section. In the following and in the
sequel,weidentifyt ∼= t∗ usinganinnerproductont,sothemomentmapµcanbeconsideredas
amaptakingvaluesint.
Theorem3.1. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M → t∗.
Suppose there exists a component of the moment map which is proper and bounded below, and further
supposethatMT hasonlyfinitelymanyconnectedcomponents. Let
(3.1) Z := µ(C) C aconnectedcomponentof Crit(kµk2) ⊆ M ⊆ t∗ ∼= t
bethesetofimagesund(cid:8)erµof(cid:12)componentsofthecriticalsetofkµk2. Supposeth(cid:9)atT actsfreelyonthelevel
(cid:12)
setµ−1(0),andletM//T bethesymplectic quotient. Forξ ∈ t,define
M := x ∈ M hµ(x),ξi ≤ 0 ,
ξ
Kξ := (cid:8)α ∈ KT∗(cid:12)(cid:12)(M) α|Mξ =(cid:9)0 , and
K := (cid:8) K . (cid:12) (cid:9)
ξ (cid:12)
ξ∈Z
X
Thenthereisashortexactsequence
0 // K // K∗(M) κ // K∗(M//T) // 0,
T
whereκ :K∗(M) → K∗(M//T)istheKirwanmap.
T
WenotethatTheorem3.1isbothaK-theoreticanalogueofthekernelcomputationinH∗(−;Q)
T
by Tolman and Weitsman [28, Theorem 4] and a slight refinement of their statement, as we now
explain. In[28,Theorem4],thekernelisexpressedasasumovertheinfinitesetofallelementsin
t. InthecasewhereM iscompact,theycommentthattheiralgorithmisinfactfinitelycomputable
in[28,Remark5.3],buttheydonotexplicitlyaddresswhathappensinthenon-compactsituation.
In Theorem 3.1 we have expressed K as a sum over a certain set Z, which is shown to be finite
in Lemma3.2below. Thepointis thatourK-theoretickernelK is explicitly computedbyafinite
algorithmeveninthenon-compactcase.
We now show that Z is finite. Recall that under our properness hypotheses, both µ and kµk2
areproper.
1
Forinstance, althoughhyperka¨hlerquotientsarerarelycompact, theyoftenadmitnaturalHamiltoniantorusac-
tions which do satisfy(1); for example, for hypertoricvarieties, this fact was exploitedin[15] to give combinatorial
descriptionsoftheirequivariantcohomologyrings,andforhyperpolygonspaces,KonnousesaHamiltonianS1-action
satisfying(1)inhisproofofKirwansurjectivityforthesespaces[24].(Seealso[17]formoreonexamplesofthistype.)
8 MEGUMIHARADAANDGREGORYD.LANDWEBER
Lemma3.2. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M → t∗.
Suppose there exists a component of the moment map which is proper and bounded below, and further
supposethatMT hasonlyfinitelymanyconnectedcomponents. Thenthereareonlyfinitelymanyconnected
components of the critical set of the norm-square kµk2 : M → R≥0 ofthe momentmap. In particular, the
setZ in(3.1)isfinite.
Proof. ForasubgroupH ⊆ T ofthetorusT,letM denotethesubsetofM consistingofpointsp
H
withStab(p) = H.Let
(3.2) M = M
H
H⊆T
G
betheorbittypestratificationofM asin[12,Section3.5]. SinceT iscompact,theactionisproper.
By standard Hamiltonian geometry[6] and the theory of proper group actions (see e.g. [12, Ap-
(i) (i)
pendix B]), the closure M of each connected component M of M is itself a closed Hamil-
H H H
(i)
tonian T-space with moment map given by restriction. Since the restriction of Φ to M is also
H
(i)
proper and bounded below, each M contains a T-fixed point. By equivariant Darboux, any T-
H
invariant tubular neighborhood of a connected component of MT intersects only finitely many
T-orbit types. Since by assumption there are only finitely many connected components of MT,
wemayconcludethatthereareonlyfinitelymanyorbittypesinthedecomposition(3.2),andthat
each M has only finitely many connected components. Kirwan proves [21, Lemma 3.12] that
H
there is at most one critical value of kµk2 on each Φ((M )(i)), so we may conclude that there are
H
onlyfinitelymanycriticalvaluesofkµk2. Sinceµisproper,thereareonlyfinitelymanyconnected
componentsofCrit(kµk2). Thefinalassertioninthetheoremfollowsfrom[21,Corollary3.16]. (cid:3)
The proof of Theorem 3.1 closely follows the Morse-theoretic argument given in the compact
case in H∗(−;Q) in [28, Theorem 3], except that we use our K-theoretic Atiyah-Bott lemma
T
(Lemma 2.1). Hence we do not give all the details below. Instead, we only briefly indicate why
thesumcaninfactberestrictedtothesetZ.
ProofofTheorem 3.1. Wereferthereaderto[28]fordetails. First,theonlypointoftheproofrequir-
ingsubstantialargumentistoprovethatker(κ) ⊆ K.Second,weordertheconnectedcomponents
of the critical sets of the norm-square kµk2; denote these as {C }m , where C = µ−1(0). Hence
i i=0 0
α ∈ ker(κ) exactly means α| = 0. Third, by observing that all components of MT are also
C0
components of Crit(kµk2), it suffices by Theorem 2.5 to show that there exists a β ∈ K such that
α| = β| foralli. Thefinalandmostimportantstepinthisargumentisaninductiveconstruc-
Ci Ci
tion of the β, for which it suffices to show that for 1 ≤ ℓ ≤ m and α ∈ ker(κ) such that α| = 0
Ci
for all 0 ≤ i ≤ ℓ−1, there exists an element α′ ∈ K such that α′| = α| for all 0 ≤ i ≤ ℓ. The
Ci Ci
constructiongivenby Tolmanand Weitsmancan nowbeexplicitly seentoproducean elementα
which is in fact contained in K . Since µ(C ) ∈ Z by definition (3.1), this implies that α′ ∈ K.
µ(Cℓ) ℓ
Therestoftheargumentfollowsthatin[28].
(cid:3)
4. GKM THEORY IN K-THEORY
The main result of this section is to show that, under some technical conditions on the orbit
stratification, theK-theoryofa Hamiltonian T-spacehas acombinatorial description,usingdata
from the equivariant one-skeleton. Such a description is useful for explicit computations of the
kerneloftheKirwan map. Asmentionedin theIntroduction,thisresultisbestviewedas partof
THEK-THEORYOFABELIANSYMPLECTICQUOTIENTS 9
a large body of work inspired by the original paper of Goresky, Kottwitz, and MacPherson [11].
Both the statement of our theorem and its proof are K-theoretic versions of those given in [15,
Theorem2.11],withslightdifferencesinthehypotheses(explainedbelow).
We begin by defining an extra condition on the T-action that will be necessary to state the
theorem.
Definition4.1. LetT beacompacttorusand(M,ω)aHamiltonianT-space. Wesaythattheaction
is GKM if MT consistsoffinitely many isolated points,and theT-isotropyweightsat each fixed
pointp ∈ MT arepairwiselinearlyindependentint∗.
Z
Remark4.2. TheGKMconditionontheT-isotropyweightsgiveninDefinition4.1islessrestrictive
than the hypothesis in [15, Theorem 2.11] that the T-weights are relatively prime in H∗(pt;Z).
T
This is due to the difference betweenequivariant Euler classes in K∗(pt) and thosein H∗(pt;Z).
T T
A more detailed discussion of the differences between the Atiyah-Bott lemma in equivariant K-
theoryandinintegralBorelequivariantcohomologycanbefoundin[16,Section2].
We now briefly recall the construction of the combinatorial and graph-theoretic data used in
ourtheorem. LetN denotethesubsetofM givenby
N := p ∈ M codim(Stab(p)) = 1 .
Thus N consists of the points in M(cid:8)whose T(cid:12)-orbits are exactly o(cid:9)ne-dimensional. The equivariant
(cid:12)
one-skeleton ofM isthendefinedtobetheclosureN ofN. Hence
N = p ∈M codim(Stab(p)) ≤1 = N ∪MT.
The GKM condition states tha(cid:8)t at each(cid:12) fixed point p ∈ MT(cid:9), the T-isotropy weights are pairwise
(cid:12)
linearlyindependentint∗. EachT-weightspacecorrespondstoacomponentofN,theclosureof
Z
which is either an S2 ∼= P1 (with north and south poles being T-fixed points {p,q} ⊆ MT) or a
copy of C (with origin a T-fixed point). Thus, given the GKM condition, the one-skeletonN is a
collection ofprojectivespacesP1 andaffine spacesC,gluedatT-fixedpoints. Bydefinition,each
component of N is equippedwith a T-action which is specifiedby a weight in t∗ ∼= Hom(T,S1);
Z
this weight appears in the T-weight decomposition of the isotropy action on the corresponding
T-fixedpoint. (ThereisasignambiguityintheT-weightforacomponentofN whoseclosureisa
P1,dependingonthechoiceofnorthorsouthpole;however,thischoicedoesnotaffecttheGKM
computationtobedescribedbelow.)
From this data we construct the GKM graph, a labelled graph Γ = (V,E,α), associated to the
GKM T-space M. The vertices V of Γ are the T-fixed points V = MT, and there is an edge
(p,q) ∈ E exactly when there exists an embedded P1 ⊂ N containing as its two T-fixed points
{p,q} ⊂ P1. Additionally, welabel each edge(p,q) with theweightα specifyingthe T-action
(p,q)
onthecorrespondingP1 asdiscussedabove. NotethatthecomponentsofN correspondingtoan
affine space CdonotcontributetotheGKM graph sinceeach such Cequivariantly retractstoits
correspondingT-fixedpoint.
Now we return to the setting of the previous sections. Let (M,ω) be a symplectic manifold
equippedwith a Hamiltonian T-action. As before, we assume that the moment map µ : M → t∗
has a component which is proper and bounded below. In addition, we now assume that the T-
actiononM isGKM,soinparticularMT isafinitesetofisolatedpointsinM. Letµξ :M → Rbea
genericcomponentofµwhichisproperandboundedbelow,aschoseninLemma2.3. Thenµξ isa
MorsefunctiononM. OrderthecriticalpointsCrit(µξ) = MT = {p }m sothatµξ(p ) <µξ(p )if
i i=1 j k
10 MEGUMIHARADAANDGREGORYD.LANDWEBER
andonlyifj < k.SincetheMorsefunctionisT-invariant,thenegativegradientflowwithrespect
toaT-invariantmetricisT-equivariant.
LetU betheflow-downcell fromthecritical pointp . Thenthenegativegradientflow givesa
i i
T-equivariantdeformationretractionofM totheunionoftheU ,i.e.,
i
M ∼ U
i
i
[
is a T-equivariant homotopy equivalence. Hence to study the T-equivariant K-theory of M, we
may instead study that of M′ := U . Each U contains a single critical point p , and T U is
i i i i pi i
a T-representation; by the GKM assumption, T-weights occurring in T U are pairwise linearly
S pi i
independent.
Wemaynowusetheresultsof[14]toshowthat,undertheconditionsoutlinedabove,theGKM
graph Γ combinatorially encodes the equivariant K-theory of the Hamiltonian T-space M. We
firststateacruciallemmawhichinvolvesK-theoreticequivariantEulerclassesinK∗(pt)∼= R(T),
T
where R(T) is the representation ring of T; this will be the key step in the proof of the main
theorem. Let C be a 1-dimensional representation of T with weight σ ∈ t∗. Recall that the K-
σ Z
theoreticT-equivariantEulerclassoftheT-bundleC → ptis
σ
e (σ) := 1−e−σ ∈ K∗(pt) ∼= R(T).
T T
By propertiesof the Euler class, if E = C is a direct sum of such 1-dimensional representa-
i σi
tions,thentheequivariant Eulerclass ofLE istheproducteT(E) = i(1−e−σi).Thefollowingis
aspecialcaseof[29,Lemma4.9]:
Q
Lemma 4.3. Let σ,τ ∈ t∗ be linearly independent in t∗. Then the corresponding Euler classes e (σ) =
Z Z T
1−e−iσ ande (τ) = 1−e−iτ ∼= R(T)arerelatively primeinK∗(pt)= R(T).
T T
WewillusethislemmatoobtainthefollowingcombinatorialdescriptionofK∗(M). SinceMT
T
consistsoffinitelymanyisolatedpoints,
K∗(MT)∼= K∗(p)∼= h :V = MT → K∗(pt) ∼=R(T) .
T T T
p∈MT
M (cid:8) (cid:9)
WethendefinetheΓ-subringofK∗(MT)tobe
T
h(p)−h(q) ≡ 0(mode(α ))
K∗(Γ,α) := h: V → K∗(pt) ∼= R(T) (p,q) ⊆ K∗(MT).
T foreveryedge(p,q) ∈ E T
(cid:26) (cid:12) (cid:27)
(cid:12)
Wehavethefollowing: (cid:12)
(cid:12)
Theorem4.4. LetT beacompacttorusand(M,ω)aHamiltonianT-spacewithmomentmapµ : M → t∗.
Suppose there exists a component of the momentmap which is proper and bounded below, and that the T-
actiononM isGKM.Thentheinclusionı :MT ֒→ M inducesanisomorphism
ı∗ : K∗(M) → K∗(Γ,α) ⊆ K∗(MT).
T T
Proof. Wewilluseaspecialcaseof[14,Theorem3.1],whichstatesthatforT-spacessatisfyingcer-
tain assumptions,theequivariant K∗-theoryis isomorphicvia ı∗ totheΓ-subringdefinedabove.
T
Wemustthereforecheckthateachofthehypothesesnecessaryforthistheoremissatisfied.
We have already seen that it suffices to compute the K∗-theory of the subspace M′ := U
T i i
of M. Let M := U be the union of these cells up to the i-th cell. Then M = M is a
i 1≤j≤i i i Si
T-invariant stratification of M′, and by construction each quotientMi/Mi−1 is homeomorphicto
S S