Moment maps, cobordisms, and Hamiltonian group actions

356 Pages·2002·1.885 MB·English

by  Victor Guillemin

Moment maps, cobordisms, and Hamiltonian
group actions
Viktor Ginzburg
Victor Guillemin
Yael Karshon
Author address:
Department of Mathematics, University of California at Santa
Cruz, Santa Cruz, CA 95064
E-mail address: [email protected]
Department of Mathematics, Massachusetts Institute of Technol-
ogy, Cambridge, MA 02139
E-mail address: [email protected]
Institute of Mathematics, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel
E-mail address: [email protected]

Contents
Chapter 1. Introduction 1
1. Topological aspects of Hamiltonian group actions 1
2. Hamiltonian cobordism 4
3. The linearization theorem and non-compact cobordisms 5
4. Abstract moment maps and non-degeneracy 7
5. The quantum linearization theorem and its applications 8
6. Acknowledgements 10
Part 1. Cobordism 13
Chapter 2. Hamiltonian cobordism 15
1. Hamiltonian group actions 15
2. Hamiltonian geometry 21
3. Compact Hamiltonian cobordisms 24
4. Proper Hamiltonian cobordisms 27
5. Hamiltonian complex cobordisms 29
Chapter 3. Abstract moment maps 31
1. Abstract moment maps: de(cid:12)nitions and examples 31
2. Proper abstract moment maps 33
3. Cobordism 34
4. First examples of proper cobordisms 37
5. Cobordisms of surfaces 39
6. Cobordisms of linear actions 42
Chapter 4. The linearization theorem 45
1. The simplest case of the linearization theorem 45
2. The Hamiltonian linearization theorem 47
3. The linearization theorem for abstract moment maps 51
4. Linear torus actions 52
5. The right-hand side of the linearization theorems 56
6. The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas 58
Chapter 5. Reduction and applications 63
1. (Pre-)symplectic reduction 63
2. Reduction for abstract moment maps 65
3. The Duistermaat{Heckman theorem 69
4. Ka(cid:127)hler reduction 72
5. The complex Delzant construction 73
6. Cobordism of reduced spaces 81
v
vi CONTENTS
7. Je(cid:11)rey{Kirwanlocalization 82
8. Cutting 84
Part 2. Quantization 87
Chapter 6. Geometric quantization 89
1. Quantization and group actions 89
2. Pre-quantization 90
3. Pre-quantizationof reduced spaces 96
4. Kirillov{Kostantpre-quantization 99
5. Polarizations,complex structures, and geometric quantization 102
6. Dolbeault Quantization and the Riemann{Roch formula 110
7. Stable complex quantization and Spinc quantization 113
8. Geometric quantization as a push-forward 117
Chapter 7. The quantum version of the linearization theorem 119
1. The quantization of Cd 119
2. Partition functions 125
3. The character of (Cd) 130
Q
4. A quantum version of the linearization theorem 134
Chapter 8. Quantization commutes with reduction 139
1. Quantization and reduction commute 139
2. Quantization of stable complex toric varieties 141
3. Linearization of [Q,R]=0 145
4. Straightening the symplectic and complex structures 149
5. Passingto holomorphic sheaf cohomology 150
6. Computing global sections; the lit set 152
7. The C(cid:20)ech complex 155
8. The higher cohomology 157
9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 159
10. Overview of the literature 162
Part 3. Appendices 165
Appendix A. Signs and normalization conventions 167
1. The representation of G on C (M) 167
1
2. The integral weight lattice 168
3. Connection and curvature for principal torus bundles 169
4. Curvature and Chern classes 171
5. Equivariant curvature; integral equivariant cohomology 172
Appendix B. Proper actions of Lie groups 173
1. Basic de(cid:12)nitions 173
2. The slice theorem 178
3. Corollariesof the slice theorem 182
4. The Mostow{Palaisembedding theorem 189
5. Rigidity of compact group actions 191
Appendix C. Equivariant cohomology 197
1. The de(cid:12)nition and basic properties of equivariant cohomology 197
CONTENTS vii
2. Reduction and cohomology 201
3. Additivity and localization 203
4. Formality 205
5. The relation between H and H 208
(cid:3)G (cid:3)T
6. Equivariant vector bundles and characteristicclasses 211
7. The Atiyah{Bott{Berline{Vergnelocalization formula 217
8. Applications of the Atiyah{Bott{Berline{Vergnelocalization formula 222
9. Equivariant homology 226
Appendix D. Stable complex and Spinc-structures 229
1. Stable complex structures 229
2. Spinc-structures 238
3. Spinc-structures and stable complex structures 248
Appendix E. Assignments and abstract moment maps 257
1. Existence of abstract moment maps 257
2. Exact moment maps 263
3. Hamiltonian moment maps 265
4. Abstract moment maps on linear spaces are exact 269
5. Formal cobordism of Hamiltonian spaces 273
Appendix F. Assignment cohomology 279
1. Construction of assignment cohomology 279
2. Assignments with other coe(cid:14)cients 281
3. Assignment cohomologyfor pairs 283
4. Examples of calculations of assignment cohomology 285
5. Generalizations of assignment cohomology 287
Appendix G. Non-degenerate abstract moment maps 289
1. De(cid:12)nitions and basic examples 289
2. Global properties of non-degenerate abstract moment maps 290
3. Existence of non-degenerate two-forms 294
Appendix H. Characteristic numbers, non-degenerate cobordisms, and
non-virtual quantization 301
1. The Hamiltonian cobordism ring and characteristicclasses 301
2. Characteristic numbers 304
3. Characteristic numbers as a full system of invariants 305
4. Non-degenerate cobordisms 308
5. Geometric quantization 310
Appendix I. The KawasakiRiemann{Roch formula 315
1. Todd classes 315
2. The Equivariant Riemann{Roch Theorem 316
3. The KawasakiRiemann{Roch formula I: (cid:12)nite abelian quotients 320
4. The KawasakiRiemann-Roch formula II: torus quotients 323
Appendix J. Cobordism invariance of the index of a transversally elliptic
operator by Maxim Braverman 327
1. The SpinC-Dirac operator and the SpinC-quantization 327
2. The summary of the results 329
viii CONTENTS
3. Transversally elliptic operatorsand their indexes 331
4. Index of the operator B 333
a
5. The model operator 335
6. Proof of Theorem 1 336
Bibliography 339
Index 349
CHAPTER 1
Introduction
1. Topological aspects of Hamiltonian group actions
The objects we will be concerned with in this monograph, such as symplectic
forms and moment maps, are easy to de(cid:12)ne and yet are of great complexity and
depth. This complexity manifests itself in a variety of ways. For example the Dar-
boux theorem is one of the most elementary theorems of symplectic geometry, but
(cid:12)nding Darboux coordinates for a particular symplectic structure (for instance, a
coadjoint orbit) can give rise to intricate problems in linear algebra and ordinary
di(cid:11)erential equations. To deal with this complexity one is often forced to narrow
one’s focus and concentrate on speci(cid:12)c aspects of these objects, for example, in-
variance properties of moment maps (energy or momentum conservation laws) or
topological data (the cohomology class of the form or the homotopy class of the
associated almost complex structure).
In this monograph,ourfocuswill be onglobalpropertiesof Hamiltoniantorus
actions and their connection with topology.
1.1. GlobalinvariantsofHamiltoniangroupactions. Overthecourseof
thelasttwentyyearsithasgraduallybecomeclearthatthistopic,globalproperties
of Hamiltonian group actions, has more to do with topology, and less to do with
symplectic geometry, than was previously realized.
The(cid:12)rstinklingofthiscamefromtheDuistermaat{Heckmantheorem,[DH1].
This theorem states that the oscillatory integral for the moment map of a torus
actiononasymplecticmanifoldisexactlyequaltotheleadingtermofitsasymptotic
expansion. Hence, allterms of higher orderin the expansion vanish. This provides
a formula for the Fourier transform of the push-forward of the Liouville measure
by the moment map in terms of the (cid:12)xed points of the action. More explicitly
consider an action of the circle, G, on a compact symplectic manifold (M2n;!)
with isolated (cid:12)xed points and moment map (cid:8): M R. Then the Duistermaat{
!
Heckman formula reads
1 1 n e(cid:8)(p)
e(cid:8)!n = :
n! 2(cid:25) (cid:11)
(cid:18) (cid:19) ZM p2XMG j;p
Here MG is the (cid:12)xed point set and (cid:11) ;::: ;(cid:11) arQe the weights of the linearized
1;p n;p
action of G on T M at a (cid:12)xed point p.
p
AlthoughtheDuistermaat{Heckmantheoremwasoriginallyprovedinthesym-
plectic setting, it wassoondiscoveredbyBerline andVergneand AtiyahandBott,
[BV1, AB1], that the theorem is just a particular case of a general localization
formula in equivariant cohomology for torus actions. This formula, of a purely
topologicalnature,equatesthe integralof anequivariantcohomologyclass overM
andthesumoftheintegralsofthisclassoverthecomponentsofthe(cid:12)xedpointset,
1
2 1. INTRODUCTION
withcorrectionscomingfromtheactionsonthenormalbundlestothecomponents.
Explicitly,
u
F
u= j :
e( )
ZM F ZF VF
X
Here u is an equivariant cohomology class on M, the summation runs over the
components F of the (cid:12)xed point set, and e( ) is the equivariant Euler class of
F
V
the normal bundle to F. When applied to the class u = e (! (cid:8)), the localization
(cid:0) (cid:0)
theorem turns into the Duistermaat{Heckman formula.
A second example of this \topologizing" of the global theory of Hamiltonian
torus actions involves geometric quantization and indices of Dirac operators asso-
ciated with symplectic structures. The Atiyah{Singerindex theorem expresses the
index of such an operator as the integral of a certain characteristic class. (See,
e.g., [ASe, ASi] and [BGV, Du, Gil].) When the operator is invariant under a
compactgroupaction,thisindexbecomesavirtualrepresentationanditscharacter
canbeevaluatedasthe integralofanequivariantcohomologyclass. Thegeometric
quantization (M;!) is de(cid:12)ned as the virtual representation, i.e., index, of the
Q
Dirac operator. In this case, the index theorem takes the form
(M;!)= e!=2(cid:25)Td(M);
Q
ZM
where Td(M) is the Todd class of M. In the equivariant situation Td(M) should
bereplacedbytheequivariantToddclassand!bytheequivariantsymplecticform
! (cid:8),where(cid:8)isthemomentmap. Notethatinthisformulatheintegranddepends
(cid:0)
only on the cohomology class of the symplectic structure and the Chern classes of
the almost complex structure associated with the symplectic form. This Chern
classisactually justaninvariantofthe \stablecomplexstructure"associatedwith
this almost complex structure.1 Moreover, a cohomology class, a stable complex
structure, and an orientation are su(cid:14)cient to de(cid:12)ne a Dirac operator with the
\correct" index. A variant of this is \Spinc quantization", which only depends on
the cohomologyclass and the orientation.
When the group acting on the symplectic manifold is abelian, the index is
determinedbythe(cid:12)xedpointdata. Anexplicitformula,theAtiyah{Bott{Lefschetz
(cid:12)xed point theorem, [AB1], can be obtained by applying the localization theorem
to the integrand above.
A third example is the quantization commutes with reduction theorem. This
asserts that the G-invariant part of the quantization of a symplectic G-manifold is
equal to the quantization of the reduction at the zero level of the moment map.
Variousversionsof thistheorem, oftenreferredtoas[Q,R]=0,havebeenprovedin
the lastdecade. (See Section10of Chapter8fora detailedsurveyandreferences.)
This result is also essentially of a topological nature although the situation is now
more subtle. As we have seen, geometric quantization only requires a cohomology
class and an equivariant stable complex structure. However, symplectic reduction
requiresmore. Namely,weneedamomentmap. Wegetthatfromanequivariantly
closed two-form. (Recall that an equivariant two-form is a pair consisting of an
invarianttwo-formandanequivariantmaptog . Thisformisequivariantlyclosed
(cid:3)
ifandonlyifthetwo-formisclosedandthemapsatis(cid:12)esHamilton’sequation. The
1SincewewillneedthisclassinsituationswhenMisnotsymplecticornoteven{dimensional,
we recall that a stable complex structure is just a complex structure on the sum TM Rk for
(cid:8)
somek.
1. TOPOLOGICAL ASPECTS OF HAMILTONIAN GROUP ACTIONS 3
form is not assumed to be non-degenerate, but we will still refer to the map as a
moment map.) Under certain additional assumptions, purely topological proofs of
the quantization commutes with reduction theorem are obtained in [Met3, Par2]
using equivariant K-theory. In this book we give an alternative topological proof,
in the case where G is a torus and the (cid:12)xed points are isolated.
Afourthexampleisthenon-abelianlocalizationtheoremofJe(cid:11)reyandKirwan,
[JK1]. This result, which is closely related to the quantization commutes with
reduction theorem, expresses the integral over the reduced space of an equivariant
cohomology class coming from a Hamiltonian G-manifold in terms of the (cid:12)xed
points of the action.2 For G a torus, a topological interpretation of the Je(cid:11)rey{
Kirwan theorem can be found, for example, in [GGK1].
1.2. Geometry of Hamiltonian group actions as a branch of equivari-
ant topology. To summarize, the four theorems which we described above are
theoremsabout objectswhich oneencountersinsymplectic geometrybut arebasi-
cally theorems in equivariant topology. Moreover,it would not be an exaggeration
to say that the area of symplectic geometry which deals with global properties of
Hamiltonian group actions is, to a large extent, a branch of equivariant topology.
This topology, however, involves manifolds that are equipped with somewhat un-
conventional structures. The choice of structures depends on the kind of question
one is asking, but some common traits are already clear. For the Duistermaat{
Heckmanformula,itissu(cid:14)cienttoconsidermanifoldsequippedwithanequivariant
cohomologyclass of degree two. In questions of geometric quantization, one needs
an additional structure, such as an equivariant stable complex or Spinc structure.
In the \quantization commutes with seduction" theorem, symplectic reduction is
involved, and for this one needs a moment map. Thus one has to replace the co-
homology class by a closed equivariant two-form. In other words, the structure
considerednow is a triple (M;!;(cid:8)) consisting of an oriented G-manifold (not nec-
essarily of even dimension), a G-invariant closed two-form ! (not necessarily of
maximal rank), and an equivariant moment map (cid:8), such that ! and (cid:8) are related
bytheHamiltonequations(whichmeansthattheformaldi(cid:11)erence! (cid:8)isaclosed
(cid:0)
equivariant two-form). We will henceforth refer to such a triple as a Hamiltonian
G-manifold.
These are not the only possible choices of structures, and, perhaps, not the
optimal ones. Below, in Section 4, we propose yet one more re(cid:12)nement, which
allows one to separate the moment map from the cohomologyclass.
Note that the cohomological data and the stable complex data re(cid:13)ect the two
di(cid:11)erentrolesthatasymplecticformplays: aclosedformdeterminesacohomology
class,andanon-degenerateformdetermines analmostcomplex(and hence, stable
complex) structure. As a result of separatingthe two roles, a considerableamount
of information is lost, but the setting becomes simpler.
There is one more implication of non-degeneracy which we have entirely ig-
nored so far. Namely, the non-degeneracy of a symplectic form implies a non-
degeneracy condition on the moment map. This condition can also be looked at
from a topologicalpoint of view, and we will do so in Section 4.2. Non-degeneracy
is often essential in the local study of Hamiltonian actions, e.g., in singular re-
duction, [ACG, BL, SL]. The local questions are closely connected with (and
2Inspiteofitsname,theJe(cid:11)rey{Kirwannon-abelianlocalizationtheoremisnotanextension
oftheBerline{Vergne{Atiyah{Bott abelianlocalizationtonon-abeliangroups.
4 1. INTRODUCTION
to some extent motivated by) the investigation of stability of relative equilibria of
mechanical systems. Here, the reduced energy-momentum and Lagrangian block
diagonalization methods, [Lew, MSLP, SLM], are among the most e(cid:14)cient and
versatile. This is apparently due to the fact that these methods use only moment
maps and Hamiltonians rather than symplectic structures.
We conclude this section by pointing out that we are not attempting to make
theindefensibleclaimthatall oftheglobaltheoryofHamiltoniantorusactionscan
be reduced to questions in topology. There are many genuinely symplectic results,
inwhichthesymplecticformitself,notjustthetopologicalstructuresonecande(cid:12)ne
fromit,playsafundamentalrole. Forexample,classifyingHamiltonianactionsand
determining whether they admit a Ka(cid:127)hler structure are symplecto-geometric, not
topological, questions. See e.g., [Ka4, KT2, To, Ww]. Finally, there are many
topics that we will not touch on in this book: topics in symplectic topology and
Poissongeometry which address problems of a di(cid:11)erent nature from those that we
will be considering below and which require methods di(cid:11)erent from ours.
2. Hamiltonian cobordism
The Duistermaat{Heckman formula and the index formula involve integration
over the underlying manifold. By Stokes’ formula, the Duistermaat{Heckman in-
tegral and the index are both cobordism invariant, provided that the structures
necessary for producing these invariants extend over the cobording manifold. The
same applies to many other global invariants of Hamiltonian group actions which
are obtained by integrating characteristic classes over the manifold or the reduced
manifold.
Thishasledustoconsider\cobordismtheories"oforientedmanifoldsequipped
with G-actions and equivariant cohomology classes of degree two (or closed equi-
variant two-forms) and, if necessary, G-equivariant stable complex structures,3 see
[GGK1].
Tobespeci(cid:12)c,let(M ;! ;(cid:8) ),forr =0;1,beoriented2n-dimensionalcompact
r r r
HamiltonianG-manifolds.4 Thesemanifoldsaresaidtobecobordant ifthereexists
a compact oriented 2n+1-dimensional Hamiltonian G-manifold with boundary,
(W;!;(cid:8)), such that
(i) @W = M M , and
0 1
(cid:0) t
(ii) i ! =! and i (cid:8)=(cid:8) ,
(cid:3)r r (cid:3)r r
where i : M W is the inclusion map. We call such a cobordism a Hamiltonian
r r
!
cobordism. Inthisde(cid:12)nition,onlytheequivariantcohomologyclasses[! (cid:8) ]mat-
r r
(cid:0)
ter: the same cobordism theory would result from considering manifolds equipped
withequivariantcohomologyclassesratherthantwo-forms. If,inaddition,eachM
r
is equipped with a G-equivariant stable complex structure, we assume that these
structures extend to W.
3A word of warning on terminology: what is traditionally called \symplectic cobordism"
in algebraic topology is cobordisms between pairs(Xr;Jr), for r =0;1, where Xr is a compact
orientedmanifoldandJrisastablereductionofthestructuregroupofthetangentbundleofXrto
thecomplexsymplecticgroup. Thisisrelated,butonlyremotelyso,tothetypeofcobordismwe
areconsideringhere. Thiscobordismisalsodi(cid:11)erentfromsymplecticcobordismbetweencontact
manifolds, which is a partial order rather than an equivalence relation, studied in symplectic
topology.
4If!r issymplectic,theorientationofMr neednotcoincidewiththeorientationde(cid:12)nedby
!n.
r