Moment maps, cobordisms, and Hamiltonian group actions

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 Technology, 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. Topological aspects of Hamiltonian group actions 2. Hamiltonian cobordism 3. The linearization theorem and non-compact cobordisms 4. Abstract moment maps and non-degeneracy 5. The quantum linearization theorem and its applications 6. Acknowledgements

1 1 4 5 7 8 10

Part 1.

13

Cobordism

Chapter 2. Hamiltonian cobordism 1. Hamiltonian group actions 2. Hamiltonian geometry 3. Compact Hamiltonian cobordisms 4. Proper Hamiltonian cobordisms 5. Hamiltonian complex cobordisms

15 15 21 24 27 29

Chapter 3. Abstract moment maps 1. Abstract moment maps: definitions and examples 2. Proper abstract moment maps 3. Cobordism 4. First examples of proper cobordisms 5. Cobordisms of surfaces 6. Cobordisms of linear actions

31 31 33 34 37 39 42

Chapter 4. The linearization theorem 1. The simplest case of the linearization theorem 2. The Hamiltonian linearization theorem 3. The linearization theorem for abstract moment maps 4. Linear torus actions 5. The right-hand side of the linearization theorems 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas

45 45 47 51 52 56 58

Chapter 5. Reduction and applications 1. (Pre-)symplectic reduction 2. Reduction for abstract moment maps 3. The Duistermaat–Heckman theorem 4. K¨ ahler reduction 5. The complex Delzant construction 6. Cobordism of reduced spaces

63 63 65 69 72 73 81

v

vi

CONTENTS

7. Jeffrey–Kirwan localization 8. Cutting Part 2.

Quantization

82 84 87

Chapter 6. Geometric quantization 1. Quantization and group actions 2. Pre-quantization 3. Pre-quantization of reduced spaces 4. Kirillov–Kostant pre-quantization 5. Polarizations, complex structures, and geometric quantization 6. Dolbeault Quantization and the Riemann–Roch formula 7. Stable complex quantization and Spinc quantization 8. Geometric quantization as a push-forward

89 89 90 96 99 102 110 113 117

Chapter 7. The quantum version of the linearization theorem 1. The quantization of Cd 2. Partition functions 3. The character of Q(Cd ) 4. A quantum version of the linearization theorem

119 119 125 130 134

Chapter 8. Quantization commutes with reduction 1. Quantization and reduction commute 2. Quantization of stable complex toric varieties 3. Linearization of [Q,R]=0 4. Straightening the symplectic and complex structures 5. Passing to holomorphic sheaf cohomology 6. Computing global sections; the lit set ˇ 7. The Cech complex 8. The higher cohomology 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 10. Overview of the literature

139 139 141 145 149 150 152 155 157 159 162

Part 3.

165

Appendices

Appendix A. Signs and normalization conventions 1. The representation of G on C∞ (M) 2. The integral weight lattice 3. Connection and curvature for principal torus bundles 4. Curvature and Chern classes 5. Equivariant curvature; integral equivariant cohomology

167 167 168 169 171 172

Appendix B. Proper actions of Lie groups 1. Basic definitions 2. The slice theorem 3. Corollaries of the slice theorem 4. The Mostow–Palais embedding theorem 5. Rigidity of compact group actions

173 173 178 182 189 191

Appendix C. Equivariant cohomology 1. The definition and basic properties of equivariant cohomology

197 197

CONTENTS

2. 3. 4. 5. 6. 7. 8. 9.

Reduction and cohomology Additivity and localization Formality The relation between H∗G and H∗T Equivariant vector bundles and characteristic classes The Atiyah–Bott–Berline–Vergne localization formula Applications of the Atiyah–Bott–Berline–Vergne localization formula Equivariant homology

vii

201 203 205 208 211 217 222 226

Appendix D. Stable complex and Spinc -structures 1. Stable complex structures 2. Spinc -structures 3. Spinc -structures and stable complex structures

229 229 238 248

Appendix E. Assignments and abstract moment maps 1. Existence of abstract moment maps 2. Exact moment maps 3. Hamiltonian moment maps 4. Abstract moment maps on linear spaces are exact 5. Formal cobordism of Hamiltonian spaces

257 257 263 265 269 273

Appendix F. Assignment cohomology 1. Construction of assignment cohomology 2. Assignments with other coefficients 3. Assignment cohomology for pairs 4. Examples of calculations of assignment cohomology 5. Generalizations of assignment cohomology

279 279 281 283 285 287

Appendix G. Non-degenerate abstract moment maps 1. Definitions and basic examples 2. Global properties of non-degenerate abstract moment maps 3. Existence of non-degenerate two-forms

289 289 290 294

Appendix H. 1. 2. 3. 4. 5.

Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Hamiltonian cobordism ring and characteristic classes Characteristic numbers Characteristic numbers as a full system of invariants Non-degenerate cobordisms Geometric quantization

301 301 304 305 308 310

Appendix I. The Kawasaki Riemann–Roch formula 1. Todd classes 2. The Equivariant Riemann–Roch Theorem 3. The Kawasaki Riemann–Roch formula I: finite abelian quotients 4. The Kawasaki Riemann-Roch formula II: torus quotients

315 315 316 320 323

Appendix J. Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 1. The SpinC -Dirac operator and the SpinC -quantization 2. The summary of the results

327 327 329

viii

CONTENTS

3. 4. 5. 6.

Transversally elliptic operators and their indexes Index of the operator Ba The model operator Proof of Theorem 1

331 333 335 336

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 define and yet are of great complexity and depth. This complexity manifests itself in a variety of ways. For example the Darboux theorem is one of the most elementary theorems of symplectic geometry, but finding Darboux coordinates for a particular symplectic structure (for instance, a coadjoint orbit) can give rise to intricate problems in linear algebra and ordinary differential equations. To deal with this complexity one is often forced to narrow one’s focus and concentrate on specific aspects of these objects, for example, invariance 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, our focus will be on global properties of Hamiltonian torus actions and their connection with topology. 1.1. Global invariants of Hamiltonian group actions. Over the course of the last twenty years it has gradually become clear that this topic, global properties of Hamiltonian group actions, has more to do with topology, and less to do with symplectic geometry, than was previously realized. The first inkling of this came from the Duistermaat–Heckman theorem, [DH1]. This theorem states that the oscillatory integral for the moment map of a torus action on a symplectic manifold is exactly equal to the leading term of its asymptotic expansion. Hence, all terms of higher order in 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 fixed points of the action. More explicitly consider an action of the circle, G, on a compact symplectic manifold (M 2n , ω) with isolated fixed points and moment map Φ : M → R. Then the Duistermaat– Heckman formula reads  n Z X eΦ(p) 1 1 Q eΦ ω n = . αj,p n! 2π M G p∈M

Here M G is the fixed point set and α1,p , . . . , αn,p are the weights of the linearized action of G on Tp M at a fixed point p. Although the Duistermaat–Heckman theorem was originally proved in the symplectic setting, it was soon discovered by Berline and Vergne and Atiyah and Bott, [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 topological nature, equates the integral of an equivariant cohomology class over M and the sum of the integrals of this class over the components of the fixed point set, 1

2

1. INTRODUCTION

with corrections coming from the actions on the normal bundles to the components. Explicitly, Z X Z u|F . u= M F e(VF ) F

Here u is an equivariant cohomology class on M , the summation runs over the components F of the fixed point set, and e(VF ) is the equivariant Euler class of the normal bundle to F . When applied to the class u = e−(ω−Φ) , the localization 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 associated with symplectic structures. The Atiyah–Singer index 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 compact group action, this index becomes a virtual representation and its character can be evaluated as the integral of an equivariant cohomology class. The geometric quantization Q(M, ω) is defined as the virtual representation, i.e., index, of the Dirac operator. In this case, the index theorem takes the form Z Q(M, ω) = eω/2π Td (M ), M

where Td (M ) is the Todd class of M . In the equivariant situation Td (M ) should be replaced by the equivariant Todd class and ω by the equivariant symplectic form ω−Φ, where Φ is the moment map. Note that in this formula the integrand depends 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 class is actually just an invariant of the “stable complex structure” associated with this almost complex structure.1 Moreover, a cohomology class, a stable complex structure, and an orientation are sufficient to define a Dirac operator with the “correct” index. A variant of this is “Spinc quantization”, which only depends on the cohomology class and the orientation. When the group acting on the symplectic manifold is abelian, the index is determined by the fixed point data. An explicit formula, the Atiyah–Bott–Lefschetz fixed 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. Various versions of this theorem, often referred to as [Q,R]=0, have been proved in the last decade. (See Section 10 of Chapter 8 for a detailed survey and references.) 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 requires more. Namely, we need a moment map. We get that from an equivariantly closed two-form. (Recall that an equivariant two-form is a pair consisting of an invariant two-form and an equivariant map to g∗ . This form is equivariantly closed if and only if the two-form is closed and the map satisfies Hamilton’s equation. The

1 Since we will need this class in situations when M is not symplectic or not even–dimensional, we recall that a stable complex structure is just a complex structure on the sum T M ⊕ R k for some k.

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 fixed points are isolated. A fourth example is the non-abelian localization theorem of Jeffrey and Kirwan, [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 fixed points of the action.2 For G a torus, a topological interpretation of the Jeffrey– Kirwan theorem can be found, for example, in [GGK1]. 1.2. Geometry of Hamiltonian group actions as a branch of equivariant topology. To summarize, the four theorems which we described above are theorems about objects which one encounters in symplectic geometry but are basically 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 unconventional structures. The choice of structures depends on the kind of question one is asking, but some common traits are already clear. For the Duistermaat– Heckman formula, it is sufficient to consider manifolds equipped with an equivariant cohomology class 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 cohomology class by a closed equivariant two-form. In other words, the structure considered now is a triple (M, ω, Φ) consisting of an oriented G-manifold (not necessarily of even dimension), a G-invariant closed two-form ω (not necessarily of maximal rank), and an equivariant moment map Φ, such that ω and Φ are related by the Hamilton equations (which means that the formal difference ω −Φ is a closed 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 refinement, which allows one to separate the moment map from the cohomology class. Note that the cohomological data and the stable complex data reflect the two different roles that a symplectic form plays: a closed form determines a cohomology class, and a non-degenerate form determines an almost complex (and hence, stable complex) structure. As a result of separating the two roles, a considerable amount of information is lost, but the setting becomes simpler. There is one more implication of non-degeneracy which we have entirely ignored so far. Namely, the non-degeneracy of a symplectic form implies a nondegeneracy condition on the moment map. This condition can also be looked at from a topological point 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 reduction, [ACG, BL, SL]. The local questions are closely connected with (and 2 In spite of its name, the Jeffrey–Kirwan non-abelian localization theorem is not an extension of the Berline–Vergne–Atiyah–Bott abelian localization to non-abelian groups.

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 efficient 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 the indefensible claim that all of the global theory of Hamiltonian torus actions can be reduced to questions in topology. There are many genuinely symplectic results, in which the symplectic form itself, not just the topological structures one can define from it, plays a fundamental role. For example, classifying Hamiltonian actions and determining whether they admit a K¨ ahler 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 Poisson geometry which address problems of a different nature from those that we will be considering below and which require methods different 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 integral 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. This has led us to consider “cobordism theories” of oriented manifolds equipped with G-actions and equivariant cohomology classes of degree two (or closed equivariant two-forms) and, if necessary, G-equivariant stable complex structures, 3 see [GGK1]. To be specific, let (Mr , ωr , Φr ), for r = 0, 1, be oriented 2n-dimensional compact Hamiltonian G-manifolds.4 These manifolds are said to be cobordant if there exists a compact oriented 2n + 1-dimensional Hamiltonian G-manifold with boundary, (W, ω, Φ), such that (i) ∂W = −M0 t M1 , and (ii) i∗r ω = ωr and i∗r Φ = Φr , where ir : Mr → W is the inclusion map. We call such a cobordism a Hamiltonian cobordism. In this definition, only the equivariant cohomology classes [ωr −Φr ] matter: the same cobordism theory would result from considering manifolds equipped with equivariant cohomology classes rather than two-forms. If, in addition, each M r is equipped with a G-equivariant stable complex structure, we assume that these structures extend to W . 3 A 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 oriented manifold and Jr is a stable reduction of the structure group of the tangent bundle of Xr to the complex symplectic group. This is related, but only remotely so, to the type of cobordism we are considering here. This cobordism is also different from symplectic cobordism between contact manifolds, which is a partial order rather than an equivalence relation, studied in symplectic topology. 4 If ω is symplectic, the orientation of M need not coincide with the orientation defined by r r ωrn .

3. THE LINEARIZATION THEOREM AND NON-COMPACT COBORDISMS

5

Now it is clear that the Duistermaat–Heckman integral and the equivariant index are invariants of the cobordism. The same is true for any other invariant obtained by integrating the product of equivariant Chern classes with the equivariant cohomology class of the two-form. We refer to these invariants as mixed characteristic numbers. These are the invariants we are primarily concerned with in this book. Note that mixed characteristic numbers are not really numbers but elements of H ∗ (BG), i.e., polynomials in dim G variables. This cobordism theory was studied in [GGK1] and applied to the problems that we discussed in Section 1. Additionally, we proved in [GGK1] that “cobordism commutes with reduction”, i.e., that if Hamiltonian G-manifolds are cobordant then so are the reduced spaces, with the induced structures. (Since a regular reduced space is only an orbifold, the cobordism involved here is an orbifold cobordism, cf. [Dr].) Having defined Hamiltonian cobordisms we face the problem of calculating the Hamiltonian cobordism ring. This problem is tangential to the main purpose of this book and we defer its detailed discussion to Appendix H. We note that the results of [GGK1] were obtained simultaneously with the development of Shaun Martin’s cobordism technique [Mart1] to study the cohomology ring of reduced spaces. 3. The linearization theorem and non-compact cobordisms 3.1. The linearization theorem. In this section we will focus on the case where G is a torus. The fixed point data for a G-manifold M are the fixed point set M G and the linearized G-action on the normal bundle VM G to M G . If M is equipped with an equivariant stable complex structure, M G inherits this structure and the normal bundle is a genuine complex bundle with the G-action preserving the complex structure. In addition, when M is a Hamiltonian G-manifold, the restriction of the equivariant cohomology class to M G is also assumed to be a part of the fixed point data. The Berline–Vergne–Atiyah–Bott localization theorem ensures that every equivariant characteristic number of a (stable complex, Hamiltonian) G-manifold can be expressed in terms of the fixed point data. These characteristic numbers are also invariants of cobordism and, conjecturally, determine the cobordism class. Thus one may expect that the cobordism class of M is determined by the fixed point data. This is indeed true, as was shown by Gusein-Zade in the early 70s, [GZ1]. This fact can be expressed more succinctly as asserting that M is cobordant to its fixed point data: Theorem 1.1 (Linearization Theorem, I). Let G be a torus, let M be a compact (stable complex) Hamiltonian G-manifold, and let Fi , i = 1, . . . , k, be the connected components of the fixed point set M G . Then M is equivariantly cobordant to the disjoint union of the normal bundles to Fi : G (1.1) M∼ VF i . In particular, if M G is finite, M is equivariantly cobordant to the disjoint union of the tangent spaces Tp M , for p ∈ M G .

A word of warning: For this statement to be meaningful a somewhat unorthodox definition of the symbol “∼” is required. Indeed, the right-hand side of (1.1) is

6

1. INTRODUCTION

non-compact and the introduction of non-compact objects into a cobordism theory usually has a trivializing effect. For example, every compact manifold M is cobordant to the empty set via the non-compact cobordism M × [0, 1). This difficulty is fundamental and not easy to fix. For instance, one might think that one can get around this difficulty by equipping manifolds and cobordisms with proper functions. However, this does not solve the problem because a pair (M, f ) consisting of a compact manifold and a proper function is cobordant to the empty set via the cobordism (M × [0, 1), f + 1/(1 − t)2 ), where t ∈ [0, 1).5 Thus we have to impose additional constraints to make non-compact cobordisms non-trivial and thus make sense of the right-hand side of (1.1). The rest of this section is devoted to doing this. The first version of Theorem 1.1 was proved in [GGK1]. In that version we used the moment map on M and symplectic cutting, [Ler1], to compactify the spaces VFi or, to be more accurate, find compact approximations of these spaces. When the fixed points are isolated, this version of the theorem asserts that M is cobordant to a union of certain twisted projective spaces associated with the fixed point data. Here by a twisted projective space we mean the quotient of the sphere by an action of S 1 with arbitrary non-zero weights. (Note also that a similar theorem is obtained in [Mart1].) Thus in [GGK1], (1.1) is not yet a literal identity. The cobordism is still a cobordism of Hamiltonian stable complex compact G-orbifolds. Notice also that if n f is one of the twisted projective spaces occurring in this cobordism, it comes CP equipped with a stable complex structure which can be incompatible with its natural K¨ ahler structure (even if M is symplectic). This shows that stable complex structures in our theory are a matter not of convenience but of necessity. The first version of Theorem 1.1 in which (1.1) becomes a literal cobordism appeared in [Ka3] and involved abstract moment maps. This notion we will describe in Section 4. First, however, we outline a different way to make rigorous sense of (1.1). 3.2. Non-compact Hamiltonian cobordism. Two Hamiltonian G-manifolds with proper moment maps are properly cobordant if there exists a Hamiltonian cobordism between them such that the moment map on the cobording manifold is proper. Clearly, we have a map from the set of compact cobordism classes to the set of proper cobordism classes. For manifolds with stable complex structures, this map is one-to-one; see [GGK2].6 Hence by passing from compact to proper cobordisms we lose no information about compact cobordism classes. Now we are in a position to state a rigorous version of Theorem 1.1. For the sake of simplicity we restrict our attention to the case of isolated fixed points. Theorem 1.2 (Linearization Theorem, II). Let G be a torus and let (M, ω, Φ) be a compact (stable complex) Hamiltonian G-manifold with isolated fixed points. Then there exist equivariant closed forms ωp − Φp on Tp M for p ∈ M G with Φp 5 Although in such a theory all compact manifolds are cobordant to zero, the theory does detect the ends of a non-compact manifold. Thus the theory is not completely trivial. 6 Strictly speaking the result of [GGK2] concerns stable complex cobordisms with abstract moment maps. However, it is not hard to modify the proof to obtain the Hamiltonian version of this result.

4. ABSTRACT MOMENT MAPS AND NON-DEGENERACY

7

proper (and equivariant stable complex structures) such that (M, ω, Φ) is properly cobordant to the disjoint union of (Tp M, ωp , Φp ). The forms ωp can always be chosen symplectic. However, even when ω is symplectic, the stable complex structure on Tp M induced by ω may fail to be compatible with ωp . We prove several versions of the linearization theorem in Chapter 4. Finally, we note that cobordism techniques discussed here are used in [MW2, MW3] to study Hamiltonian loop group actions and moduli spaces of flat connections. 3.3. Reduction and cobordism. The “cobordism commutes with reduction” principle asserts that reductions of cobordant manifolds are cobordant. Combined with the linearization theorem this implies that the reduced space Mred at a regular value of the moment map is cobordant in the class of orbifolds to a union of toric orbifolds, provided that the fixed points are isolated: G Mred ∼ (Tp M )red . p∈M G

We emphasize here that these toric orbifolds (Tp M )red may carry non-standard stable complex structures even when M is symplectic. 4. Abstract moment maps and non-degeneracy 4.1. Abstract moment maps. An abstract moment map is an equivariant mapping M → g∗ which has some of the formal properties of genuine moment maps but without a two-form. More explicitly, let M be a G-manifold and Φ a G-equivariant mapping of M into g∗ . For a closed subgroup H of G, let h be the Lie algebra of H and ΦH : M → h∗ the composition of Φ and the natural projection g∗ → h∗ . Note that if Φ is a genuine moment map, ΦH is the moment map for the H-action on M . Hence, ΦH is constant on the connected components of M H . This property of genuine moment maps is taken as the definition of an abstract moment map. Definition 1.3. An equivariant map Φ : M → g∗ is an abstract moment map if for every closed subgroup H of G, the map ΦH is constant on connected components of M H . We emphasize that this definition does not require M to be equipped with a two-form. However, as we showed in [GGK3], if G is a torus and Φ an abstract moment map, there always exists, at least locally, an equivariant closed two-form whose moment component is Φ. We also gave a necessary and sufficient condition for the existence of a global such two-form. As we have pointed out above, abstract moment maps were introduced in [Ka3] to make the statement of the linearization theorem precise. Namely, consider oriented G-manifolds equipped with proper abstract moment maps and, if necessary, G-equivariant stable complex structures and/or equivariant cohomology classes of degree two. A proper cobordism between two such manifolds is defined in the obvious way: the moment map on the cobording manifold is required to be proper. We refer to this cobordism as a cobordism of proper abstract moment maps. The linearization Theorem 1.2 holds with proper cobordisms of Hamiltonian G-manifolds replaced by cobordisms of this type.

8

1. INTRODUCTION

The main advantage of working with abstract moment maps rather than with Hamiltonian G-manifolds is that the moment map is no longer attached to the equivariant cohomology class. One can still carry out reduction at a regular value of an abstract moment map. Moreover, an equivariant cohomology class induces an ordinary cohomology class on the reduced space, and an equivariant stable complex structure descends to a stable complex structure on the reduced space. Thus, the “cobordism commutes with reduction” principle still holds for proper cobordism of abstract moment maps. Examples of abstract moment maps have come up in other contexts in symplectic geometry not related to cobordisms or global properties of Hamiltonian group actions. For instance, the locked momentum map introduced by Lewis et al, [Lew, SLM], in geometric mechanics in the early nineties is an abstract moment map in our sense, i.e., it is not associated with a symplectic form; see Example 3.11. Abstract moment maps are implicitly used in [LT2] in intersection cohomology calculations. 4.2. Non-degenerate abstract moment maps. A moment map associated with a genuine symplectic form satisfies an additional non-degeneracy condition which comes from the non-degeneracy of the symplectic form. We say that an abstract moment map Φ : M → g∗ is non-degenerate if for every ξ ∈ g the ξcomponent Φξ is a Morse–Bott function and the critical set of Φξ is exactly the zero set of the vector field ξM induced on M by ξ. It turns out that when G is a torus this condition is necessary and sufficient for Φ to be locally associated with a symplectic form; see Appendix G. The well-known convexity theorem asserts that when G is a torus and M is compact and symplectic, the moment map image Φ(M ) is a convex polytope and the level sets of Φ are connected, [At3, GS2]. The proof of this theorem is purely Morse–theoretic and relies on the fact that moment maps for symplectic forms are non-degenerate. Hence the convexity theorem also holds for non-degenerate abstract moment maps. Likewise, the Kirwan theorem (which asserts that a symplectic Hamiltonian G-manifold is equivariantly formal, i.e., ∗ HG (M ) = H ∗ (M ) ⊗ H ∗ (BG)) is also true for manifolds with non-degenerate abstract moment maps since formality is a consequence of the Morse–theoretic properties described above. This circle of questions is discussed in detail in Appendix G. Furthermore, it is reasonable to expect that some of the singular reduction theorems, [BL, SL], will have counterparts for manifolds with non-degenerate abstract moment maps. For example, it appears to be true that Φ−1 (a)/G is a stratified space for all a ∈ g∗ . We conclude this section by pointing out that in the non-abelian case the theory of abstract moment maps is entirely undeveloped (see, however, [Brad]) and it is not clear at all which of the results discussed above can be generalized to actions of non-abelian groups. 5. The quantum linearization theorem and its applications 5.1. The quantum linearization theorem. The linearization theorem says that a compact Hamiltonian manifold M with isolated fixed points is cobordant to the disjoint union of the tangent spaces Tp M at the fixed points p of the action.

5. THE QUANTUM LINEARIZATION THEOREM AND ITS APPLICATIONS

9

The quantum analogue of the linearization theorem simply asserts that the geometric quantization Q(M ) is equal, as a virtual representation, to the sum of the quantizations of the tangent spaces: X Q(Tp M ). Q(M ) = p∈M G

Here, as in the linearization theorem, M is equipped with an equivariant two-form and an equivariant stable complex structure. The tangent spaces Tp M are also equipped with such structures, inherited from M as in Theorem 1.2. Since the linear spaces Tp M are not compact, extra care should be exercised when their quantizations are defined. Here we adopt the definition of Q(Tp M ) using L2 -Dolbeault cohomology as in, for example, [Man1, Man2]. With this definition, every irreducible representation of G occurs in Q(Tp M ) with finite multiplicity. The quantum linearization theorem asserts that the sum of these multiplicities over all p ∈ M G is equal to the multiplicity with which the representation occurs in Q(M ). As stated, the quantum linearization theorem is a simple consequence of the Atiyah–Bott Lefschetz theorem. A more general version of the quantum linearization theorem is proved by Braverman, [Brav4]. Braverman’s theorem does not require M to be compact: the theorem applies, for example, to a stable complex G-manifold with finite fixed point set and proper abstract moment map. 5.2. Quantization commutes with reduction. As an application of cobordism techniques we will give in Chapter 8 a detailed proof of the “quantization commutes with reduction” theorem in the case where G is a torus and the fixed points are isolated. We prove two versions of this theorem. One version is purely topological. To state it, consider a stable complex compact Hamiltonian G-manifold M and let α be an integral regular value of the moment map Φ : M → g∗ . Then the theorem asserts that Q(M )α = Q(Mα ). Here Q(M )α is the multiplicity with which α occurs in Q(M ) and Mα is the reduction of M at α. The second version of the theorem is symplectic. In this version we do not require α to be regular but M is then required to be symplectic; the quantization of Mα is defined using a suitable desingularization. The proofs of both of these theorems follow the same path. The idea is to reduce the problem to the case of a linear space by using the cobordism invariance of quantization and the linearization theorem. For the sake of simplicity, we suppose that the fixed points of the action are isolated. To begin with, let us also assume that α is a regular value for the moment maps on all the Tp M s. Then the linearization theorem yields the cobordism G Mα ∼ (Tp M )α . p∈M G

Hence, since the quantization is an invariant of cobordism, X Q(Mα ) = Q((Tp M )α ). p∈M G

10

1. INTRODUCTION

On the other hand, the quantum linearization theorem asserts that X Q(M ) = Q(Tp M ). p∈M G

Thus it suffices to show that quantization commutes with reduction for the linear spaces Tp M : Q(Tp M )α = Q((Tp M )α ).

The quantization of a linear space is well-understood and an explicit expression for Q(Tp M )α is given in terms of a partition function. The reduced space (Tp M )α is a toric orbifold equipped with its standard symplectic structure and a stable complex structure. These structures are not necessarily compatible with each other, and one of the main complications in the proof will be dealing with this incompatibility. We show that Q((Tp M )α ) is equal to the quantization of (Tp M )α with respect to the standard complex structure and a new two-form ω obtained from the standard symplectic form by a certain shift. Hence, X Q((Tp M )α ) = (−1)k dim H 0,k ((Tp M )α ; L),

where L is a holomorphic line bundle over (Tp M )α whose first Chern class is [ω]. (The cohomology spaces on the right-hand side are computed in [Od] for an arbitrary line bundle L.) The first term H 0,0 ((Tp M )α ; L) is just the space of global holomorphic sections of L. These can be identified with those holomorphic functions on a certain open and dense subset W ⊂ Tp M which transform correctly under the torus action. We calculate dim H 0,0 ((Tp M )α ; L) combinatorially, in terms of partition functions, and conclude that this dimension is exactly Q(Tp M )α . The next step is to show that dim H 0,k ((Tp M )α ; L) = 0 if k > 0; this is one of the complications we alluded to above. (Notice that this result is akin to the Kodaira vanishing theorem.) We argue combinatorially again. Roughly speaking, the cohomology in question is identified with the relative simplicial cohomology of a pair (∆, L), where ∆ is the moment polytope and L is a contractible subset of its boundary, and by elementary topology this cohomology vanishes. When α is regular for M but not necessarily for the tangent spaces Tp M , we perturb the moment map so that the above construction goes through. (The “right” perturbation is obtained by shifting Φ by a small abstract moment map rather than by a constant.) A more general version of the “quantization commutes with reduction” theorem for non-compact manifolds is obtained by Braverman in [Brav4]. 6. Acknowledgements The first version of this monograph was a set of lecture notes for the Rademacher lectures at the University of Pennsylvania in the spring of 1995, and we thank Alexander Kirillov for the suggestion of making these notes into an AMS monograph. This version was the first of many, and we are deeply grateful to Sergei Gelfand for shepherding this project through its several stages of near completion and allowing us the latitude of innumerable rewrites and alterations. During the early stages of this work we had a fruitful exchange of ideas with Shaun Martin. Others who have helped significantly in numerous discussions are

6. ACKNOWLEDGEMENTS

11

Sue Tolman (several pivotal ideas in the combinatorial part of the [Q,R]=0 proof are hers), Moshe Baruch, Phil Bradley, Maxim Braverman, Ana Cannas da Silva, Emmanuel Farjoun, Megumi Harada, Allen Knutson, Eugene Lerman, Debra Lewis, Assaf Libman, Eckhard Meinrenken, David Metzler, Paul-Emile Paradan, Elisa Prato, Ofer Ron, Yuli Rudyak, Reyer Sjamaar, Shlomo Sternberg, Avishay Vaaknin, Michele Vergne, Jonathan Weitsman, Chris Woodward, Siye Wu, and Moshe Zadka. We would like to express our deepest gratitude to each and every one of them. We are deeply grateful to Maxim Braverman, who wrote Appendix J, for making this generous contribution to the book. Yael Karshon adds: I am greatly indebted to Dror B.-N. for his partnership which enables my mathematical work; I thank him with all my heart. Sue Tolman’s inspiring mathematical collaboration has had a huge impact on my contribution to this book. Finally, it was my late grandfather Refa’el Saporta, master of rhymes, from whom I learned to appreciate writing. The financial support of the National Science Foundation and of the United States Israel Binational Science Foundation, grant number 96–210, have been crucial to our collaboration.

Part 1

Cobordism

CHAPTER 2

Hamiltonian cobordism 1. Hamiltonian group actions 1.1. Definitions. Let G be a Lie group and g its Lie algebra. A smooth G action on a manifold M gives rise to a Lie algebra anti-homomorphism, g → Vect(M ),

associating to a Lie algebra element ξ ∈ g the vector field ξM that generates the action on M of the one-parameter subgroup {exp(tξ) | t ∈ R} of G. Let ω be a closed G-invariant two-form on M . A moment map is a smooth map Φ : M → g∗ that is equivariant with respect to the G-action on M and the coadjoint action on g∗ and such that the components Φξ = hΦ, ξi satisfy Hamilton’s equation, (2.1)

dΦξ = ι(ξM )ω,

for all ξ ∈ g. In other words, for every tangent vector η we have ηΦξ = ω(ξM , η), where we let η act as a derivation on the function Φξ . When G is the circle group S 1 , we identify g and g∗ with R, and the mo∂ . (See Appendix A for our ment map equation becomes dΦ = ι(ξM )ω for ξ = ∂θ conventions.) Example 2.1. Consider the plane R2 with the standard symplectic form dx ∧ dy = rdr ∧ dθ. Let the circle group S 1 act by rotations, with generating vector field ∂ ∂ ∂ x ∂y − y ∂x = ∂θ . Then − 21 (x2 + y 2 ) = − 21 r2 is a moment map. More generally, on P n 2 n 1 C = (R ) with ω = dxP i ∧ dyi , the S -action with weights P m1 , .2. . , m2n ∈ Z has ∂ the generating vector field mj ∂θj and moment map − 21 mj (xj + yj ).

A G-action on (M, ω) is called Hamiltonian if it admits a moment map Φ. The moment map is then determined by (2.1) uniquely up to translation. A Hamiltonian

Figure 2.1. Rotations of the plane 15

16

2. HAMILTONIAN COBORDISM

G-manifold is a triple (M, ω, Φ), where M is an oriented manifold with a G action, ω is a closed two-form, and Φ : M → g∗ is a moment map. Example 2.2. Consider the torus S 1 × S 1 = R2 /2πZ2 with the symplectic form induced from R2 . Let the circle group S 1 act by rotations of the first factor. This action is not Hamiltonian: in the periodic coordinates (x mod 2π) and (y ∂ )dx ∧ dy = dy, which has no mod 2π), the moment map equation is dΦ = ι( ∂x global periodic solution. 1.2. Hamiltonian mechanics. The two-form ω is often assumed to be symplectic, i.e., not only closed but also non-degenerate. If ω is non-degenerate, the vector fields ξM (and hence the action, if G is connected) are determined by the moment map through (2.1). Non-degeneracy implies that the dimension of M is even, and is equivalent to the condition that the top wedge product ω n , where n = 21 dim M , never vanishes. This induces an orientation on M . The Liouville measure is defined by the integration of the volume form ω n /n! with respect to the symplectic orientation. For example, the standard symplectic form (2.2)

ωstd =

n X i=1

dxi ∧ dyi

n on R2n is non-degenerate, and ωstd = n!dx1 ∧ dy1 ∧ . . . ∧ dxn ∧ dyn . Locally, any symplectic form can be brought to the form (2.2) by an appropriate choice of local coordinates by Darboux’s theorem. (See [McDSa, Section 3.2].) Writing xi = qi and yi = pi , the moment map equation (2.1) becomes Hamilton’s equations from mechanics: the vector field ξM generates the flow that solves the differential equations

q˙i = ∂Φξ /∂pi

(2.3)

and p˙i = −∂Φξ /∂qi .

When Φξ is replaced by the classical Hamiltonian 1 2 p + U (q) = kinetic + potential energy, 2m the equations below become the time flow in phase space: H =

q˙ =

p m

and

p˙ = −

∂U . ∂q

More generally, the phase space of a classical mechanical system is modeled by a symplectic manifold (M, ω), and the dynamics of the system is the R-action on M whose moment map is the Hamiltonian (total energy) function H ∈ C ∞ (M ). The kinematics of the system, namely, the “shape” of phase space, often involve symmetries, which mathematically are given by a proper group action (see Appendix B). The “conjugate momenta” for these symmetries are no other than the moment map coordinates for this action. Noether’s theorem asserts that these quantities are conserved by the time flow if the Hamiltonian is preserved by the symmetry. In this formulation, the proof is immediate: if ηM is the vector field that generates the time flow, so that ω(ηM ) = dH(·), then dΦξ = ηM Φξ = ω(ξM , ηM ) = −ξM H = 0. dt

1. HAMILTONIAN GROUP ACTIONS

17

Example 2.3. For N particles in R3 of masses m1 , . . . , mN , the velocity phase space is T (R3N ) = R6N , with position coordinates xi , y i , z i , and velocity coordinates x˙ i , y˙ i , z˙ i , and with a symplectic form ω=

N X i=1

mi (dxi ∧ dx˙ i + dy i ∧ dy˙ i + dz i ∧ dz˙ i ).

The group R acts by translations with generating vector fields P ∂ ∂z i , and moment map 3

Φ(x1 , . . . , z˙N ) =

N X

P

∂ ∂xi ,

P

∂ ∂y i ,

and

mi (x˙ i , y˙ i , z˙ i ),

i=1

which associates to each state of the system its total linear momentum. The group SO(3) acts by rotations with generating vector fields ξ1 , ξ2 , ξ3 , where ξ1 = PN i i i i i i i i i=1 y ∂/∂z − z ∂/∂y + y˙ ∂/∂ z˙ − z˙ ∂/∂ y˙ , etc., and moment map Φ=

N X i=1


mi y i z˙ i − z i y˙ i , z i x˙ i − xi z˙ i , xi y˙ i − y i x˙ i ,

which associates to each state of the system its total angular momentum. 1.3. The Duistermaat–Heckman measure. In this book we do not assume that ω is symplectic. When we write a Hamiltonian G-manifold as a triple (M, ω, Φ), we assume that the orientation and G-action on M are also given. Even if ω happens to be non-degenerate, we do not insist that it be compatible with the orientation on M . We define the Liouville measure, as before, by integrating the form ω n /n! with respect to the given orientation; this is now a signed measure. Definition 2.4. The Duistermaat–Heckman measure on g∗ is the push-forward of Liouville measure by the moment map. We denote it DH M . Remark 2.5. It is convenient to work with the differential form (of mixed degree) 1 1 exp ω = 1 + ω + ω ∧ ω + ω ∧ ω ∧ ω + . . . . 2! 3! R With the convention that M β = 0 if the degree of β is different than the dimension of M , Liouville measure is given by integration of exp ω.

Example 2.6. Take the unit sphere S 2 = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} with its standard area form. Let the circle group act by rotations, fixing the north and south poles. The moment map is the height function Φ(x, y, z) = z. The Duistermaat–Heckman measure is 2π times Lebesgue measure on the interval [−1, 1]. This amounts to Archimedes’ observation, that the area of the strip {(x, y, z) ∈ S 2 | a ≤ z ≤ b} is 2π(b − a), assuming −1 ≤ a < b ≤ 1. If we fix the height difference b − a, this can be a short but wide strip near a pole of S 2 , or it can be a thin but long strip near the equator. Equivalently, the standard area form on S 2 is given in cylindrical coordinates by ω = dθ ∧ dz. In some contexts, one can forget the two-form and just keep track of the action and the moment map, or, more specifically, the moment map values at the fixed points. For instance, if M is compact and G is a torus acting with isolated fixed

18

2. HAMILTONIAN COBORDISM

→

Figure 2.2. Rotations of S 2 and moment map points, the moment map values at the fixed points determine global invariants such as the Duistermaat–Heckman measure (see Chapter 4) and the geometric quantization (see Chapter 6). The following important example illustrates the role of fixed points: Example 2.7. Let ω be a rotation-invariant two-form on S 2 . The Duistermaat– Heckman measure is 2π times Lebesgue measure on the interval [Φ(q), Φ(p)], where p and q are the north and south poles and Φ is a moment map. In particular, the total area is 2π(Φ(p) − Φ(q)). We leave the proof to the reader as an exercise. 1.4. Existence of moment maps. We turn to the question of whether a moment map exists. Example 2.8 (Exact moment maps). Let ω = −dµ, where µ is a G-invariant one-form. Then the function Φ : M → g∗ defined by Φξ = µ(ξM ) is a moment map. Proof. Because µ is invariant, its Lie derivative along ξM vanishes, LξM µ = 0, for all ξ ∈ g. By Cartan’s homotopy formula for the Lie derivative, LξM = dι(ξM )+ ι(ξM )d, this implies the moment map equation d (ι(ξM )µ) = −ι(ξM )dµ. In particular, if ω is exact and invariant and G is compact, a moment map always exists. (The reason is that there always exists µ as in Example 2.8; see Corollary B.13.) What happens if we remove the exactness assumption? Just the fact that the two-form ω is closed and invariant implies that the oneforms ι(ξM )ω are closed. Indeed, by Cartan’s formula, dι(ξM )ω = (dι(ξM ) + ι(ξM )d) ω = LξM ω = 0. If the one-forms ι(ξM )ω are exact, one can choose functions Φξ that satisfy the moment map equation (2.1) for basis elements ξ of g and take their linear combinations. Therefore, if we ignore the equivariance requirement, a moment map always exists locally, and the obstruction for it to exist globally lies in the first de Rham cohomology of M . We will be particularly interested in torus actions. For these, the equivariance condition amounts to the moment map being invariant (because the coadjoint action is trivial), and this condition automatically follows from Hamilton’s equation (2.1): Proposition 2.9. Suppose that G is a torus and Φ : M → g∗ satisfies dΦξ = ι(ξM )ω for all ξ ∈ g. Then Φ is G-invariant.

1. HAMILTONIAN GROUP ACTIONS

19

Proof. The tangent space at p to the orbit G · p is {ξM |p | ξ ∈ g}. Hence, it is enough to show that LξM Φη = 0 for all ξ, η ∈ g. Let us first show that LξM Φη is constant along an orbit. For this we need to show that LζM LξM Φη = 0 for all ζ ∈ g. Indeed, L ζM L ξM Φ η

= LζM ι(ξM )dΦη = LζM ι(ξM )ι(ηM )ω = LζM ω(ηM , ξM ) = (LζM ω)(ηM , ξM ) + ω([ζM , ηM ], ξM ) + ω(ηM , [ζM , ξM ]),

where the last equality is Leibniz’s rule. All three summands are zero. This shows that LξM Φη is constant on each orbit. By compactness, each orbit contains a point where Φη is maximal. At this point, the derivative of Φη at any direction along the orbit must be zero. In particular, LξM Φη = 0. In general, the obstruction to the existence of a moment map lies in the Lie algebra cohomology. This implies that a moment map always exists if G is semisimple, even when H 1 (M ) 6= 0. Concretely, the obstruction to the existence of a map Φ satisfying Hamilton’s equation (2.1) is the element of H 1 (g)⊗H 1 (M ) = H 1 (g; H 1 (M )) represented by the cocycle c : ξ → [ι(ξM )ω]. (The fact that this is a cocycle, that is, that c([ξ, η]) = 0, follows from the formula ι([ξM , ζM ]) = LξM ι(ζM ) − ι(ζM )LξM .) If this obstruction is zero, the obstruction to the existence of an equivariant moment map is the element of H 2 (g, R) represented by the cocycle Φ[ξ,η] − {Φξ , Φη } where Φ is any map satisfying (2.1) and {, } is the Poisson bracket (see Section 1.5). See, e.g., [CW, Section 7.2], [Can, Section 2.6], and [Gin4] for the treatment of this question in the more general context of actions on Poisson manifolds, 1.5. The Poisson algebra. If ω is non-degenerate, it associates to each smooth function f ∈ C ∞ (M ) a vector field ξf ∈ Vect(M ) such that df (·) = ω(ξf , ·). The Poisson bracket {f, g} := ξg f

defines a Lie algebra structure on C ∞ (M ), such that the map f 7→ ξf is an antiLie homomorphism from C ∞ (M ) to Vect(M ). If G is connected, the equivariance condition on a moment map Φ : M → g∗ is equivalent to the map (2.4)

g → C ∞ (M ),

ξ 7→ Φξ

being a Lie algebra homomorphism. Let us briefly recall the proofs of these facts. We have ω(ξf , ξg ) = ξg f = {f, g}. This implies that the Poisson bracket is anti-symmetric. We have (2.5)

ξf ω(ξg , ξh ) = ξf {g, h} = {{g, h}, f }

and (2.6)

ω([ξf , ξg ], ξh ) = −[ξf , ξg ]h = −ξf ξg h + ξg ξf h

= −{{h, g}, f } + {{h, f }, g}.

For any three vector fields a, b, and c, we have (2.7)

dω(a, b, c) = aω(b, c) + cyclic permutations − ω([a, b], c) − cyclic permutations .

20

2. HAMILTONIAN COBORDISM

By (2.5), (2.6) and (2.7), = {{g, h}, f } + cyclic permutations + {{h, g}, f } + cyclic permutations − {{h, f }, g} − cyclic permutations = C − C + C,

dω(ξf , ξg , ξh ) (2.8) where

C := {{f, g}, h} + cyclic permutations.

Since ω is closed, (2.8) implies C = 0, so the Poisson bracket satisfies the Jacobi identity. Also, as in (2.6), [ξf , ξg ]h + ξ{f,g} h = ({{h, g}, f } − {{h, f }}, g) + {h, {f, g}} = −C = 0,

showing that f 7→ ξf is an anti-Lie homomorphism. Finally, because {Φξ , Φη } = −ξM Φη , the map (2.4) is a Lie algebra homomorphism if and only if ξM Φη = Φ−[ξ,η] , which exactly means that Φ : M → g∗ is equivariant with respect to the action of the Lie algebra of G (see Section 1.6 of Appendix B). 1.6. Poisson algebras for degenerate two-forms. The notion of the Poisson algebra of a symplectic manifold generalizes to manifolds (M, ω) equipped with closed two-forms. The motivation for this generalization comes from quantization. Given an (integral) symplectic manifold, the Kirillov–Kostant “pre-quantization recipe” produces a “(pre-)quantization” of the entire Poisson algebra C ∞ (M ). Given an (integral) Hamiltonian G-action, this recipe produces a “(pre-)quantization” of the moment map components Φξ , even if the two-form is degenerate. See Chapter 6 for details. An attempt to unite these two notions of “quantization” naturally leads to an infinite-dimensional algebra which is canonically associated to any manifold M and closed two-form ω. We denote this algebra P(M, ω). As a vector space, (2.9)

P(M, ω) = {(f, v) ∈ C ∞ (M ) × Vect(M ) | df = ιv ω}.

Multiplication is defined by (2.10)

(f, v) · (g, u) = (f g, f u + gv).

A Lie bracket is defined by (2.11)

[(f, v), (g, u)] = (Lu f, −[u, v]) .

Note that Lu f = −Lv g = ω(v, u). With these structures, P(M, ω) is a Poisson algebra, in the sense that it is simultaneously a commutative algebra and a Lie algebra and that the Lie bracket is a derivation with respect to the multiplication operation. We leave the details as an exercise to the reader. Example 2.10. When ω is symplectic, P(M, ω) ∼ = C ∞ (M ) via (f, v) 7→ f. When ω = 0, P(M, 0) = R × Vect(M ).

For any ω, there is a short exact sequence

0 → R → P(M, ω) → Ham(M ) → 0,

2. HAMILTONIAN GEOMETRY

21

where Ham(M ) = {v ∈ Vect(M ) | ιv ω is exact }. As in the symplectic case, any (f, v) ∈ P(M, ω) generates a flow on M (through v), and we have the energy conservation law Lv f = 0 and Poincar´e’s integral invariant Lv ω = 0. In fact, the canonical homomorphism P(M, ω) → Vect(M ),

(f, v) 7→ v

gives rise to an infinitesimal P(M, ω)-action on (M, ω). In Section 4 of Chapter 6 we will identify P(M, ω) as the algebra of infinitesi1 [ω] is mal symmetries of a “pre-quantization data” (P, Θ) for (M, ω), assuming 2π integral. Example 2.11. For any Hamiltonian G-manifold (M, ω, Φ), we have a homomorphism of Lie algebras g → P(M, ω),

ξ 7→ (Φξ , ξM ).

Remark 2.12. This construction generalizes to Dirac structures. A Dirac structure is a sub-bundle V of T ∗ M ⊗ T M that satisfies a certain integrability condition. Two forms, Poisson structures, and foliations are special cases of Dirac structures. For a Dirac structure, the Hamilton equation in the definition (2.9) has to be replaced by the condition that (df, v) be a section of V. We leave the details as an exercise to the reader. (See, [Co] for the definition of Dirac structures and [Wei3] for further references.) 2. Hamiltonian geometry Geometry of moment maps for torus actions on symplectic manifolds is rich in beautiful results, some of which we will survey here. Throughout this section, G is a torus acting on a compact symplectic manifold (M, ω) with moment map Φ : M → g∗ . We begin with the convexity theorem of Atiyah, Guillemin, and Sternberg: Theorem. The image Φ(M ) is a convex polytope in the vector space g ∗ . We call Φ(M ) the moment polytope. More explicitly, the fixed point set has finitely many connected components, and the moment map takes a constant value on each such component. The moment map image is the convex hull of these values: (2.12)

Φ(M ) = conv{Φ(p) | p ∈ M G }.

There is a companion theorem of the convexity theorem, involving the set Φ(M )reg of regular values of Φ in Φ(M ). We call its connected components the alcoves of the moment polytope. Theorem. The alcoves are open convex polytopes. A closely related result is the following connectedness theorem: Theorem. The level sets Φ−1 (α), α ∈ g∗ , are connected.

22

2. HAMILTONIAN COBORDISM

Another related result is the stability theorem: Theorem. As a map to Φ(M ), the moment map is open. Equivalently, as p varies within a moment fiber Φ−1 (α), the moment map image of a neighborhood of p remains the same and equal to a small neighborhood of α ∈ g∗ in Φ(M ). This is the reason for the name “stability”. We emphasize that this theorem is not local: it asserts that the moment map can have “corners” and “foldings” only on the boundary of Φ(M ). For example, if G = S 1 , the end-points of Φ(M ) are the only local maxima and minima of Φ. The convexity and connectedness theorems are due to Atiyah and Guillemin and Sternberg. The stability theorem for compact manifolds is proved in [Sj3, Theorem 6.5]. For proper moment maps to open convex sets, see [LMTW] or [CDM]. We will discuss the convexity theorem, as well as other consequences of non-degeneracy, in Appendix G. From the local normal form [GS7, Marl], which completely describes a Hamiltonian space near an orbit, one deduces local versions of the above theorems: a neighborhood of an orbit maps to an open subset of a convex polytope; the level sets are locally connected; the moment map image of a neighborhood of p remains the same as p varies continuously within Φ−1 (α). Then, one argues one way or another that the result remains true globally. For instance, one may employ Morse theory, as in [GS2], using the following result: Theorem. If G = S 1 , the moment map Φ : M → R is a perfect Morse-Bott function. If the fixed points are isolated, Φ is a Morse function. This result, which follows from the local normal form (cf. [GS6, §32]), essentially goes back to Frankel [F]. For G a torus, each component of Φ is Morse-Bott. Note that the critical points of Φ are exactly the fixed points for the action, as follows from (2.1). Alternatively, to prove the convexity and related theorems, one may argue by induction on the dimension of the torus, as in [At3], or one may look for shortest paths in M with respect to the pull-back of a metric on g∗ , as in [CDM]. We note that “patching together” the local results is similar to the following elementary but non-trivial theorem: Let A be a subset of Rn which is closed, connected, and locally convex (every point has a neighborhood U such that A ∩ U is convex). Then A is convex. We state an important special case of the normal form theorem: Theorem. Any fixed point p ∈ M has a neighborhood which is isomorphic to P a neighborhood of the origin in Cn , with the standard symplectic form dxi ∧ dyi , and with the torus acting through an inclusion into (S 1 )n . This theorem is a version of the equivariant Darboux theorem (see, e.g., [GS6]) and can be proved by the standard Moser’sP homotopy argument. |zi |2 αi,p where −αi,p ∈ Z∗G are the The moment map on Cn is Φ(p) + 12 isotropy weights or just weights at p (see Appendix A). Its image is the moment cone, X Cp = Φ(p) + R+ αi,p .

2. HAMILTONIAN GEOMETRY

23

Now suppose that M is compact. The stability theorem and the local normal form theorem imply that there exists a neighborhood U of Φ(p) such that (2.13)

Φ(M ) ∩ U = Cp ∩ U.

The moment polytope is equal to the intersection of the moment cones: \ Φ(M ) = (2.14) Cp . p∈M G

This follows from (2.13), together with the facts that Φ(M ) is a convex polytope and that its vertices are a subset of {Φ(p) | p ∈ M G }. Equations (2.12) and (2.14) are important characterizations of the moment polytope Φ(M ) which play a role in the proof of the “quantization commutes with reduction” theorem; see Chapter 8. When ω is not symplectic, the moment image Φ(M ) is not a polytope and appears to be of little interest. However, in the presence of a stable complex structure (see Appendix D), the isotropy weights, and hence the moment cones, are well defined. The intersection ∩Cp and the convex hull conv Φ(M G ) might now be different from each other, and they both provide meaningful replacements for the moment polytope. The Morse theoretic properties of the moment map imply that, for a torus action with isolated fixed points, the cohomology class [ω] is determined by the moment map values at the fixed points. To see this, restrict to a sub-circle action with the same fixed points. Because its moment map is a perfect Morse function, the second homology is generated by invariant two-spheres descending from fixed points. The value of [ω] on such a two-sphere is determined by the moment map values at the fixed points as in Example 2.7. Often, the moment map encodes much more than the cohomology class of the two-form. For instance, the moment map plays a central role in classification theorems. Theorem (Delzant, [De]). If dim G = 21 dim M , the moment polytope Φ(M ) determines the Hamiltonian manifold (M, ω, Φ) up to isomorphism. Here, isomorphism means equivariant symplectomorphism that preserves the moment map. For a torus G, compact Hamiltonian G-manifolds with dim G = 1 2 dim M are called Delzant spaces. The moment polytope ∆ = Φ(M ) is regular, meaning that the edge vectors at each vertex can be normalized to be a Z-basis of the lattice Z∗G in g∗ . An explicit construction of a Delzant space (M, ω, Φ) from a given regular polytope ∆ is described in Section 5 of Chapter 5. In this Delzant situation, the level sets of Φ are precisely the G-orbits; moreover, the moment map gives a diffeomorphism M/G → ∆, meaning that a function on ∆ extends to a smooth function on g∗ exactly if it pulls back to a smooth invariant function on M , as can be seen from the local normal form. The set of regular values of Φ consists of only one alcove: the interior of ∆. The Duistermaat–Heckman measure is equal to Lebesgue measure on ∆ times (2π)n (cf. Example 2.6). Hence, the Liouville volume of a Delzant space is proportional to the Euclidean volume of its moment polytope: Z exp ω = (2π)n volEuc ∆.

M

Delzant’s theorem was generalized to orbifolds by Lerman and Tolman [LT1]: a Hamiltonian G-orbifold is determined by the moment polytope ∆, together with

24

2. HAMILTONIAN COBORDISM

−

+

+

Figure 2.3. Twisted polytope for a Delzant space an integer for each facet F of ∆, which encodes the orbifold singularity along the codimension 2 submanifold Φ−1 F ⊂ M . The complexity of a Hamiltonian G-manifold is 21 dim M − dim G, which measures how far we are from the Delzant situation. For actions of complexity one, classification results can be found in [Ka4, KT2, KT3]. The complexity is equal to half the dimension of a non-empty regular reduced space Φ−1 (α)/G. It also effects the Duistermaat–Heckman measure, whose density function is given on each alcove by a polynomial of degree at most the complexity; see Section 3 of Chapter 5. Delzant’s theorem has an interesting generalization in the case that ω is degenerate. Over the set g∗reg of regular values of Φ, the quotient M = Φ−1 g∗reg /G is a manifold, and the map Φ : M → g∗reg is a proper local diffeomorphism. The degree function associates to each regular value α ∈ g∗reg the number of points in the preimage of α, counted with appropriate signs. If M is a toric variety, M/G is homeomorphic to a polytope with boundary homeomorphic to S n−1 , the moment map restricts to a map Φ : S n−1 → g∗ r{α} ∼ = Rn r{point}, and the winding number of this map is equal to the degree at α. The Duistermaat–Heckman measure is equal to Lebesgue measure times the degree function times (2π)n . For example, in Figure 2.3, the polygon on the right corresponds to some Delzant space, and the “twisted polygon” on the left corresponds to a non-symplectic two-form on the same space. The Duistermaat–Heckman measure is equal to Lebesgue measure on the bottom triangle and minus Lebesgue measure on the top triangle, times (2π)2 . The same degree function also gives the multiplicities of weights for the quantization of M . See [KT1, GK, Masu, Hat3]. 3. Compact Hamiltonian cobordisms Let G be a connected Lie group and (M, ω, R Φ) a compact Hamiltonian Gmanifold. The Duistermaat–Heckman integral is M exp(iΦ + ω), that is, the function I : g → C given by Z ξ (2.15) I(ξ) = eiΦ ω n /n!, M

where n =

1 2

dim M . Passing to the variable α = Φ(p), Z dM (ξ) I(−ξ) = e−ihα,ξi DHM = DH α∈g∗

is the Fourier transform of the Duistermaat–Heckman measure (see Definition 2.4). By the method of stationary phase, the integral of eitϕ is the sum over the critical points, {p | dϕ|p = 0}, of certain expressions that depend on the Hessian d2 ϕ|p , plus a remainder term that decays as t → ∞ at least as fast as t−n−1 . If ω is symplectic, the critical points of ϕ = Φξ are exactly the fixed points for the G-action, as follows from (2.1). The astounding discovery of Duistermaat and Heckman is that in this case the remainder term is identically zero.

3. COMPACT HAMILTONIAN COBORDISMS

25

In Section 6 of Chapter 4 we give the resulting Duistermaat–Heckman exact stationary phase formula (which involves summation over fixed points and is valid for all invariant closed two-forms). We deduce it from the “Hamiltonian linearization theorem”. In Section 7 of Appendix C we deduce the Duistermaat–Heckman formula from the Atiyah-Bott-Berline-Vergne localization formula in equivariant cohomology. Perhaps the first indication that cobordisms may be relevant for Hamiltonian geometry was the realization by Guillemin and Sternberg, in the mid-eighties [GS5], that the Duistermaat–Heckman integral is an invariant of cobordism: Lemma 2.13. Let (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ) be Hamiltonian G-manifolds that are cobordant in the following sense: there exists a compact Hamiltonian Gmanifold with boundary (W, ω, Φ) such that ∂W = −M0 t M1 , ω|∂W = ω0 t ω1 ,

and

Φ|∂W = Φ0 t Φ1 . Then their Duistermaat–Heckman integrals are equal. The minus sign in −M0 indicates reversal of orientation. Proof. The equality Z (2.16)

ξ

M0

eiΦ0 ω0n /n! =

Z

ξ

M1

eiΦ1 ω1n /n!

follows from Stokes’ theorem, noting that the integrands extend to a closed 2n-form, ξ eiΦ ω n /n!, on W . The closedness of this 2n-form is a special case of the following lemma.

Lemma 2.14. Let (W, ω, Φ) be a (2n+1)-dimensional Hamiltonian G-manifold. Then for any smooth function ϕ : g∗ → R, the 2n-form (ϕ ◦ Φ)ω n on W is closed.

Proof. By the chain rule, the total derivative of this form is hdϕ ◦ Φ, dΦ ∧ ω n i, where h, i denotes the pairing between g and g∗ . Note that dϕ ◦ Φ is a g-valued function and dΦ ∧ ω n is a g∗ -valued (2n + 1)-form. For any ξ, dΦξ ∧ ω n =

1 ι(ξM )ω n+1 = 0, n+1

because ω n+1 is a (2n + 2)-form on a (2n + 1)-manifold. Lemma 2.13 was proved by Guillemin and Sternberg [GS5] in the context of of equivariant cohomology. We recall a relevant definition: Definition 2.15. An equivariant differential two-form on a G-manifold M is, by definition, a formal sum ω + Φ, where ω is an invariant two-form on M and Φ is a smooth equivariant function from M to g∗ . This equivariant form is equivariantly closed if (d + ι(ξM ))(ω + Φξ ) = 0 for all ξ ∈ g. The equivariant form ω + Φ is equivariantly exact if there exists an invariant one-form µ such that ω + Φ ξ = 2 (d + ι(ξM ))µ for all ξ ∈ g. The second equivariant cohomology, denoted HG (M ), is the quotient of the space of equivariantly closed two-forms by the subspace of equivariantly exact two-forms. See Appendix C.

26

2. HAMILTONIAN COBORDISM

Example 2.16. ω − Φ is equivariantly closed if and only if Φ is a moment map for ω; it is equivariantly exact if and only if Φ is an exact moment map for ω. (See Example 2.8). There is a simple relationship between equivariant cohomology and Hamiltonian cobordism: Lemma 2.17. Let ω0 − Φ0 and ω1 − Φ1 be equivariantly closed two-forms in the same equivariant cohomology class. Then the Hamiltonian G-manifolds (M, ω 0 , Φ0 ) and (M, ω1 , Φ1 ) are cobordant. Proof. Let µ be a one-form such that (ω1 − Φ1 ) − (ω0 − Φ0 ) = (d + ι(ξM ))µ. Take the cobording manifold M = [0, 1] × M with the two-form ω0 + d(tµ) and moment map Φξ = Φξ0 − tµ(ξM ). If we only assume that ω0 and ω1 are in the same ordinary de Rham cohomology class, we again obtain a cobordism, but only if we choose the moment maps carefully: Lemma 2.18. Let ω0 and ω1 be closed two-forms on M in the same cohomology class. If the action is Hamiltonian for ω0 , it is also Hamiltonian for ω1 , and we can choose the moment maps Φ0 and Φ1 so that (M, ω0 , Φ0 ) and (M, ω1 , Φ1 ) are cobordant. Proof. Suppose that ω0 − ω1 = dβ. Let Φ0 be a moment map for (M, ω0 ). Then Φξ1 = ω0 + β(ξM ) is a moment map for (M, ω1 ). They are cobordant through W = [0, 1] × M with ω = ω0 − d(tβ) and Φξ (t, m) = Φξ0 (m) + tβ(ξM ). The reason that we cannot specify the moment maps in advance in Lemma 2.18 is that in general a moment Φ is not cobordant to the Rmoment map Φ+α for α ∈ g∗ . If the two-form is symplectic, or, more generally, if M ω n 6= 0, one can eliminate this ambiguity by normalizing moment maps as in [GS5]: Lemma 2.19. Let M be a compact oriented 2n-dimensional manifold with a Gaction. Let R ω0 nand ω1 be closed two-forms in the same cohomology class such that R n ω 6= 0. Let Φ0 and Φ1 be corresponding moment maps that satisfy ω = 0 M 1 M the normalization condition Z Z (2.17) Φ0 ω0n = Φ1 ω1n = 0. M

M

Then (M, ω0 , Φ0 ) and (M, ω1 , Φ1 ) are cobordant. Proof. Consider the product [0, 1] × M with the two-form ω = ω0 − d(tβ) and moment map Φξ = Φξ0 +tβ(ξM ), where dβ = ω0 −ω1 . Denote by it : M → [0, 1]×M the inclusion map m 7→ (t, m). Then i∗0 ω = ω0 , i∗1 ω = ω1 , and i∗0 Φ = Φ0 . It remains to show that i∗1 Φ = Φ1 . Since both of these R are moment Rmaps for ω1 , they differ by a constant, so it is enough to show that M (i∗1 Φ)ω1n = M Φ1 ω1n . By assumption, termR is zero. Because i1 is homotopic to i0 , the left term is equal to Rthe right R R (i∗ Φ)ω1n = M i∗1 (Φω n ) = M i∗0 (Φω n ) = M Φ0 ω0n = 0. M 1

4. PROPER HAMILTONIAN COBORDISMS

27

4. Proper Hamiltonian cobordisms Let (M, ω, Φ) be a Hamiltonian G-manifold. Assume that the moment map Φ : M → g∗ is proper (the preimage of any compact set is compact); the manifold M need not be compact. Many properties of compact Hamiltonian G-manifolds continue to hold for proper moment maps on non-compact manifolds. For instance, G is a torus and ω is symplectic, the moment image Φ(M ) is a convex polyhedron; see [CDM, LMTW]. We would also like to refer to the Liouville measure on M and the Duistermaat– Heckman measure on g∗ . However, the measure of an unbounded set might not be defined, because the set might be a union of two unbounded sets on which these measures are +∞ and −∞ respectively. To be more precise, one must define the Liouville measure and the Duistermaat–Heckman measure DH M as distributions. Similarly, the Duistermaat–Heckman integral I(ξ) can be replaced (up to sign) d M . Concretely, the Liouville distribution and the by the Fourier transform DH Duistermaat–Heckman distribution are the continuous linear functionals Z Z Z n f 7→ f ω /n! and ϕ 7→ ϕ DHM = (ϕ ◦ Φ) ω n /n! g∗

M

Cc∞ (M )

M

Cc∞ (g∗ )

on the spaces and of compactly supported smooth functions. The d Fourier transform DHM associates to a function S on g the value Z Z Z Z b S(ξ)e−ihα,ξi dξ DHM = “ S(ξ)I(−ξ)dξ”. S(α) DHM = g∗

g∗

g

g

This is a distribution (in fact, a tempered distribution) if DHM is a tempered distribution (see [Rudi]). This is the case if M is tame (diffeomorphic to the interior of a G-manifold with boundary), because DHM is then “piecewise polynomial with finitely many pieces”. As in the compact case, if G is a torus and Φ is proper, the Duistermaat– Heckman measure has a piecewise polynomial density function with respect to Lebesgue measure on g∗ ; see [Pr, PW]. The Duistermaat–Heckman formula, expressing the Duistermaat–Heckman integral in terms of fixed point data, holds under conditions that are slightly more restrictive than the moment map being proper: first, the fixed point set must be finite. Second, some component Φη of the moment map must be proper and bounded from below. Such a moment map is called polarized and is automatically proper. The generalization of the Duistermaat–Heckman formula to this case is due to Prato and Wu [PW]. Polarized moment maps play a major role in our theory, particularly in the linearization theorem of Chapter 4. This discussion leads us to the definition of proper Hamiltonian cobordism: Definition 2.20. A proper Hamiltonian cobordism between two Hamiltonian G-manifolds with proper moment maps, (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ), is a Hamiltonian G-manifold (W, ω, Φ), where W is a manifold with boundary, and an orientation preserving equivariant diffeomorphism from −M0 t M1 to the boundary ∂W that carries ω to ω0 t ω1 and Φ to Φ0 t Φ1 . Here, −M0 denotes the manifold M0 equipped with the opposite orientation. The properness assumption on the moment maps is crucial; its omission can have disastrous consequences. For example, every manifold M is cobordant to the empty manifold via the cobordism W = (0, 1] × M . It is easy to make this a

28

2. HAMILTONIAN COBORDISM

Hamiltonian cobordism if M is a Hamiltonian manifold, but the cobording moment map (t, m) 7→ Φ(m) will not be proper. Lemma 2.21. Proper Hamiltonian cobordism is an equivalence relation. Proof. The only non-obvious part of this assertion is that proper Hamiltonian cobordisms are transitive. In other words, assume that (Mi , ωi , Φi ), for i = 1, 2, 3, are proper Hamiltonian G-manifolds, such that the first is properly cobordant to the second, and the second is properly cobordant to the third. Then the first is properly cobordant to the third. This can be shown as follows. Let (W, ω, Φ) be a proper cobordism between the first and the second manifold and (W 0 , ω 0 , Φ0 ) between the second and the third. We first modify these cobordisms so as to make them cylindrical near the boundary piece M2 , and then attach W to W 0 along this boundary piece to obtain a proper Hamiltonian cobordism from M1 to M3 . Being cylindrical means that a neighborhood of M2 in W is isomorphic to a product M2 × (−, 0], where G only acts on the M2 factor and where the two-form and moment map are the pull-backs from M2 . A tubular neighborhood of M2 in W is equivariantly diffeomorphic to U = (−1, 0] × M2 . The proof of this in the non-equivariant case (see [BJ], [MiSt], or [La1]) goes through in the presence of a proper group action. Consider the projection map π : U → M2 and the inclusion map i : M2 = ∂U → U . Because ω and Φ coincide with π ∗ ω2 and π ∗ Φ2 on ∂U , which is a strong deformation retract of U , there exists an invariant one-form β such that ω|U = π ∗ ω2 + dβ

and

Φ|U = π ∗ Φξ2 − β(ξW ).

Let V be an open neighborhood of ∂U which is contained in (−1/2, 0] and on which the difference between Φ and π ∗ Φ2 is, say, less than one (with respect to some metric on g∗ ). Let ρ : U → [0, 1] be a smooth function such that ρ = 1 outside V and ρ = 0 on some neighborhood of ∂U . The two-form π ∗ ω2 + d(ρ(t)β) and moment map π ∗ Φξ2 − ρ(t)β(ξW ) extend to W and provide a proper Hamiltonian cobordism as required. One may also work with η-polarized Hamiltonian cobordisms, meaning that the η-component of all moment maps is polarized (proper and bounded from below), for some fixed η ∈ g. Such cobordism is also an equivalence relation; this is proved exactly like Lemma 2.21. However, the composition of an η-polarized cobordism and an η 0 -polarized cobordism might not be polarized if η 6= η 0 . (As a trivial counterexample with G = S 1 , a disjoint union W t W 0 can be viewed as a composition of cobordisms along an empty boundary and is not polarized if the moment map is unbounded from above on W and unbounded from below on W 0 .) In the rest of this section we assume that G is a torus. Remark 2.22 (Locality of Hamiltonian cobordisms). We may fix a convex open subset U of g∗ and assume that the moment image is contained in U and that the moment map is proper as a map to U . The convexity, connectedness, and stability theorems continue to hold; see [LMTW]. We can also work with proper Hamiltonian cobordisms over U . An interesting observation is that cobordism is a local notion: if two Hamiltonian spaces are properly cobordant over U and over V , then they are properly cobordant over U ∪ V (possibly through an orbifold).

5. HAMILTONIAN COMPLEX COBORDISMS

29

The idea of the proof is to “cut” (a la Lerman [Ler1]) each manifold into pieces, each of which maps properly to U or V , and note that the result of the cutting is cobordant to the original manifold. Allowing proper Hamiltonian cobordisms enriches the compact theory with new objects without degenerating it with any new equivalences: Theorem 2.23. Suppose that two Hamiltonian manifolds are compact and are properly cobordant. Then they are also compactly cobordant (possibly through an orbifold). This is proved in [GGK2] for abstract moment maps (cf. Remark 3.29). A similar proof holds for Hamiltonian manifolds. Hence, the theory of proper Hamiltonian cobordisms is at least as non-trivial as the theory of compact Hamiltonian cobordisms. Non-triviality is also illustrated by the following result (cf. Lemma 2.13): Theorem 2.24. Two Hamiltonian G-manifolds that are properly cobordant have the same Duistermaat–Heckman measure. Proof. Let (W, ω, Φ) be a proper Hamiltonian cobordism between (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ). It is enough to prove that for any compactly supported smooth function ϕ : g∗ → R, the integrals of ϕ against the two Duistermaat–Heckman measures are equal; equivalently, that Z Z (ϕ ◦ Φ0 ) ω0n /n! = (ϕ ◦ Φ1 ) ω1n /n!. M0

M1

This equality follows from Stokes’ theorem, once we notice that the integrands extend to the differential form (ϕ ◦ Φ)ω n on the cobording manifold W , and that this form is closed (by Lemma 2.14) and compactly supported. 5. Hamiltonian complex cobordisms The notion of Hamiltonian cobordism captures invariants which are determined by the equivariant cohomology class [ω−Φ], such as the Duistermaat–Heckman measure. Other invariants, like the geometric quantization considered in Chapter 6, require a richer structure. There are various additional structures which enable one to define geometric quantization. We will work with stable complex Hamiltonian G-manifolds, i.e., Hamiltonian G-manifolds equipped with G-equivariant stable complex structures.1 Recall that a stable complex structure is given by a complex structure J on the fibers of a Whitney sum T M ⊕ Rk ; see Section 1 of Appendix D for details. The manifolds that we care to quantize are symplectic manifolds, to which there are naturally associated almost complex structures. (See Example D.12.) However, we choose to work with stable complex, not just almost complex structures, for several reasons. One advantage of stable complex structures is that one can consider their cobordisms. This is in contrast to almost complex structures, which can only exist on even dimensional manifolds. 1 There are strong arguments in favor of working with a slightly different structure, namely, a Spinc -structure, in order to define geometric quantization. See [CKT], [Par3], or Section 3.5 of Appendix D.

30

2. HAMILTONIAN COBORDISM

Definition 2.25. A Hamiltonian complex cobordism between two stable complex Hamiltonian G-manifolds is a Hamiltonian cobordism between these manifolds, together with a G-equivariant stable complex structure on the cobording manifold which restricts to the given structures on the boundary. For a more precise formulation, see Proposition D.14 and Definition D.22. As before, we may consider compact cobordisms, proper cobordisms, or ηpolarized cobordisms, and each of these is an equivalence relations. The geometric quantization for an “integral” stable complex Hamiltonian Gmanifolds, (defined in Section 7 of Chapter 6,) is invariant under Hamiltonian complex G-cobordism. See Section 7 of Chapter 6 and Appendix J. Another motivation for considering stable complex structures comes from the notion of reduction. A G-invariant almost complex structure which is incompatible with a symplectic form does not descend to an almost complex structure on the reduced space Φ−1 (α)/G, but only to a stable complex structure. (See Section 2.3 of Chapter 5.) Incompatible almost complex structures arise naturally in the cobordism linearization theorem (see Chapter 4) and in its quantum version (see Chapter 7). See also [GGK1]. Finally, equivariant stable complex structures enable one to “resolve” the orbifold singularities of a reduced space; cf. [GGK2]. In this case, the “locality of Hamiltonian cobordisms” (Remark 2.22) and the fact that “compact cobordisms classes inject into proper cobordism classes” (Theorem 2.23) are true without introducing orbifolds.

CHAPTER 3

Abstract moment maps Many aspects of Hamiltonian group actions involve the moment map and not the two-form. Hence, it is often convenient to focus entirely on the properties of moment maps and to ignore two-forms. For this purpose we introduce the notion of an abstract moment map. 1. Abstract moment maps: definitions and examples Let G be a Lie group, g its Lie algebra, and g∗ the dual space. For any map Ψ which takes values in g∗ and any subgroup H of G with Lie algebra h we denote by Ψh or ΨH the composition of Ψ with the natural projection g∗ → h∗ . Similarly, for any Lie algebra element ξ ∈ g, we denote by Ψξ the ξth component of Ψ, i.e., the real valued function hΨ, ξi. Definition 3.1 ([Ka3]). Let M be a G-manifold. An abstract moment map on M is a smooth map Ψ : M → g∗ with the following two properties: 1. Ψ is G-equivariant; 2. For any subgroup H of G, the map ΨH : M → h∗ is locally constant on the submanifold M H of points fixed by H. The values of ΨH on the connected components of M H form the assignment associated with Ψ; see Appendix E. The second requirement in Definition 3.1 can be replaced by (2’) For any Lie algebra element ξ ∈ g, the function Ψξ is locally constant on the zero set M ξ of the corresponding vector field ξM . Example 3.2. The constant function zero is an abstract moment map. Example 3.3. On a manifold with a circle action with isolated fixed points, any invariant function is an abstract moment map. Example 3.4. Let X be a vector field on M that generates a circle action and let h, i be an invariant Riemannian metric on M . Then the function hX, Xi is an abstract moment map with zero assignment. Example 3.5. If Ψ : M → g∗ is an abstract moment map and α ∈ g∗ is fixed by the coadjoint action, the translation Ψ + α is also an abstract moment map. In the special case G = S 1 , the composition F ◦ Ψ is an abstract moment map for any smooth F : R → R.

Example 3.6. Let M be a G-manifold and let Ψ : M → g∗ be a moment map for an invariant closed two-form ω on M . Recall, that this means Ψ is G-equivariant and satisfies Hamilton’s equation, (3.1)

ι(ξM )ω = dΨξ 31

for all ξ ∈ g.

32

3. ABSTRACT MOMENT MAPS

Then Ψ is an abstract moment map. Indeed, the coordinates of ΦH are Φξ for ξ ∈ h. These satisfy dΦξ |M H = ι(ξM )ω|M H = 0 because ξM vanishes on M H . If equation (3.1) holds, we say that ω is compatible with Ψ, or that Ψ is associated with ω. An abstract moment map associated with some two-form is called a Hamiltonian (abstract) moment map. Not every abstract moment map is Hamiltonian. See Example E.36. In Theorem E.37 we give a condition for an abstract moment map to be Hamiltonian. Moment maps associated with true symplectic forms must additionally satisfy some non-degeneracy requirements, which we analyze in Appendix G. It is worth pointing out that a moment map on a Poisson manifold (see, e.g., [CW]) might not be an abstract moment map even when the Poisson structure is preserved by the action. The reason is that, since a moment map is defined only up to addition of Casimir functions, and since on a Poisson manifold Casimir functions often exist in abundance, a moment map on a Poisson manifold might not be constant on the fixed point set. Example 3.7. Let a Lie group G act on a manifold M and let µ be any invariant one-form. Then the function Ψ : M → g∗ defined by Ψξ = µ(ξM )

(3.2)

is an abstract moment map. Such moment maps have been studied on contact manifolds M , when µ is a contact form, in [Al, Ge, Wt, LW, Lo], specifically in the context of reduction. A definition of a contact moment map which is independent of a choice of contact one-form was given by Lerman in [Ler3]: Example 3.8. Following [Ar, Appendix 4] and [Ler3], we consider a contact manifold (M, ξ) with a G-action preserving the contact structure ξ. Recall that ξ is a maximally non-integrable codimension one sub-bundle of the tangent bundle T M . Let ξ 0 be its annihilator in T ∗ M . The corresponding contact moment map is the exact moment map on ξ 0 corresponding to the canonical one-form Ψ : ξ 0 → g∗ ,

Ψξ (q, p) = p(ξM |q ).

More generally, an abstract moment map that arises by equation (3.2) is called exact (cf. Example 2.8). A compatible two-form is then given by ω = −dµ. An exact moment map has the property that ΨH vanishes on M H for all H ⊆ G. This property characterizes exact moment maps among all abstract moment maps; see Corollary E.27. Many “classical” moment maps, such as the canonical moment map on a cotangent bundle, or the moment map on a pre-quantization circle bundle, are exact. See Examples 3.11 and 3.12. The following two examples exhibit functoriality properties of abstract moment maps with respect to G and M , respectively. Example 3.9. Let M be a G-manifold with an abstract moment map Ψ : M → g∗ . Let a Lie group H act on M through a group homomorphism ϕ : H → G followed by the G-action. Then the composition of Ψ with the natural map ϕ∗ : g∗ → h∗ is an abstract moment map for the H-action. If Ψ is a Hamiltonian, respectively, exact, moment map, so is ϕ∗ ◦ Ψ. Example 3.10. Let M be a G-manifold with an abstract moment map Ψ : M → g∗ . Let N be another G-manifold and let f : N → M be a G-equivariant map.

2. PROPER ABSTRACT MOMENT MAPS

33

Then the pullback f ∗ Ψ = Ψ ◦ f is an abstract moment map for the G-action on N . This map is proper if f and Ψ are proper. If Ψ is a Hamiltonian, respectively, exact, moment map, so is f ∗ Ψ. Example 3.11. Following [SLM] and [Lew], consider a G-manifold Q, and denote by J : T ∗ Q → g∗ the canonical moment map: J ξ (α) = α(ξQ (x)), where α ∈ Tx∗ Q. The action map F : Q × g → T Q is defined by F (x, ξ) = ξQ (x) (cf. Section 1.6 of Appendix B). Consider a Lagrangian on Q with Legendre transformation L : T Q → T ∗ Q; see, e.g., [Ar]. By Example 3.10, the composition F

L

J

I = (LF )∗ J : Q × g → T Q → T ∗ Q → g∗

is an exact abstract moment map. For example, assume that L arises from a Riemannian metric h , i on Q so that L(v) = hv, ·i for a tangent vector v. Then Iζ (x, ξ) = hξQ (x), ζQ (x)i for all ζ ∈ g. The map I, called the locked inertia tensor, is used in the analysis of relative equilibria. (See [SLM] and [Lew].) Note that in general Q × g is not a symplectic manifold and G, in this example, does not have to be commutative. Example 3.12. Let (M, ω, Ψ) be a Hamiltonian G-manifold. Let π : P → M be a pre-quantization circle bundle. This means that P is a G-equivariant principal circle bundle with a connection one-form Θ, such that dΘ = −π ∗ ω, and such that the moment map Ψ is given by π ∗ Ψξ = Θ(ξP ) for all ξ ∈ g. (For more details, see Chapter 6.) Then the pullback π ∗ Ψ : P → g∗ is an exact moment map. Example 3.13. A linear combination of abstract moment maps is an abstract moment map. The advantage of working with exact moment maps over Hamiltonian ones is that these can be patched together: Example 3.14. If Ψ0 and Ψ1 are exact moment maps, then so is (1−ρ)Ψ0 +ρΨ1 for any smooth function ρ : M → [0, 1]. More generally, if Ψ0 and Ψ1 arise from one-forms µ0 and µ1 , then f Ψ0 + gΨ1 arises from the one-form f µ0 + gµ1 , for any smooth functions f and g. Example 3.15. A direct sum of abstract moment maps on the same manifold is not necessarily an abstract moment map. Namely, suppose that G1 and G2 act on M with abstract moment maps Ψ1 and Ψ2 , that the two actions commute with each other, and that Ψ1 and Ψ2 are invariant under both actions. The sum Ψ1 ⊕ Ψ2 is not necessarily an abstract moment map for the G1 × G2 -action as it would be if Ψ1 and Ψ2 were associated with the same closed two-form. For example, let G1 = G2 be the circle and assume that the G1 -action coincides with the G2 -action. Then the anti-diagonal H ⊂ G1 × G2 acts trivially on M . The H-component of Ψ1 ⊕ Ψ2 is just Ψ1 − Ψ2 . The direct sum is an abstract moment map if and only if Ψ1 = Ψ2 + const. 2. Proper abstract moment maps In this section we focus on abstract moment maps that are proper. Recall, a map is proper if the preimage of any compact set is compact. Example 3.16. The zero map is a proper abstract moment map if and only if the manifold is compact.

34

3. ABSTRACT MOMENT MAPS

Example 3.17. If the fixed point set M G has a non-compact component, M does not admit a proper abstract moment map. Example 3.18. Let V be a complex vector space with a linear circle action that fixes only the origin. For any invariant Hermitian inner product, h·, ·i, the function Ψ(v) = hv, vi is an abstract moment map. It is proper if and only if h·, ·i is positive or negative definite. P mj |zj |2 for the Hamiltonian circle action In particular, the moment map 12 on Cn with weights −mj ∈ Z is proper if and only if all the mj ’s are positive or all are negative. More for a torus action on Cn with weights −αj ∈ Z∗G , the moment P generally, map 21 |zj |2 αj is proper if and only if there exists η ∈ g such that αj (η) > 0 for all j. See Proposition 4.15. Definition 3.19. Let η ∈ g be a Lie algebra element. A function Ψ : M → g∗ is said to be η-polarized if its ηth component, Ψη : M → R, is proper and bounded from below. It is polarized if it is η-polarized for some η. A polarized function is always proper, but a proper function (e.g., id : g∗ → g∗ ) is not necessarily polarized. If M is compact, all functions M → g∗ are η-polarized for all η ∈ g. A linear combination of proper abstract moment maps is an abstract moment map, but it is not necessarily proper. However, a positive linear combination of η-polarized abstract moment maps is again an η-polarized abstract moment map. This technical fact plays an important role in our theory and is the key reason for the introduction of polarized abstract moment maps. More specifically, polarized abstract moment maps possess the following two properties, which may in generally fail for proper abstract moment maps: • A positive linear combination of η-polarized abstract moment maps on the same manifold is again an η-polarized (hence proper) abstract moment map. • Let Ψj : Mj → g∗ , j = 1, 2, be η-polarized abstract moment maps. Consider the product M1 × M2 with the diagonal G-action. Then Φ(m1 , m2 ) := Φ1 (m1 ) + Φ2 (m2 ) is an η-polarized (hence proper) abstract moment map. The first of these two properties allows us to carry out “patching” arguments. (See Appendix E.) The second of the two properties gives a ring structure on the cobordism classes of η-polarized abstract moment maps. 3. Cobordism Recall, a cobordism between two manifolds, M0 and M1 , is a manifold with boundary W , and a diffeomorphism from the disjoint union of M0 and M1 to the boundary of W : (3.3)

∂W = M0 t M1 .

For oriented manifolds, one insists that boundary orientation on ∂W transports to the given orientation on M1 and the opposite orientation on M0 . In ordinary cobordism theory, one only considers compact manifolds; otherwise, the theory becomes trivial: every manifold M is non-compactly cobordant to the empty set via W = (0, 1] × M .

3. COBORDISM

35

An equivariant cobordism is one that carries a group action, i.e., a cobordism in which M0 , M1 , and W are G-manifolds, and the diffeomorphism (3.3) is equivariant. Example 3.20. A compact manifold that admits a free S 1 -action is (equivariantly) cobordant to the empty set. Indeed, such a manifold, M , is a circle bundle. Take the associated disc bundle W = M ×S 1 D2 . It still admits an S 1 -action because S 1 is abelian. Its boundary is diffeomorphic to M via m 7→ [m, 1]. Notice, 1 however, that the fixed point set W S is always non-empty, as it contains the zero section M/S 1 . In fact, if the action on M is free, M may fail to be cobordant to the empty set in the class of manifolds with free S 1 actions. This is already the case when M = S 1 , as follows from the fact that M/S 1 = point is not the boundary of any compact manifold. We now consider cobordisms between abstract moment maps. We do not require our manifolds to be compact; the compactness assumption is replaced by the demand that the abstract moment maps be proper. In what follows, all abstract moment maps will be proper unless otherwise specified. Definition 3.21. Let M0 and M1 be oriented G-manifolds with proper abstract moment maps Ψ0 : M0 → g∗ and Ψ1 : M1 → g∗ . A proper cobordism between (M0 , Ψ0 ) and (M1 , Ψ1 ) is a (possibly non-compact) equivariant oriented cobordism W between M0 and M1 , and a proper abstract moment map Ψ : W → g∗ , such that the equivariant diffeomorphisms (3.3) carries the abstract moment map on W to those on M0 and M1 : Ψ|∂W = Ψ0 t Ψ1 . Remark 3.22. We may refer to the above as a cobordism between M0 and M1 if it is clear from the context what moment maps on M0 and M1 we consider. For instance, on a proper Hamiltonian G-manifold (M, ω, Φ) (see Chapter 2) we take the moment map Φ. A compact G-manifold is by default equipped with the zero moment map unless specified otherwise. Example 3.23. Every compact equivariant cobordism becomes a proper cobordism by setting all abstract moment maps to be identically zero. Example 3.24. A compact manifold M with a free S 1 -action is properly cobordant to the empty set via W = (0, 1] × M and Ψ(t, m) = 1−t t . It is absolutely crucial for the abstract moment maps to be proper; otherwise, every manifold M would be cobordant to the empty set via the non-compact cobordism M × (0, 1], where the G action and abstract moment maps are induced from those on M . It is also crucial to assume Condition 2 of Definition 3.1, that the abstract moment map components must be locally constant on appropriate fixed point sets. Dropping this condition would result in new identifications; for instance, every compact S 1 -manifold would then be cobordant to the empty set via the noncompact cobordism (0, 1] × M and the proper function (t, m) = Φ(m) + 1−t t . Lemma 3.25. Cobordism of proper abstract moment maps is an equivalence relation. Proof. As in the proof of Lemma 2.21, the crucial point is to modify a cobording G-manifold and proper abstract moment map Ψ : W → g∗ so as to make it

36

3. ABSTRACT MOMENT MAPS

cylindrical near a connected component M2 of the boundary ∂W . A tubular neighborhood of M2 is equivariantly diffeomorphic to U = (−1, 0]×∂M2; cf. [MiSt]. Let Ψ2 denote the restriction of Ψ to M2 . Let π : U → M2 denote the projection map. Let V be a neighborhood of ∂U which is contained in (−1/2, 0] and on which the difference between Ψ and π ∗ Ψ2 is, say, less than one (with respect to some metric on g∗ ). Let ρ : U → [0, 1] be a smooth function such that ρ = 1 outside V and ρ = 0 on some neighborhood of the boundary M2 . Then π ∗ ∂Ψ + ρ(t)(Ψ − π ∗ ∂Ψ) extends to a proper abstract moment map on W which is cylindrical near M2 . Remark 3.26. Various extra structures can be incorporated in the notion of cobordism of abstract moment maps. For example, may consider stable complex cobordisms (see Section 5 of Chapter 2). Finally, the assumption that all abstract moment maps be proper can be replaced by a stronger requirement that they be η-polarized with respect to a fixed vector η ∈ g. Note that η-polarized cobordism is also an equivalence relation; the proof of Lemma 3.25 still goes through. Example 3.27. A proper Hamiltonian cobordism becomes a cobordism of proper abstract moment maps once the two-forms are discarded. (See Definition 2.20.) The relationship between Hamiltonian cobordisms and cobordisms of abstract moment maps is discussed in Section 5 of Appendix E. Let us now turn to the relationship between proper cobordism and ordinary compact equivariant cobordism. The following example shows that we do not introduce new relations between non-equivariant compact cobordism classes by taking their equivariant non-compact cobordism classes with zero moment maps: Example 3.28. Let M0 and M1 be compact manifolds with trivial G-actions. Let W be a G-manifold with boundary ∂W = M0 t M1 and a proper abstract moment map Ψ : W → g∗ . Then M0 and M1 are compactly cobordant. Indeed, because G acts trivially on ∂W , it acts trivially on its neighborhood, and hence it acts trivially everywhere, after discarding any connected components which are disjoint from ∂W . A G-manifold with a trivial action and proper moment map must be compact, because its moment map must be constant. Any proper cobordism of manifolds with trivial G-actions and abstract moment maps is automatically compact. Indeed, a connected group acts on a connected manifold with boundary and the action is trivial on the boundary, (or, in fact, on any submanifold of codimension one), it acts trivially everywhere. A manifold with a trivial action and a proper abstract moment map must be compact, because its moment map must be constant. Remark 3.29 (Compact versus non-compact cobordism). If two compact Gmanifolds with (possibly zero) abstract moment maps are properly cobordant through a non-compact manifold W , and if W admits an equivariant stable complex structure, then the two manifolds are also cobordant via some other, compact, W 0 . The way to show this is to first use Lerman’s cutting procedure [Ler1] to replace a non-compact cobordism by a compact orbifold cobordism, and then to get rid of the orbifold singularities by successive equivariant surgery along singular strata. The details of this argument can be found in [GGK2]. If the group G is the circle group, the presence of a stable complex structure can be replaced by a restriction on the stabilizer subgroups which are allowed to occur in the cobordism; see [GGK2].

4. FIRST EXAMPLES OF PROPER COBORDISMS

37

Finally, we note that the set of proper cobordism classes of oriented G-manifolds equipped with proper abstract moment maps forms an abelian group under the operation of disjoint unions. Moreover, for any Lie algebra element η ∈ g, the set of η-polarized cobordism classes forms a ring. This follows from the fact, noted in Section 2, that if M1 and M2 are equipped with η-polarized abstract moment maps, so is M1 × M2 . 4. First examples of proper cobordisms The following uniqueness lemmas assert that an abstract moment map is determined by its values at the fixed points and its behavior at infinity. We restrict our attention to circle actions, and, first to linear circle actions: Lemma 3.30. Let M = V be a vector space with a linear circle action. Let Ψ 0 and Ψ1 be two proper abstract moment maps on V which take the same value at the origin and such that Ψ0 (v) and Ψ1 (v) either both approach ∞ or both approach −∞ as kvk goes to ∞. Then Ψ0 and Ψ1 are properly cobordant. We note that a proper abstract moment map Ψ : V → R must satisfy either Ψ(v) → ∞ or Ψ(v) → −∞ as kvk → ∞. Indeed, the preimage of the closed interval [−N, N ] is contained in some large ball BN in V ; the complement V rBN maps to either [N, ∞] or [N, −∞], because it is connected. We also note that the sub-space of fixed points is compact (because Ψ is con1 stant on it and is proper), and hence equal to the origin: V S = {0}. More generally, Lemma 3.31. Let Ψ0 and Ψ1 be proper abstract moment maps on a manifold M with a circle action. Suppose that Ψ0 and Ψ1 take the same values at the fixed points and are both bounded from below or both bounded from above. Then Ψ 0 is properly cobordant to Ψ1 . Yet more generally, if M is a tame S 1 -manifold (equivariantly diffeomorphic to the interior of a compact S 1 -manifold with boundary), two proper abstract moment maps are properly cobordant if they take the same values at the fixed points and, at each end of M , either both approach +∞ or both approach −∞. Note that since M is a two-dimensional S 1 -manifold the requirement that M is tame is equivalent to that M has a finite number of connected components. Proofs. Take the trivial cobordism [0, 1] × M with the abstract moment map Ψ(t, m) = (1 − t)Ψ0 (m) + tΨ1 (m). Example 3.32. Consider the plane C = R2 with the standard S 1 -action and the standard orientation. We denote by C(a, +∞) the cobordism class of an abstract moment map Ψ which takes the value a at the origin and which is bounded from below. This is well defined by Lemma 3.30. Similarly, we denote by C(a, −∞) the cobordism class of the plane with the standard action and orientation and a proper abstract moment map which takes the value a at the origin and which is bounded from above. For instance, we may assume that the moment maps on C(a, ±∞) are just Ψ(z) = a ± |z|2 , z ∈ C. Let C(a, ±∞) denote the space C(a, ±∞) with the opposite circle action; let −C(a, ±∞) denote C(a, ±∞) with the opposite orientation. Then

38

3. ABSTRACT MOMENT MAPS

Figure 3.1. Linearization Theorem: S 2 is cobordant to two planes. (3.4)

C(a, ±∞) = −C(a, ±∞)

via complex conjugation: z 7→ z.

Example 3.33. Denote by S 2 (a, b) the cobordism class of the sphere S 2 ⊂ R3 with the standard orientation, the standard circle action by rotations about the z-axis, and an abstract moment map Ψ which takes the values a and b on the north and south poles. (All such Ψ’s are cobordant by Lemma 3.31.) Then (3.5)

S 2 (a, b) = C(a, +∞) + C(b, +∞).

The cobordism giving rise to (3.5) is shown in Figure 3.1. As the cobord˜ that satisfies the following ing abstract moment map we can take any function Ψ conditions: ˜ is proper and bounded from below. For example, on the complement of a • Ψ ˜ to be the projection to the z-axis. sufficiently large ball we may take Ψ ˜ • Ψ takes the values a and b at the north and south poles, respectively. ˜ is constant on the segments of the z-axis contained in the cobording • Ψ manifold. (This is necessary because these segments form the fixed point set for the S 1 -action.) The restrictions of this function to S 2 and to the two copies of C are cobordant to the given abstract moment maps on these spaces by Lemmas 3.30 and 3.31. The cobordism (3.5) is the first example of the Linearization Theorem: the sphere is properly cobordant to the disjoint union of its tangent planes at the fixed points, equipped with the induced orientations and S 1 -actions. In a similar way, we have (3.6)

S 2 (a, b) = C(a, −∞) + C(b, −∞),

which in combination with (3.5) implies the relation (3.7)

C(a, +∞) + C(b, +∞) = C(a, −∞) + C(b, −∞).

Example 3.34. Consider the cylinder C = S 1 ×R with coordinates θ ∈ R/2πZ and t, the standard orientation, given by dθ ∧ dt, and the standard S 1 -action, generated by ∂/∂θ. Consider the following proper abstract moment maps: • Ψ+ (eiθ , t) = t; • Ψ− (eiθ , t) = −t; • Ψ1 (eiθ , t) = t2 ;

5. COBORDISMS OF SURFACES

39

Figure 3.2. A cylinder is cobordant to two planes. • Ψ2 (eiθ , t) = −t2 . We have (C, Ψ− ) = −(C, Ψ+ ),

where the minus sign denotes the opposite orientation, via (eiθ , t) 7→ (eiθ , −t). Both (C, Ψ1 ) and (C, Ψ2 ) are cobordant to the empty set via the trivial cobordism [0, ∞) × C with the abstract moment maps Ψ(x, t) = ±(x + t2 ). Example 3.35. Consider the cylinder C = S 1 ×R with the standard S 1 -action, the standard orientation, and the abstract moment map Ψ+ (eiθ , t) = t. Let C+ denote its cobordism class. Then for any a ∈ R, (3.8)

C+ = C(a, +∞) + C(a, −∞).

˜ on A cobordism giving (3.8) is sketched in Figure 3.2. The moment map Ψ the cobordism can be taken to be equal to the projection to the z-axis on the complement to a sufficiently large ball. On the “middle part” of the cobordism ˜ ≡ a, so that Ψ ˜ is constant on the fixed point set (the segment of the zwe set Ψ axis contained in the cobording manifold). The abstract moment map obtained by ˜ to C is cobordant to Ψ+ . An explicit construction of this cobordism restricting Ψ (a formula rather than a picture) is given in Section 8 of Chapter 5. Note that (3.8) implies that (3.9)

C(a, +∞) + C(a, −∞) = C(b, +∞) + C(b, −∞),

which is equivalent to (3.7) by (3.4). 5. Cobordisms of surfaces In this section we will compute the cobordism group of surfaces with S 1 -actions and proper abstract moment maps. We note that in an oriented surface with an effective S 1 -action each orbit is a fixed point or the action on the orbit is free. This follows from the local normal form and the fact that an effective action is locally effective if the manifold is connected (see Appendix B). We also note that a connected surface S 1 -manifold is automatically tame, meaning, equivariantly diffeomorphic to the interior of a compact S 1 -manifold. This follows, by looking at M/S 1 , from the local normal form and the classification of one-dimensional manifolds.

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3. ABSTRACT MOMENT MAPS

Recall from Example 3.32 that C(a, ±∞) denotes the cobordism class of C = R 2 with the standard orientation and circle action and with the moment map Ψ(z) = a ± |z|2 . Theorem 3.36. Let G = S 1 and let (M, Ψ) be a connected oriented surface with an effective G-action and a proper abstract moment. Then (M, Ψ) is properly cobordant to a finite (possibly empty) disjoint union of planes C(a, ±∞) and C(a, ±∞). Let Γeff be the group of cobordism classes of (M, Ψ) where M is a tame oriented surface with a locally effective circle action and a proper abstract moment map. (Equivalently, M has finitely many connected component and the action is effective on each component.) Theorem 3.37. The group Γeff is the abelian group generated by C(a, ±∞), a ∈ R, with the relations generated by (3.10)

C(a, +∞) − C(a, −∞) = C(b, +∞) − C(b, −∞).

Corollary 3.38. The group Γeff is the free abelian group generated by the spaces C(a, +∞), a ∈ R, and C(0, −∞). Alternatively, Γeff is the free abelian group generated by C(a, +∞), a ∈ R, and C+ . (See Example 3.35.) Proof of Theorem 3.36. Step 1. The following is the list of orientable (although not oriented) connected surfaces with effective circle actions: • • • •

T2 = (S 1 )1 × (S 1 )2 with the standard action by rotations on the first factor; S 2 with the standard action by rotations about the z-axis; R2 = C with the standard action by rotations about the origin; the cylinder C = S 1 × R with the standard action by rotations on the first component.

Exercise. Prove that an orientable connected surface with an effective S 1 action is equivariantly diffeomorphic to one from the above list. It is clear that if Theorem 3.36 holds for a surface with one orientation, it also holds for the same surface with the opposite orientation. It suffices therefore to prove that the theorem for G-surfaces from the above list equipped with the standard orientation and an arbitrary abstract moment map. Step 2. An abstract moment map Ψ on T2 is just a smooth function on the second factor, T2 /G = (S 1 )2 → R. Let N = T2 ×G D2 = D2 × (S 1 )2 , where G ˜ on acts on D2 in the standard way. Extend Ψ to a smooth G-invariant function Ψ G 1 ˜ N which is constant on N = {0} × (S )2 . The pair (N, Ψ), where N is oriented according to the orientation of T2 , gives a cobordism between (T2 , Ψ) and zero. Step 3. Let Ψ be an abstract moment map on S 2 , i.e., Ψ is just a G-invariant function. As shown in Example 3.33, (S 2 , Ψ) is cobordant to C(a, +∞)+C(b, +∞), where a and b are the values of Ψ at the north and the south poles, respectively. Step 4. By Lemma 3.30 and Example 3.32, R2 with an abstract moment map Ψ is cobordant to C(a, ±∞) for the standard orientation and to C(a, ±∞) otherwise, where a = Φ(0). Step 5. A proper abstract moment map Ψ on the cylinder C factors through a proper function, also denoted by Ψ, on R = C/G. The following three functions are of particular interest to us: Ψ+ (t) = t, Ψ− (t) = −t, and Ψ0 (t) = t2 where t ∈ R.

5. COBORDISMS OF SURFACES

41

Exercise. Prove that for any proper Ψ, the pair (C, Ψ) is cobordant to (C, Ψ± ) or (C, ±Ψ0 ). Let C+ denote the cobordism class of (C, Ψ+ ). By Example 3.34, (C, Ψ) is either cobordant to zero or to ±C. By Example 3.35, C+ = C(a, −∞) + C(a, +∞). This completes the proof of the theorem. Proof of Theorem 3.37. As is immediately clear from Theorem 3.36, the planes C(a, ±∞), a ∈ R, generate Γeff . The relations (3.10) between C(a, ±∞) follow from (3.7) and (3.4). It remains to show that these are the only relations. To this end, let us assume that (3.11)

m X i=1

− λ+ i C(ai , +∞) + λi C(ai , −∞) = 0

for some distinct ai ∈ R and λ± i ∈ Z, and show that (3.11) follows from (3.10). Let (M, Ψ) be the disjoint union of appropriately oriented copies of C with quadratic moment maps as in Example 3.32 such that (M, Ψ) realizes the left-hand side of (3.11). For x ∈ R not equal to a1 , . . . , am , denote by Nx the algebraic number of points in Ψ−1 (x)/G (counted with orientations). Since (M, Ψ) is cobordant to zero by (3.11), Ψ−1 (x)/G is cobordant to zero as a zero-dimensional oriented manifold. Thus Nx = 0 for all x. On the other hand, − Nai + − Nai − = λ+ i − λi ,

when  > 0 is sufficiently small. Since the left-hand side is zero, we conclude that − λ+ i = λi .

Set λi = λ+ i , so that (3.11) turns into the equation (3.12)

m X i=1


λi C(ai , +∞) − C(ai , −∞) = 0.

Let x > max{a1 , . . . , am }. Then X X (3.13) λi = λ+ i = Nx = 0. i

i

Fix b ∈ R. As follows from (3.9),

C(ai , +∞) − C(ai , −∞) = C(b, +∞) − C(b, −∞).

By multiplying each of these equations by λi , adding them up, and taking into account (3.13), we obtain (3.12) or equivalently (3.11). Thus (3.11) does follow from (3.9). Let us now describe the full group Γ of cobordism classes of proper abstract moment maps on tame oriented G-surfaces. Denote by Γm , m > 0, the group of such cobordism classes with G-actions which factor through the natural projection G = S 1 → S 1 /Zm and an (S 1 /Zm )-action effective on every connected component. The following result is nearly obvious. Theorem 3.39. The identification of G = S 1 with S 1 = S 1 /Zm gives rise to an isomorphism Γm → Γeff . The full group of cobordism classes Γ is isomorphic to

42

3. ABSTRACT MOMENT MAPS

M

(3.14)

Γm = Γ⊕N eff ,

m∈N

where N denotes the set of positive integers. Exercise. Prove Theorem 3.39. Notice that (3.14) is based on two facts: (1) There are no relations between groups Γm for different m’s. (2) The group of cobordism classes with trivial G-actions is zero. The latter readily follows from the observation that the domain of a proper abstract map for a trivial G-action must be compact. Finally, let us consider surfaces (M, Ψ) which are not necessarily tame. This means that we allow infinitely many connected components. Consider linear combinations X X mi C(ai , +∞) + (3.15) nj C(bj , −∞) i∈I

j∈J

where mi , nj ∈ Z and ai , bj ∈ R. For the moment map to be proper, we require that • either I is finite or ai → ∞; • either J is finite or bj → ∞; • the moment maps are standard, that is, the moment map on the cylinder C+ is Ψ+ (eiθ , t) = t, and the moment map on C are quadratic. Moreover, we may assume that ai , i ∈ I, are all different and that bj , j ∈ J, are all different. With these assumptions, we have Theorem 3.40. Every oriented surface with a locally effective circle action and proper abstract moment map is properly cobordant to a space of the form (3.15). Moreover, two surfaces of the form (3.15) are cobordant if and only if their difference is a finite combination of the relations (3.10). Finally, in allowing noneffective actions, there is a copy of the above group for each quotient S 1 /Zm , m ∈ N. 6. Cobordisms of linear actions

In Chapter 3 we will prove that (polarized) cobordism classes are generated by linear actions on vector spaces and on vector bundles. We will now show that there are no non-trivial relations between polarized cobordism classes of linear actions on vector spaces. (Note that cobordisms on the right- or left-hand side of (3.10) are not simultaneously polarized.) Let G be a torus. Fix an element η of the Lie algebra g. Let Γηlin denote the subgroup of the cobordism group generated by η-polarized cobordism classes of oriented vector spaces V , equipped with linear G-actions and η-polarized abstract moment maps Ψ. Let [V, α] denote the cobordism class of (V, Ψ) when Ψ(0) = α. (In Proposition 4.18 we will show that any linear G action with V η = {0} admits an η-polarized moment map.) Proposition 3.41. The group Γηlin is freely generated by the classes [V, α], for all α ∈ g∗ and all isomorphism classes of linear G-actions with V η = {0}. In other words, there no relations among these classes in Γηlin .

6. COBORDISMS OF LINEAR ACTIONS

43

Proof. Take any η-proper cobordism between [Vi , αi ], i = 1, . . . , N . The fixed point set in the cobording manifold is a union of compact one-dimensional manifolds with boundary, whose boundary points are the origins of the Vi ’s, and possibly some additional compact manifolds without boundary that are irrelevant for this argument. Because a compact one-manifold with boundary is an interval, the vector spaces Vi are arranged in pairs. Since the cobording abstract moment map is constant on the fixed point set, αi = αj for each such pair. For each component of the cobording fixed point set, the G-action on normal bundle is the same on all fibers, therefore, the representations Vi and Vj are isomorphic for each pair.

CHAPTER 4

The linearization theorem The simplest group actions on manifolds are linear representations on vector spaces. These are not legitimate objects for ordinary equivariant cobordisms, because they are not compact. However, they are legitimate objects for proper cobordisms. In this chapter we will give the linearization theorem, which, roughly, “decomposes” a G-manifold, when G is a torus, into linear G-representations. For a torus action with isolated fixed points on a compact manifold, this theorem can be informally read as G M∼ Tp M, p∈M G

where M G is the fixed point set and Tp M is the tangent space to M at p. For non-isolated fixed points, on the right-hand side we get the normal bundles to the connected components of M G . The linearization theorem holds for manifolds with η-polarized (abstract, or Hamiltonian) moment maps, for any pre-chosen η ∈ g. It implies that the ηpolarized cobordism classes are generated, as a group, by vector spaces and vector bundles. Remark 4.1. Essentially the same is true with proper (not polarized) cobordisms: a proper moment map can be “cut” (` a la Lerman, [Ler1]) so that each resulting piece is polarized with respect to some (not the same) η; the linearization theorem can then be applied to the pieces. See Section 8 of Chapter 5. 1. The simplest case of the linearization theorem To capture the main idea of the linearization theorem, let us concentrate on the moment map and ignore the two-form. The notion of a “moment map without a two-form” was made precise in Chapter 3, where we introduced abstract moment maps. For an action of the circle group S 1 , an abstract moment map is simply a real valued S 1 -invariant function which is locally constant on the fixed point set; it is polarized if it is proper and bounded from below. Proposition 4.2. Let G = S 1 be the circle group. Let M be a manifold with a G-action with isolated fixed points, and let Ψ : M → R be a polarized abstract moment map. Then there exists a cobordism of manifolds with G-actions and polarized abstract moment maps, G (M, Ψ) ∼ (4.1) (Tp M, Ψ# p ), p∈M G 45

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4. THE LINEARIZATION THEOREM

ρ(t)

0

1 t

1 2ε

Figure 4.1. A proper function on (0,1] where for each fixed point p ∈ M G , the vector space Tp M is equipped with the linear isotropy G-action induced from M , and Ψ# p : Tp M → R is a polarized abstract (0) = Ψ(p). moment map with Ψ# p Remark 4.3. The map Ψ# p on the right-hand side of (4.1) is not specified. However, it is unique up to proper cobordism. Indeed, two such maps are cobordant through the trivial cobordism [0, 1]×Tp M with the polarized abstract moment map # ˜ v) = (1 − t)(Ψ# Ψ(t, p )1 (v) + t(Ψp )2 (v). (See also Section 4 of Chapter 3.) Proof of Proposition 4.2. Consider the product (0, 1] × M , with the circle action induced by that on M . This provides a non-compact cobordism of M with the empty set. The function (t, m) 7→ Ψ(m) is an abstract moment map, but it is not proper. We correct this by adding a function ρ(t) that approaches ∞ as t approaches 0, and that vanishes for 21  ≤ t ≤ 1, for some 0 <  < 1. See Figure 4.1. The sum ˜ m) = Ψ(m) + ρ(t), Ψ(t, (t, m) ∈ (0, 1] × M, is polarized. This follows from the fact that ρ(t) is polarized on (0, 1] and Ψ(m) ˜ on the product is polarized on M . If the fixed point set is empty, the function Ψ (0, 1] × M provides a proper cobordism of (M, Ψ) with the empty set. This finishes the proof. ˜ is polarized, but it is not an If the fixed point set is non-empty, the function Ψ abstract moment map. The reason is that the components of the fixed point set in (0, 1] × M are the sets (0, 1] × {p}, where p is a fixed point in M , and the function ˜ is not constant on these components. However, in this case we can take the same Ψ ˜ on the manifold function Ψ G Bp , W = ((0, 1] × M ) r p∈M S 1

where Bp is the set of points in (0, 1] × M which are -close to the point (0, p), i.e., o n 2 Bp = (t, m) ∈ (0, 1] × M | t2 + (distance(m, p)) < 2 ,

with respect to some metric on M . Here  is the same as in the definition of the function ρ(t). Consider the fixed point set W G . The following observation is crucial: If (t, p) ∈ W G , then ρ(·) vanishes on a neighborhood of t.

2. THE HAMILTONIAN LINEARIZATION THEOREM

47

p

Bp

M

(0, 1] Figure 4.2. Proof of the linearization theorem For if ρ(t) 6= 0, then t < , which implies that (t, p) ∈ Bp if p is a fixed point. ˜ to W is an abstract moment map. This map It follows that the restriction of Ψ ˜ to the closed is polarized because it is the restriction of the polarized function Ψ subset W of (0, 1] × M . The removal of Bp creates a boundary component of W that is equivariantly diffeomorphic to the -neighborhood of p in M via the projection map (t, m) 7→ m. We may assume that the metric is chosen so that this -neighborhood is equivariantly diffeomorphic to the tangent space Tp M and the closures of these neighborhoods are disjoint from each other for different p’s. Then there exists an equivariant diffeomorphism G (4.2) Tp M t M. ∂W ∼ = p∈M G

˜ by the inclusion map Tp M → ∂W is a polarized abstract moment The pullback of Ψ # map Ψ# : T M → R such that Ψ# p p p (0) = Ψ(p) and Ψp (v) → ∞. ˜ is a cobordism with the desired properties. The pair (W, Ψ) 2. The Hamiltonian linearization theorem Let G be a torus, g its Lie algebra, and g∗ the dual space. Recall that a map to g is η-polarized, for η ∈ g, if its η-component is proper and bounded from below. A map from a compact space to g∗ is automatically η-polarized for all η. Let G act on a manifold M . For each Lie algebra element η ∈ g, let M η = {ηM = 0} denote the zero set of the corresponding vector field. For a generic η, this set coincides with the fixed point set M G of G. Indeed, M η 6= M G only if η belongs to the infinitesimal stabilizer of a point in M rM G , and these infinitesimal stabilizers form a countable union of proper subspaces of g (see Corollary B.46). For any η, the connected components of the set M η are closed submanifolds F of M . (See Corollary B.40.) Note that these components are compact if Φ is ηpolarized (because Φη |F is both proper and constant). For each component F of M η let N F denotes its normal bundle in M . The group G acts on N F by bundle automorphisms; this action is induced from the G-action on M . The total space of N F is oriented; the orientation is induced from the orientation on M . We can now state the Hamiltonian linearization theorem. ∗

Theorem 4.4 (Hamiltonian linearization theorem). Fix a torus G and an element η ∈ g of its Lie algebra. Let (M, ω, Φ) be a Hamiltonian G-manifold whose

48

4. THE LINEARIZATION THEOREM

moment map Φ is η-polarized. Then there exists an η-polarized Hamiltonian cobordism G (4.3) (N F, ωF# , Φ# (M, ω, Φ) ∼ F ), F ∈π0 (M η )

where for each connected component F of M η , its normal bundle N F is equipped with an invariant closed two-form ωF# and an η-polarized moment map Φ# F , and the pullbacks of ωF# and Φ# to the zero section coincide with the pullbacks of ω and Φ F to F .

We stress that the two-forms ωF# are not induced from the manifold. Even in the case that the two-form ω is identically zero, the linearization theorem is interesting and non-trivial. Proof of Theorem 4.4. Choose an invariant Riemannian metric on M and an 0 <  < 1, such that the -neighborhood of each connected component F of M η is equivariantly diffeomorphic to the normal bundle N F and the closures of these neighborhoods are disjoint for different F ’s. (The exponential map identifies a neighborhood of F with a disc bundle in N F , which, in turn, is isomorphic to all of N F .) Let BF be the set of points in (0, 1] × M which are -close to {0} × F . Let G W = ((0, 1] × M )r BF , (4.4) F

so that

(4.5)

∂W =

G F

∂BF t ({1} × M ) ∼ =

G F

NF t M.

Let ρ : (0, 1] → R be a function such that ρ(t) approaches infinity as t approaches 0 and vanishes for 12  ≤ t ≤ 1. Note that, if ηW = 0 at (t, m), then ρ = 0 on a neighborhood of t.

This guarantees that the one-form β on W defined by  ρ(t) h · , ηW i ηM 6= 0, hηW , ηW i β( · ) =  0 ηM = 0

is smooth. We take the two-form and moment map on W given by ˜ ξ = π ∗ Φξ + β(ξW ), (4.6) ω ˜ = π ∗ ω − dβ and Φ M

M

where πM is the projection to M . It is easy to verify the moment map equation ˜ ξ = ι(ξW )˜ dΦ ω. Substituting ξ = η in (4.6), we see that the η-component of the cobording moment map is ˜ η (t, m) = Φη (m) + ρ(t). (4.7) Φ Because ρ(·) is polarized (proper and bounded from below) on (0, 1] and Φη (·) is polarized on M , the sum (4.7) is polarized on (0, 1] × M , and hence on the closed ˜ is an η-polarized cobordism. subset W . Therefore, (W, ω ˜ , Φ) ˜ to M by the inclusion Because β = 0 near {1} × M , the pullbacks of ω ˜ and Φ map M ∼ {1} × M → ∂W are equal to ω and Φ. = For each component F of M η , the normal bundle N F embeds into ∂W as the boundary component ∂BF , and the zero section F ⊂ N F embeds into ∂W as the

2. THE HAMILTONIAN LINEARIZATION THEOREM

49

˜ to F via the set {} × F . Because β = 0 near this set, the pullbacks of ω ˜ and Φ inclusions F → N F → ∂W are equal to the pullbacks of ω and Φ via the inclusion F → M. ˜ is an η-polarized cobordism with the The Hamiltonian G-manifold (W, ω ˜ , Φ) required properties. Remark 4.5. For future reference we note that the boundary piece M ⊆ ∂W is ∗ a strong deformation retract of W and that T W = πM T M ⊕R where πM : W → M is the projection. Moreover, inclusion map iF : N F → ∂W can be chosen so that the composition πM ◦iF : N F → M , when differentiated along F in the fiber directions, induces the identity map on N F . The two-form and moment map on the right-hand side of (4.3) are not specified. However, they are unique up to cobordism by the following lemma: Lemma 4.6. Let π : E → F be a vector bundle and let a torus act on it by bundle automorphisms. Let ω0 and ω1 be closed invariant two-forms on the total space of E, with moment maps Φ0 and Φ1 that are η-polarized, such that i∗F ω0 = i∗F ω1 and i∗F Φ0 = i∗F Φ1 , where iF : F → E is the inclusion of F into E as the zero section. Then the trivial cobordism [0, 1] × E carries a two-form and an η-polarized moment map which gives rise to a cobordism between (E, ω0 , Φ0 ) and (E, ω1 , Φ1 ). We also note the following special case: Lemma 4.7. Let V be a vector space with a linear torus action. Let ω0 and ω1 be closed invariant two-forms with moment maps Φ0 and Φ1 which are η-polarized and which take the same value at the origin. Then the trivial cobordism [0, 1] × V carries a two-form and an η-polarized moment map giving a cobordism connecting (V, ω0 , Φ0 ) and (V, ω1 , Φ1 ). Proof of Lemma 4.6. Because iF is a homotopy equivalence and i∗F ω0 = the forms ω0 and ω1 are in the same cohomology class. Let β be a one-form such that ω1 = ω0 − dβ. Without loss of generality, we may assume that i∗F β vanishes. (Otherwise, we replace β by β − π ∗ i∗F β.) Furthermore, by the averaging argument, we can assume that β is invariant. The sum Φξ0 + β(ξE ) is a moment map for ω1 which coincides with Φ0 , and therefore with Φ1 , along F . On the trivial cobordism [0, 1] × E, we take the two-form ω0 − d(tβ) and moment map Φξ0 + tβ(ξE ) = (1 − t)Φξ0 + tΦξ1 . i∗F ω1 ,

Remark 4.8. If M η consists of isolated fixed points (which implies M η = M ), equation (4.3) turns into G (4.8) (Tp M, ωp# , Φ# (M, ω, Φ) ∼ p ). G

p∈M G

If ω is symplectic, each Tp M is a symplectic vector space, with the two-form ωp = ω|Tp M induced from M . However, the corresponding quadratic moment map, which is equal to the Hessian of Φ at p, might not be polarized. Still, we can choose the forms ωp# to be symplectic, although different from the forms ωp . See Proposition 4.18. In contrast with this, when there are non-isolated fixed points, the forms ωF# cannot be in general chosen symplectic. For example, if the Duistermaat–Heckman measure for (N F, ωF# , Φ# F ) is positive in one region and negative in another, it cannot arise as a push-forward of a symplectic measure. See Example 4.19.

50

4. THE LINEARIZATION THEOREM

Various structures on the manifold M naturally extend to the linearization theorem cobordism (4.3). Let W denote the cobording manifold (4.4) and πM : W → M the projection map. Remark 4.9. (1) Let L → M be an equivariant complex line bundle. This bundle pulls back to an equivariant complex line bundle L|F on F , and further pulls back to an equivariant complex line bundle LF on the normal bundle N F . Then G L∼ LF F

∗ πM L

via the line bundle on the cobording manifold W . This argument applies also to vector bundles, principal bundles, etc. (2) Let [β] be an equivariant cohomology class on M . (See Appendix C for the relevant definitions.) This class pulls back to an equivariant cohomology class [β|F ] on F , and further pulls back to an equivariant cohomology class [βF ] on the normal bundle N F . Then G [β] ∼ [βF ] F

∗ πM [β]

via the equivariant cohomology class on the cobording manifold W . (3) Recall that an equivariant stable complex structure on M is a fiberwise complex structure on T M ⊕ Rk for some k. For each connected component F of M η , for any η ∈ g, the normal bundle N F inherits a stable complex structure JF . Then G JF ∼ J F

√ ∗ ∗ J ⊕ −1 on T W ⊕ R = πM via the stable complex structure J˜ = πM T M ⊕ R2 ∼ = ∗ πM T M ⊕ C. For more details, see Section 1.3 of Appendix D. We will explicitly state two special cases. The first of these is the linearization theorem for stable complex Hamiltonian G-manifolds with isolated fixed points. We will use it in our cobordism proof of “quantization commutes with reduction”, in Chapter 8. Theorem 4.10. Let G be a torus and (M, ω, Φ) a compact Hamiltonian Gmanifold. Suppose that G acts with isolated fixed points. Let η ∈ g be a generic Lie algebra element, (generic in the sense that M η = M G). Let J be an equivariant stable complex structure on M . For each fixed point p ∈ M G , consider the tangent space Tp M with the orientation, linear isotropy G-action, and complex structure J p that are induced from M . Then there exists an η-polarized Hamiltonian complex cobordism, G (4.9) (Tp M, ωp# , Φ# (M, ω, Φ, J) ∼ p , Jp ), p∈M G

# where ωp# and Φ# p are a two-form and moment map on Tp M such that Φp (0) = Φ(p).

The second special case is the linearization theorem with equivariant cohomology classes. We will use it in our topological version of the Jeffrey-Kirwan theorem, in Section 7 of Chapter 5.

3. THE LINEARIZATION THEOREM FOR ABSTRACT MOMENT MAPS

51

Theorem 4.11. Let (M, ω, Φ) be a Hamiltonian G-manifold with isolated fixed ∗ points. For each equivariant cohomology class [β] ∈ HG (M ), there exists an ηpolarized equivariant cobordism G (M, ω, Φ, [β]) ∼ (Tp M, ωp# , Φ# p , [β(p)]). p∈M G

3. The linearization theorem for abstract moment maps In some contexts it is useful to work with a variant of the linearization theorem which involves abstract moment maps instead of ordinary moment maps. (See [Ka3].) Theorem 4.12 (Linearization theorem for abstract moment maps). Let a torus G act on a manifold M and let Ψ : M → g∗ be an η-polarized abstract moment map, for some η ∈ g. For each component F of the zero set M η = {ηM = 0}, equip its normal bundle N F with the G-action induced from M . Then there exists an η-polarized cobordism G (M, Ψ) ∼ (4.10) (N F, Ψ# F ), F ∈π0 (M η )

where Ψ# F is an η-polarized abstract moment map whose restriction to the zero section coincides with the restriction of Ψ to F . In addition, (1) If M is oriented and N F is equipped with the orientation induced from M , (4.10) is an oriented cobordism. (2) If L → M is an equivariant complex line bundle and LF → N F is the induced equivariant complex line bundle on N F , the cobordism (4.10) carries a line bundle whose restrictions to M and the components N F are L and LF , respectively. (3) If [β] is an equivariant cohomology class on M and [βF ] is the induced equivariant cohomology class on N F , the cobordism (4.10) carries an equivariant cohomology class whose restrictions to M and the components N F are [β] and [βF ], respectively. (4) If J is an equivariant stable complex structure on M and JF is the induced equivariant stable complex structure on N F , the cobordism (4.10) carries an equivariant stable complex structure whose restrictions to M and the components N F are J and JF , respectively. Remark 4.13. The abstract moment map Ψ# F on the right-hand side of (4.10) is unique up to cobordism. Indeed, suppose that Ψr : N F → g∗ , for r = 0, 1, are both η-polarized and coincide on the zero section. Then they are cobordant through the trivial cobordism [0, 1] × N F with the η-polarized abstract moment map (1 − t)Ψ0 + tΨ1 . Proof of Theorem 4.12. Choose an invariant Riemannian metric on M and an 0 <  < 1 such that the -neighborhood of each connected component F of M η is equivariantly diffeomorphic to the normal bundle N F and such that the closures of these neighborhoods are disjoint for different components F . Let BF be F the set of points in (0, 1] × M that are -close to {0} × F . Then W := ((0, 1] × M )r F BF F provides a non-compact equivariant cobordism between M and F N F .

52

4. THE LINEARIZATION THEOREM

Let ρ : (0, 1] → R be a function such that ρ(t) approaches infinity as t approaches 0 and vanishes for 12  ≤ t ≤ 1. For any m ∈ M rM η , let αm ∈ g∗ be such that αm (η) = 1 and αm (ξ) = 0 for all ξ in the infinitesimal stabilizer gm = {ξ | ξM (p) = 0}. For each (t, m) ∈ W rW η , let Ut,m be an open invariant neighborhood which retracts to the orbit of (t, m). The constant function αm is an abstract moment map on this neighborhood. Let ϕi (t, m) be an invariant partition of unity on W rW η , with the support of ϕi contained in the open set Uti ,mi . Define X e m) = Ψ(m) + ρ(t) ϕi (t, m)αmi . Ψ(t, i

e is well defined and Because ρ(·) vanishes on a neighborhood of W η , the function Ψ e is an abstract moment map and is smooth on W . It is not hard to check that Ψ η-polarized. This function restricts to Ψ on M and to Ψ# F on N F . An orientation, line bundle, equivariant cohomology class, and/or stable complex structure on M naturally extend to M ×(0, 1] and restrict to the subset W , exactly as in Remark 4.9. 4. Linear torus actions

The right-hand side of the linearization theorem identity is comprised of linear representations of tori. Let us recall standard facts about such representations. We refer the reader to [Ad] for proofs of these and to Appendix A for our conventions on weights. Every complex linear representation of a torus G splits into complex one– dimensional representations, and these are parametrized by the elements of the weight lattice Z∗G . So every complex linear representation of G is equivalent to the following representation of G on Cd : (4.11)

exp ξ : (z1 , . . . , zd ) 7→ (e−

√ −1α1 (ξ)

z1 , . . . , e−

√ −1αd (ξ)

zd ),

ξ ∈ g.

The weights for the representation, −α1 , . . . , −αd are unique up to permutation. A representation of G on a real vector space is isomorphic to a direct sum of a trivial representation and a representation of the form (4.11) with all αj 6= 0. The weights αj are now determined up to permutation and sign: the one-dimensional representations of G on C with weights αj and −αj are isomorphic through z 7→ z. The elements ±αj ∈ (Z∗G r{0}) /±1 are called the real weights of the representation; see [Ad]. Symplectic G-representations are similar to complex representations: any symplectic vector space with a linear symplectic G-action is isomorphic to Cd with the standard symplectic form and with the action (4.11). The moment map for the action (4.11) is d

(4.12)

Φ(z) = Φ(0) +

1X |zj |2 αj . 2 j=1

This is most easily seen in polar coordinates, where the standard symplectic form is X ω= rj drj ∧ dθj ,

4. LINEAR TORUS ACTIONS

53

the generating vector fields are ξM = − Hence, the moment map

X

αj (ξ)

∂ , ∂θj

Φ(z) = Φ(0) + satisfies dΦξ =

X

ξ ∈ g.

1X 2 r j αj 2

(rj drj ) αj (ξ).

In the rest of this section we assume that Φ(0) = 0. The image of the moment map is the convex polyhedral cone o nX sj αj | s ∈ Rd+ . C(α1 , . . . , αd ) =

The moment map itself is the composition of the map
1 J : Cd → Rd+ , J(z1 , . . . , zd ) = (4.13) |z1 |2 , . . . , |zd |2 2 with the map X (4.14) π : Rd+ → g∗ , π(s) = s j αj , where Rd+ is the positive orthant

Rd+ = {s ∈ Rd | sj ≥ 0, j = 1, . . . , d}. Notice that the G-action (4.11) is given by the homomorphism (4.15)

G → (S 1 )d ,

exp ξ 7→ (eiα1 (ξ) , . . . , eiαd (ξ) ),

followed by the action of (S 1 )d on Cd by −1 (a1 , . . . , ad ) : (z1 , . . . , zd ) 7→ (a−1 1 z1 , . . . ad zd ).

The map (4.13) is the moment map for this (S 1 )d action, and the projection map X Rd → g ∗ , s 7→ s j αj

in (4.14) is the dual to the inclusion (4.15). For α ∈ C(α1 , . . . , αd ) we consider the polytope X (4.16) sj αj = α}. ∆α := {s ∈ Rd+ |

This polytope is the intersection of the positive orthant Rd+ with the affine subspace X A(α) = {x ∈ Rd | xj αj = α}. Equivalently, the polytope ∆α is a level set of the map (4.14): ∆α = π −1 (α). We say that the elements αj of g∗ are polarized if there exists a vector η ∈ g such that

(4.17)

αj (η) > 0 for j = 1, . . . , d.

54

4. THE LINEARIZATION THEOREM

Let us recall some standard facts from convex geometry that relate properties of the αj ’s to properties of the map π, its image C(α1 , . . . , αd ), and its level sets ∆α . Proposition 4.14. Let α1 , . . . , αd be elements of a vector space g∗ . The following conditions are equivalent to each other: (1) The αj are polarized. (2) The convex hull of the αj ’s does not contain the origin 0 ∈ g∗ . (3) The projection map (4.14) is proper. (4) The set ∆α is compact for every α ∈ g∗ . (5) When α = 0, the set ∆α contains only the origin: ∆0 = {0}.

(6) The set ∆α is compact for some α ∈ C(α1 , . . . , αd ). (7) The cone C(α1 , . . . , αd ) is proper, that is, it does not contain a line. Proof. Suppose that the elements αj are polarized. Let η ∈ g be a polarizing vector, so that αj (η) > 0 for all j. Then is positive. If α = π(s) =

P

m := min{αj (η)} si αi , then

X α(η) ≥ m si . P In particular, we cannot have α = 0 when sj = 1. This proves (4.14). If K ⊂ g∗ is compact and π(s) ∈ K, then X M := max α(η) ≥ m sj . (4.18)

α∈K

Hence, the preimage π

−1

(K) is contained in the simplex   X M s ∈ Rd+ | si ≤ . m

Therefore, (4.14) implies (4.14) and (4.14). Let us now assume that D := conv{α1 , . . . , αd } does not contain the origin. Then there exists a closed ball B around the origin which is still disjoint from D. By standard convexity theory, (see, e.g., [Gr¨ u, §2.2]), there exists a hyperplane which separates B from D, that is, there exists a vector η and a real number  such that α(η) ≥  for all α ∈ D and α(η) ≤  for all α ∈ B. Such  must be positive. Hence, α(η) > 0 for all α ∈ D, and, in particular, for all α = αj . This shows that the αj ’s are polarized. Hence, (4.14) implies (4.14). Clearly, if the projection map π is proper, then its level sets, ∆α , are compact. Now, if s = (s1 , . . . , sd ) is in the set n o X (4.19) ∆0 = s | sj αj = 0, sj ≥ 0 , j = 1, . . . , d ,

then so is (ts1 , . . . , tsd ) for all t ≥ 0. So ∆0 is a cone in g∗ . If this cone is also compact, it must be the zero cone. Condition (4.14) is proved. Condition (4.14) follows immediately. We have established the implications (4.14) ⇔ (4.14) ⇒ (4.14) ⇒ (4.14) ⇒ (4.14) ⇒ (4.14). Let us now show that the negation of (4.14) implies the negation of (4.14). Suppose that the convex hull ofP α1 , . . . , αd contains the origin. Let P sj be non-negative real numbers such that sj = 1 and sj αj = 0. For any

4. LINEAR TORUS ACTIONS

55

P α, if σ = (σ1 , . . . , σd ) satisfies σj αj = α and σj ≥ 0 for all j, then so does (σ1 + ts1 , . . . , σd + tsd ) for all t ≥ 0. In other words, the entire ray σ + ts, t ≥ 0, is contained in ∆α , and so ∆α is not compact. Having shown the equivalence of Conditions (4.14)–(4.14), it remains to show that these conditions are equivalent to Condition (4.14). If the cone C(α1 , . . . , αd ) is not proper, then there exists β 6= 0 such that both β and −β are in the cone. 0 Let sP = (s01 , . . . , s0d ) andPs00 = (s001 , . . . , s00d ) be non-negative coefficients such that β = s0j αj and −β = s00j αj . Let s = s0 + s00 = (s1 , . . . , sd ). Then all sj are non-negative and not all zero, and X sj αj = 0. Dividing this equation by the sum s1 + . . . + sd , we conclude that the origin 0 is in the convex hull of α1 , . . . , αd . Hence, the negation of (4.14) implies the negation of (4.14). Finally, assuming the negation P Pof (4.14), let s1 , . . . , sd be non-negative real numbers such that sj αj = 0 and sj = 1. Up to permutation we may assume that s1 6= 0, so that d 1 X −α1 = s j αj s1 j=2

is a non-negative combination of α2 , . . . , αd . The cone C(α1 , . . . , αd ) then contains both α1 and −α1 , so it contains the entire line through α1 . This proves the negation of (4.14). For our purposes, the most important property of moment maps is properness. Proposition 4.15. The moment map d

Φ(z) =

1X |zj |2 αj 2 j=1

is η-polarized if and only if αj (η) > 0 for all j. The moment map Φ is proper if and only if it is polarized. Proof. The moment map is Φ = π ◦ J, where J and π are given by (4.13) and (4.14). Because J is proper and onto, Φ = π ◦ J is proper if and only if π is proper and is η-polarized if and only if π is η-polarized. So we need to show that π is η-polarized if and only if the collection of αj ’s is η-polarized and that π is proper if and only if it is polarized. By Proposition 4.14, applied to R instead of g∗ and αj (η) instead of αj , the map π η is proper if and only if all the αj (η) have the same sign. In this case, clearly, π η is bounded from below if and only if the αj (η) are positive. This proves the first claim. By Proposition 4.14, π is proper if and only if the αj are η-polarized for some η, hence, if and only if π is η-polarized for some η. This proves the second claim. Recall that the Duistermaat–Heckman measure is the push-forward of the Liouville measure via the moment map. We will need the following formula for the Duistermaat–Heckman measure of a linear space. Proposition 4.16. Suppose that the weights αj are polarized and generate Z∗G . Then the density of the Duistermaat–Heckman measure on g∗ (with respect to the

56

4. THE LINEARIZATION THEOREM

Lebesgue measure) is the function α 7→ (2π)d vol(∆α ), where ∆α = {s ∈ Rd+ |

(4.20)

X

sj αj = α},

Here, volumes are normalized as follows. The affine P plane A(α) in which ∆ α lies contains a parallel shift of the lattice {m ∈ Zd | mj αj = 0}. The volume is normalized so that a fundamental chamber for this lattice has volume one. In g ∗ , Lebesgue measure is normalized so that the volume of g∗ /Z∗G is one.
Proof. The map J(z1 , . . . , zd ) = 12 |z1 |2 , . . . , |zd |2 pushes the Liouville measure on Cd to the Lebesgue measure on Rd+ , multiplied by the constant coefficient (2π)d . Indeed, the Liouville measure can be written as |ds1 · · · dsd dθ1 · · · dθd |, where sj = 21 rj2 and rj and θj are polar coordinates. The map to Rd is the projection (s1 , . . . , sd , θ1 , . . . , θd ) 7→ (s1 . . . , sd ),

and the Lebesgue measure on Rd+ is ds1 · · · dsd . Hence, the Duistermaat–Heckman measure on g∗ is equal to the push-forward of the Lebesgue measure on Rd+ (up to (2π)d ) via the projection map (4.14). The proposition follows. We will also need the following fact concerning the further push-forward, to R, by a single component of the moment map. Proposition 4.17. Suppose that the weights αi are η-polarized. Then the measure on R obtained as the push-forward of the Liouville measure on C d via Φη : Cd → R has polynomial growth. Proof. Because the weights αj are η-polarized, the number m := min{α1 (η), . . . , αd (η)} is positive. For all z, d

Φη (z) =

1X 1 |zj |2 αj (η) ≥ mkzk2 . 2 j=1 2

Thus, if |Φη (z)| ≤ x, then kzk2 ≤ 2x/m. Hence, the push-measure, evaluated on the interval [−x, x], is no larger than the volume of the ball {z ∈ Cd | kzk2 ≤ 2x/m}. Since this volume has polynomial growth in x, Proposition 4.17 follows. 5. The right-hand side of the linearization theorems The Hamiltonian linearization theorem involves certain data that is not given explicitly: the symplectic form and moment map on the right-hand side of (4.3). The theorem becomes more useful once we show that this data can be given by an explicit formula. In this section we give explicit formulas for two-forms and η-polarized moment maps on vector spaces and vector bundles. These can be used, by Lemma 4.6, as formulas for the right-hand side of the linearization theorem. Our first formula applies to the case when a torus acts with isolated fixed points and the vector η is generic. In particular, we show that the forms ωp# can be chosen symplectic.

5. THE RIGHT-HAND SIDE OF THE LINEARIZATION THEOREMS

57

Proposition 4.18. Consider the action of a torus G on Cd with weights −α1 , . . . , −αd ∈ Z∗G :   √ √ (4.21) exp ξ : (z1 , . . . , zd ) 7→ e− −1α1 (ξ) z1 , . . . , e− −1αd (ξ) zd .

Let η ∈ g be a Lie algebra element such that αj (η) < 0 for 1 ≤ j ≤ r and αj (η) > 0 for r + 1 ≤ j ≤ d. Let 1 = . . . = r = −1 and r+1 = . . . = d = 1. Set α# j =  j αj . Then d X j dxj ∧ dyj ω# = j=1

is a symplectic form with η-polarized moment map d

Φ# (z) = Φ# (0) +

1X |zj |2 α# j . 2 j=1

The vector η is called a polarizing vector ; the weights α# j are called the polarized weights. Proof. The proof is by a straight-forward computation. See Section 4. We recall that every linear torus action on a vector space V is isomorphic to an action of the form (4.21); see Section 4. Also, if the vector field ηV vanishes only at the origin (as is the case for V = Tp M when F = {p} on the right-hand side of (4.3)), then αj (η) 6= 0 for all j, and we may assume that αj < 0 exactly if 1 ≤ j ≤ r. Therefore, any linear action that occurs on the right-hand side of (4.3) can be brought to the form as in Proposition 4.18. We now consider the case when the set M η = {ηM = 0} is not discrete. (This happens if G acts with some non-isolated fixed points, or, more generally, M G is discrete, but η is not generic). Suppose that the manifold M is equipped with an equivariant stable complex structure. Then N F becomes a complex vector bundle. In the rest of this section, we derive an explicit formula for the twoform ω # and η-polarized moment map Φ# on the total space of a G-equivariant complex vector bundle π : E → F , where F is compact and connected, E = N F , and η ∈ g is a Lie algebra element such that ηE vanishes exactly on the zero section: E η = F . This provides an explicit formula for the right-hand side of the linearization theorem identity by Propositions 4.6 and D.19 (due to which an equivariant complex structure on N F carries the same information as a stable complex structure on F and a fiberwise complex structure on N F ). We construct ω # and Φ# using Sternberg’s “minimal coupling” procedure, following [GLS]. Let H ⊆ G be the closure of the one-parameter subgroup of G generated by η. By decomposing the fiberwise action of H into isotypical components, we obtain a decomposition E = E1 ⊕ . . . ⊕ Es such that each Ej is a G-equivariant complex vector bundle over F and H acts on Ej fiberwise as multiplication by the inverse of a character ρj : H → S 1 . Let αj = dρj ∈ h∗ be the corresponding weight. We may assume that αj (η) < 0 exactly if 1 ≤ j ≤ r. Let 1 = . . . = r = −1 and r+1 = . . . = s = 1. Let α# j = j αj be the “polarized weights”. Choose an invariant fiberwise Hermitian product on each Ej , and let Pj → F be the corresponding unitary frame bundle. Then Pj is a principal bundle with

58

4. THE LINEARIZATION THEOREM

structure group U(mj ), where mj is the rank of Ej , and Ej = Pj ×U(mj ) Cmj is the associated bundle. Let P → F be the fiberwise product P = P1 ×F . . . ×F Ps . This is a principal bundle for the group K = U(m1 ) × . . . × U(ms ), and E is the associated bundle P ×K Cm with m = m1 + . . . + ms . We will obtain the two-form ω # by realizing E as a (pre-)symplectic reduction. Let Θ be a connection one-form on P ; it associates to each u ∈ T P an element Θ(u) of the Lie algebra k of K. Pairing with the dual space k∗ , we get a real valued ˜ on P × k∗ given by (u, α) 7→ hΘ(u), βi for (u, α) ∈ T(p,β) (P × k∗ ). one-form Θ Consider the product P × k∗ × Cm with the closed two-form s X ˜+ σj# + π ∗ ωF , ω ˜ = −dΘ j=1

where σj is the standard symplectic form on the jth factor in Cm = Cm1 ×. . .×Cms and σj# = j σj . Let K act diagonally through the principal action on P . More precisely, we let k ∈ K act by k −1 (to convert the principal right action into a left action), the coadjoint action on k∗ , and the standard action on Cm . The moment map for this action is (p, β, z) 7→ −β + ΦK (z), where ΦK : Cm → k∗ is the moment map for the linear K-action on Cm with symplectic form σ1# + . . . + σs# . The zero level set is Z = {(p, β, z) | β = ΦK (z)} ∼ = P × Cm and the reduced space is m Z/K = P ×K C = E. The two-form ω ˜ on Z descends to a two-form on the reduced space, whose pullback to P × Cm , which is obtained by substituting β = ΦK (z) in the formula for ω ˜ , is ω # = −dhΘ, ΦK (z)i + #

s X

σj# + π ∗ ωF .

j=1

The restriction of ω to the zero section is ωF . The group G acts on P on the left by bundle automorphisms, with generating vector fields ξP , ξ ∈ g. This induces an action on P ×K Cm = E whose moment map for the reduced two-form ω # is
ξ Φ# ([p, z]) = hΘ(ξP ), ΦK (z)i + ΦξF (π(p)), ξ ∈ g. The restriction of this moment map to the zero section is ΦF . The η-component of Φ# is
η Φ# = ΦηK (z) + ΦηF (π(p)), Ps η = 21 j=1 kzj k2 α# j (η) + ΦF (π(p)),

for z = (z1 , . . . , zs ) where zj ∈ Cmj . Because α# j (η) > 0 for all j and F is compact, the η-component is proper and bounded from below. In other words, the moment map that we have constructed is η-polarized, as required. 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas In Chapter 2 we introduced the Duistermaat–Heckman measure, which is the push-forward of the Liouville measure via the moment map, and its Fourier transform, the Duistermaat–Heckman oscillatory integral. There are explicit formulas

6. THE D-H AND G-L-S FORMULAS

59

which express these invariants in terms of infinitesimal data at the fixed points: these are the Duistermaat–Heckman formula for the oscillatory integral, and the Guillemin–Lerman–Sternberg formula for the push-forward measure. In this section we derive these formulas from the Hamiltonian linearization theorem. The Duistermaat–Heckman measure is invariant under proper Hamiltonian cobordism by Theorem 2.24. Therefore, the Hamiltonian linearization theorem implies that the Duistermaat–Heckman measure is a sum of Duistermaat–Heckman measures associated to vector spaces, or to vector bundles. For instance, for a torus action with isolated fixed points, we get X DH(M,ω,Φ) = (4.22) DH(Tp M,ωp# ,Φ# . p ) p∈M G

We can derive explicit formulas for the summands in (4.22) from the explicit formulas for (Tp M, ωp# , Φ# p ), which were obtained in Section 5. Namely, let −α1,p , . . . , −αd,p be the isotropy weights for the G-action on Tp M , and let α# j,p be the corresponding polarized weights (see Proposition 4.18). Let  p be equal to 1 or −1 according to whether the number of j’s such that α# j,p = −αj,p is even or odd. At this point we note that the αj,p are defined if M is stable complex (in particular, if M is symplectic), but otherwise each αj,p is defined only up to a sign. However, p is well defined if we require the isomorphism Tp M ∼ = Cd giving the form (4.21) to preserve the complex orientation. Moreover, the product Qd j=1 αj,p is also well defined (as a function on g). Let us now apply Proposition 4.16 to each Tp M and adjust for the orientations. Then (4.22) turns into the Guillemin–Lerman–Sternberg formula X (4.23) DH(M,ω,Φ) = Lebesgue measure × (2π)d p vol(∆p,α ), p∈M G

where

n o X ∆p,α = s ∈ Rd+ | Φ(p) + s j α# j,p = α .

This formula, when M is compact and ω is symplectic, was obtained in [GLS] by Guillemin, Lerman, and Sternberg. We allow M to be non-compact, as long as the moment map Φ is η-polarized. Each summand of (4.23) is a measure on g∗ whose push-forward via η : g∗ → R has polynomial growth as was proved in Proposition 4.17. This implies that the function ethx,ηi , x ∈ g∗ , can be integrated against this measure when t < 0. If the number of fixed points is finite, integrating ethx,ηi against the measures on the leftand right-hand sides of (4.23) gives the equality  d Z X 2π ethΦ(p),ηi 1 (4.24) ethΦ,ηi ω d = p − . Qd # d! M t j=1 αj,p (η) p∈M G Qd Qd Because j=1 α# j,p = p j=1 αj,p , the signs p cancel, and (4.24) turns into Z X  2π d ethΦ(p),ηi 1 (4.25) ethΦ,ηi ω d = . − Qd d! M t α (η) j,p G j=1 p∈M

We proved the equality (4.25) for all η ∈ g such that Φ is η-polarized and for all t < 0. The left- and right-hand sides of (4.25) are well defined when η is replaced by any ξ ∈ g ⊗ C which is not in the kernel of αj,p for any j, p and where t is any

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4. THE LINEARIZATION THEOREM

nonzero complex number. Moreover, they are analytic functions of such ξ and t. Let us inspect more closely for which ξ and t the equation (4.25) holds. We return to the equality (4.23) of measures on g∗ . For any ξ near η and t with negative real part, the function ethx,ξi can still be integrated against each summand on the right-hand side of (4.23). Since there are finitely many summands, this function can also be integrated against the measure on the left-hand side of (4.23). This integration results in (4.25) with η replaced by ξ and now the formula holds for an open set of t’s and ξ’s. By analytic continuation, (4.25), with η replaced by ξ, S holds for all t ∈ Cr{0} and for all ξ ∈ g ⊗ Cr j,p ker αj,p . Setting t = i, we get the exact stationary phase formula of Duistermaat and Heckman: 1 d!

(4.26)

Z

eiΦ ω d = (2πi)d M

X

p∈M G

eiΦ(p) , Qd j=1 αj,p

S as an equality between analytic functions on gr j,p ker αj,p . We recall the relevant conventions: the isotropy weights are −αj,p , and the moment map definition is dΦξ = ι(ξM )ω. Duistermaat and Heckman originally stated and proved their formula (4.25) for M compact and under the additional assumption that the two-form ω is symplectic. With this assumption, the fixed points for the torus action are precisely the critical points for the moment map, and, if the action has isolated fixed points, the components Φξ of the moment map, for generic ξ, are non-degenerate Morse functions. The stationary phase approximation then gives 1 d!

Z

eithΦ,ηi ω d = M



2πi t

d X ithΦ(p),ηi an error term e Q + of order t−d−1 αj,p (η) p∈M G as t → ∞.

See [GS1, Chapter I]. The Duistermaat–Heckman formula (4.24) asserts that the error term is identically zero, so that the stationary phase approximation gives an exact expression for the oscillatory integral on the left. Berline and Vergne, [BV1], showed that the Duistermaat–Heckman formula is a special case of the localization theorem for equivariant differential forms. Atiyah and Bott [AB2] interpreted the Duistermaat–Heckman formula in the context of topological equivariant cohomology. These arguments clarified that for compact manifolds the Duistermaat–Heckman formula is purely topological and holds without any non-degeneracy assumption on ω. (See Section 7 of Appendix C.) Prato and Wu [PW] proved the Duistermaat–Heckman formula in the form (4.26) for non-compact symplectic manifolds whose fixed point set is finite and whose moment map has some component that is proper and bounded from below. In fact, their work served as one of the motivations for us to develop this theory. These results of Prato and Wu have been further extended by Paradan, [Par1], to the case where the fixed point set may have an infinite number of connected components. Let us now consider the case where the fixed point set M G is not discrete, or, more generally, where the polarizing vector η ∈ g is not generic. Again, the Hamiltonian linearization theorem, together with the cobordism invariance of the Duistermaat–Heckman measure (Theorem 2.24), immediately imply a formula for

6. THE D-H AND G-L-S FORMULAS

61

−

=

+

Figure 4.3. The Guillemin-Lerman-Sternberg formula for CP2 =

+

+ −

Figure 4.4. The Guillemin-Lerman-Sternberg formula for CP2 , with a non-generic choice of “polarizing vector” the Duistermaat–Heckman measure: (4.27)

DH(M,ω,Φ) =

X

F ∈π0 (M η )

DH(N F,ω# ,Φ# ) . F

F

The terms on the right-hand side were computed explicitly by Guillemin and Cannas da Silva in [CG]. Similarly to the case where the fixed points are isolated, integrating the measures on the left- and right-hand sides of (4.27) against the function eth·,ηi and taking analytic continuation, one obtains a version of the Duistermaat– Heckman formula (4.26) which applies when fixed points are not isolated or η is not generic. A version of the Duistermaat–Heckman formula for non-isolated fixed points was already given in [DH2] and a version of the Guillemin–Lerman–Sternberg formula for non-isolated fixed points and for orbifolds was obtained by Guillemin and Cannas da Silva in [CG]. Our formulas generalize those to the cases where M is not compact or when M η is larger than M G . See Example 4.19. Example 4.19. Take M = CP2 with the Fubini–Study symplectic form, whose P3 pullback to S 5 ⊂ C3 is the standard two-form j=1 dxj ∧ dyj . The linear action of G = S 1 × S 1 on C3 by scalar multiplication on the first two coordinates induces an action on CP2 . The image of the moment map is the Lebesgue measure on a triangle. The action on CP2 has three isolated fixed points; their images are the vertices of the triangle. Each of the summands DH Tp M is the Lebesgue measure on the region between two rays and vanishes outside. The Guillemin-LermanSternberg formula exhibits the triangle as a combination of three such “wedges”; see Figure 4.3. Notice that this involves a choice of direction in which the infinite rays are pointing; this is the choice of the “polarizing” vector η. Let η be the generator of the first factor in G = S 1 × S 1 . The zero set M η consists of one copy of CP1 and one isolated fixed point. The (generalized) GuilleminLerman-Sternberg formula gives the combination illustrated in Figure 4.4, in which the first summand is the Duistermaat–Heckman (signed) measure for the normal bundle of CP1 in CP2 .

CHAPTER 5

Reduction and applications We begin this chapter by describing “symplectic” reduction of closed but not necessarily non-degenerate two-forms. We then generalize this procedure to an abstract moment map, while keeping track of various additional structures. This treatment is partially taken from [CKT], where we carefully reduced orientations, almost complex structures, stable complex structures, and Spinc structures. (In this book we postpone Spinc reduction to Section 3.4 of Appendix D.) We then prove the Duistermaat–Heckman theorem about the variation of the reduced symplectic form. Next, we explain the reduction of K¨ ahler structures and consider an important special case—toric varieties, as reductions of Cd . We prove that cobordism commutes with reduction, i.e., cobordant spaces have cobordant reductions. Together with the linearization theorem of Chapter 4, this implies that, for a Hamiltonian torus action with isolated fixed points, the reduced space is cobordant to a disjoint union of (stable complex) toric varieties. This important fact was observed by Shaun Martin in his study of the cohomology ring of reduced spaces [Mart1]. We use this fact in Chapter 8, in our cobordism proof of “quantization commutes with reduction”. Finally, we give a topological interpretation of the Jeffrey–Kirwan localization, which was one of the motivations for the development of this cobordism theory. 1. (Pre-)symplectic reduction Let (M, ω, Φ) be a Hamiltonian G-space for a torus G. Recall that by definition the moment map Φ is G-invariant. Hence, G acts on each level set of the moment map Φ. The reduced space is the quotient Z = Φ−1 (α).

Mα = Z/G,

It is useful to keep in mind the inclusion-quotient diagram (5.1)

Z π↓ Mα

i

,→ M

More generally, we may consider the reduced space Mα = Φ−1 (α)/G whenever G is a Lie group acting properly on M and the value α is in the subspace (g∗ )G which is fixed under the coadjoint action (e.g., α = 0), so that G acts on Φ−1 (α). (If in the non-abelian case α is not preserved by the coadjoint action, the reduction procedure requires some modifications. For example, the G-action on Z should be replaced by the Gα -action.) Suppose that α is a regular value for g∗ . Then the level set Φ−1 (α) is a manifold of dimension dim M − dim G, by the implicit function theorem. Also, the 63

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5. REDUCTION AND APPLICATIONS

G-action on Φ−1 (α) is locally free. This means that the stabilizers are discrete, or, equivalently, that the infinitesimal stabilizer gp = {ξ ∈ g | ξM (p) = 0} (see Section 1.5 of Appendix B) is trivial for all p. Indeed, since α is a regular value, dΦξp 6= 0 for any ξ ∈ g. Then, since dΦξ |p = ι(ξM )ω|p ,

we see that ξM (p) 6= 0 for any ξ ∈ g which exactly means that the action is locally free, i.e., gp = 0. The quotient Mα = Z/G is then an orbifold and the map Z → Mα is a principal G-orbi-bundle. (See Section 3.1 of Appendix B.) In the special case that G acts on Z freely, Mα is a manifold and Z → Mα is a principal G-bundle. In particular, we have dim Mα = dim M − 2 dim G

if α is regular.

(Pre-)symplectic reduction asserts that the two-form ω on M descends to a two-form ωα on the quotient Mα . Originally symplectic reduction was introduced for symplectic forms (see, e.g., [MW]); hence the name. However, as was later observed, it is sufficient to assume that ω is closed and G-invariant: Theorem 5.1 (Symplectic reduction). Let (M, ω, Φ) be a Hamiltonian G-space. Suppose that α ∈ (g∗ )G is a regular value for Φ; or, more generally, that the level set Φ−1 (α) is a manifold and G acts on it locally freely. Then there exists a unique closed two-form ωα on Mα such that π ∗ ωα = i∗ ω, where π : Z → Mα is the quotient map and i : Z → M is the inclusion map. The reduced form ωα is non-degenerate on Mα if and only if the form ω is non-degenerate on M at the points of Φ−1 (α). If a Lie group G acts properly and locally freely on a manifold Z, a differential form β on Z is the pullback of some form on the orbifold G/Z if and only if β is basic, meaning that β satisfies the following two conditions. First, β is G-invariant. Second, β is horizontal, meaning that ι(ξZ )β = 0 for all the vector fields ξZ , ξ ∈ g, that generate the action. Proof of Theorem 5.1. The two-form i∗ ω is G-invariant because ω is Ginvariant; it is horizontal because ι(ξM )i∗ ω = dΦξ ◦ i, which vanishes because Φ is constant on Φ−1 (α). Hence, i∗ ω is basic, and there exists an ωα such that π ∗ ωα = i∗ ω. To prove the non-degeneracy assertion of the theorem, first note that the null space of ω in Z is equal to the sum of the null space of ω in M and the tangent space to the orbit. In other words, for m ∈ Z, {u ∈ Tm Z | ω(u, v) = 0 for all v ∈ Tm Z} = {u ∈ Tm M | ω(u, v) = 0 for all v ∈ Tm M } + {ξM (m) | ξ ∈ g}.

We leave the proof of this fact to the reader as an exercise. Furthermore, a vector u belongs to the null space of ω in Z if and only if its image π∗ u in T Mα belongs to the null space of ωα in Mα . This implies that ω is non-degenerate at the points of Z if and only ωα is non-degenerate. Example 5.2. Let S 1 act on Cn+1 by scalar Pnmultiplication. The moment map for the standard symplectic form is Φ(z) = 12 i=0 |zi |2 . Its level set Z = Φ−1 (α)

2. REDUCTION FOR ABSTRACT MOMENT MAPS

65

√ is the (2n + 1)-sphere in Cn+1 of radius 2α. The quotient Mα = Z/S 1 is CPn . The reduced symplectic form ωα is the Fubini–Study form on CPn . Example 5.3. Take the phase space of a system of N particles, R6N , with R3 acting by translations, as described in Example 2.3. The reduced space Mα is obtained by fixing the total linear momentum and ignoring simultaneous translations of the N particles. Hence, Mα describes the system relative to the center of mass. Symplectic reduction provides a notion of a quotient in the symplectic category. Namely, suppose that we have a symplectic manifold M with an action of a group G. The naive set theoretic quotient M/G is not naturally symplectic; it might even be odd-dimensional. Instead, one defines the symplectic quotient to be the reduced space Φ−1 (0)/G. This is denoted M//G. In spite of its ambiguity (if the moment map is allowed to shift), this definition works well for many purposes. If α is a singular value of the moment map Φ, the reduced space Mα = Φ−1 (α) is a stratified space, and the two-form ω still reduces to a two-form ωα on Mα , in an appropriate sense. See [ACG, SL] and references therein. 2. Reduction for abstract moment maps Let G be a torus. Recall that an abstract moment map on a G-manifold M is a smooth invariant map Ψ : M → g∗ such that for any subgroup H of G, the H-component ΨH : M → h∗ is locally constant on the H-fixed point set M H . (See Chapter 3.) As for ordinary moment maps, we define the reduced space Z = Φ−1 (α),

Mα = Z/G,

and consider the inclusion-quotient diagram Z π↓ Mα

i

,→ M

2.1. Regular reduced spaces. If (M, ω) is a symplectic manifold with a torus action and Φ is a moment map, the torus acts locally freely on the regular level sets of Φ. The same holds for abstract moment maps; the proof is only slightly harder: Lemma 5.4. Let a torus G act on a manifold M with an abstract moment map Ψ. If α is a regular value of Ψ, the G-action on the level set Ψ−1 (α) is locally free. Proof. It is enough to prove that every point whose stabilizer is not discrete is a critical point for Ψ. First, let us assume that G is a circle. Let F be a component of its fixed point set. Since Ψ is constant on F , the restriction of dΨ to the tangent bundle of F is zero. Since Ψ is invariant under the circle action, the restriction of dΨ to the normal bundle of F is zero. Indeed, at each p ∈ F the normal space to F is a vector space with a circle action that only fixes the origin. On such a vector space, for every invariant function, the origin is a critical point. Hence, every fixed point for the circle action is a critical point for the abstract moment map.

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5. REDUCTION AND APPLICATIONS

Now, let G be a torus of any dimension. We need to show that any point m ∈ M whose stabilizer has positive dimension is a critical point for Ψ. Let H be a circle subgroup of the stabilizer of m. The previous paragraph, applied to the action of H, implies that m is a critical point for the H-component of Ψ, hence for Ψ. By Lemma 5.4, if α is a regular value of Ψ, the reduced space Mα is an orbifold. If Φ is proper, Mα is compact. 2.2. Orientations on reduced spaces. Let M be a manifold with a proper action of a torus G and an abstract moment map Ψ : M → g∗ .

Let α ∈ g∗ be a regular value for Ψ. Recall that if G is not a torus, we assume for the sake of simplicity that α = 0 or, more generally, that α is G-invariant. As above, consider the reduced space Mα = Z/G ,

Z = Φ−1 (α).

In this section we show that an orientation on M naturally induces an orientation on Mα and that an equivariant stable complex structure on M naturally induces a stable complex structure on Mα . We have an exact sequence of vector bundles over Z, (5.2)

0 → T Z → T M |Z → N Z → 0,

which follows immediately from the definition of the normal bundle. By the implicit function theorem, the normal bundle is trivial: (5.3)

N Z = Z × g∗

and an isomorphism Tp M/Tp Z ∼ = g∗ is given by the map dΨ. The tangent spaces to the G-orbits form a sub-bundle of T Z which we denote T O. This bundle is also trivial: (5.4)

T O = Z × g,

where an isomorphism g ∼ = Tp (G · p) is obtained by sending ξ to ξM (p). By the definition of the reduced space, the sequence (5.5)

0 → T O → T Z → π ∗ T Mα → 0

is exact. From (5.2)–(5.5), we obtain an isomorphism of G-equivariant vector bundles, ∼ π ∗ T Mα ⊕ g ⊕ g ∗ . (5.6) T M |Z =

This isomorphism is not canonical: it depends on the choices of splittings of the exact sequences. However, different choices lead to equivariantly homotopic isomorphisms (5.6). Indeed, the difference between two equivariant splittings of (5.2) is an invariant section of the vector bundle over Z whose fiber over p ∈ Z is formed by linear maps from N Zp to Tp Z. The space of such sections is contractible. A similar argument applies to the splittings of (5.5). The vector space g⊕g∗ is naturally oriented: a basis for g together with the dual basis in g∗ determine a canonical (i.e., independent of the basis of g) orientation of g ⊕ g∗ . Then by (5.6), an orientation on M induces an orientation on Mα .

2. REDUCTION FOR ABSTRACT MOMENT MAPS

67

2.3. Stable complex reduction. In this section we will show that a Gequivariant stable complex structure J on M induces a stable complex structure Jα on the reduced space Mα . By definition, J is an invariant complex structure on the fibers of a Whitney sum T M ⊕ Rk , where Rk is equipped with the trivial G-action. We use two equivalence relations on such structures. Suppose that J0 and J1 are complex structures on T M ⊕ Rk0 and T M ⊕ Rk1 . These stable complex structures are bundle equivalent if T M ⊕ Rk0 and T M ⊕ Rk1 become equivariantly isomorphic complex vector bundles after adding some trivial complex vector bundles. The structures J0 and J1 are homotopic if the complex structures on T M ⊕ Rk0 and T M ⊕ Rk1 become equivariantly homotopic after adding some trivial complex vector bundles. Homotopic structures are always bundle equivalent, but not vice versa. We refer the reader to Section 1 of Appendix D for a detailed discussion of these notions. From (5.6) we obtain an isomorphism (5.7)

π ∗ T Mα ⊕ g ⊕ g ∗ ⊕ R k ∼ = TZ M ⊕ R k .

The structure J transports through this isomorphism to a G-invariant complex structure on the fibers of π ∗ T Mα ⊕ g ⊕ g∗ ⊕ Rk . This complex structure descends to a complex structure on the fibers of T Mα ⊕ g ⊕ g∗ ⊕ Rk . We identify (5.8)

g ⊕ g ∗ ⊕ Rk ∼ = Rs

where s = 2 dim G + k, and obtain a complex structure on the fibers of T Mα ⊕ R s , i.e., a stable complex structure Jα on Mα . The isomorphism (5.7) is canonical up to homotopy. We insist that the linear isomorphism (5.8) respect the orientations, hence, it is also determined uniquely up to homotopy. Consequently, the reduced stable complex structure Jα is canonical up to homotopy. Homotopic equivariant stable complex structures on M reduce to homotopic stable complex structures on Mα , and bundle equivalent equivariant stable complex structures on M reduce to bundle equivalent stable complex structures on Mα . 2.4. Relation with symplectic reduction. We recall that an (invariant) symplectic structure naturally determines an (invariant) almost complex structure, unique up to homotopy, and hence an (equivariant) stable complex structure, unique up to homotopy. (See Section 1.3 of Appendix D.) In this case, the two notions of reductions are consistent with each other. Namely, let (M, ω) be a symplectic manifold and Ψ a genuine moment map for a G-action on M . The reduced space Mα is again a symplectic manifold, with symplectic form ωα . Let J be an equivariant stable complex structure on M that is associated with ω, and let J0 be a stable complex structure on Mα that is associated with ωα . We claim that J0 is homotopic to the stable complex structure Jα that is obtained from J by reduction. Choose a basis of g and the dual basis in g∗ . Then the isomorphism (5.6) takes the form (5.9)

TZ M ∼ = π ∗ T Mα ⊕ R2 dim G .

We choose this isomorphism in such a way that ω descends to the form ωα ⊕ σ, where σ is the standard symplectic form on R2 dim G . (The fact that this is possible

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can be seen, e.g., from the proof of the symplectic reduction theorem using local coordinates. We leave the details to the reader as an exercise.) Let Jstd be the standard complex structure on R2 dim G = Cdim G . This structure is compatible with the standard symplectic form. Let Jα0 be an almost complex structure on Mα that is compatible with ωα . Then π ∗ Jα0 ⊕ Jstd transports via (5.9) to an invariant almost complex structure J 0 on M that is compatible with ω. From the definition of reduction, (J 0 )α = Jα0 . Because J 0 is compatible with ω, J 0 is homotopic to J, and hence Jα0 is homotopic to Jα , as required. In contrast with the symplectic case, in general an almost complex structure on M need not descend to an almost complex structure on Mα . In fact, starting with an almost complex structure J on M , let us reduce J to a stable complex structure Jα on Mα with respect to a moment map that is unrelated to J. Then Jα may fail to be homotopic, or even bundle equivalent, to an almost complex structure on Mα . This is the main reason that stable complex structures arise in the context of this book. Example 5.5.P Consider M = Cn with the diagonal action of G = S 1 and moment map Ψ(z) = |zj |2 . The reduction at Ψ = 1 is the complex projective space CPn−1 . Let J be the non-standard complex structure on Cn whose complex coordinates are (z 1 , . . . , z r , zr+1 , . . . , zn ) for 0 ≤ r ≤ n. Note that J is not compatible with the standard symplectic structure on Cn for which Ψ is a genuine moment map. We claim that unless r = 0, or r = n and is odd, the reduced stable complex structure Jα is not induced by any almost complex structure on CPn−1 . Indeed, an easy calculation shows that the total Chern class of (T CPn−1 , Jα ) is (1 − u)r (1 + u)n−r , where u is a suitably chosen generator of H 2 (CPn−1 ; Z). The integral of the top Chern class cn over CPn−1 is (−1)r−1 r + (−1)r (n − r). Recall that for any complex vector bundle the top Chern class is equal to the Euler class; see, e.g., [MiSt]. Therefore, if Jα is induced by an almost complex structure, this Chern number must be equal to the Euler number χ(CPn−1 ) = n. In other words, we must have (−1)r−1 r + (−1)r (n − r) = n, which can happen only when r = 0 or when r = n and is odd.

2.5. Reduction of cohomology. A two-form on M does not naturally descend to Mα . However, an equivariant cohomology class does. Here we just outline the construction of reduction in cohomology; for more details and relevant definitions we refer the reader to Section 2 of Appendix C. If the G-action on Z is free, the natural pullback map ∗ (Z) π ∗ : H ∗ (Z/G) → HG

is an isomorphism (for cohomology with real coefficients). The inverse map ∗ HG (Z) → H ∗ (Z/G)

is called the Cartan map. To each equivariant cohomology class c on M we associate an ordinary cohomology class cα on Mα by applying the Cartan map to the restriction of c to Z: (5.10)

π ∗ cα = i∗ c.

3. THE DUISTERMAAT–HECKMAN THEOREM

69

The map c 7→ cα

(5.11)

is the famous Kirwan map; it is a ring homomorphism from the equivariant cohomology of M to the ordinary cohomology of the reduced space Mα . Kirwan’s surjectivity theorem asserts that if Ψ is a moment map for a symplectic form on a compact manifold, then the map (5.11) is onto. The reduction of the triple (M, Ψ, c) is defined to be the pair (Mα , cα ). 3. The Duistermaat–Heckman theorem Let (M, ω, Φ) be a Hamiltonian G-space with a proper moment map Φ : M → g∗ .

Denote by g∗reg the subset of g∗ consisting of the regular values of Φ. The set g∗reg is open (because Φ is proper) and dense (by Sard’s theorem). The theorem of Duistermaat and Heckman is concerned with the following question: How does the pair (Mα , ωα ) vary as α varies in g∗reg ? The answer depends on the topology of the orbifold fibration π : Z → Mα and, in particular, on the curvature class of this fibration. Let us first recall the definition of this class. 3.1. Variation of the reduced form. From now on, we assume that G is a torus. Let α be a regular value of Φ. Over a neighborhood of α, the moment map Φ is a proper submersion. Hence, Φ is a fibration (by Ehresmann’s lemma, which asserts that a proper submersion is locally a projection). This remains true equivariantly: the G-spaces Φ−1 (α + ε), for ε ∈ g∗ sufficiently small, are equivariantly diffeomorphic to each other. Therefore, the reduced forms ωα+ε can be considered as all living on Mα . Let [ωα+ε ] ∈ H 2 (Mα ) denote the cohomology class of ωα+ε . Theorem 5.6 ([DH1]). For ε ∈ g∗ near 0, 2

[ωα+ε ] = [ωα ] − hε, ci

where c ∈ H (Mα ) ⊗ g is the curvature class of the fibration π : Z → Mα . Proof. Let W be a small neighborhood of α, so that we may identify its preimage as Φ−1 (W ) = Z × W . Let τ be a (g-valued) connection one-form on the principal G-orbi-bundle (5.12)

p : Φ−1 (W ) → Φ−1 (W )/G.

The curvature is a g-valued closed two-form F on (Z/G)×W such that p∗ F = −dτ . Its restriction, Fα+ to (Z/G) × {α + }, represents the curvature class c of the fibration Z → Z/G, for each α +  ∈ W . We need to show that the cohomology class [ωα+ ] − h, ci = [ωα+ − h, Fα+ i] on Z/G is independent of . This would follow from homotopy invariance if we show that the forms ωα+ − h, Fα+ i

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5. REDUCTION AND APPLICATIONS

are the restriction to the fibers of a closed form on (Z/G) × W . Indeed, these forms are the restrictions to the fibers of the form which pulls back to the closed form ω + d hΦ − α, τ i

on Z × W .

3.2. Volume of reduced spaces. Consider the regular reduced spaces Mα = Φ−1 (α)/G,

α ∈ g∗reg .

The Duistermaat–Heckman function associates to every regular value α the Liouville volume of the corresponding reduced space Z ωαd , ϕ(α) = volume(Mα ) = Mα d!

where d = 12 dim M − dim G is half the dimension of the reduced space, ωα is the reduced two-form, and we integrate with respect to the reduced orientation. An important aspect of the Duistermaat–Heckman theorem is the following result: Theorem 5.7. The Duistermaat–Heckman function is a polynomial in α of degree at most d = 12 dim M − dim G on each connected component of g∗reg .

Proof. Fix a regular value α in Φ(M ) and consider values of ε ∈ g∗ near 0. By the Duistermaat–Heckman theorem (see Section 3.1), identifying Mα with Mα+ε , we have Z Z d ωα+ 1 (5.13) = (ωα − hε, ci)d , vol(Mα+ε ) = d! d! Mα Mα where

1 1 dim Mα = dim M − dim G 2 2 and c is the curvature class of the fibration Z → Z/G. This is a polynomial in ε of degree at most d. d=

Recall that the Duistermaat–Heckman measure DH M on g∗ is the push-forward via Φ of the Liouville measure. Theorem 5.8. Assume that the action is effective. Then the Duistermaat– Heckman measure can be written as DHM = (2π)dim G ϕ(x)|dx|, where |dx| denotes the Lebesgue measure on g∗ and ϕ(x) is the Duistermaat– Heckman function. Remark 5.9. The Lebesgue measure on g∗ is normalized in such a way that a fundamental chamber for the lattice Z∗G = Hom(ZG , 2πZ) in g∗ has total volume 1. Example 5.10. Let a torus G act on a compact symplectic manifold (M, ω) with a moment map Φ : M → g∗ . Let us assume that 1 dim G = dim M, (5.14) 2 and also that the action is effective. Then (M, ω, Φ) is called a Delzant space. (See Section 2 of Chapter 2.) By the convexity theorem, the image ∆ = Φ(M ) is a convex polytope. The dimension assumption (5.14) implies that the regular reduced spaces Mα are zero-dimensional, and hence finite. Because the reduced

3. THE DUISTERMAAT–HECKMAN THEOREM

71

spaces are connected, they are single points, and their Liouville volumes are all equal to one. The corresponding Duistermaat–Heckman measure is (2π)dim G |dx| on ∆ and zero outside ∆, by Theorem 5.8. In particular, vol(M ) = (2π)dim G vol(∆), R where the volume on the left is the Liouville volume, M ω n /n!, and the volume on the right is the Euclidean volume of ∆. In particular, for the standard S 1 -action on S 2 with moment map the height function, the Duistermaat–Heckman measure is 2π times the Lebesgue measure on the interval [−1, 1]; cf. Example 2.6.

(5.15)

Proof of Theorem 5.8. Consider first a neighborhood of a free orbit in M . On such a neighborhood there exists a coordinate system θ1 , . . . , θk , x1 , . . . , xk , y1 , . . . , y2d , where k = dim G, θi ∈ R/2πZ, and xi are coordinates on g∗ , such that the G-action is generated by the vector fields ∂θ∂ 1 , . . . , ∂θ∂k and the moment map is (x1 , . . . , xk ). The coordinates yj descend to coordinates on each reduced space Mα . (The fact that the components xi of the moment map can be taken as coordinates follows from the fact that the points of a free orbit are regular points for the moment map. The existence of the coordinates θi and yi on a level set for the moment map follows from the slice theorem. These coordinates can be extended to neighboring level sets; this follows from Ehresmann’s lemma, which guarantees that a proper submersion is locally a projection.) Hamilton’s equation,   ∂ ι ω = dxi , ∂θi implies that ω must have the form X X ω= dθi ∧ dxi + eij dxi ∧ dxj (5.16) X X + fij dxj ∧ dyj + gij dyi ∧ dyj , where the sums are over the appropriate indices i, j, and where the coefficients are functions of the xi ’s and yj ’s only. The reduced form on Mα = Φ−1 (α)/G is ωα = gij (α, y)dyi ∧ dyj .

Taking the top wedge product of (5.16), we see that ω n /n! = ±dθ1 ∧ . . . ∧ dθk ∧ dx1 ∧ . . . ∧ dxk ∧ (ωα )d /d!.

When we push ω n /n! forward, by Fubini’s theorem we can first integrate with respect to the θi ’s. This contributes the factor (2π)k to the measure. Then we integrate the form ωαd /d!, which leads to the (contribution of the neighborhood to the) Duistermaat–Heckman function, and then integrate with respect to the xi ’s over g∗ . To finish the proof, we argue as follows. Let %j be an invariant partition of unit, defined on the union of the free orbits in M , such that each %j is supported in a neighborhood with coordinates as described above. Because the action is free on and dense set, the measure defined by integrating the differential form P an open n % ω /n! on this union is equal to the Liouville measure on M . For each j, j j let fj be the function which associates to every regular value α of Φ the integral

72

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5. REDUCTION AND APPLICATIONS

P %j ωαd /d!. Then fj is equal to the Duistermaat–Heckman function on regular values. Finally, a slight modification of the above computation shows that the push-forward to g∗ of %j ω n /n! is equal to fj |dx| on g∗ . The theorem follows. Mα

Theorems 5.8 and 5.7 imply the following theorem of Duistermaat and Heckman: Theorem 5.11. The Duistermaat–Heckman measure on g∗ has a piecewise polynomial density with respect to the Lebesgue measure. The polynomials on g∗ equal to the density function of the Duistermaat– Heckman measure on the connected components of g∗reg are called the Duistermaat– Heckman polynomials. The book [GLS] contains explicit recipes for computing these polynomials, examples, pictures, and relations with representation theory. When the manifold is a coadjoint orbit for a compact Lie group with the action of the maximal torus of the group, the values of the Duistermaat–Heckman polynomials approximate the multiplicities of weights in irreducible representations of the group. This was, in fact, Heckman’s original motivation for introducing this push-forward measure; see [He]. 4. K¨ ahler reduction Let (M, ω, Φ) be a Hamiltonian G-manifold, equipped with an invariant K¨ ahler structure. Then every regular reduced space is a K¨ ahler orbifold. This was shown by Guillemin and Sternberg in [GS3]. In this section we sketch their argument. Recall that a K¨ ahler structure on a symplectic manifold is a compatible complex structure. A complex structure is given by complex local coordinates such that transition functions are holomorphic. This induces an almost complex structure on M , i.e., a fiberwise complex structure, given by a bundle map J : TM → TM such that J 2 = −identity. Most almost complex structures are not integrable, i.e., do not come from genuine complex structures. The compatibility condition is that (5.17)

h·, ·i = ω(·, J·)

is a (positive definite) Riemannian metric on M . Suppose that the group G is compact. Let GC be its complexification. For example, if G is a subgroup of (S 1 )n , then its Lie algebra g is contained in iRn , and the complexification of G is the subgroup GC = G · exp(ig) of (C× )n . Suppose also that the moment map Φ is proper. The G-action naturally extends to a holomorphic GC action, with the additional generating vector fields JζM , ζ ∈ g. The definitions of the moment map and the metric (5.17) imply that dΦξ (·) = h·, −JξM i, that is, the action of the “non-compact part” of GC is generated by the gradient vector fields of Φξ , for ξ ∈ g. This, in turn, implies that for any regular level set Z = Φ−1 (α) the map g × Z → M given by (ζ, z) 7→ exp(iζ) · z is open. Therefore, W = GC · Z is an open subset of M . This is the “stable” subset of M with respect to the Gaction and the level α ∈ g∗ , in the sense of Geometric Invariant Theory (G.I.T.).

5. THE COMPLEX DELZANT CONSTRUCTION

73

The group GC ∼ = g × G acts properly and locally freely on W ∼ = g × Z, hence, the G.I.T. quotient (5.18)

M//G := W/GC

is a complex orbifold. (See Corollary (B.32).) For every p ∈ Z, the stabilizer group of p in GC is finite and contained in G. Moreover, every GC -orbit in W intersects Z transversely, in a unique G-orbit. Hence, the natural inclusion map Z ⊂ W descends to a diffeomorphism Z/G → W/GC . In this way, the reduced space Z/G becomes a complex orbifold. Remark 5.12. The complex structure on the reduced space is compatible with the reduced symplectic structure (see Section 2.4), i.e., the entire K¨ ahler structure descends to the reduced space. However, note that the Riemann metric associated with the K¨ ahler structure on the reduced space is not “induced” from the Riemannian metric on Z. For example, the metric on Cn is flat, and the metric on the reduced space CPn−1 is not flat. We refer the reader to [Kir] for a broader discussion of the G.I.T. in algebraic geometry and its relation with the symplectic quotient. Remark 5.13. K¨ ahler reduction is consistent with stable complex reduction. Namely, if J is the stable complex structure associated to the complex manifold M , and Jα is its reduction (as described in Section 2.3), then Jα is (homotopic to) the stable complex structure associated to the complex orbifold Z/G. 5. The complex Delzant construction 5.1. Symplectic reduction of Cd . Let a compact abelian Lie group G act effectively on Cd with weights −α1 , . . . , −αd ∈ g∗ and moment map d

(5.19)

Φ(z) =

1X |zi |2 αi . 2 i=1

Assume that the weights are polarized, so that the moment map is proper. (See Proposition 4.15.) Let α ∈ g∗ be a regular value for Φ. Consider the level set Z = Φ−1 (α)

and the reduced space Mα = Z/G. We also consider the action of Td = (S 1 )d on Cd with moment map
1 (5.20) |z1 |2 , . . . , |zd |2 , J(z) = 2

so that G acts as a subgroup of Td . Because the action of Td commutes with the action of G, there is a residual Hamiltonian action of Td on Mα . This action is not effective: the subgroup G acts trivially on Mα . However, the quotient group K = Td /G acts faithfully on Mα .

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5. REDUCTION AND APPLICATIONS

Since the Td -moment map (5.20) is G-invariant, it descends to a map J : Mα → R . This map is a moment map for the (non-effective) action of Td on Mα : Hamilton’s equation for J follows immediately from Hamilton’s equation for J. The image of this map is the polytope X (5.21) ∆α = {s ∈ Rd | sj ≥ 0 for all j, and sj αj = α}, d

which is the intersection of the orthant and the affine space

Rd+ = {s ∈ Rd | sj ≥ 0 for j = 1, . . . , d} A(α) = {x ∈ Rd |

X

xj αj = α}.

By applying an appropriate translation, we can ensure that the moment map J takes values in the target space k∗ . Since K is a quotient group of Td , the target space k∗ is a subspace of (Rd )∗ = Rd . In fact, this subspace is just n o X (5.22) A(0) = x ∈ Rd | xj α j = 0 .

To identify the polytope (5.21) with a polytope that actually lies in the space (5.22), one has to specify the value of the K-moment map at some point, p ∈ Mα , and subtract from it the value of the Td -moment map at p. Let this difference be ν. Then, as a subset of k∗ , i.e., of the space (5.22), the image of the K-moment map is ∆ = ∆α − ν. Let us describe this image as a polytope in Rn . The quotient K = Td /G is a compact connected abelian Lie group. Hence it is isomorphic to Tn for n = d − dim G. Let us fix an isomorphism between K and Tn . Then we have an exact sequence ρ π 1 −→ G −→ Td −→ Tn −→ 1, where ρ : G → Td is the inclusion map and π : Td → Td /K = Tn is the quotient map. On the infinitesimal level, this becomes (α1 ,... ,αd )

π

∗ 0 −→ g −−−−−−−→ Rd −→ Rn −→ 0,

where −αj are the weights for the G-action on Cd , and, dually, π

L

∗ 0 −→ Rn −→ Rd −→ g∗ −→ 0,

where

L(x1 , . . . , xd ) =

X

xi α i .

Let us translate the moment polytope ∆α ⊂ Rd+ into Rn , as described above: pick any value ν ∈ ∆α ; then the moment polytope in Rn is which is equal to

∆ = {y ∈ Rn | π ∗ (y) + ν ∈ Rd+ },

{y ∈ Rn | hei , π ∗ (y) + νi ≥ 0 for all i = 1, . . . , d},

where ei are the standard basis elements. Finally, this polytope is equal to the intersection of half-planes: (5.23)

∆u,λ =

d \

i=1

{y ∈ Rn | hui , yi + λi ≥ 0},

5. THE COMPLEX DELZANT CONSTRUCTION

where λi = hei , νi and

75

ui = π∗ (ei ).

5.2. The symplectic Delzant construction. Every convex polytope ∆ in Rn can be written in the form (5.23), where ui are the normal vectors to the facets. Hence, every convex polytope can be expressed as an “affine slice of R d+ ”, by reversing the above construction. Moreover, if the polytope is integral, that is, if the ui can be chosen in Zn , the polytope actually arises as the moment polytope for a reduction of Cd . Namely, define G to be the kernel of the projection map π : Td → T n whose linearization π∗ : R d → R n

sends the basis element ei to the vector ui . Let Φ : Cd → g∗

be the corresponding moment map. Let α ∈ g∗ be the image of λ ∈ Rd under the natural projection L : Rd → g∗ that is dual to the inclusion map G ,→ Td . Then the quotient Td /G ∼ = Tn acts on the reduced space Mα = Φ−1 (α)/G with a moment ∼ image ∆α = ∆u,λ . We will now describe the conditions on the polytope ∆ which guarantee that the reduced space Mα is actually a manifold or an orbifold. Denote by F∆ the collection of subsets I of {1, . . . , d} which correspond to faces of ∆. In other words, I ∈ F∆ if and only if there exists y ∈ ∆ such that hui , yi + λi = 0 exactly if i ∈ I. The polytope (5.23) is called simple if for every I ∈ F∆ , the normal vectors ui , for i ∈ I, are linearly independent. If, in addition, these normal vectors can be extended to a Z-basis of the lattice Zn , the polytope (5.23) is called a regular polytope, or, equivalently, a Delzant polytope. Remark 5.14. To check whether or not a polytope is simple, or is Delzant, it is enough to check the above condition for I’s which correspond to vertices of ∆. Proposition 5.15. If the integral polytope ∆ is simple, the corresponding reduced space Mα is an orbifold. If the polytope is Delzant, the reduced space is a manifold. Proof. In the construction described above, the polytope ∆ = ∆u,λ is translated by the map y 7→ π ∗ (y) + λ to

∆α = {s ∈ Rd+ |

X

sj αj = α}.

For y ∈ R , we have hui , yi + λi = 0 if and only if π ∗ (y) + λ lies on the coordinate hyperplane {si = 0} in Rd+ . The moment map pre-image of such a point y consists of elements (z1 , . . . , zd ) of Φ−1 (α) such that zi = 0 exactly if i ∈ I. The Td -stabilizer of such an element is TI = {λ ∈ Td | λi = 1 ∀i 6∈ I}. n

Hence, the Td -stabilizers of points z ∈ Φ−1 (α) are exactly the subtori TI for I ∈ F∆ .

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The G-stabilizers which occur in the level set Φ−1 (α) are the intersections G ∩ TI for I ∈ F∆ . The intersection G ∩ TI is the kernel of the map TI → Tn whose linearization sends ei to ui for all i ∈ I. If ui , for i ∈ I, are linearly independent, then G ∩ TI is finite. Therefore, if the polytope is simple, G acts on Φ−1 (α) with finite stabilizers, and the quotient Φ−1 (α)/G is an orbifold. If the vectors ui , for i ∈ I, extend to a Z-basis of Zn , then G ∩ TI is trivial. Therefore, if the polytope is Delzant, G acts on Φ−1 (α) freely, and the quotient Φ−1 (α)/G is a manifold. We have shown that the reduction Mα of Cd by a closed subgroup of Td at a regular value α is a Hamiltonian K-orbifold for K = Td /G ∼ = Tn . We have 1 dim Mα , 2 that is, Mα is a Delzant space. (See Section 2 of Chapter 2.) Suppose that G acts freely on the level set Φ−1 (α), so that Mα is a manifold, not an orbifold. Then we have shown that the moment map image of Mα is a Delzant polytope. Moreover, starting from any Delzant polytope ∆ in Rn , we showed how to produce, by reduction of Cd , a Delzant manifold whose moment polytope is ∆. By Delzant’s classification theorem, a Delzant manifold is determined by its moment map image, which is always a Delzant polytope. This implies that, up to isomorphism, every Delzant manifold can be obtained by the procedure that we described above. dim Tn = d − dim G =

Remark 5.16. The Delzant classification theorem has been generalized to orbifolds by Lerman and Tolman. (See Section 2 of Chapter 2.) It follows that every Delzant orbifold can also be obtained from Cd by reduction. 5.3. The complex Delzant construction. In this section we describe the complex Delzant construction, i.e., the construction of a complex toric variety starting from a Delzant polytope ∆; see [Au, Cox1, Cox2] for further details and applications. Given ∆, one may construct a subgroup G of Td as described above, and take the reduced space Mα = Z/G with Z = Φ−1 (α) where Φ : Cd → g∗ is the Gmoment map. Let GC = G · exp(ig) be the complexification of G. Recall from Section 4 that the “semi-stable” set W = GC · Z

is an open subset of Cd , and that we have a natural diffeomorphism (5.24)

∼ =

Mα = Z/G → W/GC

which shows that Mα as a complex orbifold. The resulting complex toric variety is the quotient W/GC . We will now give an explicit description of the open set W of Cd in terms of the combinatorics of the polytope ∆. First, let us record which sets of coordinates may vanish simultaneously at the points of Z: (5.25)

Fα = {I(z) | z ∈ Z = Φ−1 (α)},

where I(z) = {i | zi = 0}.

We note that this collection is the same as the set of faces of the moment polytope (5.21): Fα = F ∆ .

5. THE COMPLEX DELZANT CONSTRUCTION

77

Recall that an alcove is a connected component of the set of regular values of Φ contained in the image Φ(Cd ). Lemma 5.17. The collection Fα is independent of α in an alcove. Proof. Over an alcove, the moment map Φ is an (S 1 )d -invariant proper submersion, and hence an (S 1 )d -invariant fibration. In particular, the (S 1 )d orbit types that occur in Φ−1 (α) are the same for all α in the alcove. The (S 1 )d orbit type of z is determined by the set {i | zi = 0}. The complex torus (C× )d acts on Cd by coordinate-wise multiplication. The orbits for this action are the sets OI = {z ∈ Cd | zi = 0 if and only if i ∈ I} = {0}I × (C× )drI for all subsets I of {1, . . . , d}. We also consider the open sets WI = CI × (C× )drI , where C× = Cr{0}. The following characterization of the set W = GC · Z is crucial: Theorem 5.18. (5.26)

W =

[

WI =

I∈Fα

[

OI .

I∈Fα

Here will reproduce the proof from [Guil1, Appendix 1]. Proof of the easy part of Theorem 5.18. By definition, [ (C× )d · Z = OI . I∈Fα

Because GC ⊆ (C× )d , this implies that W ⊆

[

OI .

I∈Fα

Also, since OI ⊆ WI for each I, we have [ [ OI ⊆ WI . I∈Fα

I∈Fα

The remaining inclusions WI ⊆ W will be proved in Section 5.6. The above characterization of the set W of semi-stable points completes the complex Delzant construction. Namely, given a polytope ∆ in Rn with d facets, the combinatorics of its facets determines an open subset W of Cd by (5.26), the slopes of its facets determine the subgroup G of Td as described above, and the corresponding toric variety is the quotient W/GC .

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5.4. Orbicharts on a toric variety. An important consequence of Theorem 5.18 is that it provides an explicit open covering of the toric variety W/G C which is convenient to work with. Namely, Mα is the union of the open sets WI /GC , for I ∈ Fα . Moreover, it is enough to consider only WI /GC such that |I| = n. We already know that Mα is an orbifold. We will now exhibit each element of the above covering as an orbichart on Mα , that is, the quotient of Cn by a finite group. Moreover, the intersections of these sets will be finite quotients of products of copies of C and C× . We will deduce, in Chapter 8, that this is a “good covering” for the purpose of computing the Dolbeault cohomology of a toric variety. Suppose that I ∈ Fα and |I| = n. Consider the map ψ : CI → WI /GC , obtained as the composition
CI → CI × {1}drI ,→ WI → WI /GC .

This map descends to a smooth one-to-one immersion (5.27)

CI /ΓI → WI /GC ,

where ΓI = {g ∈ GC | gi = 1 for all i ∈ drI}.

We argue that this group is finite. Indeed, because I ∈ Fα and by the definition of Fα , there exists z ∈ Z = Φ−1 (α) such that zi = 0 exactly if i ∈ I. The stabilizer of z in GC is precisely ΓI . Since α is regular, this stabilizer is finite. Next, consider the group homomorphism (5.28)

GC → (C× )drI

obtained as the composition of the inclusion GC ,→ (C× )d with the projection (C× )d → (C× )drI . Its kernel is precisely ΓI , and, hence, is finite. By dimension count, the homomorphism (5.28) is also onto. This implies that the map (5.27) is onto. Consequently, this map is a diffeomorphism, and hence an orbichart on Mα , as claimed. For |J| < n, there exists I ∈ Fα such that |I| = n and J ⊂ I. This follows, e.g., from the fact that the polytope ∆α is simple. The
diffeomorphism (5.27) identifies the set WJ /GC with the subset CJ × (C× )IrJ /ΓI of CI /ΓI .

5.5. Regular moment values. Let us digress to examine the moment map (5.19) more closely. Recall that a point z is regular for Φ exactly if its stabilizer is discrete. When checking this condition we may restrict our attention to the identity component of G. The action of an element exp ξ of G, for ξ ∈ g, is given by   (exp ξ) : (z1 , . . . , zd ) 7→ e−iα1 (ξ) z1 , . . . , e−iαd (ξ) zd . The stabilizer of z is

(5.29)

{exp ξ | eiαj (ξ) = 1 whenever zj 6= 0}.

Let I be the subset of {1, . . . , d} such that zj = 0 if and only if j ∈ I. The stabilizer (5.29) is discrete if and only if the weights αj , for j ∈ I, span g∗ . This implies the following characterization of the set of regular values of Φ. Set sj = |zj |2 .

5. THE COMPLEX DELZANT CONSTRUCTION

79

Lemma 5.19. An element α ∈ g∗ is a regular value of Φ if and only if for every subset J of {1, . . . , d}, if the equation X α= (5.30) s j αj , sj > 0 j∈J

has a solution, then the weights αj , j ∈ J, span g∗ .

Recall that α is regular if and only if G acts locally freely on the level set Z = Φ−1 (α), and that, in this case, the reduced space Mα = Z/G is an orbifold. One is sometimes interested in the “super-regular” values of Φ, defined as the values α such that G acts freely on Z = Φ−1 (α), so that Mα = Z/G is a manifold. A similar argument gives the following characterization of super-regular values, when G is a torus: Lemma 5.20. An element α ∈ g∗ is a super-regular element of Φ if and only if for every subset J of {1, . . . , d}, if the equation (5.30) has a solution, then the weights αj , j ∈ J, generate the weight lattice Z∗G . We warn the reader that, whereas the set of regular values is open and dense in g∗ (by Sard’s theorem combined with the properness of Φ), this is not true for the set of super-regular values. This set is open, but it might even be empty. In fact, the mere existence of a super-regular value imposes rather severe restrictions on the weights α1 , . . . , αd . 5.6. The semi-stable points in Cd . In this section we complete the proof of Theorem 5.18. We first show the second equality of (5.26): Lemma 5.21.

[

WI =

I∈Fα

Proof. Because WI =

[

OI .

I∈Fα

[

OJ ,

J⊆I

it suffices to prove that if I ∈ Fα and J ⊆ I, then also J ∈ Fα . Indeed, suppose I ∈ Fα . Let z ∈ Φ−1 (α) be such that zi = 0 exactly if i ∈ I. By Lemma 5.17 and because the alcove is open, there exists a neighborhood U of z such that for all α0 ∈ Φ(U ) we have Fα0 = Fα . By perturbing the ith coordinates of z for i ∈ IrJ, we obtain an element z 0 in U such that zi0 = 0 exactly if i ∈ J. Then for α0 = Φ(z 0 ) we have J ∈ Fα0 = Fα . The rest of this section is devoted to completing the proof of Theorem 5.18. Proof of Theorem 5.18. It remains to prove that WI ⊆ W for all I ∈ Fα . By Lemma 5.21, it is enough to prove that OI ⊆ W for all I ∈ Fα . The argument below follows closely [Guil1, Appendix 1]. Pick any I ∈ Fα and any z ∈ OI , so that zj = 0 exactly if j ∈ I. Because W = GC · Φ−1 (α), we need to show that there exists a ∈ GC such that Φ(a · z) = α. The action of the element exp(iη) of GC , for η ∈ g, is given by (exp(iη) · z)j = e−ihαj ,iηi zj = ehαj ,ηi zj .

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Writing a = b exp(iη) with b ∈ G and η ∈ g, we get (5.31)

d

Φ(a · z) =

d

1X 1X 1 X 2hαj ,ηi 2 2 |(a · z)j | αj = |(exp(iη) · z)j | αj = e s j αj , 2 j=1 2 j=1 2 j∈J

1 2 2 |zj | 0

where sj = > 0 for each j ∈ J := {1, . . . , d}rI. By the definition of Fα , there exists z ∈ Z such that zj0 = 0 exactly if j ∈ I. Hence, d

α = Φ(z 0 ) =

(5.32)

X 1X 0 2 s0j αj , |zj | αj = 2 j=1 j∈J

where

s0j

=

1 0 2 2 |zj |

(5.33)

> 0 for each j ∈ J. Thus, α is in the open polyhedral cone nX o s0j αj | s0j > 0 , j ∈ J .

Hence, it is enough to show that the image of the map X (5.34) ehαj ,ζi sj αj f (ζ) = j∈J

contains the cone (5.33). (The image of (5.34) is clearly contained in the cone (5.33); we need the opposite inclusion.) Our first observation is that f (ζ) is the Legendre transformation of the function X ehαj ,ζi sj . F : g → R, F (ζ) = j∈J

In other words, (5.35)

∗

Tζ∗ g.

f (ζ) = dF |ζ ,

when we identify g = Next, we note that the function F is strictly convex, i.e., that the Hessian d2 F is positive definite everywhere. Indeed, X (5.36) d 2 |ζ F = ehαj ,ζi sj αj ⊗ αj . j∈J

hαj ,ζi

Because the coefficients e sj are positive, to show that the bilinear form (5.36) is positive definite, it suffices to show that the weights αj , j ∈ J, generate g. This follows immediately from Lemma 5.19. Lemma 5.22. Suppose that F is strictly convex. Then F has a critical point if and only if F (ζ) −→ ∞. ζ→∞

Proof. See [Guil1, Appendix 1, Theorem 3.2]. The value α is in the image of (5.34) if and only if there exists ζ such that dF |ζ = α, by equation (5.35). Set Fα (ζ) = F (ζ) − α(ζ). Then dFα = dF − α

and d2 Fα = d2 F.

From the first equality we see that α is in the image of (5.34) if and only if Fα has a critical point. From the second equality we conclude that Fα is strictly convex, because F is. By (5.22), Fα has a critical point if and only if (5.37)

Fα (ζ) −→ ∞. ζ→∞

6. COBORDISM OF REDUCED SPACES

81

Substituting (5.32) into the expression for Fα , we have X Fα (ζ) = ehαj ,ζi aj − bj αj (ζ). j∈I

The jth summand approaches ∞ as ζ escapes ker αj , by the following fact. The function g : R → R, g(x) = aex − bx satisfies g(x) −→ ∞ if a and b are positive numbers. x→±∞

Because ∩j∈J ker αj = {0} (by Lemma 5.19), the desired asymptotic behavior (5.37) holds. 6. Cobordism of reduced spaces As we have already pointed out, the fact that “cobordism commutes with reduction” can be used to establish a number of results on the global geometry of Hamiltonian torus actions on symplectic manifolds (cf. [GGK1]). A similar fact holds for abstract moment maps: Theorem 5.23. Let (M, Ψ) and (M 0 , Ψ0 ) be cobordant manifolds with Gactions and proper abstract moment maps. Let α ∈ g∗ be a value that is regular for both Ψ and Ψ0 . Then the corresponding reductions, Mα and Mα0 , are cobordant (through orbifolds). ˜ be a cobording manifold with proper moment map. Proof. Let (W, Ψ) ˜ the quotient Ψ ˜ −1 (α)/G gives a cobordism If α is also a regular value for Ψ, between Mα and Mα0 . ˜ and If α is not a regular value, we choose a β which is a regular value for Ψ which is close enough to α so that the entire interval [α, β] consists of values that are regular for both Ψ and Ψ0 . This is possible because the set of regular values of ˜ is dense (by Sard’s theorem) and the sets of regular values of Ψ and Ψ0 are open Ψ (because these maps are proper). ˜ the orbifold Ψ ˜ −1 (β)/G is a cobordism between Mβ Since β is regular for Ψ, 0 and Mβ . Because the interval [α, β] consists of regular values, the level sets Ψ−1 (α) and −1 Ψ (β) are equivariantly diffeomorphic. This follows from Ehresmann’s lemma (that a proper submersion is a fibration), which remains valid in the presence of a group action. Dividing by the G-action, we see that the reduced spaces Mα and Mβ are diffeomorphic. Similarly, Mα0 and Mβ0 are diffeomorphic. As a consequence, Mα ∼ = Mβ ∼ Mβ0 ∼ = Mα0 . Because diffeomorphic spaces are cobordant, and because the cobordism relation is transitive, the reduced spaces Mα and Mα0 are cobordant. Theorem 5.23 continues to hold in the presence of additional structures. More specifically, we may consider one or more or the following additional structures on (M, Ψ): (1) An orientation; (2) An equivariant stable complex structures, J; (3) A closed two-form, ω, for which Ψ is a genuine moment map; (4) An equivariant cohomology class, c;

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(5) An equivariant complex line bundle, L. Each of these structures gives rise to a structure of the same type on the reduced space Mα , as follows: (1) The reduced orientation (see Section 2.2); (2) The reduced stable complex structure, Jα , (see Section 2.3); (3) The reduced two-form ωα (see Section 1); (4) The reduced cohomology class cα (see Section 2); (5) The reduced line bundle (L|Z )/G, or, more generally, the reduced line bundle Lγ = (L|Z ⊗ C−γ ) /G where γ ∈ Z∗G is any integral weight. Suppose that M 0 carries a structure of the same type. Moreover, suppose that the cobordism between M and M 0 also carries a structure of the same type, extending the given structures on M and M 0 . Then the cobordism of Theorem 5.23, between Mα and Mα0 , will carry the same type of a structure, giving a cobordism between the structures on Mα and Mα0 . This is seen by easy adaptations of the proof of Theorem 5.23. (In the case (3) of a two-form, one should also apply the Duistermaat–Heckman theorem (Theorem 5.6) to express [ωα ] through [ωβ ] and use the fact that the cobordism class of ωα only depends on its cohomology class.) Corollary 5.24. Consider triples (M, Ψ, c), where M is a manifold with a G∗ action, Ψ : M → g∗ a proper abstract moment map, and c ∈ HG (M ) an equivariant ∗ cohomology class. Let cα ∈ H (Mα ) be the reduced cohomology class for a regular value α ∈ g∗ . Then the integral Z cα

Mα

is an invariant of cobordism of the triple (M, Ψ, c).

Proof. This follows from Stokes’ theorem applied to the orbifold that gives a cobordism of the reduced spaces. Corollary 5.24, applied to the equivariant cohomology classes of (ω −Φ−α) d /d! on M , implies that the Duistermaat–Heckman function is an invariant of cobordism. Alternatively, this fact follows from Theorems 2.24 and 5.8. 7. Jeffrey–Kirwan localization ∗

Let Φ : M → g be a moment map for a G-action on a compact symplectic manifold. Suppose that Mred = Φ−1 (0)/G is a regular reduced space. An equi∗ variant cohomology class [c] ∈ HG (M ) descends to an ordinary cohomology class ∗ [cred ] ∈ H (Mred ); see Section 2. An important property of the Kirwan map ∗ κ : HG (M ) → H ∗ (Mred ),

[c] 7→ [cred ]

is that it is onto. Therefore, ∗ H ∗ (Mred ) = MG (M )/ ker κ.

Often, we want to compute the cohomology ring of a space which arises by symplectic reduction, Mred , in terms of the cohomology of M which is often more tractable. For instance, the moduli space of flat connections of a principal bundle arises in this way (cf., for instance, [Jef, Hue].) In fact, the study of this moduli space has been one of the motivations for the investigation of cohomological properties of Hamiltonian group actions. Here we concentrate on situations where Mred is

7. JEFFREY–KIRWAN LOCALIZATION

83

a manifold, or, at worse, an orbifold. By the Poincar´e duality, the kernel of the Kirwan map is determined by the linear functional Z (5.38) [cred ]. I : [c] 7→ Mred

Indeed, [c] is in the kernel if and only if I([c] ∪ [c0 ]) = 0 for every [c0 ]. With this motivation, one would like to obtain a formula for the Kirwan numbers Z ∗ (5.39) (M ). [cred ] , [c] ∈ HG Mred

Jeffrey and Kirwan [JK1] derived such a formula explicitly. They worked with Hamiltonian actions of compact non-abelian groups on compact symplectic manifolds, and expressed the integral (5.39) by an explicit formula that only involves the data at the set of points that are fixed under the action of a maximal torus. Then Guillemin gave a topological interpretation of the Jeffrey–Kirwan formula in the case that G is a torus. Shaun Martin showed, by topological means, that the Jeffrey–Kirwan localization formula for non-abelian groups would follow from such a formula for abelian groups. See [Mart2]. We will now present an approach to the Jeffrey–Kirwan localization based on our cobordism techniques; cf. [GGK1]. Suppose that M is compact, or, more generally, that the moment map Φ is η-polarized for some η ∈ g, i.e., hΦ, ηi is proper and bounded from below. By the Linearization Theorem (Theorem 4.11), G (N F, ωF# , Φ# (M, ω, Φ, [c]) ∼ F , [cF ]). F ∈π0 (M η )

The Kirwan number (5.39) is an invariant of cobordism (Corollary 5.24). This follows from the “cobordism commutes with reduction” theorem (Lemma 5.23) combined with Stokes’ theorem. Hence, Z X Z (cF )red . cred = Mred

F ∈π0 (M η )

(N F )red

Notice that the F -summand is nonzero only if hΦ(F ), ηi < ha, ηi. For this formula to be useful, we need to be able to compute the integrals on the right-hand side. The simplest case is when the torus G acts with isolated fixed points p, so that N F = Tp M is a vector space. If the polarizing vector η is chosen generic, we have Z X Z cred = (cp )red . Mred

p∈M G

(Tp M )red

The space (Tp M )red , being the reduction of a linear space under a linear torus action, is a toric variety, corresponding to the polytope ∆p . (See Section 5). One can explicitly express the Kirwan numbers (5.39) in terms of the polytopes ∆p . This method was used in [Guil2] to compute the Riemann–Roch number of Mred . Metzler applied this method in [Met1, Met2] to give explicit formulas for topological invariants of Mred such as the signature, the Poincar´e polynomial, and the Euler characteristic. In fact, the cobordism method we just outlined applies under more general assumptions. Namely, let Ψ : M → g∗ be an abstract moment map, associated with an action of a torus G and let α ∈ g∗ be a regular value for Ψ. Consider ∗ the reduced space Mred = Ψ−1 (α)/G. The Kirwan map HG (M ) → H ∗ (Mred ) is

84

5. REDUCTION AND APPLICATIONS

no longer onto, however, it is well defined, as are the Kirwan numbers (5.39). The above arguments still allow one to compute these numbers. (Note that in this case one should invoke the Linearization theorem 4.12 for abstract moment maps rather than the Hamiltonian Linearization theorem.) 8. Cutting Lerman’s symplectic cutting [Ler1] is a simple and versatile operation on Hamiltonian G-manifolds. Given a Hamiltonian S 1 -manifold and moment map Φ : M → R with regular value α, the Lerman construction “cuts” M into two Hamiltonian S 1 -manifolds, M+ and M− . Topologically, M+ (resp., M− ) is obtained from the pre-image Φ−1 (α, ∞) (resp., Φ−1 (−∞, α)) by collapsing its boundary, which is the level set Z = Φ−1 (α), along the S 1 -orbits. As a result, we have decompositions (5.40)

M+ = Φ−1 (α, ∞) t Mα

and M− = Φ−1 (−∞, α) t Mα ,

where Mα = Z/S 1 is the reduced space. In fact, in each of the decompositions (5.40), the pieces on the right-hand side fit together smoothly, i.e., M+ and M− are smooth orbifolds. Moreover, M+ and M− inherit the structure of Hamiltonian S 1 -manifolds, and, in the presence of a G-action on M that commutes with the S 1 -action, the G-action descends to M+ and M− . The most common variety of symplectic cutting is that for a Hamiltonian Gmanifold M , where G a torus and Φ is the moment map for a sub-circle S 1 ⊂ G. We may then repeat the construction using other sub-circles and end up with an orbifold which coincides with M on the preimage of a polyhedral region ∆ in g∗ over the interior of ∆ and with appropriate “collapsing” of M over the boundary of ∆. In most applications, one starts with a compact polytope ∆ ⊂ g∗ . Then taking larger and larger ∆’s, one obtains successive “compact approximations” of M . Our original version of the linearization theorem, given in [GGK1], was stated in terms of such compact approximations; this was before non-compact cobordisms were introduced. If ∆ is a proper convex polyhedral cone, then there exists a linear projection ∆ → R, given by β 7→ hβ, ηi for some η ∈ g, which is proper and bounded from below. The resulting cut manifold is then η-polarized. By appropriately dividing g∗ into a finite number of proper convex polyhedral cones, we can cut any manifold M with proper moment map into finitely many pieces, each of which is η-polarized for some η. (This fact was used in Remark 4.1.) Let us assume, for simplicity, that the regular value at which we cut is α = 0. To define the symplectic forms on M+ and M− one can invoke symplectic reduction. For instance, M+ is constructed as the reduction of M × C with respect to the diagonal S 1 -action  M+ = (m, z) ∈ M × C | Φ(m) − |z|2 = 0 /S 1 .  p  The inclusion maps of Φ−1 (0, ∞) and M0 into M+ are given by m 7→ m, Φ(m)

and [m] 7→ [m, 0].

8. CUTTING

85

Cutting can also be performed with an abstract moment maps and with various kinds of additional structures: an orientation, a stable complex structure, a compatible two-form, an equivariant cohomology class, or a line bundle. Recall that in the previous sections we showed that the reduction procedure applies to each of the these structures. Furthermore, each of these structures on M naturally extends to M × C, and, hence, induces a similar kind of a structure on the cut-spaces M+ and M− . Remark 5.25. Reduction and cutting can also be defined for Spinc structures. The reduction of a Spinc structure is rather straight-forward once we have the “destabilization” procedure of Proposition D.55. Cutting of a Spinc structure is more involved and gives a slightly different result than might be expected: let L and L+ be the determinant line bundles for the Spinc structures on M and on M+ . Their restrictions to the open set Φ−1 (0, ∞) coincide. However, the restriction of L+ to M0 is not equal to Lred := L|Z /S 1 , but, rather, to Lred ⊗ N , where N := Z ×S 1 C. See [CKT, Section 6]. In the context of this book, it is important to note that we have a cobordism M ∼ M + t M− .

(5.41)

To see this, we first focus on one orbit in Z. Assume, for simplicity, that this is a free orbit. Its neighborhood in M can be identified with U × S 1 × R, where U is a neighborhood in Z/S 1 = M0 , where S 1 acts on the second factor only, and where R is the coordinate in the normal direction to Z. The orbit in Z gives a point in M0 . Its neighborhood in M+ (resp., M− ) is U × C (resp., U × C). On a neighborhood of a free orbit in Z, the cobordism (5.41) takes the form U × C ∼ (U × C) t (U × C),

where C = S × R. In Example 3.35 of Chapter 3 we showed that 1

(5.42)

C ∼ C t C.

The cobordism (5.41) is obtained by performing the cobordism (5.42) fiberwise over M0 . We now give an explicit construction for the cutting cobordism (5.41). Let  ˜ = (h, m, z) ∈ R × M × C | −1 ≤ h2 − |z|2 ≤ 1 and Φ(m) = ρ(h) , (5.43) W

where ρ(h) is a pre-chosen even function such that ρ(h) = 0 for −1 ≤ h ≤ 1 and ρ0 (h) > 0 for all h > 1. It is not hard to check that (1, 0) and (−1, 0) are regular ˜ is an orbifold with values of the functions h2 − |z|2 and Φ(m) − ρ(h). Hence, W boundary o n oGn oGn p p p ˜ = h = |z|2 + 1 ∂W h = − |z|2 + 1 |z| = h2 + 1 . Let S 1 act on W diagonally, on the M and C factors. It is not hard to check that this action is locally free. On the quotient orbifold ˜ /S 1 W =W we take the S 1 -action induced by the S 1 -action on the M -factor of (5.43) and the abstract moment map [h, m, z] 7→ Φ(m). It is not hard to see that the boundary components of W are isomorphic to M+ , M− , and M , respectively.

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5. REDUCTION AND APPLICATIONS

Finally, we warn the reader that the cutting cobordism does not carry an equivariant cohomology class or a stable complex or Spinc -structure. In the presence of such structures on M , one has 2

M ∼ M + t M− t B

where B is an S -bundle over the reduced space M0 , equipped with a constant abstract moment map.

Part 2

Quantization

CHAPTER 6

Geometric quantization Our goal in the second part of the book is to prove that “quantization commutes with reduction” by means of cobordisms. In this chapter we explain what we mean by “quantization”. For more comprehensive accounts of geometric quantization, see [Ki4, Kost, Wd]. 1. Quantization and group actions By “quantization”, mathematicians refer to a process which associates to a “classical system” its corresponding “quantum system”. This is not a precise notion: there is freedom in the mathematical interpretation of the notions of classical or quantum systems. Also, one requires different axioms in different contexts. One tries, on the one hand, to axiomatize actual physical phenomena, and on the other hand, to impose requirements which are natural (or, at least, realistic) from the mathematical point of view. A discussion of various concepts of quantization can be found in [BW]. For a physical point of view, see, e.g., [Dc]. Usually, one associates to a symplectic manifold (M, ω) a Hilbert space H, and one associates to smooth functions f : M → R skew-adjoint operators Af : H → H. One would like the following Dirac axioms to hold: (1) Poisson bracket of functions passes to commutator of operators: A{f,g} = [Af , Ag ];

(2) Linearity: Aaf +bg = √ aAf + bAg for a, b ∈ C; (3) Normalization: 1 7→ −1 · identity; (4) “Minimality”: any complete family of functions passes to a complete family of operators. In Axiom (4), a family of functions is complete if it separates points almost everywhere on M ; a family of operators is complete if it acts irreducibly on H. It is impossible to quantize the entire Poisson algebra C ∞ (M ) while satisfying these four axioms. This was proved by Gr¨ oenewald and van Hove [Gr¨ oe, VH1, VH2] for R2n , and was later extended to other symplectic manifolds and Poisson algebras; see [GGG] and references therein. One possible compromise is to quantize only a sub-algebra of C ∞ (M ). Recall that the Poisson algebra (C ∞ (M ), {, }), and hence any subalgebra, acts on (M, ω) as a Lie algebra (see Chapter 2). Thus, by (1) and (1), quantization transforms this Lie algebra action on (M, ω) into a Lie algebra action on H. Now suppose that a Lie group G acts on M with a moment map Φ : M → g∗ . The moment map components Φξ , ξ ∈ g, form a Lie sub-algebra of C ∞ (M ), because Φ[ξ,η] = {Φξ , Φη }. Suppose that this Lie sub-algebra is quantizable by the skewadjoint operators Aξ , ξ ∈ g. Then these operators satisfy the commutation relations [Aξ , Aη ] = A[ξ,η] , and hence define an action of the Lie algebra g on H. This action 89

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6. GEOMETRIC QUANTIZATION

may integrate to an action of G, or of a covering of G, on H. (A skew-adjoint operator Aξ generates a unitary flow via Schr¨ odinger’s equation v˙ = Aξ v. These flows generate the G-action.) This property of quantization, that it associates to a Hamiltonian group action on M a linear group action on H, can also be interpreted as functoriality of the quantization, which is a desirable property from the mathematical point of view. Conversely, suppose that to a Hamiltonian G-action we associate a representation of G on H. Infinitesimally, this gives a representation of the Lie algebra g by operators Aξ , ξ ∈ g. Let us associate the operator Aξ to the moment map component Φξ , for each ξ ∈ g. This “quantization”, defined on the Lie subalgebra {Φξ | ξ ∈ g} of C ∞ (M ), satisfies the Dirac axioms (1)–(3). We will view quantization as a process which transforms a Hamiltonian Gaction on M into a representation of G on H, even in cases where this does not follow from the Dirac conditions, for instance, when G is not connected, or when G is a torus (in which case a g-action generally does not integrate to a G-action). The minimality axiom, (4), then becomes (4’) The quantization of a transitive Hamiltonian G-action is an irreducible Grepresentation. By a theorem of Kirillov and Kostant [Ki4, Kost], a transitive Hamiltonian G-manifold is (a cover of) an orbit for the coadjoint action of G on g∗ . The celebrated “orbit method” is a program for constructing representations of Lie groups geometrically by “quantizing” their coadjoint orbits. See [Ki5] for an up-to-date review. 2. Pre-quantization In geometric quantization, the Hilbert space H that one takes is, essentially, a subspace of the space of sections of a certain line bundle L over the symplectic manifold M . The construction of this line bundle is the first and simplest part of the geometric quantization program, and is called pre-quantization. Remark 6.1. An alternative recipe, using Spinc structures instead of line bundles, leads to a quantization that is “better behaved” in several ways but is slightly harder to define. See Section 7.3. 2.1. Pre-quantization line bundles. Consider a manifold M and a closed two-form ω. A pre-quantization line bundle for (M, ω) is a complex line bundle L whose curvature class is the cohomology class [ω]. Equivalently, its first Chern 1 class c1 (L) maps to 2π [ω] under the natural homomorphism (6.1)

i : H 2 (M ; Z) −→ H 2 (M ; R).

See Appendix A for the relevant definitions and conventions. In particular, our 1 convention R is such that the tautological line bundle over CP has curvature class [ω] with CP1 ω = −2π. Since complex line bundles are classified by H 2 (M ; Z) via L 7→ c1 (L), the man1 ifold (M, ω) is pre-quantizable if and only if 2π [ω] is integral (see Proposition C.46). Since ker i is exactly the torsion subgroup of H 2 (M ; Z), the pre-quantization line bundle L is determined by [ω] uniquely up to torsion elements of H 2 (M ; Z). We refer the reader to Section 6 of Appendix C or to [Ki4, Kost] for more details.

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Remark 6.2. In contrast, (M, ω) is “Spinc -pre-quantizable” if and only if 1 − 2π [ω0 ] is integral, for some fixed cohomology class [ω0 ] ∈ H 2 (M ; R). See Section 7.3. 1 2π [ω]

Remark 6.3. Our notion of a pre-quantization line bundle should not be confused with the notion of a “quantum line bundle” used by Vergne and Duflo (see [Ve2]). Their notion is essentially equivalent to a Spinc structure. 2.2. Pre-quantization data. Let (M, ω) be a manifold with a closed two– form. A pre-quantization of (M, ω), or pre-quantization data for (M, ω), is a Hermitian line bundle (L, h, i) equipped with a Hermitian connection ∇ whose curvature is ω. Thus, if (L, h, i , ∇) is a pre-quantization, then L is a pre-quantization line bundle on (M, ω). Equivalently, a pre-quantization of (M, ω) is a principal U(1)-bundle π : P → M and a connection form Θ on P with curvature ω. Recall that Θ is an U(1)-invariant one-form on P satisfying (6.2)

Θ(∂/∂θ) = 1,

where ∂/∂θ is the vector field which generates the principal U (1)-action, and (6.3)

π ∗ ω = −dΘ.

Recall that the one-to-one correspondence between Hermitian line bundles and principal U(1)-bundles associates to (L, h, i) its unit circle bundle P = {v ∈ L | hv, vi = 1},

and, conversely, associates to P the line bundle

L = P ×U(1) C.

The pre-quantization (L, h, i, ∇) uniquely determines the pre-quantization (P, Θ), and vice versa. Explicitly, the covariant derivative satisfies the equation

∇ : Γ(L) → Ω1 (M ; L)

∇s √ = −1s∗ Θ, s whenever s is a section of P ⊂ L. This relation can also be seen geometrically if we think of connections in terms of parallel transport and note that a connection on L is Hermitian if and only if the unit circle bundle P is preserved by parallel transport. In Chapter 7 we will be quantizing symplectic manifolds whose symplectic forms are exact. For such manifolds the pre-quantization picture is particularly simple:

(6.4)

Example 6.4. Let ω be an exact two-form on M and let β be a one-form on M with dβ = −ω. Then (M, ω) can be pre-quantized by the trivial U(1)-bundle P = M × U(1) and the connection one-form Θ = dθ + π ∗ β, where θ is the angle coordinate on U(1). We have already seen that a necessary and sufficient condition for a pre1 quantization line bundle to exist is that the cohomology class 2π [ω] be integral. Now that we also require a connection, we can ask: • When does a pre-quantization exist? • To what extent is it unique?

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We will be interested in pre-quantization data up to gauge equivalence. (A gauge equivalence is an isomorphism of bundles that respects the additional structures.) Two pre-quantizations (L, h, i, ∇) are gauge equivalent if and only if the corresponding pre-quantizations (P, Θ) are gauge equivalent. Proposition 6.5. Let ω be a closed two-form on M such that the cohomology 1 [ω] is integral. Let L → M be any pre-quantization line bundle. Then L class 2π supports a pre-quantization (L, h, i, ∇) for (M, ω). If M is simply connected, the pre-quantization structure on L is unique up to gauge equivalence. More generally, the pre-quantization structures on L are classified by H 1 (M ; R)/H 1 (M ; Z), up to gauge equivalence. The uniqueness question for pre-quantization is discussed in greater detail in [Kost, pages 142-143]. Proof of Proposition 6.5. Fix a Hermitian structure h, i on L and a connection form Θ on the unit circle bundle P ⊂ L. Its curvature lies in the cohomology class [ω], hence, dΘ + π ∗ ω descends to an exact two-form, dα, on M . The difference Θ0 := Θ − π ∗ α is a connection form on P with curvature ω. Hence, (P, Θ0 ) provides a pre-quantization for (M, ω). Two connection forms, Θ and Θ0 , with the same curvature differ by Θ − Θ0 = π ∗ α, where α is a closed one-form on M . These connections differ by a gauge 1 [α] is integral. (The gauge transformation if and only if the cohomology class 2π transformation is then given by a map M → U(1) which pulls back the angle form dθ to α.) Since integral one-forms include all exact one-forms, (6.5)

pre-quantization structures (P, Θ) for (M, ω) on P ∼ H 1 (M ; R) . = 1 gauge equivalence H (M ; Z)

If M is simply connected, H 2 (M ; Z) has no torsion by the universal coefficients theorem, and so P is uniquely determined by [ω], up to isomorphism. Since also H 1 (M ; R) = 0, it follows from (6.5) that the connection form Θ is unique up to a gauge transformation. As a consequence, if M is simply connected, all connection forms Θ with dΘ = −π ∗ ω are equivalent by gauge transformations. Corollary 6.6. Let (L, h, i, ∇) be pre-quantization data for (M, ω). Suppose that M is simply connected and ω is exact. Let β be a one-form on M with dβ = −ω. Then there exists a trivializing section s of L such that (6.6)

∇s √ = −1β. s

The section s is unique up to a non-zero multiplicative constant. The norm of s is constant. Proof. By Proposition 6.5 and Example 6.4, there exists a trivialization P = M × U(1) such that Θ = dθ + π ∗ β. By (6.4), a constant section s satisfies (6.6). To prove the uniqueness, write an arbitrary section of L in the form f s, where f ∈ C ∞ (M ), and observe that the condition (6.6) for f s implies that df = 0, and hence that f is constant.

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2.3. Equivariant pre-quantization line bundles. Recall that a Hamiltonian G-manifold (M, ω, Φ) is, by definition, an oriented G-manifold M equipped with a G-invariant two-form ω and a moment map, i.e., an equivariant map Φ : M → g∗ satisfying (6.7)

dΦξ = ι(ξM )ω

for all ξ ∈ g.

We assume that G is a compact Lie group. A G-equivariant complex line bundle L over M is called a G-equivariant pre-quantization line bundle for (M, ω, Φ) if its first equivariant curvature class is [ω − Φ]. (See Appendix A.) When such a line bundle exists, we say that (M, ω, Φ) is pre-quantizable. The following result is an immediate consequence of Theorem C.47 (also see [HY, Ri]). Theorem 6.7. A G-manifold (M, ω, Φ) is pre-quantizable if and only if the 1 equivariant cohomology class 2π [ω − Φ] is integral, i.e., lies in the image of the natural homomorphism (6.8)

2 2 HG (M ; Z) → HG (M ; R).

Isomorphism classes of equivariant pre-quantization line bundles for (M, ω, Φ) are 2 in a one-to-one correspondence with the elements of HG (M ; Z) which are sent to 1 [ω − Φ] under (6.8). 2π Remark 6.8. The existence of a map Φ satisfying (6.7) only depends on the cohomology class of ω in H 2 (M ; R), for if ω is replaced by ω + dα, where α is a G-invariant one-form, then (6.7) is satisfied with Φξ replaced by Φξ − α(ξM ). In particular, if L a pre-quantization line bundle for (M, ω), the G-action on M might not lift to a G-action on L. A cautionary example is the manifold G itself. The only line bundle for which the G-action lifts is the trivial line bundle; for, if there were such a lift, every G orbit other than the zero section would be a trivializing section. Example 6.9. Let G be compact, connected, and simply connected. Then a 1 [ω] is integral. Hamiltonian G-manifold (M, ω, Φ) is pre-quantizable if and only if 2π Proof. Let (P, Θ) be a (non-equivariant) prequantization circle bundle. The G-action on M lifts to a g action on P . See Corollary 6.18. Because G is connected and simply connected, this g action integrates to a G-action on P . Alternatively, 2 it is not hard to show that in this setting HG (M ) = H 2 (M ), which, by Theorem 6.7, implies the assertion of the example. Example 6.10. Let G be a torus. Then a compact connected Hamiltonian G-manifold (M, ω, Φ) with M G 6= ∅ (e.g., (M, ω) is symplectic) is pre-quantizable 1 if and only if 2π [ω] is integral and the values of Φ on the fixed point set all belong 1 to the weight lattice Z∗G . Moreover, it is enough to assume that 2π [ω] is integral ∗ and that Φ(p) ∈ ZG for just one fixed point p. Proof. Let (P, Θ) be a (non-equivariant) pre-quantization circle bundle. Recall that the lift of the g action to P is given by the vector fields ∂ ξP = (ξM )hor + Φξ ∂θ ∂ where ∂θ generates the principal circle action. Define g ξ : P → P , for ξ ∈ g, by the condition that g tξ , for t ∈ R, is the flow generated by ξP . The G-action on P , if it

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exists, is obtained by letting exp(ξ) act by g ξ . This is well defined exactly if g ξ is the identity map whenever ξ ∈ ZG = ker exp. The map g ξ lifts the action of exp(ξ) on M . In particular, if ξ ∈ ZG , then g ξ lifts the identity map on M . Hence, there exists a function R ξ : M → S 1 such that g ξ (p) = p · R(π(p))

for all p. We have a G-action on P exactly if R ξ is the constant function with value 1 for every ξ ∈ ZG . ∂ , which integrates to Rξ (p) = Over a fixed point p ∈ M G we have ξP = Φξ ∂θ ξ eiΦ (p) . Thus Rξ (p) = 1 exactly if hΦ(p), ξi ∈ 2πZ. This holds for all ξ ∈ ZG exactly if Φ(p) ∈ Z∗G . Because the fixed point set is non-empty, to complete the proof it is enough to show that Rξ is a constant function for every ξ ∈ ZG . We fix ξ ∈ ZG and will show that the function Rξ is constant along any curve in M . We fix a curve γ(s), for s ∈ [0, 1]. We will pull back the bundle P to the cylinder R/Z×[0, 1] by the map ϕ(t, s) = exp(tξ) · γ(s). Consider the function on the cylinder given by Ψ(t, s) = Φξ (γ(s)). Then ϕ∗ (P, Θ) is a pre-quantization bundle for the two-form ∂Ψ dt ∧ ds. ϕ∗ ω = ∂s Let g t , for t ∈ R, be the flow on ϕ∗ P generated by the vector field ∂ (∂t )hor + Ψ ∂θ ∂ ∂ , and ∂θ denotes the where ( )hor denotes the horizontal lift, ∂t is shorthand for ∂t t generator of the principal circle action. Then ϕ intertwines g with the flow g tξ on P . In particular, g 1 is the fiberwise rotation by R(s) = Rξ (γ(s)). To finish, we need to show that R(s) is independent of s. It is enough to show that g t commutes ∂ with the flow hs that is generated by (∂s )hor , the horizontal lift of ∂s := ∂s . For t s this it is enough to show that the vector fields that generate g and h commute. Indeed, by (6.19), ∂Ψ ∂ ∂ =− . [(∂s )hor , (∂t )hor ] = [∂s , ∂t ]hor + ω (∂s , ∂t ) ∂θ ∂s ∂θ Also,     ∂ ∂ ∂Ψ ∂ ∂Ψ ∂ + Ψ (∂s )hor , . (∂s )hor , Ψ = = ∂θ ∂s ∂θ ∂θ ∂s ∂θ   ∂ Summing these, we get the required commutation (∂s )hor , (∂t )hor + ψ ∂θ = 0. In the next section we examine in more detail the constructions involved in the statement of Theorem 6.7.

2.4. Equivariant pre-quantization data. Let L → M be a G-equivariant pre-quantization line bundle for (M, ω, Φ) and π : P → M its unit circle bundle with respect to some Hermitian metric. The pair (P, Θ), where Θ is a connection form on P , is said to be a G-equivariant pre-quantization for (M, ω, Φ) if Θ is G-invariant and ω − Φ is the equivariant curvature of Θ:

(6.9)

π ∗ (ω − Φ) = −dG Θ.

This condition means that π ∗ ω = −dΘ, and, additionally,

(6.10)

π ∗ Φξ = Θ(ξP )

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for all ξ ∈ G. Namely, the “vertical component” of the vector field ξP is Φξ . (The “horizontal component” is determined by π∗ ξP = ξM .) Condition (6.9) implies that [ω −Φ] is the equivariant curvature class of L. (See 1 Appendix A.) Alternatively, the class 2π [ω − Φ] is the image of the equivariant first Chern class of P under the homomorphism (6.8). The following result is an equivariant analog of Proposition 6.5. Proposition 6.11. Let (M, ω, Φ) be a Hamiltonian G-manifold such that the 1 equivariant cohomology class 2π [ω − Φ] is integral. Let L → M be any equivariant pre-quantization line bundle. Then L supports equivariant pre-quantization data (L, h, i , ∇) for (M, ω, Φ). If G is connected, the pre-quantization structures on L 1 1 are classified up to G-equivariant gauge equivalence by HG (M ; R)/HG (M ; Z). The proof of this proposition follows the same lines as the proof of Propo1 [ω − Φ] must be integral for a G-equivariant sition 6.5. Note that the class 2π pre-quantization bundle to exist, by Theorem 6.7. Proof of Proposition 6.11. By Theorem 6.7, there exists a G-equivariant U(1)-principal bundle π : P → M with curvature class [ω − Φ]. Fix a G-invariant connection form Θ on P . (See Appendix A.) Its equivariant curvature lies in the equivariant cohomology class [ω − Φ], so π ∗ (ω − Φ) = −dG Θ + π ∗ dG α

for some G-invariant one-form α on M . Then Θ0 = Θ − π ∗ α is a G-invariant connection on P which satisfies (6.9), and (P, Θ0 ) provides a G-equivariant prequantization for (M, ω, Φ). Let Θ and Θ0 be two G-equivariant pre-quantization structures on P . It is easy to see that Θ − Θ0 = π ∗ α where α is a closed G-invariant one-form on M such 1 α is integral, the connections Θ and Θ0 are that α(ξ) = 0 for all ξ ∈ g. When 2π equivalent by means of a gauge transformation, namely, by a function f : M → U(1) that satisfies the equation f ∗ dθ = α or, equivalently, df = α. This gauge transformation is automatically G-equivariant. Indeed, LξM f = α(ξM ) = 0 which means that f is G-equivariant, for G is assumed to be connected. Conversely, assume that two pre-quantization structures on P with the same equivariant curvature are G-equivariantly gauge equivalent. Then these structures 1 differ by a closed basic one-form, π ∗ α, such that 2π α is integral. 1 Since the first basic cohomology is equal to HG (M ; R) (see Example C.8) the proposition follows. 2.5. The representation of G on the space of sections. To a G-action on (M, ω) we have associated a G-action on a line bundle L → M by bundle automorphisms. From this we obtain a linear G-representation on the vector space of sections of L in the following way. The graph of a section f : M → L is the subset of L defined as graph(f ) = {(m, l) | m ∈ M , l = f (m) ∈ Lm }.

A group element a ∈ G, acting on the total space of L, transforms this graph to the graph of another section, a · f , given by (6.11)

(a · f )(m) = a · (f (a−1 · m)),

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where we use the dot to denote the actions of G on the manifold M , on the bundle L, and on the space of sections. For the trivial bundle L = M × C, the action (6.11) becomes the action (6.12)

(a · f )(m) = f (a−1 · m)

on functions f : M → C, discussed in Appendix A. Infinitesimally, the action (6.11) on the space of sections is given by Kostant’s formula, √ ξ 7→ −∇ξM + −1π ∗ Φξ for all ξ ∈ g, (6.13)

in the presence of a pre-quantization structure on L. Note that the operators on the right-hand side of (6.13) satisfy the quantization axioms (1)–(3) of Section 1, for the functions f = Φξ . When M is equipped with a G-invariant measure µ, the space of L2 -sections of L is a unitary representation of G: the inner product is given by Z hs1 , s2 i = hs1 , s2 iµ, M

where hs1 , s2 i on the right-hand side is the pointwise Hermitian product of sections s1 and s2 of L. A G-invariant measure µ always exists if G is compact or, more generally, the G-action is proper. Finally, if ω is symplectic, a natural choice for µ is the Liouville measure on M . To summarize, given an equivariant pre-quantization for a Hamiltonian Gaction, we can quantize the moment map components Φξ by the operators (6.13) acting on the Hilbert space of sections of L, and the Dirac axioms (1)–(1) are satisfied. 3. Pre-quantization of reduced spaces

3.1. Pre-quantization and reduction. Let G be a torus and (M, ω, Φ) is a pre-quantizable Hamiltonian G-manifold. Recall that if α ∈ g∗ is a regular value of the moment map Φ : M → g∗ , the reduced space at α is the orbifold Mα = Φ−1 (α)/G. This orbifold carries a closed two-form ωα which arises as the reduction of the form ω. To formulate the “quantization commutes with reduction” theorem in Chapter 8, we will need to know when this reduced space is pre-quantizable. A sufficient (but not necessary) condition for this is given in Proposition 6.12 below. Before stating the proposition, we must explain what we mean by the pre-quantization of an orbifold. We prefer to avoid a formal treatment of orbifolds and orbi-bundles, which would have forced us to work with orbi-atlases throughout. For our purposes, it is sufficient to restrict our attention to orbifolds which are presented, i.e., given by the quotient of a manifold by a Lie group which acts with finite stabilizers. If Q → Z is a principal U(1)-bundle on which G acts with finite stabilizers, the quotient Q/G is a principal U(1)-orbi-bundle over the orbifold Z/G. Similarly, if L → Z is a complex line bundle on which G acts with finite stabilizers, the quotient L/G is a complex line orbi-bundle over Z/G, and a G-invariant Hermitian structure on L induces a Hermitian structure on L/G. The differential forms on Z/G can be viewed simply as the G-basic differential forms on Z. (A differential form β on Z is said to be G-basic, if β is G-invariant and horizontal, i.e., ι(ξZ )β = 0 for all ξ ∈ g. See Appendix B for more details.) From this point of view, the

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97

reduced form ωα can be interpreted as just the form ω|Z where Z = Φ−1 (α). With these definitions, the pre-quantization construction described in Section 2.2 easily generalizes to orbifolds. Proposition 6.12. The reduced space (Mα , ωα ) is pre-quantizable if α lies in the weight lattice Z∗G of G. Remark 6.13. When Mα is a manifold, a necessary and sufficient condition for (Mα , ωα ) to be pre-quantizable is stated in Section 2.1: the cohomology class [ωα ] must be integral. It is not hard to show that this condition holds when Mα is smooth and α ∈ Z∗G . However, this criterion does not apply directly to orbifolds. Although the results of Sections 2.1 and 2.2 can be generalized to orbi-bundles over orbifolds, this generalization is non-trivial and here we use a different approach: we explicitly construct a pre-quantization for (Mα , ωα ) as the “reduction” of prequantization data for (M, ω, Φ). Proof of Proposition 6.12. Let (P, Θ) be a pre-quantization for the Hamiltonian G-manifold (M, ω, Φ), i.e, π : P → M is a G-equivariant principal U(1)bundle, dΘ = −π ∗ ω, and Θ(ξP ) = π ∗ Φξ for all ξ ∈ g. Let α be a regular value of Φ and let Q be the restriction of P of the level set Z = Φ−1 (α), i.e., Q = π −1 (Z). Let us first consider the case α = 0. The orbifold P0 = Q/G is a U(1)-bundle over the reduced space M0 = Z/G. These spaces and fibrations form the following commutative diagram: P ←- Q −−−−→ P0       y y y

M ←- Z −−−−→ M0 where the vertical arrows denote taking quotients by G. Since Θ(ξP ) = π ∗ Φξ = 0 on Z, the restriction to Q ⊂ P of the connection form Θ is horizontal with respect to the fibration Q → P0 , and hence is basic. Thus it descends to a connection form Θ0 on the principal orbi-bundle P0 → M0 . The pair (P0 , Θ0 ) provides us with a pre-quantization of (M0 , ω0 ). We leave the details of this argument to the reader as an exercise. This construction does not apply when α is not zero. In this case, the connection one-form on Q does not descend to Q/G. Moreover, the curvature class of the U(1)bundle Q/G → Mα is not [ωα ]. Example 6.14. Consider M = C2 with the standard symplectic structure ω, the action of G = S 1 by a · (z1 , z2 ) = (a−1 z1 , a−1 z2 ), and the moment map Φ = |z1 |2 + |z2 |2 . For α > 0, the reduced space is Mα = S 2 , and the circle bundle Q/G is the (negative of the) Hopf bundle over S 2 . In particular, it is independent of α, whereas the reduced symplectic form does depend on α. The pre-quantization for an arbitrary α ∈ Z∗G is obtained from the case of −1 α = 0 by the “shifting” trick. Let Φ0 = Φ − α, so that Z = Φ−1 (α) = Φ0 (0). The reduction of the Hamiltonian manifold (M, ω, Φ) at the value α is the same as the reduction of the Hamiltonian manifold (M, ω, Φ0 ) at the value 0:

(6.14)

(Mα , ωα ) = (M00 , ω00 ).

To find a pre-quantization of (M, ω, Φ0 ), we only need to change the G-action on the pre-quantization circle bundle P for (M, ω, Φ). Namely, denote by γα the character

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of G associated with α ∈ Z∗G : γα (exp ξ) = e

√ −1hξ,αi

.

We let P 0 denote the circle bundle P with the new G-action obtained by letting g ∈ G act as the element (g, γα (g)−1 ) of G × U(1). (Here we use the assumption that α ∈ Z∗G .) For this new action,

∂ , ∂θ where ∂/∂θ is the infinitesimal generator of the principal U(1)-action. Because Θ(ξP 0 ) = Θ(ξp ) − α(ξ), this action gives rise to a pre-quantization for Φ0 = Φ − α. We let Q0 denote the restriction of P 0 to Z; then Pα := Q0 /Z is a pre-quantization circle bundle for the reduced space (6.14). ξP 0 = ξP − α(ξ)

We note that the pre-quantization line bundle for (Mα , ωα ) is (6.15)

Lα = (L|Z ⊗ C−α )/G.

Example 6.15. In Chapter 8 we will encounter an important special case, when L|Z is the trivial line bundle with fiberwise G-action given by a weight δ ∈ Z∗G , i.e., L|Z = Z × Cδ . In this case, the pre-quantization line bundle (6.15) is Lα = Z ×G Cα−δ .

3.2. Pre-quantization of singular reduced spaces. We continue to assume that G is a torus, but would now like to consider singular values α of the moment map. For this, we adopt an approach which is due to Meinrenken and Sjamaar, [MeSj]: we desingularize the reduced space by passing to a nearby regular reduced space as follows. In the construction of Section 3.1, the value α plays two roles: it is used to define the reduced space Mα , and it is used to define the pre-quantization line bundle Lα . If we pass to nearby regular values α + h, the bundles Lα = (L|Z ) ⊗ C−α → Mα+h , for Z = Φ−1 (α + h), are all isomorphic. (This follows from the rigidity of compact group actions. See Section 5 of Appendix B.) If α ∈ Z∗G is a singular value for the moment map, we can attempt to prequantize the reduced space in a similar way. Namely, choose a regular value α +h ∈ Φ(M ) near α (not necessarily in Z∗G ), and let Z = Φ−1 (α + h), so that the quotient Mα+h = Z/G is an orbifold. Consider the line bundle Lα = (L|Z ⊗ C−α )/G over this orbifold. We declare this to be a pre-quantization line bundle for the singular reduced space (Mα , ωα ). Let us use the symbol ωα to also denote the curvature of Lα → Mα+h , with respect to some connection. This is a closed two-form on Mα+h which is generally different from the reduced two-form ωα+h . On the quotient, we declare (Mα+h , ωα ) to be a desingularization of (Mα , ωα ), We may also define ωα directly, using the Duistermaat–Heckman theorem. Let us fix a chamber (i.e., a connected component of regular values of Φ) which contains α in its closure and consider nearby values α + h which lie in this same chamber. We may identify the reduced spaces Mα+h with each other, (e.g., with the help of a connection for the fibration Φ−1 (A) → A, where A is the chamber). Thus we have an orbifold Mred and diffeomorphisms Mα+h ∼ = Mred for various h’s. Moreover, the “presentation” for this orbifold, Z/G, also remains isomorphic as h varies. Whereas these diffeomorphisms are not canonical, the induced isomorphisms in

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cohomology are canonical. By the Duistermaat–Heckman formula (Theorem 5.6), the cohomology class of the reduced symplectic form [ωα+h ] depends linearly on h. Hence, there exist cohomology classes [ωα ] and c in H 2 (Mred ) such that ωα+h = [ωα ] − hh, ci. Explicitly, c is the curvature class of the fibration Z → Mred . We take the class [ωα ] and pretend that Mred is the reduced space at α, even if α is singular. The orbifold (Mred , ωα ) provides a desingularization of the singular reduced space (Mα , ωα ). Note that if α is regular we recover the same reduced two-form ωα . Also note that these constructions do not determine the two-form ωα but only its cohomology class. However, this will be enough to define quantization (see Section 6). In subsequent sections of this chapter we show how to define quantization, given pre-quantization data. For our desingularization to be useful, the resulting quantization should be independent of the choice of h as α + h varies between different chambers. Unfortunately, this is not true for arbitrary ω and α (see Example 8.9). However, if ω is symplectic and α+h is chosen in the image Φ(M ), the quantization is independent of the choice of desingularization. This was shown by Meinrenken and Sjamaar, [MeSj]. It also follows from the “quantization commutes with reduction” theorem that we will prove in Chapter 8. We will need a yet more general way to desingularize a reduced space. As before, let (M, ω, Φ) be a Hamiltonian G-manifold, for G a torus, and let α ∈ Z∗G be a singular value for Φ. We may desingularize the reduced space by deforming Φ instead of deforming α. Explicitly, let Φ0r be a smooth family of abstract moment maps, parametrized, say, by r ∈ R (or we may take a more general parameter space). Suppose that when r = 0 we have Φ00 = Φ. We may define the desingularization of (Mα , ωα ) to be the space Mα0 := Φ0r

−1

(α)/G,

where r near 0 is such that α is a regular value for Φ0r . We take the pre-quantization line bundle −1 L0α = (L|Z ⊗ C−α )/G, Z = Φ0 (α),

and the two-form ωα0 obtained as the curvature of L0α with respect to some connection. The Meinrenken-Sjamaar desingularization is the special case Φ0 = Φ − h. As before, this construction may or may not be useful. The important feature needed is that when α is a regular value of Φ, taking any deformation of Φ and following the above recipe with Φ0 = Φ0r for small enough r, we recover a space that is isomorphic to the regular reduced orbifold (Mα , ωα ) with its pre-quantization line bundle Lα . 4. Kirillov–Kostant pre-quantization 4.1. Equivariant pre-quantization: infinitesimal constructions. Our exposition would not be complete without a review of a more traditional treatment of equivariant pre-quantization, due to Kostant and Kirillov; see [Ki1, Ki5, Kost]. In this approach, one starts with a G-action on (M, ω) and a non-equivariant prequantization (P, Θ), and tries to lift the G-action to P using a moment map Φ, by the relation (6.10): (6.16)

π ∗ Φξ = Θ(ξP ).

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Usually, this approach only gives a lift to L of an infinitesimal action, which is often sufficient as far as the physical motivation is concerned (see Section 1). An advantage of this method is that one lifts to L the action of a much larger Lie algebra, namely, the entire Poisson algebra C ∞ (M ). More generally, if ω is degenerate, we obtain a lift of the Poisson algebra associated with (M, ω) which was defined in Section 1.6 of Chapter 2. Let us briefly recall some notions concerning Lie algebra actions on manifolds, from Section 1.6 of Appendix B. An action of a Lie algebra g on a manifold M is a Lie algebra anti-homomorphism g → Vect(M ). A lift of this action to a principal U(1)-bundle π : P → M is a g action on P which commutes with the principal U(1)-action (a∗ ξP = ξP for all a ∈ U(1)) and which lifts a g-action on M , i.e., ξM = π∗ ξP for all ξ ∈ g. Here, ξM and ξP are the vector fields generated by ξ ∈ g on M and P , respectively. We recall that the Poisson algebra associated to a manifold M and closed twoform ω is P(M, ω) = {(f, v) ∈ C ∞ (M ) × Vect(M ) | df = ιv ω}.
Its Lie algebra structure is [(f, v), (g, u)] = 21 (Lu f − Lv g), −[u, v] . If ω is symplectic, P(M, ω) ∼ = C ∞ (M ). The Lie algebra P(M, ω) acts on (M, ω) by (f, v) 7→ v.

(6.17)

See Section 1.6 of Chapter 2 for details. Let (P, Θ) be pre-quantization data for (M, ω), i.e., a U(1)-principal bundle P and a connection one-form Θ with curvature ω. For a vector field v on M , denote ∂ be the vector field on P that generates by vhor the horizontal lift of v to P . Let ∂θ the principal U(1)-action. Below, by abuse of notation, we use the same symbol to denote a function on M and its pull-back to P . Proposition 6.16. The map ∂ ∂θ is a lift of the infinitesimal P(M, ω)-action on M to an infinitesimal P(M, ω)-action on P . (6.18)

f 7→ vhor + f ·

Proof. As in the symplectic case, we have Lv g = −Lu f = ω(u, v) for (f, v) and (g, u) in P(M, ω). Furthermore,

∂ ∂θ since Θ([vhor , uhor ]) = dΘ(vhor , uhor ) = ω(v, u); cf. (A.10). Using these identities, it is not hard to show that ∂ ∂ 1 ∂ [vhor + f , uhor + g ] = [v, u]hor − (Lu f − Lv g) · , ∂θ ∂θ 2 ∂θ which means that (6.18) is a Lie algebra anti-homomorphism. (6.19)

[vhor , uhor ] = [v, u]hor + ω(v, u)

Proposition 6.17. The algebra P(M, ω) is isomorphic via (f, v) 7→ vhor + ∂ f ∂θ to the Lie algebra of infinitesimal symmetries of the principal bundle P with connection Θ.

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Proof. A vector field ξ on P is an infinitesimal symmetry if and only if Lξ Θ = ∂ 0 and ξ is U(1)-invariant, i.e., [ ∂θ , ξ] = 0. It is easy to see that (6.18) is injective ∂ and that every vector field of the form vhor + f ∂θ , where (f, v) ∈ P(M, ω), is an infinitesimal symmetry. Hence, we only need to show that all infinitesimal symmetries have this form. Let ξ be an infinitesimal symmetry. Since ξ is U(1)-invariant, there exists a ∂ . It remains to prove function f and a vector field v on M such that ξ = vhor + f ∂θ that (f, v) ∈ P(M, ω), i.e., ιv ω = df . This follows immediately from the condition Lξ Θ = 0. We are now in a position to determine which G-actions on (M, ω) lift to (P, Θ) on the infinitesimal level. Corollary 6.18. A g-action on M which preserves ω lifts to a g-action on P which preserves Θ if and only if the action is Hamiltonian. Moreover, there is a natural bijection, given by (6.16), between such lifts and moment maps. Proof. When the action is Hamiltonian with moment map Φ, the map ξ → (Φξ , ξM ) is a Lie algebra homomorphism g → P(M, ω). By Proposition 6.16, P(M, ω) acts on (P, Θ) by symmetries via (6.18). When this action is composed with g → P(M, ω), it becomes a g-action by symmetries via (6.16). Conversely, a g-action by symmetries on (P, Θ) gives rise to a Lie algebra homomorphism g → P(M, ω), by Proposition 6.17. The first component of this lift is a moment map. In this way we obtain a one-to-one correspondence between moment maps and lifts. Until now, in this section, we only considered Lie algebra actions. Now, let P be a (non-equivariant) pre-quantization circle bundle for (M, ω), and suppose that a compact group G acts on (M, ω). We have shown that a (non-equivariant) pre-quantization structure on P determines an infinitesimal lift of the action to a g-action on P . As seen in Section 2.4, this lift integrates to a genuine G-action for some choice 1 of pre-quantization structure if the equivariant cohomology class 2π [ω − Φ] is integral. However, for a general pre-quantization structure, the infinitesimal lift need not integrate to a genuine G-action even if this cohomology class is integral. Example 6.19. Let M = S 1 so that ω = 0 and let G = S 1 act on M by translations. Then P = S 1 × U(1). Consider Θ = dθ + a dt, where θ is the angle coordinate on U(1) and t is the angle coordinate on M . Then the infinitesimal lift of the G-action to P integrates to a lift of the G-action if and only if a ∈ Z. 4.2. Uniqueness of the lift. Let (M, ω, Φ) be a Hamiltonian G-manifold, for a compact Lie group G, and let π : P → M be a pre-quantization U(1)-bundle for (M, ω). In general, the infinitesimal lift ξP depends on the choice of connection Θ on P . However, sometimes the infinitesimal lift is canonically determined by ω and Φ: Lemma 6.20. The lift ξP is independent of the connection if one of the following conditions hold: (1) The action on M of any maximal torus in G has a fixed point. (2) M is compact and H 1 (M ) = 0 (or, more generally: H 1 (M ) = 0 and every G-invariant function on M has a critical point).

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Remark 6.21. The lemma extends to proper actions of Lie groups. The compactness of M should then be replaced by the requirement that M/G be compact. Proof. The connection form Θ is determined by ω up to addition of the pullback of a G-invariant closed form α. Denote by ξP the lift for a connection form ∂ ∂ Θ and let ξP0 be the lift for Θ + π ∗ α. Then ξP0 = ξP − α(ξM ) ∂θ , where ∂θ is the vector field generating the principal U(1)-action on P . Thus, it suffices to prove that α(ξM ) ≡ 0 under either of the above conditions. First, observe that α(ξM ) is constant on M . Indeed, d(α(ξM )) = LξM α = 0, since α is closed and G-invariant. Clearly, α(ξM ) = 0 at the points where ξM = 0. Thus if a maximal torus containing exp(ξt) acts with fixed points, α(ξM ) = 0 everywhere on M . If H 1 (M ) = 0, the form α is exact and hence α(ξM ) = LξM f for some smooth (G-invariant) function f on M . Then α(ξM ) = 0 at a critical point of f , and, therefore, everywhere on M . 5. Polarizations, complex structures, and geometric quantization 5.1. Polarizations. In Hamiltonian mechanics, the state space is the cotangent bundle T ∗ X of the configuration space X. In quantum mechanics, the state space is the Hilbert space of complex valued “wave functions” L2 (X). The theory of geometric quantization attempts to understand the correspondence T ∗X

L2 (X)

within the context of symplectic geometry and to define analogues of L2 (X) for symplectic manifolds other than cotangent bundles. Functions in L2 (X) can be thought of as functions on T ∗ X which are independent of the fiber variables. More generally, Kostant’s notion of a polarization 1 allows us to consider functions (or sections of a line bundle) on a symplectic manifold which are “independent of half of the variables”. With a “real polarization”, one has local coordinates x1 , y1 , . . . , xd , yd , such that a function f is polarized if ∂ ∂ f = ... f = 0. ∂y1 ∂yd We will work with a “complex polarization”, which is nothing but a complex structure compatible with ω in a suitable sense. One then has local complex coordinates z1 , . . . , zd , and a function f is polarized if ∂ ∂ f = ... f = 0, ∂z 1 ∂z d i.e., f is holomorphic. An advantage of restricting to a subspace of “polarized sections” is that the resulting pre-quantization may satisfy the “minimality axiom” (see Section 1). The entire space of sections of a line bundle L → M is simply too big for this axiom to hold. Although we could work exclusively with complex structures, we prefer to start with the general definition of a polarization. 1 We warn the reader that the notion of a polarization introduced in this section is entirely unrelated to the notions of “η-polarized weights” and “η-polarized moment maps” used throughout this book.

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Recall that a (complex) distribution is a complex sub-bundle V ⊂ T M ⊗ C. If (M, ω) is symplectic, V is called Lagrangian if for every p ∈ M the subspace Vp of Tp M ⊗ C is Lagrangian, i.e., dimC Vp = 21 dim M and the complex-valued two-form induced from ω vanishes on Vp . Furthermore, V is said to be integrable if for any two vector fields v1 and v2 which are sections of V, the Lie bracket [v1 , v2 ] is also a section of V. Definition 6.22. A polarization of a symplectic manifold (M, ω) is a complex distribution, V ⊂ T M ⊗ C, which is Lagrangian and integrable. In geometric quantization, one starts with a symplectic manifold (M, ω), which is equipped with both a pre-quantization (L, h, i , ∇) and a polarization V. No compatibility condition is required between the pre-quantization and the polarization. Definition 6.23. A distribution V ⊂ T M ⊗ C of rank d = 21 dim M is called totally real if VR = V ∩ V is a real subbundle of T M of rank d. The distribution V is called totally complex if VR = {0}. A polarization is called real, resp., complex, if it is given by a totally real, resp., totally complex, distribution. Remark 6.24. The terminology “totally complex polarization” is often used for what we call “complex polarization”. We safely omit the adjective “totally” because we will not be considering polarization of “mixed” type. Example 6.25. For a real polarization, VR is an integrable subbundle of T M , and hence, by Frobenius’ theorem (see, e.g., [Sg]), defines a foliation of M , whose leaves are Lagrangian submanifolds of M . A fundamental example of such a polarization is the foliation of the cotangent bundle into the cotangent spaces. Conversely, every Lagrangian foliation defines a real polarization. Example 6.26. A complex structure on M gives rise to a totally complex integrable distribution V whose fibers are spanned by ∂ ∂ (6.20) ,... , , ∂z 1 ∂z d where z1 , . . . , zd are local complex coordinates. The distribution V is Lagrangian if and only if the symplectic form ω is of type (1, 1). Conversely, a totally complex distribution V gives rise to an almost complex structure J on M (i.e., a bundle automorphism J : T M → T M such√that J 2 = −identity) with the property that at a point x ∈ M , the vectors u + −1Ju, for all u ∈ Tx M , span the space Vx . By the Newlander–Nirenberg theorem, [NN], the distribution V is integrable if and only if J comes from a genuine complex structure on M . Under these assumptions, ω(·, J·) defines a (possibly indefinite) metric on M . The structures J and ω are said to be compatible if this metric is positive-definite. (See Example D.12.) This means that (M, ω, J) is a K¨ ahler manifold. The reader interested in more details should consult [McDSa, Section 4.1], [Ki4], [KN, Chapter 8], or [Wel]. Example 6.27. If V is a complex polarization, so is V. If V corresponds to a complex structure J with local coordinates zi , then V corresponds to the complex structure −J, with local coordinates z i . Suppose now that a (pre-)symplectic manifold (M, ω) is equipped with prequantization data (L, h, i, ∇) and with a polarization V. A section s of L over an open set U ⊆ M is polarized if ∇v s = 0 for all sections v of V|U . As a first

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approximation to a quantization of M we take the space of all polarized sections of L. Example P 6.28. Consider the cotangent bundle M = T ∗ X with its Pcanonical one-form, γ = pi dqi , and the canonical symplectic form, ω = −dγ = dqi ∧ dpi . By Example 6.4, T ∗ X can be quantized by the trivial Hermitian line bundle L = T ∗ X × C or the trivial circle P = T ∗ X × U(1) with the connection one-form Θ = dθ + γ. The sections of L are functions g : T ∗ X → C. For each vertical vector field v we have γ(v) = 0, and hence ∇v g = Lv g. Therefore, the polarized sections are functions on T ∗ X which are constant on fibers, i.e., pullbacks to T ∗ X of functions on X. Remark 6.29. To obtain a Hilbert space, one might be tempted to take the space of L2 integrable polarized sections. However, this does not give the right answer: for instance, a function on T ∗ X which is constant on the fibers is never L2 integrable with respect to the Liouville form on T ∗ X. Some “fine tuning” of the theory is achieved by working with “half-densities” or with “half-forms”. See, e.g., [Bl, RR, GS1]. 5.2. Quantization for compact complex manifolds. Let us now focus on quantization for complex polarizations. This quantization is particularly easy to describe, for it fits completely in the framework of complex geometry. Let (M, ω) be equipped with a complex polarization. In other words, M is a complex manifold and ω is a closed (1, 1)-form, which need not even be symplectic from now on. Fix a pre-quantization (L, h, i , ∇) for (M, ω). As we will show, L then becomes a holomorphic line bundle, its polarized sections are holomorphic sections, and our first approximation to a quantization Q(M ) is the space H 0 (M ; OL ) of holomorphic sections. However, it turns out more convenient to work P with the sheaf O L of holomorphic sections and to take the alternating sum (−1)i H i (M ; OL ). In the presence of a G-action which preserves the complex structure and pre-quantization structure, this quantization becomes a (virtual) representation for G. Finally, when M is not compact, one can carry out this construction with the space of holomorphic L2 -sections. In this section we recall the notions involved in the definition of the quantization space Q(M ) for a compact M . Proposition 6.30. The line bundle L admits a unique holomorphic structure such that the local holomorphic sections of L are exactly its local polarized sections. Alternatively, this holomorphic structure is uniquely determined by the condition ∇0,1 = ∂.

An explicit construction of this holomorphic structure in based on the following key observation. Lemma 6.31. For every point p ∈ M there exist local coordinates z 1 , . . . , zd on a neighborhood U of p and a non-vanishing section s of L over U such that (6.21)

∇

∂ ∂zi

s = 0,

i = 1, . . . , d.

Proof. Since ω has type (1, 1) and is closed, there exists a (1, 0)-form β on a neighborhood U of p which satisfies ω = dβ. Let us recall the proof of this fact. First note that ∂ω = 0 and ∂ω = 0, for dω = ∂ω +∂ω = 0. By the local exactness of the Dolbeault complex (see [GH, page 25] or [Wel]), there exists a (1, 0)-form β such that ω = ∂β on a neighborhood of

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p. We claim that ∂β = 0 is satisfied automatically. Indeed, since ∂(∂β) = ∂ω = 0, there exists a form α such that ∂β = ∂α on a neighborhood of p. Since ∂β has type (2, 0) and ∂α has type (1, 1), this implies that ∂β = 0. Without loss of generality, we may take U to be simply connected. By Corol√ lary 6.6, there exists a section s of L over U such that ∇s = − −1βs. Since β is of type (1, 0),   √ ∂ s = 0, ∇ ∂ s = − −1β ∂zj ∂zj as required. Proof of Proposition 6.30. Let s be the section from Lemma 6.31. We declare s to be a local holomorphic section of L over U . Since, by (6.21), every polarized section of L over U is the product of s with a holomorphic function, this gives a well-defined holomorphic structure on L. It is worth noticing that the holomorphic transition functions for L are usually different from its Hermitian transition functions, for no non-constant holomorphic function can have constant absolute value. The space of holomorphic sections of L is often trivial. For instance, if M is compact, the Kodaira vanishing theorem implies that there are no non-zero holomorphic sections if the curvature form ω is sufficiently negative. To get around this difficulty, one further refines the quantization procedure: instead of considering the space of holomorphic sections of L, one considers the sheaf OL of holomorphic sections and its cohomology groups H k (M ; OL ). We declare the quantization of M to be the virtual vector space X (6.22) Q(M ) = (−1)k H k (M ; OL ).

(See, e.g., [Wel] for definitions.) If ω is sufficiently positive, then the higher cohomology H >0 (M ; OL ) vanishes by Kodaira’s theorem, and we get back the space H 0 (M ; OL ) of global holomorphic sections. The idea to replace the space of holomorphic sections of L by the alternating sum of the cohomology groups can be traced to as early as 1957, see, e.g., [Bot1]; in the context of quantization, see [Kost]. Also see Example 6.39 in this chapter. The sheaf cohomology H k (M ; OL ) is equal to the cohomology H 0,k (M ; L) of the twisted Dolbeault complex

(6.23)

∂

. . . → Ω0,k (M ; L)→Ω0,k+1 (M ; L) → . . . .

(See, e.g., [Wel]). (More precisely, the differential in this complex is ∇0,1 ⊗ id + id ⊗∂. In the local trivializations given by the section s from Lemma 6.31, the differential becomes just ∂.) Hence, X (6.24) Q(M ) = (−1)k H 0,k (M ; L). Suppose now that M is equipped with an action of a Lie group G which is prequantizable and which preserves the polarization. This means that the action is holomorphic and, moreover, lifts to a holomorphic action on L. Such an action gives rise to a representation of G on the space of holomorphic sections of L and, more generally, on the cohomology H k (M ; OL ) = H 0,k (M ; L). The quantization Q(M ) defined by (6.22) or, equivalently, (6.24) then becomes a virtual representation of G.

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Finally, note that the notation Q(M ) is somewhat misleading. The quantization Q(M ) depends on ω and on the complex structure J although this is not apparent from our notation. When we need to emphasize this dependence (as, e.g., in section 4 of Chapter 7), we will incorporate these structures in the notation. The effect of a change of J on Q(M ) is touched upon in the later sections and in Appendix H. 5.3. L2 cohomology. The definition (6.24) is still not completely satisfactory when M is non-compact. For instance, the quantization should be a (virtual) Hilbert space, not just a (virtual) vector space. In the non-compact case we replace (6.24) by the alternating sum of the L2 -cohomology groups: X (6.25) Q(M ) = (−1)k HL0,k 2 (M ; L).

(See, however, Remark 6.36 below.) Let us recall the definition of Dolbeault L2 cohomology. More detailed accounts can be found in [At2, Brav1, GHS]. As above, let M be a complex manifold, ω a closed two-form of type (1, 1), and (L, h, i , ∇) a pre-quantization for (M, ω). The definition of L2 cohomology requires additional structures: let us fix a Hermitian metric on M and a measure µ. (This metric and the measure need not be related to ω nor to each other; although µ is often the Liouville measure, when ω is symplectic, or the volume form of the Hermitian metric.) The Hermitian metrics on M and L give rise to a pointwise inner product on Ω0,k (M ; L) = Γ(Λ0,k T ∗ M ⊗ L), which, when integrated against the measure µ, defines an inner product on the space of square-integrable (0, k)forms. Denote by W k the set of the (0, k)-forms α such that both α and ∂α are 2 square-integrable. Since ∂ = 0, we get a complex (W ∗ , ∂). The L2 cohomology 2 HL0,∗ 2 (M ; L) is the quotient of ker ∂ by the L -closure of the image of ∂. Remark 6.32. If U is a coordinate patch and L is trivial over U , the restriction of α to U is of the form X α= fI dz I .

The form α is L2 integrable if, for every such coordinate patch, the fI ’s are measurable and locally L2 integrable with respect to µ and the global L2 norm is finite. For such a form, ∂α is well defined as a distribution, and, by definition, α is in W k if and only if α and ∂α are L2 integrable.

Remark 6.33. In some sources (see, e.g., [GHS]) the term “L2 cohomology” is reserved for the quotient by the image of ∂, while the spaces introduced above are referred to as the reduced L2 cohomology. Remark 6.34. These are not the only interesting L2 cohomology groups of M . For example, if we replace ∂ by the de Rham differential d we obtain de Rham L2 -cohomology.

Remark 6.35. In the “orbit method”, L2 -cohomology is used to obtain geometrical constructions for representations of non-compact semi-simple Lie groups. See, e.g., [Schm]. (Also see [Brav1].) Remark 6.36. Making sense of the alternating sum (6.25) can be problematic when the summands are infinite-dimensional. It might be tempting to consider this sum as an element of the Gr¨ othendieck group of formal differences of Hilbert spaces H H 0 modulo the equivalence relations (H ⊕ H 00 ) (H 0 ⊕ H 00 ) = H H 0 .

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However, this group is trivial: H ⊕ H 00 is isomorphic to H 00 whenever the dimension of H 00 is infinite and, as a cardinal, is larger than or equal to the dimension of H. This problem also arises when one tries to define the Gr¨ othendieck group of formal differences of arbitrary unitary G-representations. It is possible to get around this problem by considering only those unitary representations in which every irreducible representation occurs with finite multiplicity. Alternatively, one can work with those representations whose distributional character is of trace class (see Section 3 of Chapter 7). In Section 1 of Chapter 7 we will compute the L2 cohomology for the simple example where M = Cd and will make sense of the quantization (6.25) in this particular example. Remark 6.37 (Dependence of the metric). The L2 -cohomology of a manifold strongly depends on the metric even when the metric is complete. Consider for example the de Rham L2 -cohomology of the plane R2 . When the plane is equipped with a flat metric, its de Rham L2 -cohomology is zero. However, if the plane is equipped with a hyperbolic metric (a complete metric with curvature K = −1), its de Rham L2 cohomology is infinite–dimensional. The L2 -cohomology of a manifold also depends on the choice of the volume form. 5.4. Relation to representation theory of Lie groups. Geometric quantization provides a link between physics, symplectic geometry, and the representation theory of Lie groups. We illustrate its role in the latter by two examples. Example 6.38. Let us consider the standard action of SU(n) on CPn−1 . We equip CPn−1 with its standard complex structure and the symplectic form ω = rω0 , where r > 0 and ω0 is the standard K¨ ahler (Fubini–Study) symplectic structure on CPn−1 , normalized so that its integral over CP1 is 2π. (See, e.g., [McDSa, Example 4.21] or [Wd, Section 5.4].) Since CPn−1 is simply connected, the pre-quantization data is unique up to an isomorphism by Proposition 6.5. The pre-quantization line bundle in this case is L = L⊗r 0 , where L0 is the dual to the tautological (Hopf) line bundle. The SU(n)-action has a unique moment map because SU(n) is semisimple (see, e.g., [CW, Chapter 7]). Thus the quantization of this data gives a well-defined virtual representation of SU(n). Since H 0,k (CPn−1 ; L) = 0 for k > 0 (see, e.g., [GH]), the quantization is a genuine representation of SU(n) on the space of holomorphic sections of L. The latter space can be identified (see, e.g., [GH]) with the space of homogeneous polynomials of degree r on Cn with the natural action of SU(n). Example 6.39. Let G be a compact semi-simple group and let M be a regular integral coadjoint orbit of G equipped with its standard Kirillov–Kostant symplectic structure (see, e.g., [McDSa, Example 5.24]) and a G-invariant complex structure compatible with ω (see Example D.12, and, specifically for coadjoint orbits, see [Wd, Section 5.5]). Similarly to the previous example, the pre-quantization data and the moment map are unique and we again obtain a well–defined quantization space Q(M ). As above, the higher cohomology spaces vanish, and hence Q(M ) is just the representation of G on the space of holomorphic sections of the prequantization line bundle L. The Borel–Weil theorem asserts that Q(M ) is an irreducible representation of G and all irreducible representations result from this construction. (See, e.g., [Wd, Section 9.2] and [Vo, Theorem 1.24] for references and a more detailed discussion.)

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As shown by Bott, [Bot1], in the Borel–Weil theorem the compatibility condition on J and ω can be omitted. If k is the number of negative squares of the indefinite metric ω(·, J·), then H 0,i (M ; L) = 0 for all i 6= k, and (−1)k Q(M ) = H 0,k (M ; L) is the corresponding irreducible representation. In Section 1 of Chapter 7 we will observe a similar “shift by k” phenomenon in the quantization of Cd . The Borel–Weil–Bott theorem has been extended to other classes of Lie groups. Analogues of this theorem for nilpotent, solvable, reductive, and real semi-simple groups can be found in [Ki1, AK, KR, Duf, HC, Pu]. The construction of irreducible representations from coadjoint orbits, known as the orbit method, occupies a central place in the method of geometric quantization and serves as a tie between symplectic geometry and representation theory. The reader interested in a more detailed treatment of the orbit method and further references may consult, for example, [Ki2, DPV, BBGK, Ki5]. 5.5. The effect of complex conjugation. Quantization depends on the choice of the polarization. More specifically, for complex polarizations, the cohomology groups H k (M ; OL ) = H 0,k (M ; L) and their alternating sum Q(M ) depend on the choice of complex structure. In this section we focus on a particular aspect of this dependence: we analyze what happens to Q(M ) when a complex structure J is replaced by −J or, equivalently, a complex polarization V is replaced by V. (See Remark 6.27.) The answer is a special case of the “Spinc shift formula” that we touch upon in Section 7.1 and discuss in detail in Appendix D. Remark 6.40. The complex quantization of a compact manifold remains unchanged if the complex structure is deformed continuously. (This follows from the Fredholm invariance of the index.) More generally, it is unchanged if the complex structure is replaced by another complex structure with the same Chern classes. (This follows from the Riemann–Roch formula. For more details, see Section 6.) As a consequence, when ω is symplectic, the quantization is independent of the complex structure J, as long as J is compatible with ω. (See Example D.12.) Other aspects of the dependence of the quantization on polarizations are analyzed in [Ki4, Section 3] and [GM]; see also Appendix H. Let (M, ω) be a (pre-)symplectic manifold with fixed pre-quantization data (L, h, i , ∇). Let J be a complex structure on M . Recall that a complex structure and pre-quantization data make L into a holomorphic line bundle (see Proposition 6.30). We denote by M the complex manifold obtained from M by taking the opposite complex structure, −J, on M . Lemma 6.41. There exists a canonical anti-complex-linear isomorphism from H 0,k (M ; L) to H 0,k (M ; L∗ ).

Recall that the Dolbeault cohomology H 0,k is equal to the cohomology of the sheaf of holomorphic sections. The new complex structure on M induces an entirely different holomorphic structure on L: the local sections of L that are holomorphic with respect to the new holomorphic structure are exactly the local “antiholomorphic sections” of L over M , i.e., those sections s that satisfy (6.26)

∇

∂ ∂zj

s = 0 j = 1, . . . , n

in local coordinates on M . Therefore, to prove Lemma 6.41, it is enough to show that the anti-complex-linear isomorphism s 7→ h·, si from L to L∗ sends local holomorphic sections of L over M to local holomorphic sections of L∗ over M .

5. POLARIZATIONS AND COMPLEX GEOMETRIC QUANTIZATION

109

Proof of Lemma 6.41. Let s be a local holomorphic section of L over M (with respect to the new holomorphic structure). For any local holomorphic section σ of L over M ,



 ∂ hσ, si = ∇ ∂ σ, s + σ, ∇ ∂ s = 0 ∂zj ∂zj ∂z j

by (6.26). Therefore, h·, si is a local holomorphic section of L∗ over M .

Example 6.42. For the trivial bundle L = M × C with its natural flat connection, holomorphic sections of (M, L∗ ) are holomorphic functions, and holomorphic sections of (M , L) are anti-holomorphic functions, over any open subset of M . Complex conjugation provides an anti-complex-linear isomorphism from the sheaf of holomorphic functions to the sheaf of anti-holomorphic functions, leading to the required isomorphism in sheaf cohomology. Alternatively, the anti-complex-linear isomorphism between H 0,k (M ) and H 0,k (M ) is induced by complex conjugation on differential forms, which sends the Dolbeault complex (Ω0,∗ (M ), ∂) to the complex (Ω∗,0 (M ), ∂) = (Ω0,∗ (M ), ∂). Remark 6.43. The new holomorphic structure, with which L quantizes (M , ω), can be described in terms of the original holomorphic line bundle L, as follows. Consider the holomorphic line bundle L over M that is obtained from L by replacing the complex structure JL on the total space of L by its negative. (Note that a section of L over M is holomorphic if and only if it is holomorphic as a section of L over M , for ds ◦ (−JM ) = (−JL ) ◦ ds if and only if ds ◦ JM = JL ◦ ds.√ Also note that, as complex line bundles, L and L are different: multiplication by −1 in one √ is multiplication by − −1 in the other.) The Hermitian structure h, i induces an ∗ ∗ isomorphism of L with the dual L , as smooth complex line bundles. Therefore, L ∗ is a pre-quantization line bundle for ω. Also, L is a holomorphic line bundle over ∗ M. Finally, the isomorphism between L and L transforms the holomorphic sec∗ tions of L precisely to the “anti-holomorphic” sections of L. We leave the details, which are similar to the proof of Lemma 6.41, to the reader. Because the dual line bundle L∗ provides a pre-quantization for −ω, Lemma 6.41 says that changing J to −J has the same effect on quantization, up to complex conjugation, as changing the sign of ω: Q(M, ω) = Q(M, −ω). Alternatively, the quantization Q(M) can be described in terms of a further twist of the Dolbeault complex (6.23). Namely, let Λd,0 be the line bundle of (d, 0) forms, where d = dimC M . Consider the holomorphic line bundle Lδ = L ⊗ Λd,0 and set (6.27)

Qδ (M ) = : (−1)d

X

(−1)k H 0,k (M ; Lδ ).

Note that, since the Dolbeault complex Ω0,∗ (M ; Lδ ) is just the Dolbeault complex Ωd,∗ (M ; L). In other words, the summands in (6.27) can also be written as H d,k (M ; L).

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Proposition 6.44. When M is compact, there exists an anti-complex-linear isomorphism H 0,k (M ; L) → H 0,d−k (M ; Lδ )∗ . Thus (6.28)

∗

Q(M , ω) = (−1)d Qδ (M, ω) .

This isomorphism is canonically determined by the complex structure and the prequantization data. Proof. Consider the Dolbeault complexes (6.29) and (6.30)

∂

. . . → Ω0,k (M ; L∗ )→Ω0,k+1 (M ; L∗ ) → . . . ∂

. . . → Ω0,k (M ; Lδ )→Ω0,k+1 (M ; Lδ ) → . . . .

The kth term in the complex (6.29) can be paired with the (d − k)th term in the complex (6.30). Namely, the product of β1 = ϕ ⊗ µ1 ∈ Γ(L∗ ) ⊗ Ω0,k (M ) and β2 = s ⊗ µ2 ∈ Γ(L) ⊗ Ωd,d−k (M ) is Z ϕ(s)µ1 ∧ µ2 . By Serre’s duality (see, e.g., [Wel, Chapter V] and [GH]), this pairing induces a non-degenerate pairing in cohomology, and hence a complex linear isomorphism H 0,k (M ; L∗ ) → (H 0,d−k (M ; L ⊗ Λd,0 ))∗ .

Combining this with the isomorphism from Lemma 6.41, we obtain the required isomorphism. Proposition 6.44 sometimes remains true even when M is non-compact, with P δ Qδ (M ) = (−1)d (−1)k HL0,k 2 (M ; L ). (See, however, Remark 6.36). This follows from a version of Serre’s duality for L2 cohomology. See, e.g., [Schm, Proposition 9.1]. We will see an example of this duality in Remark 7.8. 6. Dolbeault Quantization and the Riemann–Roch formula 6.1. Almost complex quantization. Let M = M 2d be a compact manifold with a closed two-form ω and pre-quantization data (L, h, i, ∇). In addition, let M be equipped with a complex structure J such that ω is of type (1, 1). In this case, there is an alternative way of defining the cohomology groups H 0,k (M ; L), via Hodge theory. Namely, let us fix a Hermitian metric on M . This metric gives rise to Hermitian inner products on the vector bundles Λ0,k = Λ0,k T M . Furthermore, since L is by construction equipped with a Hermitian inner product, one has Hermitian inner products on the vector bundles L ⊗ Λ0,k . These Hermitian products and the volume form give rise to differential operators t ∂ : Ω0,k (M ; L) → Ω0,k−1 (M ; L) which are L2 -adjoint to ∂ (see, e.g., [Wel]). Combining these operators, we obtain a first order elliptic differential operator t

∂ + ∂ : Ω0,even(M ; L) → Ω0,odd (M ; L) P By Hodge theory, the virtual vector space Q(M ) = (−1)i H 0,i (M ; L) is just

(6.31)

t

t

Q(M ) = ker(∂ + ∂ ) − coker(∂ + ∂ ),

6. DOLBEAULT QUANTIZATION AND THE RIEMANN–ROCH FORMULA

111

t

i.e., Q(M ) is the index of ∂ + ∂ interpreted as a virtual vector space. t For example, let us show that ker(∂ + ∂ ) is the space of harmonic forms of even degree. Let β0 ⊕ . . . ⊕ βd , where deg βj = 2j, be in this kernel. Then the forms t

t

∂βj = −∂ βj+1 ,

j = 1, . . . , d − 1,

are in im ∂ ∩ im ∂ . By Hodge theory (see, e.g., [Wel]), this intersection is zero. t Thus ∂βj−1 = ∂ βj = 0 for all j and hence each βj is a harmonic form. Assume now that J is an almost complex structure on M and (L, h, i , ∇) a pre-quantization of (M, ω), where ω is again a closed two-form. The operators ∂ : Ω0,k (M ; L) → Ω0,k+1 (M ; L) can still be defined. Namely, J gives rise to a totally complex (in general, non-integrable) distribution V ⊂ T M ⊗ C so that T M ⊗ C = V ⊕ V (see Section 5). As a result, we obtain a splitting X Λl (V) ⊗ Λm (V) Λk (T M ⊗ C) = l+m=k

and hence a bi-grading on L-valued differential forms. (See, e.g., [Wel] for more details.) Now let D : Ωk (M ; L) → Ωk+1 (M ; L)

be the operator D(s ⊗ γ) = ∇s ⊗ γ + s ⊗ dγ, where s ∈ Γ(L) and γ ∈ Ωk (M ). We define ∂ to be the (0, k + 1) component of D. Because J is now only an almost 2 complex structure, the twisted Dolbeault complex is no longer a complex: ∂ 6= 0. t However, the operator ∂ + ∂ is still elliptic. We define the Dolbeault quantization Q(M ) to be the virtual vector space given as before by (6.31). When ω is a symplectic form, this procedure leads to a canonical virtual vector space Q(M ) if we take J to be compatible with ω. Recall that an almost complex structure J is compatible with ω if ω is a (1, 1)-form and the following positivity condition holds: ω(v, Jv) > 0 for all p ∈ M and v ∈ Tp M .

Equivalently, √ hu, vi = ω(u, Jv) is a Riemannian metric. When J is integrable, the sum h, i + −1ω is a K¨ ahler metric on M . See Example D.12.

Example 6.45. The positivity condition is not satisfied for the standard complex structure on Cd with respect to the non-standard symplectic form ωr =

√

−1

d X j=1

j dzj ∧ dz j ,

where 1 = . . . = r = −1 and r+1 = . . . = d = +1, unless r = 0. Theorem 6.46. On every symplectic manifold M , there exists an (equivariant) almost complex structure J compatible with the symplectic form. When M is t compact, the virtual space Q(M ) = index (∂ + ∂ ), given by (6.31) and called the virtual quantization of M , is independent of the choice of J. To construct such a J, one may, for example, start with a Riemannian metric 1 h, i on M and set J = A(−A2 )− 2 , where A is defined by the condition h·, ·i = ω(·, A·). The space of all structures J which are compatible with ω is connected. See Example D.12 and references therein. Now the assertion that Q(M ) is independent

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of J follows from the fact that the index of a Fredholm operator is constant under variations of the operator; see, e.g., [Ki3, p. 64]. These definitions and results extend word-for-word to the G-equivariant case. The additional data necessary to define the Dolbeault quantization Q(M ) of a Hamiltonian G-manifold (M, ω, Φ) is a G-invariant almost complex structure J. Remark 6.47. In general, the quantization Q(M ) depends on the choice of an almost complex structure J and on the cohomology class [ω] or on the cohomology class [ω − Φ] in the equivariant case. However, Q(M ) is independent of the particular choice of pre-quantization of (M, ω, Φ). This follows, for example, from the equivariant index theorem (Theorem 6.51). 6.2. The Riemann–Roch formula. Let L → M be a holomorphic line bundle over a complex manifold. The Riemann–Roch formula expresses the dimension X dim Q(M ) := (−1)i dim H i (M ; OL )

as the characteristic number

Z

ec1 (L) Td M M

where c1 (L) is the first Chern class of L and Td M is the Todd class of the complex vector bundle (T M, J) → M . To be more specific, this is the characteristic class associated to the Ad-invariant polynomial P on U (n) such that "x #! 0 n 1 Y xj .. . P = . 1 − e−xj 0 x j=1 n

Hence, if L is a pre-quantization line bundle for (M, ω) and M is equipped with a complex polarization, Z 1 dim Q(M ) = e 2π ω Td (M,J) . M

This formula remains valid when J is an almost complex structure and Q(M ) is taken to be the Dolbeault quantization, so that t

t

dim Q(M ) := dim ker(∂ + ∂ ) − dim coker(∂ + ∂ ). The Riemann–Roch formula is then a special case of the Atiyah-Singer index formula. We will discuss equivariant versions of this formula in the subsequent sections of this chapter, in Section 6 of Appendix C, and in Appendix I. Example 6.48. Let M = CP1 and let L be the line bundle O(k) for some positive integer k. Then c1 (L) = kA where A is the generator of H 2 (CP1 ; Z). The Chern character of L is ec1 (L) = 1 + kA (higher powers of A vanish because the degree is greater than dim M ). The tangent bundle T M is a line bundle with first Chern class c1 (T M ) = 2A. Its Todd class is 1 Td (c1 (T M )) = c1 (T M )/(1 + e−c1 (T M ) ) = 1 + c1 (T M ) = 1 + A 2 (again, the higher powers vanish). The Riemann–Roch integral is then Z (1 + kA)(1 + A) = k + 1. M

7. STABLE COMPLEX QUANTIZATION

113

The sections of L can be identified with degree k polynomials in two variables. Hence, the space of sections has dimension k + 1. The higher cohomology vanishes: H k (M ; OL ) = 0 is obvious for k > 1 and easy to show for k = 1. 7. Stable complex quantization and Spinc quantization Quantization of compact manifolds equipped with almost complex structures, defined in the previous section, can be further generalized to manifolds with stable complex structures. This is done by passing to Spinc structures. Here we recall relevant definitions and results which are used in subsequent sections; we refer the reader to Appendix D and to [Du, Fr] for further details and references. 7.1. Stable complex quantization and the quantum shift formula. We recall that the group Spin(n), for n > 2, is the simply connected double cover of SO(n). Let q : Spin(n) → SO(n) be the covering map, and let  be the non-trivial element in the kernel of q. The group Spinc (n) is the quotient of Spin(n) × U(1) by the two-element subgroup generated by (, −1). The projection of Spin(n) × U(1) onto its two factors gives rise to the maps (6.32) and (6.33)

π : Spinc (n) → SO(n) ,

[s, c] 7→ q(s)

det : Spinc (n) → U(1) ,

[s, c] 7→ c2 .

Now let E → M be an oriented n-dimensional real vector bundle, equipped with an inner product, and let SO(E) → M be the oriented orthonormal frame bundle of M . Thus SO(E) is a principle SO(n) bundle. A Spinc structure on E is a principle Spinc (n) bundle, P → M , and a bundle map, p : P → SO(E), compatible with (6.32).2 The homomorphism (6.33) gives rise to a complex line bundle, Ldet = P ×det C, over M called the determinant line bundle of the Spinc structure. We will need the following two facts about Spinc structures, both of which are proved in Appendix D. Fact 1: A Spinc structure on E ⊕ Rk induces a Spinc structure on E. (See Proposition D.55.) Fact 2: Every complex vector bundle E → M and complex line bundle L → M determine a Spinc structure on E with determinant line bundle Ldet = L2 ⊗ Λm (E), where m is the fiber dimension of E. (See Proposition D.57.) A Spinc structure on M is, by definition, a Spinc structure on the tangent bundle, T M . As pointed out above, such a structure induces a Riemannian metric and an orientation on M . A connection on the principal bundle P enables one to define an elliptic operator, the Spinc Dirac operator, which in Spinc quantization will play the role played by ∂¯ + ∂¯t in Dolbeault quantization. We will briefly recall how this operator is defined (referring, for a more detailed description, to Section 3.5 of Appendix D, or [Du] or [GK]). Let ∆+ and ∆− be the two real spin representations of Spin(n). These fit together with the scalar representation of U(1) on C to give representations of Spinc (n) on the tensor products ∆+ ⊗ C and ∆− ⊗ C. Moreover, Clifford multiplication gives rise to the map

(6.34)

Rn ⊗ (∆+ ⊗ C) → ∆− ⊗ C .

2 In Appendix D we call this a metric oriented Spinc structure when the orientation and inner product on E are specified in advance.

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Now let SC± be the vector bundles SC± = PSpinc (n) × (∆± ⊗ C) . The connection on P determines a connection (6.35)

∇ : Γ(SC+ ) → Γ(SC+ ⊗ T ∗ M )

on SC+ and the map (6.34) induces a bundle morphism (6.36)

SC+ ⊗ T ∗ M → SC− .

By composing (6.35) and (6.36), we obtain a first order differential operator D : Γ(SC+ ) → Γ(SC− ) .

This, by definition, is the Spinc Dirac operator. It is an elliptic operator. Hence, if M is compact, the index is finite. Moreover, if all the data above are acted on by a compact group G, the index of D, viewed as a virtual vector space, is a space on which G acts, and this virtual representation of G is the Spinc quantization of the Spinc G-manifold M . Strictly speaking, our main objects are oriented manifolds and the quantization we have just defined is for the orientation induced by the Spinc structure. In general, the orientation need not be induced by the Spinc structure and in this case we define the quantization to be the negative index of D. The notion of Spinc quantization is closely related to the Dolbeault quantization that we discussed in Section 6. Namely, let J be an almost complex structure on M and let L be a pre-quantization line bundle. Then, by “Fact 2” above, there is a canonical Spinc structure on M compatible with J and L, and it turns out that for this Spinc structure the vector bundles SC+ and SC− are just the vector bundles L ⊗ Λ0,even and L ⊗ Λ0,odd , and the map (6.36) is just the symbol of the operator ∂ + ∂ t . By elementary Fredholm theory the index of an elliptic operator only depends on its symbol. Thus the quantization of M corresponding to D is the same as the quantization of M corresponding to ∂ + ∂ t . (See Section 7.3 for a discussion of the two-forms being quantized in this way.) One advantage of the Spinc quantization over the Dolbeault quantization lies in the behavior of the Spinc quantization with respect to symplectic reduction for actions of semi-simple groups; see [Par3] for more details. The same procedure can be carried out when we start with a stable complex structure on M , in which case the Dolbeault quantization is not defined. Namely, by Facts 2 and 1 we obtain from (M, L, J) a Spinc structure (P, p), and the stable complex quantization of (M, L, J) is, by definition, the Spinc quantization of (P, p). (As above, if the orientation of M is opposite of the orientation induced by J, the quantization is the negative index of D. See Appendix D for details.) In the next chapters we will encounter situations in which M has two different stable complex structures, J0 and J1 , and we will need to compare the quantizations associated with them. We will show in Section 3.3 of Appendix D that there is a line bundle Lδ over M , canonically associated with this pair of complex structures, and will prove the following result. Proposition 6.49 (Shift formula). Let L → M be a pre-quantization line bundle over M . Then (6.37)

Q(M, J0 , L) = Q(M, J, L ⊗ Lδ ).

7. STABLE COMPLEX QUANTIZATION

115

In other words, the stable complex quantization associated with J0 and L is identical with the stable complex quantization associated with J and L ⊗ Lδ . This assertion is true for both ordinary quantization and equivariant quantization. Remark 6.50. In the definition of Spinc quantization we assumed that M is compact. However, one can define an L2 -version of Spinc quantization for noncompact manifolds; see [Brav4]. The constructions given so far in this section extend to orbifolds. For example, for a presented orbifold M = N/G the pre-quantization orbi-line bundle is obtained from a G-equivariant line bundle on N . To define the virtual quantization of M , one utilizes elliptic differential operators on orbifolds; see [Kaw1, Kaw2] and also [Du]. Alternatively, the quantization can be defined (when the orbifold is presented) via transversely elliptic differential operators on the presenting manifold; see [At1, Ve3, BV3, BV4]. 7.2. The index theorem and the Atiyah–Bott fixed point formula. Let us now focus exclusively on the quantization of stable complex structures. Let G be a torus, let L be a G-equivariant complex line bundle over a compact manifold M , and let J be a G-equivariant stable complex structure on M . For example, in the context of pre-quantization, L is the line bundle part of the pre-quantization data (L, h, i , ∇), and the G-action on L is its natural pre-quantization action. As above, the quantization of this data produces a virtual representation Q(M ) of G. Denote by χ the character of this representation. Recall that χ is a smooth function (in fact, a trigonometric polynomial) on G. The composition χ ◦ exp is then a smooth function on g. Denote by cG 1 (L) the equivariant first Chern class of L and by Td G (T M ) the equivariant Todd class of T M . We refer the reader to Appendices I and C for the definition and a detailed discussion of these classes. Theorem 6.51 (The equivariant index theorem). On a neighborhood of 0 ∈ g, Z G (6.38) χ ◦ exp = ec1 (L) Td G (T M ). M

A proof of this version of the equivariant index theorem case can be found in [Du] in the almost complex case; additional references and details are also given in Section 2 of Appendix I. Note the role of the orientation of M in (6.38): the reversal of the orientation of M results in a sign change. An immediate consequence of Theorem 6.51 is the cobordism invariance of quantization. (See Section 5 of Chapter 2 for the definition of Hamiltonian complex G-cobordism.) The index theorem also implies that quantization (i.e., the index) is invariant under equivalences of stable complex structures (see Section 1 of Appendix D). Corollary 6.52. For Hamiltonian complex G-manifolds, geometric quantization is an invariant of Hamiltonian complex G-cobordism. This corollary can also be proved directly, see, e.g., [Brav3, Hi, Ni] for a proof in the non-equivariant setting. Also see Appendix J. Remark 6.53. The index formula (6.38) can be deduced from the index formula for the Spinc Dirac operator. See [ASi]. In the latter, the integrand in (6.38)

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6. GEOMETRIC QUANTIZATION

is replaced by 1 G e 2 c1 (Ldet ) AˆG (T M ),

(6.39)

where AˆG (T M ) is an equivariant cohomology class on M , which is determined entirely by the equivariant Pontrjagin classes and is thus independent of the Spinc structure. When the Spinc structure arises from the pair (J, L), the integrand in (6.39) is equal to that in (6.38). The character χ can also be expressed in terms of the fixed point set M G by the Atiyah–Bott Lefschetz fixed point formula. Let us recall what this formula says in the case when the fixed points are isolated. Let p be a fixed point of the action. The isotropy representation of G on Tp M is complex linear. Hence, its weights are well defined. Let us denote these weights by −αi,p , i = 1, . . . , d. One also has a representation of G on the fiber Lp ; the weight of this representation is the moment map value Φ(p). Theorem 6.54 (Atiyah–Bott Lefschetz fixed point formula [AB1]). If ξ ∈ g satisfies (6.40)

e

√ −1αk,p (ξ)

for all k = 1, . . . , d and all p ∈ M G , then (6.41)

χ(exp ξ) =

X p

e

√ −1Φ(p)(ξ)

6= 1

d  Y

k=1

1−e

−1 √ −1αk,p (ξ)

.

In principle, (6.41) only determines χ ◦ exp on an open dense subset of G. However, since χ ◦ exp is smooth (for Q(M ) is finite–dimensional, when M is compact), it in fact determines χ◦exp everywhere. Finally, note that Theorem 6.54 can be derived from Theorem 6.51 by applying the Atiyah–Bott–Berline–Vergne localization formula (see Section 7 of Appendix C) to the integrand in (6.38). Both of these theorems are discussed in greater detail in Appendix I. 7.3. Which two-forms are we quantizing? We have described how to obtain a (virtual) representation from a line bundle L → M and an (almost, or stable) complex structure J. We took L to be the pre-quantization line bundle for a closed two-form ω and viewed J as a polarization; additional data is needed to define the quantization of (M, ω). However, the notion of a Spinc structure “merges” L and J so that we can no longer take them apart: different pairs (L, J) may yield the same Spinc structure and hence the same quantization. Moreover, in general, a Spinc structure need not arise from a (stable) complex structure and a line bundle. These observations lead to the following new definition. Definition 6.55. Let (M, ω) be a manifold with a closed two-form. A prequantization Spinc structure for (M, ω) is a Spinc structure (P, p) such that [ω] is half the curvature class of its determinant line bundle. A Spinc structure (P, p) together with a connection on P provide Spinc pre-quantization data for (M, ω) if ω is half the curvature of the induced connection on its determinant line bundle. The Spinc quantization of (M, ω) is then the index of the corresponding Dirac Spinc operator.

8. GEOMETRIC QUANTIZATION AS A PUSH-FORWARD

117

Remark 6.56. As we will soon see, this new Spinc quantization is entirely determined by the cohomology class [ω] and hence the Spinc structure plays a purely auxiliary role in it. Moreover, this version of Spinc quantization depends on [ω] differently than the quantizations introduced in the previous sections. Generally, in this book, we will merely use Spinc structures to define the quantization of (M, ω, J) where ω is the curvature of L → M and where J is a stable complex structure. When studying the index of a Spinc Dirac operator (e.g., in Appendix J) we will use the term Spinc quantization without specifying a two-form. Example 6.57. Let (P, p) be the Spinc structure associated with a line bundle L and stable complex structure J. Suppose that L is a pre-quantization line bundle for ω. Then (P, p) is a pre-quantization Spinc structure for the two-form ω + ωJ , where ωJ is half the curvature of the dual to the canonical line bundle associated with J. Example 6.58 ([CKT, Example 2.15]). A closed two-form ω on CP2 is Spinc 1 pre-quantizable if and only if its cohomology class satisfies 2π [ω] = (k + 3/2)[σ] 2 2 for some integer k, where [σ] generates H (CP ; Z). Note that this class is not integral. By the index theorem, the index of the Spinc dirac operator is Z 1 e 2 c1 (Ldet ) AˆM M

where Ldet is the associated determinant line bundle and where AˆM is a characteristic class of the tangent bundle T M which does not depend on any additional data. Therefore, if (M, ω) is Spinc -pre-quantizable, its Spinc quantization is given by Z eω AˆM .

Q(M, ω) =

M

In particular, this quantization is independent of the choice of Spinc -structure, or on any other choices. It only depends on the cohomology class [ω] and on the orientation of M . In various applications, Spinc quantization is “better behaved” than Dolbeault quantization. See [CKT, GK] and, for the Spinc quantization of coadjoint orbits and its relation to representation theory, see [Par3]. For a broader discussion and further references, see [Ve2] (where the term “quantum line bundle” is used instead of “Spinc structure”).

Remark 6.59. The notion of Spinc pre-quantization from Definition 6.55 is essentially equivalent to the pre-quantization through “quantum line bundles” used by Duflo and Vergne (see [Ve2]) and to the pre-quantization through “Mpc -structures” used by Robinson and Rawnsley on symplectic manifolds (see [RR, §7]). 8. Geometric quantization as a push-forward Geometric quantization for stable complex structures can be defined by purely topological means, as the push-forward in K-theory. Namely, let L be a line bundle pre-quantizing a G-action on a compact oriented manifold M . Assume in addition that M is equipped with a G-equivariant stable complex structure J (see Section 1 of Appendix D).

118

6. GEOMETRIC QUANTIZATION

The G-action on L turns L into a G-equivariant bundle and thus determines an element [L] in the unitary G-equivariant K-theory KG (M ). The structure J, in turn, determines an (invariant) complex structure on the stable normal bundle to M and thus a K-theoretic orientation of M . In this setting we have a push-forward map p! : KG (M ) → KG (pt), and one sets Q(M ) = p! ([L]). This push-forward is an element of the representation ring R(G) = KG (pt). See, e.g., [Met3] for more details and further references. (A word of warning: this construction requires a careful choice of orientations. The analysis of orientations given in [Met3] in the symplectic case can be easily extended to stable complex manifolds.) The motivation for this definition comes from the equivariant index theorem. Recall that the equivariant Chern character ch G is a homomorphism from KG (M ) to a completion of the even–dimensional cohomology of M (with rational coefficients), which we denote by H ∗∗ (M ). The equivariant Riemann–Roch theorem (or, more precisely, its algebraic-topological counterpart) says that Z (6.42) ch G (L) Td G (M, J). ch G (p! ([L])) = M

G

G

(Note in this connection that ch (L) = ec1 (L) = e[ω+Φ]/2π .) The right-hand side of the Riemann–Roch formula (6.42) can be interpreted purely in terms of representa∗ tion theory. Namely, let us identify HG (pt) with the ring of G-invariant polynomial ∗∗ functions on g (see Appendix C) and HG (pt) with the ring of G-invariant formal power series on g. Then, as follows from the equivariant index theorem, ch G (p! ([L])) is just the Taylor expansion at the unit of G of χ ◦ exp, where χ is the character of the virtual representation Q(M ). Thus, this push-forward quantization is equivalent to the Dolbeault quantization introduced in Section 6. In particular, when J is a complex structure, Q(M ) is equal to the quantization defined in Section 5. Theorem 6.54 holds for this push-forward quantization Q(M ); see [ASe, Se]. In fact, by using the localization formula (C.17), one can deduce the analogue of Theorem 6.54 for Q(M ) = p! ([L]) from the Riemann–Roch formula (6.42). Unlike the index theorem, formula (6.42) is a statement of a purely topological nature, based on the fact that the Todd class Td G (M, J) measures the discrepancy between the push–forward in the equivariant K-theory and the push–forward in equivariant cohomology. We refer the reader to [FF] for a detailed and illuminating discussion of the topological Riemann–Roch theorem in the non-equivariant setting. It follows immediately from the equivariant Stokes’ formula (see Appendix C) that the push-forward quantization is an invariant of compact cobordisms of Gmanifolds equipped with the following structures: an orientation, an equivariant stable complex structure (see Section 1 of Appendix D), and a pre-quantization of the G-action.

CHAPTER 7

The quantum version of the linearization theorem The main goal of this chapter is to present a quantized version of the linearization theorem. The linearization theorem “decomposes” a symplectic manifold, equipped with a Hamiltonian action with isolated fixed points, into a disjoint union of vector spaces: G M∼ (7.1) Tp M. p∈M G

Quantization of M produces a (virtual) representation Q(M ) of the group G. The Atiyah–Bott formula expresses this representation in terms of the fixed point data in M . We will show that the right-hand side of this formula can be interpreted as the quantization of the spaces Tp M which occur in (7.1). We deduce a quantum version of the linearization theorem: X Q(M ) = (7.2) Q(Tp M ). p∈M G

Remark 7.1. This “quantum linearization theorem” can also be obtained directly from the linearization cobordism (7.1). For this, one must utilize an appropriate notion of equivariant index which is defined for non-compact manifolds and which is invariant under proper cobordism. See [Brav4]. 1. The quantization of Cd Let M be the linear space Cd and let ω be the symplectic form (7.3)

ωr =

√

−1

d X j=1

j dzj ∧ dz j

with 1 = . . . = r = −1 and r+1 = . . . = d = +1. P (That is, ω = 2 j dxj ∧ dyj , the factor 2 being a mere convenience.) Since ωr is exact and Cd is simply-connected, the pre-quantization of Cd poses no problems. By Corollary 6.6, there exists a unique (up to isomorphism) prequantization triple (L, √h, i, ∇), the line bundle L is trivial, and the trivializing section s satisfies ∇s = −1βs, where ω = −dβ. Let us now describe the quantization of Cd with the symplectic form ωr and the polarization given by the standard complex structure. The quantization line bundle L = Lr then becomes a holomorphic line bundle over Cd , and the above trivialization is holomorphic if the primitive β is of type (1, 0). Furthermore, the 119

120

7. THE QUANTUM LINEARIZATION THEOREM

holomorphic structure on Lr is independent of the choice of such a form β. We choose to work with the one-form √ X β = −1 (7.4) j z j dzj . As in the previous section, consider the L2 Dolbeault cohomology spaces d HL0,k 2 (C ; Lr ),

defined with respect√to the standard Hermitian metric on Cd and the standard Lebesgue measure ( −1)d dzdz. (Note that, up to scalar factors, the Lebesgue measure is equal to both the Liouville measure on (Cd , ωr ) and to the measure induced by the Hermitian metric on Cd .) Remark 7.2. The metric on Cd is not part of the quantization data. However, any Hermitian positive-definite inner product on the linear space Cd would result in essentially the same L2 -quantization. Hence, the structure of a linear space on Cd , the complex linear structure on Cd , and the quantization data (in fact, just ωr ) d determine the spaces HL0,k 2 (C ; Lr ). The Bargmann space L2hol (Cd ; µ) is the space of entire holomorphic functions on Cd which are L2 integrable with respect to the Gaussian measure √ 2 µ = ( −1)d e−|z| dzdz.

This space is a Hilbert space, in which the monomials z1m1 · · · zdmd , for non-negative integers mi ∈ Z+ , form an orthogonal basis. This space is sometimes referred to as the Fock space; see, for instance, [Man2] or [Wd]. Similarly, the “Bargmann r space” L2hol (C ×Cd−r ; µ) is the space of functions on Cd which are anti-holomorphic with respect to the first r coordinates and holomorphic with respect to the last (d − r) coordinates and which are L2 integrable with respect to µ. The monomials md mr mr+1 1 form an orthogonal basis for this space. zm 1 · · · z r zr+1 · · · zd d Theorem 7.3. The space HL0,k 2 (C ; Lr ) is zero if k 6= r. For k = r, this space r is the tensor product of the polarized Bargmann space L2hol (C × Cd−r ; µ) with the one-dimensional Hilbert space spanned by

(7.5)

s ⊗ dz 1 ∧ . . . ∧ dz r ,

where s is the trivializing section of Lr described above. Example 7.4. When r = 0, we have the standard K¨ ahler symplectic form ω0 = Pd 0,k d d −1 j=1 dzj ∧ dz j . The spaces HL2 (C ; L0 ) vanish for k > 0, and HL0,0 2 (C ; L0 ) 2 d 2 is the Bargmann space Lhol (C ; µ) of L integrable entire functions.

√

Remark 7.5. Note that the line bundles Lr are all equivalent to each other (and are all trivial) for various values of r as holomorphic line bundles and as Hermitian line bundles, but not as holomorphic Hermitian line bundles (nor as line bundles with connections). We have already proved the holomorphic triviality. The Hermitian triviality follows immediately from the fact that Cd is contractible. The Hermitian holomorphic non-equivalence is a consequence of the fact that for different values of r the pairs (ωr , J) give rise to indefinite metrics of different signatures. Also, arguing as in the proof of Theorem 7.3 below, one can show that a trivializing section s of Lr cannot simultaneously be holomorphic and have constant norm.

1. THE QUANTIZATION OF Cd

121

d Remark 7.6. The spaces HL0,k 2 (C ; Lr ) and their kin have been extensively d studied. For example, the representations of U(r, d−r) on HL0,k 2 (C ; Lr ) are analyzed in [Man1] where Theorem 7.3 has also been proved; see also [BR, Man2].

Proof of Theorem 7.3. The plan of the proof is as follows. First we reduce the theorem to a calculation of the L2 Dolbeault cohomology of Cd with trivial coefficients, but with respect to a different measure. Then we explicitly carry out the calculations for d = 1 and r = 0, 1. Finally, the general case is reduced to the latter two via an analogue of the K¨ unneth formula. Step 1: Reduction to the untwisted L2 Dolbeault cohomology. √ Recall that Lr admits a holomorphic trivializing section s such that ∇s = −1βs, with β = √ P −1 j z j dzj . Since dhs, si

= h∇s, si + hs, ∇si √ −1(β − β)hs, si =  X  = d − j |zj |2 hs, si,

up to a multiplicative constant (which we can scale to be equal to one), (7.6) The mapping

hs, si = e−

P

j |zj |2

.

f dz i1 ∧ . . . ∧ dz iq 7→ s ⊗ f dz i1 ∧ . . . ∧ dz iq gives rise to an isomorphism of the Dolbeault complexes, Ω0,∗ (Cd ) → Ω0,∗ (Cd ; Lr ).

This map induces an isomorphism of the L2 –Dolbeault √ complexes, where the target complex is equipped with the Lebesgue measure ( −1)d dzdz and, by (7.6), the complex Ω0,∗ (Cd ) is equipped with the measure P √ 2 µr = ( −1)d e− i |zi | dzdz.

d Emphasizing its dependence on r, we will denote the latter complex by Ω0,∗ L2 (C ; µr ) 0,∗ and its cohomology by HL2 (Cd ; µr ). Hence,

(7.7)

0,k d d HL0,k 2 (C ; Lr ) = HL2 (C ; µr ),

d and to prove the theorem it suffices to compute HL0,∗ 2 (C ; µr ). Step 2: d = 1 and r = 0. In this case, the measure is √ 2 (7.8) µ0 = −1e−|z| dzdz = µ,

and the zeroth cohomology is the Bargmann space L2hol (C; µ). For k > 1, the kth cohomology is zero, because Ω0,k (C) = 0. It remains to show that HL0,1 2 (C; µ0 ) = 0. The monomial one-forms z k z l dz form an orthogonal basis of the Hilbert space Ω0,1 (C; µ0 ). (To get an orthonormal basis, one must divide by ||z k z l dz|| = ||z k z l || = pL2 π(k + l)!.) Since   1 k l+1 z z z k z l dz = ∂ , l+1

2 the image of ∂ is dense in Ω0,1 L2 (C; µ0 ). By the definition of L cohomology (see 0,1 Section 5.3 of Chapter 6), HL2 (C; µ0 ) = 0.

122

7. THE QUANTUM LINEARIZATION THEOREM

Step 3: d = 1 and r = 1. In this case, the measure is √ 2 µ1 = −1e|z| dzdz,

and the zeroth cohomology is H 0,0 (C; L1 ) = L2hol (C; µ1 ). The assertion that this space is zero is similar in spirit to Liouville’s theorem on bounded holomorphic functions. Indeed, this space is the space of holomorphic functions f on C which satisfy Z 2 (7.9) |f (z)|2 e|z| dzdz < ∞.

In essence, this is a growth condition and, as in the case of Liouville’s theorem, a holomorphic function for which this integral is finite must be zero. Formally, however, the integral condition (7.9) does not imply directly that f is bounded, and Liouville’s theorem does not apply. So let us give an elementary proof of the fact that an entire function f satisfying (7.9) is necessarily zero. Consider the integral of |f |2 over the circle of radius r: Z  √  2 X |an |2 r2n , I(r) = f re −1θ dθ = 2π P where f (z) = an z n is the Taylor expansion of f . This integral is a non-negative monotone increasing function in r. On the other hand, (7.9) readily implies that I(rk ) → 0 for some sequence rk → ∞. Therefore, I(r) = 0 for all r > 0 and hence f ≡ 0. Since the k-th cohomology vanishes for k > 1, it remains to show that there is a unitary isomorphism L2hol (C; µ) → HL0,1 2 (C; µ1 ).

(7.10)

Consider the homomorphism Q : L2hol (C; µ) → Ω0,1 L2 (C; µ1 )

2

given by Q : f 7→ f e−|z| dz. Lemma 7.7. Q induces a unitary isomorphism (7.10) in cohomology. Proof. A simple calculation shows that Q is an isometry. Hence, since the domain of Q is complete, so is its image. As a consequence, the image of Q is closed. Therefore, to show that Q induces an epimorphism in cohomology, it suffices to prove that the image of Q projects to a dense subspace in HL0,1 2 (C; µ1 ) =

Ω0,1 L2 (C; µ1 ) i. T 0,1 closure ΩL2 (C; µ1 ) ∂Ω0,0 L2 (C; µ1 ) h

Recall that the monomials z k z l form an orthogonal basis in L2 (C; µ). Therefore, z ze dz is an orthonormal basis in Ω0,1 L2 (C; µ1 ). For any positive integers k and k l −|z|2 l, if k ≤ l, the form z z e lies in the same cohomology class as a scalar multiple 2 of the form z l−k e−|z| . This follows by induction from the equality k l −|z|2

2

2

2

z k z l e−|z| dz = lz k−1 z l−1 e−|z| dz − ∂(z k−1 z l e−|z| ). 2

Similarly, when k > l, the form z k z l e−|z| dz lies in the same cohomology class as a 2 2 scalar multiple of the form z k−l e−|z| dz = −∂(z k−l−1 e−|z| ). This shows that the

1. THE QUANTIZATION OF Cd

123

2

forms z m e−|z| dz = Q(z m ), for m ≥ 0, topologically generate HL0,1 2 (C; µ1 ). Hence the homomorphism (7.10) induced by Q is onto. To show that this homomorphism is one-to-one, we will prove that the image 0,0 m of Q is orthogonal to Ω0,1 L2 (C; µ1 ) ∩ ∂ΩL2 (C; µ1 ). Recall that the vectors Q(z ), m ≥ 0, topologically span the image of Q. Hence, it is enough to show that each 2 Q(z m ) is orthogonal to ∂(f e−|z| ), where f is such that f ∈ L2 (C; µ)

(7.11) and

2

∂(f e−|z| ) ∈ Ω0,1 L2 (C; µ1 ).

(7.12)

A straightforward calculation shows that in Ω0,1 L2 (C; µ1 ) the inner product of m −|z|2 Q(z ) and ∂(f e ) is given by Z  √

2 2 ∂(f e−|z| ), Q(z m ) = −1 ∂z (z m f e−|z| ) dzdz.

In particular, this integral exists (as a Lebesgue integral) because the inner product is defined owing to (7.12). Hence, denoting by D(r) and S(r) the disc and the circle of radius r centered at the origin, we obtain Z √

 2 2 (7.13) ∂(f e−|z| ), Q(z m ) = − −1 lim ∂(z m f e−|z| dz) r→∞ D(r) Z √ 2 z m f e−|z| dz, = − −1 lim r→∞

S(r)

where the second equality follows from Stokes’ formula. We conclude that it suffices to prove that Z 2 (7.14) z m f e−|z| dz = 0. lim r→∞

S(r)

A simple calculation relying on the Cauchy–Schwarz inequality shows that Z 2 1 m −|z|2 z fe dz ≤ 2πrm+1 e−r F (r) 2 , (7.15) S(r) where

F (r) = By (7.11),

and, therefore,

Z

Z

|f |2 µ = 2

2π

0

Z

∞

0

√ f (re −1θ 2 dθ. 2

F (r)e−r r dr < ∞, 2

F (rn )e−rn rn → 0

for some sequence rn → ∞. From this it is easy to see that 2

1

rnm+1 e−rn F (rn ) 2 → 0.

Therefore, by (7.15),

Z m −|z|2 z fe dz → 0 S(rn )

124

7. THE QUANTUM LINEARIZATION THEOREM

as rn → ∞. Since the limit in (7.14) exists, we conclude from (7.13) that the inner product is zero. This completes the proof of the lemma and the third step of the proof of the theorem. Step 4: Arbitrary d and r. The general case follows from the previous cases by the K¨ unneth formula. Namely, there exists an obvious homomorphism of complexes ⊗r ⊗(d−r) d Ω0,∗ ⊗ Ω0,∗ → Ω0,∗ L2 (C; µ1 ) L2 (C; µ0 ) L2 (C ; µr ).

One can show that this homomorphism induces an isomorphism in L2 -cohomology. The only non-zero cohomology of the tensor product is in degree r and is equal to L2hol (C; µ)⊗r ⊗ L2hol (C; µ)⊗(d−r) , (the tensor product being the Hilbert tensor product; see [Ki2, Chapter 4] or [Dou, Exercise 3.21]). It is not hard to see that the r latter space is naturally isomorphic to the Bargmann space L2hol (C × Cd−r ; µ). Remark 7.8. An argument similar to the proof of Theorem 7.3 allows one to prove a version of Serre’s duality of the L2 cohomology of L. For example, one can 0,d−k ∗ ∼ show that HLd,k (M ; L) for the measure µ0 on M = Cd . Note that 2 (M ; L) = HL2 both spaces are zero unless k = d. (For a general version of Serre’s duality for L 2 cohomology see, e.g., [Schm, Proposition 9.1].) Definition 7.9. The quantization of (Cd , ωr ), with ωr given by (7.3), is the virtual vector space: (7.16)

Q(Cd , ωr ) = (−1)r H 0,r (Cd ; Lr ).

The motivation for this definition is that the quantization of Cd ought to be the alternating sum X d (7.17) Q(Cd , ωr ) = (−1)i HL0,r 2 (C ; Lr ). In general, however, this sum might not make sense if it contained more than one infinite–dimensional term. (See Remark 6.36.) However, by Theorem 7.3, we know that the only non-zero term in the sum is the r-th term and we take this term as the definition of Q(Cd , ωr ). Thus, by Theorem 7.3 we conclude

Remark 7.10. To be consistent with the definition of quantization of compact manifolds we should indicate how this quantization of Cd depends on the orientation. To this end, we declare that (7.16) gives the quantization of Cd equipped with the standard complex orientation. Then, by definition, for Cd with the opposite orientation the quantization is −Q(Cd , ωr ). This sign convention will become relevant in Section 4 in the proof of the quantum linearization theorem. Until then we will always assume that Cd carries its standard complex orientation unless specified otherwise. Theorem 7.11. The quantization (7.17) is the tensor product of the Bargmann r space L2hol (C × Cd−r ; µ) with the one-dimensional Hilbert space spanned by with the formal sign (−1)r .

s ⊗ dz 1 ∧ . . . ∧ dz r ,

We now discuss an equivariant version of these results. Let G be a torus, let αi ∈ g∗ , i = 1, . . . , d, be weights of G, and let G act on Cd as (7.18)

τ (exp ξ)z = (e−

√ −1α1 (ξ)

z1 , . . . , e−

√ −1αd (ξ)

zd ).

2. PARTITION FUNCTIONS

125

This action preserves ωr and the complex structure of Cd . Let us show that this action is pre-quantizable by describing explicitly how it lifts to an action of G on sections of Lr . First of all we note Proposition 7.12. The moment map associated with the action τ is of the form X (7.19) Φ(z) = Φ(0) + i |zi |2 αi , where Φ(0) is an arbitrary constant.

P Proof. In polar coordinates, the two-form is 2 i ri dri ∧ dθi , andPthe infini∂ tesimal action of g on Cd is given by the generating vector fields ξCd = − αi (ξ) ∂θ i P 2 for ξ ∈ g. It is easy to check that the function Φ(0) + i ri αi satisfies the moment map equation ι(ξCd )ω = dΦξ . Corollary 6.18 gives a one-to-one correspondence between infinitesimal lifts of the action to L and moment maps Φ. By Theorem 6.7, a necessary and sufficient condition for a moment map to give a lift of the G-action is that Φ(0) ∈ Z∗G . The value Φ(0) is then the weight of the representation of G on the fiber of L at the origin. Conversely, if this condition is met, a lift of the action is given by letting G act on the trivializing section s with the weight Φ(0): (7.20)

τ (exp ξ)s = e

√ −1Φ(0)(ξ)

s.

Moreover, this pre-quantization of the Hamiltonian G-action (7.18) with moment map (7.19) is unique up to isomorphism by Proposition 6.5 and Corollary 6.18. Assume now that Φ(0) ∈ Z∗G . Because the G-action on Cd is holomorphic, one obtains a representation of G on the virtual vector space Q(Cd , ωr ). We emphasize that this representation, Q(Cd , ωr , Φ), depends on Φ(0). With (7.20) defining the action of G on the trivializing section, s, it is now easy to describe Q(Cd , ωr , Φ): Theorem 7.13. Theorems 7.3 and 7.11 hold as equalities between virtual representations of G. We will describe the representation Q(Cd , ωr , Φ) in more detail in Section 3. In particular, we will show that when the weights αi satisfy a certain positivity requirement, the alternating sum (7.17) defining Q(M ) does belong to a suitable completion of the representation ring R(G). (See Remark 6.36.) 2. Partition functions Let G be a torus and let α1 , . . . , αd ∈ Z∗G be elements of its integral weight lattice. We associate with the collection α1 , . . . , αd a function N : Z∗G → Z ∪ {∞},

called the partition function. By definition, for a weight, α, the value N (α) is the number of solutions of the equation (7.21)

d X i=1

ki αi = α,

ki ∈ Z,

ki ≥ 0,

i = 1, . . . , d.

Geometrically, N (α) is the number of lattice points lying in the polytope X ∆α = {s ∈ Rd+ | si αi = α},

126

7. THE QUANTUM LINEARIZATION THEOREM

where Rd+ is the positive orthant Rd+ = {s ∈ Rd , si ≥ 0 , i = 1, . . . , d}, i.e., N (α) = #(Zd ∩ ∆α ).

As a consequence, N (α) is approximately equal to the volume of the polytope ∆ α , normalized appropriately. The polytope ∆α is the intersection of the positive orthant Rd+ with the affine subspace X (7.22) A(α) = {x ∈ Rd | xi αi = α}. Equivalently, ∆α is a level set of the projection map X (7.23) Rd+ → g∗ , s 7→ s i αi .

Recall from Chapter 4 that the weights αi ∈ g∗ are said to be polarized if there exists a vector ξ ∈ g such that

(7.24)

αi (ξ) > 0 for i = 1, . . . , d.

To proceed we need a few standard facts from convex geometry. Proposition 7.14. Let αi be elements of the integral weight lattice Z∗G ⊂ g∗ . The following conditions are equivalent to each other: (1) The weights αj are polarized. (2) The partition function N (α), α ∈ Z∗G , assumes only finite values. (3) There exists a weight α ∈ Z∗G such that N (α) is non-zero and finite. Proof. If the weights αj are polarized, the polytopes ∆α are compact, by Proposition 4.14. This implies that N (·) is finite–valued. Suppose now that the weights αj are not polarized. By Proposition 4.14, there P exist non-negative coefficients l1 , . . . , ld , not all zero, such that lj αj = 0. Because the weights αj are in a lattice, we may assume, without loss of generality, that all lj are integers. For any α, if k = (k1 , . . . , kd ) is a solution of (7.21), then so is (k1 + ml1 , . . . , kd + mld ) for any non-negative integer m. Hence, if N (α) is nonzero, the value N (α) is infinite. The equivalence of assertions (1) and (2) will be particularly important for us: Corollary 7.15. The partition function N (·) takes finite values if and only if the weights αi are polarized. We will assume from now on that the weights αj are polarized. We have the following discrete analogue of Proposition 4.17. Theorem 7.16. Assume that the weights αi ∈ Z∗G , i = 1, . . . , d, are polarized and let n = dim span{αi | i = 1, . . . , d}. Then there exists a positive constant C such that N (α) ≤ C|α|d−n for all α ∈ Z∗G .

2. PARTITION FUNCTIONS

127

Proof. Without loss of generality, we may assume that the weights αi span g∗ ; otherwise, we replace g∗ by their span. For α ∈ Z∗G , let X A(α) = {x ∈ Rd | xi αi = α}. This is an affine plane of dimension d − n. Consider the shifting map

i : (x1 , . . . , xd ) 7→ (x1 + 1, . . . , xd + 1)

from A(α) to A(β), where β = α + α1 + . . . + αd . The map i sends Zd ∩ ∆α into X ∆β = {x ∈ Rd+ | xi αi = β}.

For each k ∈ Zd ∩∆α , the ball in A(β) of radius 1/2 centered at i(k) is still contained in ∆β . (This, in particular, implies that when Zd ∩ ∆α 6= ∅, the polytope ∆β has dimension d − n, although ∆α may have a lower dimension, and that ∆β spans the affine space A(β).) Furthermore, it is easy to see that two such balls, for distinct k and k 0 in Zd ∩ ∆α , are disjoint. Comparing volumes, we see that vN (α) ≤ vol(∆β ),

(7.25)

where v > 0 is the volume of the (d − n)-dimensional ball of radius 1/2 in A(β). Let ξ be a polarizing vector, so that By definition, β =

and so

P

m := min{α1 (ξ), . . . , αd (ξ)} > 0. xi αi for all x ∈ ∆β ⊂ Rd+ . As a consequence, X X β(ξ) = xi αi (ξ) ≥ m xi , 2

||x|| = Hence,

X

x2i

≤

X

xi

2

≤



1 β(ξ) m

2

.

∆β ⊂ A(β) ∩ B(r),

1 where B(r) is the ball in R of radius r = m β(ξ). Comparing volumes again, we conclude that vol(∆β ) ≤ vol(A(β) ∩ B(r)). Since the volume on the right-hand side is smaller than that of the (d − n)1 β(ξ), it follows from (7.25) that there exists a dimensional ball of radius r = m constant, C 0 > 0, which only depends on the weights αi , such that d

N (α) ≤ C 0 (β(ξ))d−n .

Since β(ξ) = α(ξ) + const, the theorem follows.

There is a close relationship between the partition function N (·) associated with the weights α1 , . . . , αd ∈ Z∗G and the G-action on Cd with weights −αj given by (7.26)

(z1 , . . . , zd ) 7→ (e−

√ −1α1 (ξ)

, . . . , e−

The action (7.26) has a moment map 1X |zj |2 αj . Φ(z) = 2

√ −1αd (ξ)

).

128

7. THE QUANTUM LINEARIZATION THEOREM

The image of this map is the convex polyhedral cone nX o sj αj | s ∈ Rd+ . C(α1 , . . . , αd ) =

The partition function N (·) is supported in this cone. Also, the partition function N (·) is supported in the sub-lattice of Z∗G that is generated by the lattice elements α1 , . . . , αd . We have the following relation between this lattice and the action of G: Lemma 7.17. The action (7.26) is effective if and only if the weights α 1 , . . . , αd generate Z∗G . Proof. The element exp ξ ∈ G is in the kernel of (7.26) if and only if αj (ξ) ∈ 2πZ for all j. This means that ξ is in the dual lattice to the lattice spanned by α1 , . . . , αd (with the 2π-conventions of Appendix A). On the other hand, exp ξ ∈ G is trivial if and only if ξ is in the lattice ZG = ker exp. Since the dual to Z∗G is ZG , the lemma follows. We will now assume that the weights αj are generators of the weight lattice, Z∗G , over Z. Otherwise, we may replace Z∗G by the lattice generated by the αj . We recall that an element α ∈ g∗ is a regular value of Φ if and only if it satisfies the following condition (see Lemma 5.19): For every solution of the equation m X (7.27) α= s i l αi l , l=1

with all sil > 0, the weights αi1 , . . . , αil span g∗ . Note that if α is not contained in the cone C(α1 , . . . , αd ), then it is regular by default, because equation (7.27) has no solutions. If α is in the cone C(α1 , . . . , αd ), it is regular if and only if it cannot be expressed as the convex combination of fewer than dim G elements αi . We will denote the set of regular values by g∗reg . Then the set (7.28)

C(α1 , . . . , αd ) ∩ g∗reg

is the complement in C(α1 , . . . , αd ) of the lower dimensional cones generated by subsets of the αj ’s. The connected components of the set (7.28) are open convex polyhedral cones, which we will call moment cones. Moreover, if ξ ∈ g is a polarizing vector, so that αj (ξ) > 0 for all j, then the intersection of each moment cone with the hyper-plane α(ξ) = 1 is a convex polytope, ∆, and hence the moment cone itself is the cone over this polytope: {λα | α ∈ ∆, λ ∈ R+ }.

Example 7.18. Let g∗ = R3 , and let α1 , . . . , α4 be the vertices of the unit square in the z = 1 plane. Then C(α1 , . . . , α4 ) is the cone over this square. The regular points in this cone are those of its interior points that do not lie on the diagonal planes. Hence, there are four moment cones. We will next discuss the behavior of the partition function on each of the moment cones.

2. PARTITION FUNCTIONS

129

Definition 7.19. A function P : g∗ → C is a quasi-polynomial if it is of the form √ N X −1α(ξk ) − nk P (α) = e (7.29) Pk (α), k=1

where Pk are polynomials, nk are positive integers, and ξk are elements of the group lattice ZG . The degree of P is the maximum of the degree of the Pk ’s. In what follows we will be interested mainly in the restriction of P to Z∗G . Formula (7.29) means that Z∗G partitions into the union of affine sublattices, and on each of these sublattices, P is a polynomial.

Theorem 7.20. Let C be a moment cone. Then there exists a unique quasipolynomial PC of degree d − n, with the property

(7.30)

for all α ∈

PC (α) = N (α)

Z∗G

∩ C.

In Chapter 8 we will interpret the partition function N (·) as the quantization of a K¨ ahler toric variety that is obtained from Cd by reduction. This variety has orbifold singularities, and its quantization is given by the Kawasaki Riemann–Roch formula (see Appendix I). From this formula one can deduce that the quantization is quasi-polynomial as a function of α as α varies in a moment cone. Theorem 7.20 is a combinatorial expression of this same fact. For a purely combinatorial proof of Theorem 7.20, see [Stu]. Example 7.21. Let G = S 1 , Z∗G = Z, α1 = 1, and α2 = 2. Then and

{N (0), N (1), N (2), . . . } = {1, 1, 2, 2, 3, 3, . . . },

α 3 (−1)α + + . 2 4 4 It turns out that N (α) is given by the quasi-polynomial PC even for a singular value α, on the boundary of the moment cones C: N (α) =

Theorem 7.22. For all α in the closure of the moment cone C, N (α) = PC (α). For a proof, see [Stu]. In particular, we have Corollary 7.23. Let C1 and C2 be two different moment cones whose closures contain the singular value α ∈ Z∗G . Then PC1 (α) = PC2 (α).

Recall that the pre-quantization of a singular reduced space is defined by passing to a nearby regular value; see Section 3.2. If we apply this to Cd with its standard K¨ ahler form, the dimension of the quantization is equal to PC (α) where C is the moment cone which contains a nearby regular value that we chose. This fact is a consequence of the Kawasaki Riemann–Roch formula. Corollary 7.23 is a combinatorial expression of the fact that the quantization of a singular reduced space is independent of the choice of desingularization, as long as α remains in the image of the moment map; cf. Chapter 8.

130

7. THE QUANTUM LINEARIZATION THEOREM

3. The character of Q(Cd )

3.1. Partition functions and √ the quantization of Cd . Consider the space P C with the symplectic form ωr = −1 j dzj ∧ dz j with 1 = . . . = r = −1 and r+1 = . . . = d = +1 as in Section 1. Let a torus G act on it with weights −α1 , . . . , −αd as in (7.18) and moment map (7.19) such that Φ(0) ∈ Z∗G . We recall that the quantization Q(Cd , ωr ) was defined by (7.16) as d

Q(Cd , ωr ) = (−1)r H 0,r (Cd ; Lr ).

Thus (−1)r Q(Cd , ωr ) is a genuine representation of G. In this section we describe this representation in terms of a partition function. Set α# i = 1, . . . , d. i =  i αi , Let N (·) be the corresponding partition function, i.e., N (α) is the number of nonP negative integer solutions (k1 , . . . , kd ) to the equation k i α# i = α.

Proposition 7.24. The representation (−1)r Q(Cd , ωr ) of G splits into a direct sum of one-dimensional weight spaces for G, and the multiplicity with which each weight α ∈ Z∗G occurs is N (α − ν), where (7.31)

ν = Φ(0) +

r X

α# i .

i=1

r

d

Proof. The representation (−1) Q(C , ωr ) was computed in Theorem 7.11: r this is the tensor product of the Bargmann space L2hol (C × Cd−r ; µ) with the onedimensional space spanned by sdz 1 ∧ . . . ∧ dz r . This Hilbert space is the orthogonal sum of the one-dimensional spaces spanned by (7.32)

k

r+1 s ⊗ z k11 · · · z kr r zr+1 · · · zdkd dz 1 ∧ . . . ∧ dz r .

Since the representation of G on the Bargmann space induced by (7.18) transforms the coordinate function zi according to the weight αi (see Section 2.5 of Chapter 6), it transforms the vector (7.32) according to the weight (7.33)

Φ(0) +

d X

k i α# i +

i=1

r X

α# i .

i=1

Indeed, s transforms according to the weight Φ(0); for r + 1 ≤ i ≤ d, each ziki ki transforms according to the weight ki αi = ki α# i ; for 1 ≤ i ≤ r, each z i transforms # according to the weight −ki αi = ki αi ; and, finally, for 1 ≤ i ≤ r, each dz i transforms according to the weight −αi = α# i . Adding up these weights, we obtain (7.33). The multiplicity with which a weight α occurs in the direct sum is equal to the number of ways in which α can be represented as a sum of the form (7.33). This is the number of non-negative integer solutions (k1 , . . . , kd ) of the equation d X i=1

k i α# i = α − Φ(0) −

In other words, this number is N (α − ν).

r X

α# i .

i=1

Combining Proposition 7.24 and Corollary 7.15, we obtain

3. THE CHARACTER OF Q(Cd )

131

Corollary 7.25. The weights α# i , i = 1, . . . , d are polarized if and only if each weight α ∈ Z∗G occurs in the representation (−1)r Q(Cd , ωr , Φ) with finite multiplicity. P By Proposition 4.15, the moment map Φ(0) + |zj |2 α# j is proper if and only if the weights α# are polarized. Thus, we have j Corollary 7.26. The moment map Φ on (Cd , ωr ) is proper if and only if each weight α ∈ Z∗G occurs in the representation Q(Cd , ωr , Φ) with finite multiplicity. 3.2. The distributional character of Q(Cd , ωr ). Denote by τ the representation (−1)r Q(Cd , ωr ) of G. By Proposition 7.24, an element of G, which we write as g = exp ξ for ξ ∈ g, acts by an infinite diagonal matrix in which the entry √ e −1α(ξ) occurs N (α − ν) times. Taking the trace of this matrix as a formal sum, we conclude that the character of the representation τ is, formally, √ X N (α − ν)e −1α(ξ) . χτ (exp ξ) = (7.34) α

Furthermore, in the proof of Proposition 7.24 we showed that τg acts on each basis element (7.32) of (−1)r Q(Cd , ωr ) by multiplication by the character (7.35)

e

√ P −1(ν+ j kj α# j )(ξ)

.

Adding up these contributions formally, we obtain a formal expression for the character (7.34) as the sum of a geometric progression: X √ √ # # (7.36) e −1(k1 α1 (ξ)+...+kd αd (ξ)) . χτ (exp ξ) = e −1ν(ξ) k=(k1 ,... ,kd )∈Zd +

This series does not converge as a function on G. However, it does converge as a distribution. Therefore, to make the above analysis rigorous, we must make sense of χτ as a distributional character. The formula (7.34) will then become the Fourier expansion of χτ ; see Corollary 7.28 below. Let us now work out the relevant analysis in this argument. We can think of a representation τ of G on a Hilbert space H as a function on G that takes values in the space of operators on H. To a smooth function f : G → C we can then associate the operator τf on H given by Z (7.37) τf (v) = f (g)τg (v) dg, G

for v ∈ H, where dg is the normalized Haar measure.

r d Theorem √ P 7.27. Denote by τ the representation (−1) Q(C , ωr ) of G, where ωr = −1 i dzi ∧ dz i with 1 = . . . r = −1 and r+1 = . . . d = 1, and where G acts on Cd with weights −αi , i = 1, . . . , d. Assume that the weights α# i = i αi are polarized. Then for each f ∈ C ∞ (G) the operator τf is of trace class, and the linear functional χτ (f ) = trace τf on C ∞ (G) is continuous, i.e., χτ is a distribution on G. Furthermore, X χτ (f ) = (7.38) fˆ(−α)N (α − ν) α

∞

for f ∈ C (G).

132

7. THE QUANTUM LINEARIZATION THEOREM

Proof of Theorem 7.27. Recall that the Fourier transform fˆ of f is the function on the dual group Z∗G given by Z √ ˆ f (α) = f (g)e− −1α (g) dg, (7.39) G

√

where the exponential function is viewed as a function on G, i.e., e− −1α (g) = √ − −1α(ξ) e for g = exp ξ, ξ ∈ g. Suppose that the representation τ leaves fixed the one-dimensional space C · v, spanned by a vector v, and that it acts on this space with a weight −α, i.e., τg (v) = √ − −1α(ξ) v for all g = exp ξ, ξ ∈ g. If we multiply f (g) by this character and e integrate over G, we see from (7.37) and (7.39) that v is an eigenvector for τ f with ˆ an eigenvalue f(α). Hence, by Proposition 7.24, the operator τf is diagonalizable, ˆ with eigenvalues f(α), α ∈ Z∗G , and each eigenvalue fˆ(α) occurs with multiplicity N (−α − ν). To show that τf is of trace class we must prove that the sum X (7.40) |fˆ(α)|N (−α − ν) α

converges. By Theorem 7.16, since the weights α# i determining the partition function are polarized,

(7.41) Since f is C ∞ , we also have (7.42)

|N (−α − ν)| = O(|α|d−n ). |fˆ(α)| = O(|α|−m )

for all m. From these two estimates, it follows that (7.40) converges. The character of τ is then given by (7.38). By the estimates (7.42) and (7.41), this is a continuous linear functional on C ∞ (G). (See, e.g., [Rudi].) The distribution χτ defined in Theorem 7.27 is called the distributional character of the representation of G on (−1)d Q(Cd , ωr , Φ). Let us now deduce from these results a few properties of χτ which we will use in Section 4. Recall that the Fourier transform of a distribution√χ : C ∞ (G) → C on G is, by definition, the function on Z∗G given by χ(α) ˆ = χ(e− −1α ). The following result follows easily from (7.38): Corollary 7.28. The Fourier transform χ ˆτ of χτ , as a function on Z∗G , is equal to α 7→ N (α − ν). Let η ∈ g be a polarizing vector, so that α# i (η) > 0 for all i. Proposition 7.29. The Fourier transform of χτ is supported on the cone (7.43)

# {α ∈ Z∗G | α − ν ∈ C(α# 1 , . . . , αd )}

and, in particular, its support is contained in the half space (7.44)

α(η) ≥ ν(η).

Proof. Since N (α − ν) is supported on the cone (7.43), this is an immediate consequence of Corollary 7.28. Let Gj be the codimension one subgroup of G whose Lie algebra is {αj = 0}.

3. THE CHARACTER OF Q(Cd )

133

Proposition 7.30. Assume, as before, that the weights α# i = i αi are polarized and τ is the representation (−1)r Q(M, ωr , Φ) of G. Then the distribution χτ is smooth on the complement of the set d [

Gj ,

j=1

and, on this complement, χτ (exp ξ) = (−1)r e

(7.45)

d  −1 Y √ . 1 − e −1αj (ξ)

√ −1Φ(0)(ξ)

j=1

Proof. By (7.36), the distribution χτ is a sum of smooth functions (thought of as distributions). The geometric series (7.36) does not converge as a series of functions. However, (7.36) does converge in the sense of distributions on the complement to the union of the subgroups Gj to (7.46)

χτ (exp ξ) = e

√ −1ν(ξ)

d  Y

j=1

1−e

−1 √ −1α# j (ξ)

,

which is a smooth function on this complement. We will now outline the proof of this for d = 1, The proof for an arbitrary dimension d is essentially the same. Let α = α# 1 . We want to show that the distribution ∞ √ X e −1kα k=0 √ −1α −1

) on the set e converges to the function (1 − e a function supported on this set. Then

√ −1α

6= 1. Let f ∈ C ∞ (G) be

N

X √ √ f −1kα −1(N +1)α √ − e h, f = e 1 − e −1α k=0 √

where h = f (g)(1 − e −1α )−1 is also a smooth function with support on this set. Integrating this identity over G, we see that the right-hand side is the Fourier ˆ coefficient h(−(N + 1)α) of h which tends to zero as N → ∞. Recalling that r X ν = Φ(0) + α# j , j=1

we can rewrite the product in (7.46) as χτ (exp ξ) = e

√ −1Φ(0)(ξ)

r Y

j=1

or, alternatively, as χτ (exp ξ) = (−1)r e

√ −1Φ(0)(ξ)

e

1−e

r Y

j=1

α# j

Since (7.45).

√ −1α# j (ξ)

= −αj when 1 ≤ j ≤ r and

√ −1α# j (ξ)

1

1−e α# j

d Y

1

j=r+1

√ − −1α# j (ξ)

1−e

d Y

j=r+1

√ −1α# j (ξ)

1

1−e

. √ −1α# j (ξ)

= αj when r + 1 ≤ j ≤ d, we obtain

134

7. THE QUANTUM LINEARIZATION THEOREM

4. A quantum version of the linearization theorem Let (M, ω, Φ, J) be a compact 2d-dimensional Hamiltonian G-manifold. (Here G is a torus, ω is a closed G-invariant two-form, Φ is a moment map for ω, and J is an equivariant stable complex structure.) Assume that the fixed points of the G-action are isolated. Let us recall the assertion of the linearization theorem (see Chapter 4) and remind the reader of all the ingredients that have to go into this theorem for us to be able to quantize it. At each fixed point p ∈ M G there exists a G-equivariant complex linear isomorphism (Tp M, Jp ) ∼ = Cd which converts the isotropy action on Tp M to the action on Cd given by   √ √ τp (exp ξ) = e− −1α1,p (ξ) z1 , . . . , e− −1αd,p (ξ) zd ,

where −α1,p , . . . , −αd,p are the isotropy weights at p. Choose a polarizing vector, i.e., a vector η ∈ g such that αk,p (η) 6= 0 for all k and p, and let α# k,p = k,p αk,p = ±αk,p be the corresponding polarized weights, so that

α# k,p (η) > 0 √ P ∗ for all k and p. Let ωp# = −1 k,p dzk ∧ dz k , and let Φ# p : Tp M → g be the moment map for this symplectic structure: X Φ# |zj |2 α# p (z) = Φ(p) + j .

(7.47)

The linearization theorem (see Chapter 4) states that G (M, ω, Φ, J) ∼ (7.48) (Tp M, ωp# , Φ# p , Jp ). p∈M G

Recall also that in this cobordism M is oriented and the orientations of Tp M are induced by the orientation of M . Let us now assume that (M, ω) is pre-quantizable by (L, h, i, ∇), and that the action of G on M lifts to a pre-quantization action of G on L by means of the moment map Φ. Then we have the stable complex quantization Q(M ) of M and also have the quantizations of the tangent spaces Tp M described in Section 1. For compact manifolds, quantization is a cobordism invariant. This follows immediately from the Riemann–Roch formula combined with Stokes’ theorem. This argument does not apply directly to the non-compact cobordism (7.48); see, however, [Brav4]. Nevertheless, the result is still true: Theorem 7.31 (The Quantum Linearization Theorem). As virtual unitary G-representations, X (7.49) Q(Tp M, ωp# , Φ# Q(M, ω, Φ, J) = p , Jp ). p

Remark 7.32. The equality (7.49) is interpreted in the Gr¨ othendieck group of formal differences of unitary representations of G with trace-class characters. Alternatively, Theorem 7.31 can be understood as equalities of multiplicities: for all α ∈ Z∗G , X (7.50) mult(α) = multp (α),

4. A QUANTUM VERSION OF THE LINEARIZATION THEOREM

135

where mult(α) is the multiplicity with which α occurs in the virtual representation Q(M, ω, Φ, J) and multp (α) is the multiplicity with which α occurs in the # signed representation Q(Tp M, ωp# , Φ# p , Jp ). Since the weights αi are polarized, the multiplicities are finite by Corollary 7.25. The multiplicity formula (7.50), when applied to a coadjoint orbit for a compact Lie group, yields the famous Kostant multiplicity formula. Formula (7.50) can also be found, in a slightly different guise, in [GLS]. Theorem 7.31 is equivalent to the distributional identity X (7.51) χM = χp

where χM is the character of Q(M ) and where χp is the distributional character of Q(Tp M ). We note that this equality is slightly stronger than the Atiyah–Bott Lefschetz theorem: on an open dense subset of G, the distributions χp are given by smooth functions, and on this set, (7.51) becomes (6.41).

Remark 7.33. An important special case of Theorem 7.31 is the following result of Bernstein–Gelfand–Gelfand, [BGG]. Let K be a compact semi-simple Lie group having G as its maximal torus and let M be an integral coadjoint orbit of K. By a theorem of Bott–Borel–Weil–Kostant, the quantization representation ρ of K on Q(M ) is irreducible and, in fact, all irreducible representations of K can be obtained this way. (See Example 6.39.) The theorem of Bernstein–Gelfand– Gelfand asserts that the representation ρ can be decomposed into a virtual sum of representations which, as G modules, have a much simpler structure than the G module Q(M ) itself. These “Verma modules” are just the summands on the right-hand side of (7.49). Proof of Theorem 7.31. Without loss of generality, we may assume that the orientation of M is induced by J, and hence the tangent spaces Tp M carry complex orientations induced by Jp . (The reversal of the orientation of M results in that both the right- and left-hand sides of (7.49) change sign; see Remark 7.10.) Let χM be the character of the representation of G on Q(M, ω, Φ, J). For each fixed point p, let χp be the character of the virtual representation of G on Q(Tp M, ωp# , Φ# p , Jp ). By Theorem 7.27 and, in particular, by (7.38), χp determines the multiplicities multp (α) and hence the representation Q(Tp M, ωp# , Φ# p , Jp ). Since Q(M, ω, Φ, J) is finite–dimensional, it is also determined by its character. Therefore, it suffices to prove that X (7.52) χM = χp . The summands in (7.52) were computed in Section 3. Indeed, for each p, the tangent space (Tp M, ωp# , Φ# to Cd , equipped with the G-action p , Jp ) is isomorphic √ P with the weights αk,p , the symplectic form ( −1)d j k,p dzj ∧dz j , and the moment map X X Φ# k,p |zj |2 αk,p = Φ(p) + |zj |2 α# p (z) = Φ(p) + k,p , j

j

α# k,p

where the signs k,p are such that the weights are polarized. By Proposition 7.30, −1 Y √ √ 1 − e −1αk,p (ξ) χp (exp ξ) = e −1Φ(p)(ξ)

136

7. THE QUANTUM LINEARIZATION THEOREM √

on the set G0 where e −1αk,p (ξ) 6= 1 for all p ∈ M G and k = 1, . . . , d. By the Atiyah–Bott Lefschetz fixed point formula (see (6.41)), we conclude that X χM (exp ξ) = χp (exp ξ) p

on the set G0 . In other words, the distribution X  χp − χ M χ=

is supported on the complement of G0 , i.e., on the union of the sets e

√ −1αk,p (ξ)

= 1,

p ∈ M G.

k = 1, . . . , d,

This alone is not sufficient to establish the identity (7.52). However, we have some additional information about χ. Namely, we know, by Proposition 7.29, that for some constant C0 the Fourier transform χ ˆ is supported on the half-space α(η) ≥ C0 ,

(7.53)

α ∈ Z∗G .

We will show that this forces (7.52) to hold identically by proving the following result. # Lemma 7.34. Let α# j , j = 1, . . . , l, be a collection of weights satisfying αj (η) > 0, and let χ be a distribution on G. Assume that χ is supported on the union of the sets

{exp(ξ) | e

(7.54)

√ −1α# j (ξ)

= 1,

j = 1, . . . , l},

and that the Fourier transform of χ is supported on the half-space (7.53). Then χ ≡ 0. Proof. Since G is compact, χ is of the form P ν, where ν is a measure with support on the set (7.54) and P a differential operator of finite order (see [Rudi]). Hence there exist non-negative integers, N1 , . . . , Nl such that (1 − e

(7.55)

√ −1α# 1 N1

)

. . . (1 − e

√ −1α# Nl l

) χ ≡ 0.

Without loss of generality, we can assume that these integers have been chosen to be as small as possible; i.e., if Nj > 0 and we replace Nj by Nj − 1, the identity (7.55) does not hold. We claim that in this case the Nj ’s all have to be zero and, hence, χ ≡ 0. To prove the claim, we argue by contradiction. First observe that by taking the Fourier transform of (7.55) we obtain the identity b = 0, (1 − Tα# )N1 . . . (1 − Tα# )Nl χ 1

l

where Tα# is the operator which transforms a function f : Z∗G → C to the function j

Tαi f (µ) = f (µ − α# j ). In particular, if f is supported on the set (7.53), Tα# f is supported on the set j

α(η) ≥ C0 +

α# j (η).

Suppose now that Nr > 0. Let )Nr −1 . . . (1 − Tα# )Nl χ b. h = (1 − Tα# )N1 . . . (1 − Tα# r 1

l

4. A QUANTUM VERSION OF THE LINEARIZATION THEOREM

137

Then h is supported on the set α(η) ≥ C00 with Moreover, and thus

# # C00 = C0 + N1 α# 1 (η) + . . . + (Nr − 1)αr (η) + . . . + Nl αl (η).

(1 − Tα# )h = 0, r h = T α# h = Tα2# h = . . . = TαN# h r r

r

for an arbitrary N > 0. Therefore, h is supported on the set α(η) ≥ C00 + N α# r (η).

Hence, since N is arbitrary, h ≡ 0. Thus (7.55) holds with N1 , . . . , Nl replaced by N1 , . . . , Nr − 1, . . . , Nl . This completes the proof of Lemma 7.34 and the proof of Theorem 7.31.

CHAPTER 8

Quantization commutes with reduction 1. Quantization and reduction commute Our goal in this chapter is to show that “quantization commutes with reduction”. For the sake of simplicity we assume that G is a torus. Let (M, ω) be a compact symplectic manifold, equipped with a G-action and a moment map Φ : M → g∗ . Let α be a regular value of Φ. The reduced space Mα = Φ−1 (α)/G

is a symplectic orbifold, with a symplectic form ωα induced from ω. If the Hamiltonian G-manifold (M, ω, Φ) is quantizable and α is integral, the reduced space Mα is quantizable too: pre-quantization data on M gives rise to pre-quantization data on Mα . Thus, both the equivariant quantization of M and the ordinary quantization of Mα are well defined as indices of suitable Dolbeault operators. The quantization of M is a virtual representation of G and is determined by the multiplicities with which each weight β ∈ Z∗G occurs in it. The quantization of Mα is a virtual vector space determined by its dimension. The “quantization commutes with reduction” theorem, often abbreviated as [Q,R]=0, asserts that the multiplicity with which the weight α occurs in the quantization of M is equal to the dimension of the quantization of the reduced space Mα : (8.1)

Q(M )α = Q(Mα ).

We refer the reader to Chapter 6 for details on the notion of quantization that we are using. The “quantization commutes with reduction” theorem was conjectured explicitly for the first time in [GS3]. However, less precise forms of it occur much earlier in the mathematics and physics literature; see, for instance, [KKS]. In the physics literature, this statement is as a special case of a general “meta-principle” which asserts that if one quantizes a classical physical system and then sets certain quantum mechanical observables equal to constants, one obtains the same “reduced” quantum mechanical system as that which one would obtain by setting the corresponding classical observables equal to constants and then quantizing this “reduced” classical system. “Quantization commutes with reduction” has implications in representation theory, via the orbit method. For instance, it implies an old conjecture of A. Kirillov: Consider a representation ρ of G corresponding to a coadjoint orbit O ⊂ g ∗ and a representation ρ0 of K ⊂ G corresponding to a coadjoint orbit O 0 ⊂ k∗ . Then if O0 does not occur in the image of the projection O → k∗ , the representation ρ0 is not a sub-representation of ρ|K . We will give a cobordism proof of the “quantization commutes with reduction” theorem. In principle, this proof should require no assumptions on the fixed point 139

140

8.

[Q,R] = 0

set M G , but, to make the idea of the proof as transparent as possible, we will assume that M G is finite. The idea of the proof is the following: by the linearization theorem, the manifold is cobordant to the disjoint union of the linear tangent spaces at the fixed points: G (8.2) M∼ Tp M, p ∈ M G. Since cobordism commutes with reduction, the corresponding reduced spaces are cobordant: G (8.3) Mα ∼ (Tp M )α , p ∈ M G.

Cobordant spaces have the same quantization. Therefore, to prove that “quantization commutes with reduction” for M , it suffices to prove this for the linear G-actions on Tp M . Unfortunately, certain complications arise in this argument. The value α might not be regular for the moment map Tp M → g∗ . One can overcome this problem by working with nearby regular reduced spaces, which provide a desingularization of the singular reduced space. Then, another difficulty arises: the linear spaces Tp M come equipped with complex structures that are incompatible with their symplectic structure, and with such structures, “quantization commutes with reduction” is not always true at singular levels. In this chapter we tackle these problems and prove the following version of the “quantization commutes with reduction” theorem: Theorem 8.1 (Quantization commutes with reduction at regular values). Let G be a torus and let (M, ω, Φ, J) be a quantizable stable complex Hamiltonian Gmanifold. Suppose that M is compact and M G is finite. Let α ∈ Z∗G be a weight for G which is a regular value for Φ. Then Q(M, ω, Φ, J)α = Q(Mα , ωα , Jα ). Usually, one takes ω to be a symplectic structure and J a compatible almost complex structure, however, these assumptions are not necessary. “Quantization commutes with reduction” provides yet another example of a symplectic phenomenon which turns out to be of a topological nature. When considering a singular value α of Φ, we do assume that ω is symplectic and J is a compatible almost complex structure (see Example D.12). Following [MeSj], we desingularize the reduced space Mα by considering a nearby non-empty regular reduced space Mα+h = Z/G, for Z = Φ−1 (α + h), where “nearby” means that α + h belongs to an alcove (a connected component of Φ(M )reg ) whose closure contains α. On Mα+h , we take the reduced stable complex structure Jα+h , induced from J and work with a two-form ωα defined as if reduction was carried out over the value α, not α + h. The precise construction is described in Section 3 of Chapter 6. The quantization of (Mα , ωα ) is defined to be the Dolbeault index associated with this data. We denote it by (8.4)

Q(Mα )+h = Q(Mα+h , ωα , Jα+h ).

“Quantization commutes with reduction” then asserts that (8.5)

Q(M )α = Q(Mα )+h .

The fact that the right-hand side is well defined (independent of h) was shown by Meinrenken and Sjamaar, [MeSj]. This also follows from the assertion (8.5).

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When α is a regular value for Φ, we can still carry out the above “desingularization” procedure. Thus, we obtain (8.6)

Q(Mα )+h = Q(Mα ),

because the right-hand side of (8.4) does not change as α + h varies through regular values of Φ. Thus, we recover (8.1) as a special case of (8.5). Remark 8.2. The reduced space Mα = Φ−1 (α)/G can be smooth (or an orbifold) even if α is singular. This happens when α is quasi-regular, meaning that all the points in Φ−1 (α) have the same stabilizer (up to conjugacy, if G is non-abelian). In this case, we again obtain (8.6), see [MeSj]. This should also be provable by cobordism methods. To summarize, let us state the theorem precisely: Theorem 8.3 (“Singular quantization commutes with reduction”). Let a torus G act on a symplectic manifold (M, ω) with a moment map Φ : M → g∗ . Suppose that (M, ω, Φ) is quantizable. Let J be a compatible almost complex structure. Sup∗ pose that M is compact and M G is discrete. Let α ∈ ZG be a weight for G. Suppose α 6∈ Φ(M ), so that Mα = ∅. Then Q(M, ω, Φ, J)α = 0. Suppose that α ∈ Φ(M ). Let α + h belong to a component of Φ(M )reg whose closure contains α. Then Q(M, ω, Φ, J)α = Q(Mα+h , ωα , Jα+h ). Note that here we must assume that ω is symplectic and J is compatible with ω. Otherwise, the moment map image Φ(M ) is not a meaningful invariant. In Section 9 we give variants of Theorem 8.3 which do not assume non-degeneracy. 2. Quantization of stable complex toric varieties In this section we state several propositions that can be viewed as “quantization commutes with reduction” theorems for linear spaces. The reduction of a linear space is a toric variety. However, if a linear space is equipped with incompatible symplectic and complex structures, its reduction is a stable complex toric variety. Thus, our first goal is to find the quantization of stable complex toric varieties. Let us begin with the standard K¨ ahler structure on Cd . Let a torusPG act on |zj |2 αj . Cd with weights −α1 , . . . , −αd ∈ Z∗G and with a moment map Φ(z) = 21 Assume that the weights are polarized (see Definition 3.19). The quantization of Cd with this structure was determined in Section 3 of Chapter 7: the multiplicity of each weight α ∈ Z∗G is given by
α Q(Cd ) = N (α), where N (·) is the partition function associated with the weights αj : n o X mj α j = α . N (α) = # m ∈ Zd+ |

The reduced space is (Cd )α = Φ−1 (α)/G. Then the “quantization commutes with reduction” theorem takes the following form: Proposition 8.4. Let α is a regular value of Φ. Then dim Q((Cd )α ) = N (α).

The reduced space (Cd )α is the K¨ ahler toric variety corresponding to the polytope X ∆ = {s ∈ Rd+ | sj αj = α}

142

8.

[Q,R] = 0

(see Section 5 of Chapter 5) and N (α) is the number of lattice points in this polytope. Hence, Proposition 8.4 asserts: The quantization (Riemann–Roch number) of a toric variety corresponding to a polytope ∆ is equal to the number of integral lattice points in ∆. This assertion is a well-known “folk theorem” in the toric variety literature and is usually attributed to Atiyah or Danilov (see [Dan]) who worked with an algebrogeometric construction of toric varieties. We prove Proposition 8.4 in Section 6 of this chapter “by hand”, following ideas of Sue Tolman (see [KT1]). Consider now Cd equipped with a symplectic structure and a complex structure that are possibly incompatible. Explicitly, we take the standard complex structure J and a non-standard symplectic form X ω# = j dxj ∧ dyj , where 1 = . . . = r = −1 and r+1 = . . . = d = 1. We let the torus G act, as before, with complex weights −α1 , . . . , −αd . The symplectic weights for this action are −α# j = −j αj ; we assume that these weights are polarized. Let the a moment map be 1X (8.7) |zj |2 α# Φ# (z) = ν + j 2 with ν ∈ Z∗G . The quantization of this space was also calculated in Section 3 of Chapter 7: the multiplicity of each weight α ∈ Z∗G is given by
α Q(Cd , ω # , Φ# , J) = (−1)r N (α − δ − ν),

where N (·) is the partition function associated with the polarized weights α# j , for j = 1, . . . , d, and r X δ= α# (8.8) j . j=1

In this case, the “quantization commutes with reduction” principle takes the following form:

Proposition 8.5. Suppose that α is a regular value of the moment map Φ# . Then the quantization of the reduced space is given by
r # dim Q (Cd )# α , ωα , Jα = (−1) N (α − δ − ν).

Consider now a stable complex Hamiltonian G-manifold, (M, ω, Φ, J), to which we can apply the linearization theorem. Thus, G M∼ Tp M ;

see Chapter 4. Suppose that α is a regular value for the moment maps M → g∗ and Tp M → g∗ for all p ∈ M G . From [Q,R]=0 for the linear spaces Tp M , one can deduce that [Q,R]=0 for M . We will give the precise argument in Section 3. However, there may exist values α which are regular for M but singular for some Tp M . Therefore, we must also consider singular reduction of linear spaces, even if our ultimate goal is to establish that quantization commutes with reduction for a regular value on M . Suppose now that α is a singular value for Φ# : Cd → g∗ . We desingularize the reduced space as described in Section 3 of Chapter 6, by passing to a nearby

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143

regular value α + h. “Nearby” means that α + h belongs to a connected component of Φ(M )reg whose closure contains α. “Quantization commutes with reduction” asserts that the resulting quantization has dimension (−1)r N (α − δ − ν), where δ and ν are given in (8.7) and (8.8). This assertion is true if we make an additional assumption: Proposition 8.6. If h is a positive linear combination of the non-polarized weights α1 , . . . , αd , then # r dim Q((Cd )# α+h , ωα , Jα+h ) = (−1) N (α − δ − ν).

(8.9)

The proof of Proposition 8.6 will occupy Sections 4–8 of this chapter. Here we prove a special case: Proof of a special case of Proposition 8.6. Suppose that α is not in the “polarized moment cone” X # C# = ν + R + α# j = imageΦ . Then the reduced space

# −1 (Cd )# (α)/G α = (Φ )

is empty. The right-hand side of (8.9) is then zero: otherwise, if there exist mj ≥ 0 P such that α − ν − δ = mj α # j , then α=ν+

r X

(mj + 1)α# j +

j=1

#

d X

mj α # j

j=r+1

is in C , contradicting our assumption. Equation (8.9) then reads 0 = 0. In the rest of this section and in the next section we prove that quantization commutes with reduction, assuming Proposition 8.6. Let us first prove Proposition 8.5: Proof of Proposition 8.5. When α is regular, the reduced space at α, # d # # ((Cd )# α , ωα , Jα ), is isomorphic to the nearby reduced space ((C )α+h , ωα , Jα+h ). Proposition 8.5 then follows from Proposition 8.6. We will need the following two consequences of Proposition 8.6. Proposition 8.7. Let α + h is in the “non-polarized moment cone” C=ν+

d X

R + αj .

j=1

# r Then dim Q((Cd )# α+h , ωα , Jα+h ) = (−1) N (α − δ − ν).

Proof. By Proposition 8.6, it is sufficient to find a positive linear combination h0 of αj ’s such that α+h and α+h0 belong to the same connected alcove (component of the set of regular values of Φ# in its image). P By assumption, there exist coefficients sj > 0 such that α + h = ν + sj αj . Because the alcove containing α +P h is an open convex polyhedral cone with vertex ν, it contains the entire ray ν + t sj α, t > 0. For any α and a large enough t, X α+t s j αj

144

8.

[Q,R] = 0

P will also be contained in this cone. Then h0 := tsj αj is a positive linear combination of αj ’s such that α + h0 is in the same alcove as α + h. We end the section with two examples which illustrate that we cannot remove the assumption on h in Proposition 8.6: [Q,R]=0 is not always true for reductions of Cd with incompatible symplectic and complex structures, if we desingularize by h which is not a positive linear combination of complex weights. Example 8.8. Consider C with its standard complex structure and orientation, non-standard symplectic form ω # = −dx ∧ dy, and circle action with moment map Φ# (z) = − 21 |z|2 . Then r = 1, ν = 0, δ = α# j = −1, and the multiplicity of α ∈ Z in the quantization of this space is equal to ( 0 α ≥ 0, r (−1) N (α − δ) = −1 α ≤ −1. However, the quantization of the reduced space at α = 0 is a virtual vector space of dimension −1, and it remains so if we “desingularize” by passing to α + h = h < 0. Hence, for α = 0 and h < 0, “[Q,R]=0” reads 0 = 1. The next example is even more interesting; it shows that, for singular values, [Q,R]=0 might not hold even if α is in the interior of the moment map image. Because an appropriate choice of h does give “[Q,R]=0” (by Proposition 8.6), we also see from this example that singular quantization might depend on the choice of desingularization if the symplectic and complex structures are incompatible. Example 8.9. Consider C3 with the standard complex structure J, standard orientation, and the non-standard symplectic structure ω # = −dx1 ∧ dy1 + dx2 ∧ dy2 + dx3 ∧ dy3 . Let S 1 × S 1 act on (C3 , ω # ) with a and moment map Φ# (z) =


1 |z1 |2 + |z3 |2 , |z2 |2 + |z3 |2 . 2

# # Then ν = 0, δ = α# 1 = (1, 0), α2 = (0, 1), and α3 = (1, 1). Let α = (k, k) for some positive integer k. As follows from Section 1 of Chapter 7,

α = (−1)r N (α − δ − ν) Q C3 , ω # , Φ# , J = −N (k − 1, k)

= −#{m ∈ Z3+ | m1 + m3 = k − 1 and m2 + m3 = k} = −k.

On the other hand, if we desingularize by h = (1 , 2 ) with 1 6= 2 , the quantization of the reduced space is, as is clear from Example 8.12 below, (   −(k + 1) if 1 > 2 , 3 # # Q (C )α+h , ωα , Jα+h = −k if 1 < 2 .

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3. Linearization of [Q,R]=0 In this section we deduce that [Q,R]=0 for manifolds (see Section 1) from the [Q,R]=0 theorems for linear spaces, stated in Section 2. Let (M, ω, Φ, J) be a Hamiltonian G-manifold and let L → M be a prequantization line bundle. Suppose that M is compact and M G is finite. For each fixed point p ∈ M G , let −α1,p , . . . , −αd,p be the isotropy weights at p with respect # to J. Choose a polarizing vector, and let −α# 1,p , . . . , −αd,p be the corresponding polarized weights (see Proposition 4.18). We may assume that the weights are ordered so that ( −αj,p 1 ≤ j ≤ rp , # αj,p = αj,p rp < j ≤ d, for some 1 ≤ rp ≤ d. Let

δp =

rp X

α# j,p .

j=1

Z∗G

The multiplicity of each weight α ∈ in the quantization of M is given by X α Q(M, ω, Φ, J) = (8.10) (−1)rp Np (α − δp − Φ(p)), p

# where Np (·) is the partition function associated with α# 1,p , . . . , αd,p ; see Chapter 7. The Hamiltonian Linearization Theorem (Theorem 4.10) provides the following proper Hamiltonian cobordism: G
Tp M, ωp# , Φ# (8.11) (M, ω, Φ, J) ∼ p , Jp . p

By Theorem 4.12, this cobordism carries the pre-quantization line bundles: G (8.12) (M, L, J) ∼ (Tp M, Lp , Jp ) . p

Let us now use this fact to prove the easiest case of the “quantization commutes with reduction” theorem, namely the case where α is regular for Φ and all Φ# p . Proof of a special case of Theorem 8.1. Let us assume that the weight G α is regular not just for Φ, but also Φ# p for all p ∈ M . In Section 6 of Chapter 5 we showed that reduction and cobordism commute. Hence, by (8.11), G
# (Mα , ωα , Jα ) ∼ (Tp M )# (8.13) α , (ωp )α , (Jp )α . p

Dolbeault quantization is invariant under compact cobordism. (For manifolds, cobordism invariance can be deduced indirectly from the Riemann–Roch theorem and Stokes’ theorem. For orbifolds, this fact can be deduced from the Kawasaki Riemann–Roch theorem combined with Stokes’ theorem; see Theorem I.19 of Appendix I. In Appendix J the cobordism invariance of the index is proved directly for manifolds and orbifolds.) Hence, (8.13) implies that X
# (8.14) dim Q(Mα , ωα , Jα ) = dim Q (Tp M )# α , (ωp )α , (Jp )α . p

By Proposition 8.5, which we will prove in Sections 4–8,
# rp (8.15) dim Q (Tp M )# α , (ωp )α , (Jp )α = (−1) Np (α − δp − Φ(p)).

146

8.

[Q,R] = 0

α

Figure 8.1. A value that is always singular for some Tp M Equations (8.10), (8.14), and (8.15) give the desired equality Q(M, ω, Φ, J)α = dim Q(Mα , ωα , Jα ). Unfortunately, there may exist values α which are regular for M but singular for some Tp M . Choosing a different “polarization” in the Linearization Theorem often allows one to bypass this problem. However, there may exist values α which are singular for all polarizations: Example 8.10. For G = S 1 × S 1 there exists a six-dimensional K¨ ahler Hamiltonian G-manifold M whose “X-ray” (i.e., roughly speaking, the moment map images of the orbit type strata) is given in Figure 8.1. (The manifold is a CP1 -bundle over CP1 × CP1 , constructed as the fiberwise projectivization M = P(E ⊕ C), where the holomorphic line bundle E is the tensor product T CP1  T CP1 over CP1 × CP1 with the natural torus action.) Let α be the middle point of the X-ray. For any polarization, there exists a fixed point p such that α is singular for the moment ∗ map Φ# p : Tp M → g . We will treat the case that α is regular for M but singular for Tp M later in this section. Let us now assume that the closed two-form ω is symplectic and that J is an almost complex structure compatible with ω, and prove that quantization commutes with reduction for singular values. Proof of Theorem 8.3. The important consequence of the non-degeneracy of ω that we will use is the fact that the momentP map image Φ(M ) is contained in the “non-polarized moment cone” Cp = Φ(p) + j R+ αj,p for all p (see equation (2.14)): (8.16) Z∗G ,

Φ(M ) ⊆ Cp .

which is possibly singular for Φ. Fix an alcove Consider a value α ∈ (connected component of Φ(M )reg ) whose closure contains α and choose α + h in ∗ G this alcove so that α + th is a regular value for Φ# and p : Tp M → g for all p ∈ M all 0 < t ≤ 1. The Linearization Theorem, followed by reduction with respect to the abstract moment map Φ# p − h, gives a cobordism  G (Mα+h , Lα , Jα+h ) ∼ , p ∈ M G, (Tp M )# , (L ) , (J ) p α p α+h α+h

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147

where L is a pre-quantization line bundle for (M, ω, Φ). (Specifically, we use the items (1)–(3) of the Theorem 4.12 and the reduction cobordism described in Section 6 of Chapter 5.) This implies  X  # (8.17) Q (Mα+h , ωα , Jα+h ) = p ∈ MG Q (Tp M )# α+h , (ωp )α , (Jp )α+h ,

by the cobordism invariance of the quantization (see Appendix J). By (8.16), α + h ∈ Cp for all p. By Proposition 8.7, (whose proof will be complete once we prove Proposition 8.6, in Sections 4–8),   # rp Q (Tp M )# , (ω ) , (J ) p α+h = (−1) Np (α − δp − Φ(p)). p α α+h

By this, (8.17), and (8.10),

α

(Q(M, ω, Φ, J)) = dim Q (Mα+h , ωα , Jα+h ) . We now return to the case where α is regular for Φ : M → g∗ , but possibly ∗ singular for Φ# p : Tp M → g , and ω is not necessarily symplectic. We could attempt to desingularize again by passing to α + h. However, in general we cannot choose an h which is a positive linear combination of α1,p , . . . , αd,p simultaneously for all p as we could do in the symplectic case of Theorem 8.3, thanks to the inclusion (8.16). Instead, we desingularize by deforming Φ in a slightly more sophisticated way. Let L be a pre-quantization line bundle for (M, ω, Φ). Note that the reduced space of M depends on Φ and the quantization of M depends on L and Φ. We now separate their roles. Namely, in the proof below we replace Φ by an abstract moment map Φ0 and we consider the new reduced spaces (8.18)

Mα0 = (Φ0 )−1 (α)/G

with the pre-quantization line bundle (8.19)

L0α = (L|Φ−1 (α) ⊗ C−α )/G.

Proof of Theorem 8.1. The anti-canonical line bundle of a complex manifold is the top wedge of its tangent bundle. Similarly, given a complex structure J on the vector bundle E = T M ⊕ Rk , the anti-canonical line bundle for the stable Vm complex manifold (M, J) is defined to be K = C E where m = rankC E. Let ΦK be the moment map corresponding to this line bundle. Explicitly, ΦξK = ΘK (ξP ), where P is the unit circle bundle in K with respect to some G-invariant Hermitian metric and where ΘK is a connection one-form on P . The Pdvalue of ΦK at each fixed point is the sum of the complex weights: ΦK (p) = − j=1 αj,p . Consider the abstract moment maps Φ + rΦK ,

r ∈ R.

The level sets of these maps are isomorphic to each other for r near zero. (This follows from the fact that the map R × M → R × g∗ given by (r, m) 7→ (r, Φ(m) + rΦK (m)) is proper, since M is compact, and (0, α) is a regular value.) We set Φ0 := Φ + rΦK where r > 0 is such a value. Because the α-level sets of Φ and Φ0 are isomorphic, so are their quotients: (Mα0 , L0α , Jα0 ) ∼ = (Mα , Lα , Jα ),

148

8.

[Q,R] = 0

where Mα0 and L0α are given by (8.18) and (8.19). Therefore, Q(Mα , Lα , Jα ) = Q(Mα0 , L0α , Jα0 ).

(8.20)

Let us apply the linearization theorem (see Theorem 4.12) to the manifold M with the abstract moment map Φ0 , the line bundle L, and the stable complex structure J. This gives a proper cobordism G (M, Φ0 , L) ∼ (Tp M, Φ0p , Lp ).

On the right-hand side, each moment map Φ0p : Tp M → g∗ can be chosen arbitrarily as long as it is η-polarized and takes the value Φ0 (p) = Φ(p) − r(α1,p + . . . + αd,p ) at the origin. Let us choose Φ0p (·) = Φ# p (·) − r(α1,p + . . . + αd,p ).

(8.21)

If α is regular for Φ0p for all p, then X (8.22) Q((Tp M )0α , (L0p )α , (Jp0 )α ). Q(Mα0 , L0α , Jα0 ) = p∈M G

This can be easily seen from the linearization cobordism (8.12), by applying reduction with respect to the abstract moment map Φ0 . (See Chapter 5.) By (8.21), (Tp M )0α = (Tp M )# α+hp ,

where hp = r(α1,p + . . . + αd,p ).

Thus, we have shifted the moment map by a different vector on each Tp M so that for each p the shift hp is a positive linear combination of the non-polarized weights at p. By Proposition 8.6, (8.23)

Q((Tp M )0α , (Lp )0α , (Jp )0α ) = (−1)rp Np (α − δp − Φ(p))

where the prime denotes reduction with respect to Φ0 , when α is regular for Φ0 |Tp M . Summing over all p’s and applying (8.20), (8.22), and (8.10), we obtain that, if α is regular for all Φ0 |Tp M , Q(M, ω, J)α = Q(Mα , ωα , Jα ),

as required. However, α might be singular for Φ0 |Tp M . This happens exactly when α + r(α1,p + . . . + αd,p ) is singular for Φ# p . In this case, we desingularize further, by reducing Φ0 at a value α + h0 which is regular for all Φ0 |Tp M and such that h0 is so small so that r(α1,p + . . . + αd,p ) + h0 is still a positive linear combination of αj,p ’s for each p. By Proposition 8.6, the Φ0 -reduction of Tp M at α + h has quantization of dimension (−1)rp Np (α − δp − Φ(p)). Arguing as before, we obtain 0 0 0 Q(Mα , Lα , Jα ) = Q(Mα+h 0 , Lα , Jα+h0 ) X 0 Q((Tp M )0α+h0 , L0α , Jα+h = 0) p∈M G

=

X

p∈M G

Np (α − δp − Φ(p))

= Q(M, L, J)α , as required.

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149

4. Straightening the symplectic and complex structures We begin to prepare to the calculation of the Dolbeault quantization of a stable complex toric variety, as stated in Proposition 8.6. In this section we show how to replace the stable complex structure by a complex structure, and at what price. First let us recall the setting. We consider Cd with the standard complex structure J, the action of the torus G act with weights −α1 , . . . , −αd : (8.24)

exp(ξ) : (z1 , . . . , zd ) 7→ (e−ihα1 ,ξi z1 , . . . , e−ihαd ,ξi zd ) for all ξ ∈ g,

and a non-standard symplectic form ω# =

X

j dxj ∧ dyj

with 1 = . . . = r = −1 and r+1 = . . . = d = 1 and a moment map Φ# such that Φ# (0) = ν ∈ Z∗G . We assume that the weights α# j = j αj are polarized. The toric variety that we wish to quantize is the reduction of (Cd , ω # , Φ# , J) at α. We need to prove (8.9), i.e., (8.25)

goal

# r dim Q((Cd )# α+h , ωα , Jα+h ) = (−1) N (α − δ − ν),

where h is a positive linear combination of αj ’s. The reduced space on the left-hand side stays the same if we replace the moment map Φ# by Φ# − ν and the value α by α − ν. (This is the “shifting trick”.) The right-hand side also stays the same (ν gets replaced by ν − ν = 0). Hence, without loss of generality, we may assume that the moment map vanishes at the origin: 1X |zj |2 α# Φ# (z) = j . 2 The map ( zj 7→

zj zj

j = 1, . . . , r, j = r + 1, . . . , d

turns ω # into the standard symplectic structure ω, but transforms J to a nonstandard complex structure. In other words, this map provides an isomorphism (Cd , ω # , Φ# , J) ∼ = (Cd , ω, Φ# , J # ), where the G-action on the right-hand side is given by (8.26)

#

#

exp(ξ) : (z1 , . . . , zd ) 7→ (e−ihα1 ,ξi z1 , . . . , e−ihαd ,ξi zd )

for all ξ ∈ g

and where ω is the standard symplectic form on Cd and J # is a non-standard complex structure. (Note that J # is equal to −J on the first r coordinates.) Also, this transformation flips the orientation if r is odd. Hence, the toric variety that we wish to quantize can also be obtained as the reduction of of (Cd , ω, Φ# , J # ) with respect to the G-action (8.26), with the orientation flipped if r is odd. In other words,     # (Cd )# , ω # , Jα+h ∼ , ωα , J # , = (−1)r (Cd ) α+h

r

α

α+h

α+h

where the sign (−1) indicates a possible orientation flip. Then equation (8.25) becomes   goal # (8.27) = N (α − δ). dim Q (Cd )# , ω , J α α+h α+h

150

8.

[Q,R] = 0

The pre-quantization line bundle for (Cd , ω, Φ# , J # ) is L = Cd × C,

where G acts on Cd as in (8.26) and acts on C trivially (because ν = 0). The quantization associated to a stable complex structure and a line bundle is completely determined by the corresponding Spinc structure. By the “Spinc shift formula”, the G-equivariant Spinc structure on Cd that is given by J # and L is the same as the Spinc structure that is associated with the standard complex structure J and the line bundle L ⊗ Cδ . See Section 3 of Appendix D; specifically, Example D.53. Consider now the reduction of Cd with respect to some (abstract) moment map. If two equivariant stable complex structures and line bundles on Cd determine the same equivariant Spinc structure, then, on the reduced space, the induced stable complex structures and line bundles also determine the same equivariant Spinc structure, and hence the same quantization. (See Section 3.4 of Appendix D.) Therefore, equation (8.27) is equivalent to   goal Q (Cd )# , L , J (8.28) = N (α − δ). α−δ red α+h
# −1 Here, the quotient (Cd )# (α + h)/G is taken with respect to the action α+h = Φ (8.26), the stable complex structure Jred is induced from the standard complex structure on Cd , and the pre-quantization line bundle is Lα−δ = (L ⊗ Cδ )α = ((L|Z ) ⊗ Cδ ⊗ C−α ) /G = Z ×G Cα−δ ,

where Z = (Φ# )−1 (α). The function N (·) is the partition function associated with # α# 1 , . . . , αd . We need to prove (8.28) when h is a positive linear combination of α1 , . . . , α d . 5. Passing to holomorphic sheaf cohomology Again, let us start by recalling the setting to which we have reduced the problem. We consider Cd with its standard complex and symplectic structures, and let # the torus G act with weights −α# 1 , . . . , −αd that are polarized and with moment P |zj |2 α# map Φ# (z) = 21 j . We have ( r X −α# j = 1, . . . , r, # j δ= αj and αj = # αj j = r + 1, . . . , d j=1

for some 0 ≤ r ≤ d. We consider a weight α ∈ Z∗G , pick a nearby regular value α+h, such that h is a positive linear combination of αj ’s, and set Z = (Φ# )−1 (α+h). The toric variety that we wish to quantize is Z/G. This quotient is equipped with the stable complex structure Jred that is induced from the standard complex structure on Cd , and the line bundle Lα−δ = Z ×G Cα−δ .

We need to show that the quantization of (Z/G, Lα−δ , Jred ) has dimension N (α−δ). The reduced space Z/G is actually a complex, not just stable complex, orbifold, as it can be identified with the G.I.T. quotient: Z/G ∼ = W/GC ,

5. PASSING TO HOLOMORPHIC SHEAF COHOMOLOGY

151

top left alcove α+h

α

bottom right alcove

Figure 8.2. Moment map image for Example 8.12 where W = GC · Z. (See Section 4 of Chapter 5.) Then, for any weight β ∈ Z∗G , the complex line bundle Lβ = Z ×G Cβ becomes the holomorphic line bundle Lβ = W × GC C β .

The index of the corresponding Dolbeault operator is equal to the alternating sum

X k

ˇ k W/GC , OL (−1)k H β




of the cohomology groups of the sheaf of holomorphic sections; see Chapter 6. Setting β = α − δ, we get   X
r ˇ k W/GC , OL Q (Cd )# , L , J (−1)k dim H (8.29) = (−1) . α−δ red α−δ α+h k

To prove our goal, (8.28), it is enough to show that (
goal N (α − δ) k = 0, k ˇ (8.30) dim H W/GC , OLα−δ = 0 k > 0.

Remark 8.11. A formula for the sheaf cohomology of a holomorphic line bundle over a toric variety is given in [Od]. This formula implies (8.29) rather easily in the special case when δ = 0 and α is a regular value. We end this section with a holomorphic calculation that completes the proof of Example 8.9. Example 8.12. Let G = S 1 × S 1 act on C3 by (8.31)

(a, b) · (z1 , z2 , z3 ) = (a−1 z1 , b−1 z2 , (ab)−1 z3 )

with a moment map
1 |z1 |2 + |z3 |2 , |z2 |2 + |z3 |2 . 2 The moment map image with its two alcoves is shown in Figure 8.2. Let α = (k, k) for some positive integer k, and let δ = (1, 0). Recall that the space W/GC depends on the choice of α + h. We will show that if α + h is in the bottom right alcove, Φ# (z) =

(8.32)

Lα−δ → W/GC

is the line bundle O(k) over CP1 , and the dimension of its quantization is k + 1. If α + h is in the top left alcove, (8.32) is the line bundle O(k − 1) over CP1 , and the dimension of its quantization is k. Let us prove these claims.

152

8.

[Q,R] = 0

For α + h = (κ1 , κ2 ) in either alcove, the level set Z = (Φ# )−1 (α + h) is given by the equations

1 1 (8.33) |z1 |2 + |z3 |2 = κ1 and |z2 |2 + |z3 |2 = κ2 . 2 2 On the bottom right alcove, κ1 > κ2 > 0. The equations (8.33) then imply that z1 is never zero and that z2 and z3 cannot vanish simultaneously. By Theorem 5.18, W = {(z1 , z2 , z3 ) | z1 6= 0 and (z2 , z3 ) 6= (0, 0)}. The quotient W/GC is isomorphic to CP1 and an isomorphism is given by [z1 , z2 , z3 ] 7→ [z1 z2 , z3 ],

(8.34)

where the left-hand side is an equivalence class in W/GC for the GC action (8.31). Because α − δ = (k − 1, k), the line bundle Lα−δ → W/GC can be identified with the set of equivalence classes (8.35)

[z1 , z2 , z3 , u] ∼ [az1 , bz2 , abz3 , ak−1 bk u]

where (z1 , z2 , z3 ) ∈ W , u ∈ C, and (a, b) ∈ (C× )2 . The map [z1 , z2 , z3 , u] 7→ [z1 z2 , z3 , z1 u]

sends it to the line bundle over CP1 with Chern class k, whose total space is given by the equivalence classes [w1 , w2 , u] ∼ [cw1 , cw2 , ck u], 0

w1 , w2 , u ∈ C, >0

(w1 , w2 ) 6= (0, 0).

For this line bundle, dim H = k + 1 and dim H = 0. By (8.29), the dimension of its quantization is k + 1. On the top left alcove, κ2 > κ1 > 0. The equations (8.33) then imply that W = {(z1 , z2 , z3 ) | z2 6= 0 and (z1 , z3 ) 6= (0, 0)}.

As before, the quotient W/GC is isomorphic to CP1 through the formula (8.34) and the line bundle Lα−δ is given by the formula (8.35). This line bundle, however, is isomorphic to the line bundle over CP1 with Chern class k − 1; an isomorphism is [z1 , z2 , z3 , u] 7→ [z1 z2 , z3 , z2−1 u].

The dimension of its quantization is k.

6. Computing global sections; the lit set In this section we will compute the space of global holomorphic sections. More specifically, we will show that its dimension is ˇ 0 (W/GC , OL ) = N (α − δ), (8.36) dim H α−δ

proving the first part of our goal (8.30). It will be convenient to work with an “upstairs description” of OLα−δ , in terms of holomorphic functions on an open subset of Cd : the holomorphic sections of the line bundle Lα−δ = W ×GC Cα−δ → W/GC can be viewed as G-equivariant holomorphic functions W → Cα−δ , i.e., holomorphic functions on W that have the transformation property (8.37)

f ((exp ξ) · z) = eihα−δ,ξi f (z) for all ξ ∈ g.

6. COMPUTING GLOBAL SECTIONS; THE LIT SET

153

Because W contains (Cr{0})d, a holomorphic function on W can be expanded into a Laurent series X (8.38) f= am z m . m∈Zd

This function satisfies (8.37) if and only if X (8.39) mj α # j =α−δ

for every m = (m1 , . . . , md ) such that am 6= 0. Holomorphic monomials on Cd are precisely those z m for which mj ≥ 0 for all j. On W , a priori, there could be holomorphic monomials with some mj < 0. However, the following lemma says that this cannot happen:

Lemma 8.13. Suppose that m ∈ Zd satisfies (8.39) and the monomial z m is holomorphic on W . Then mj ≥ 0 for all j.

Hence, the space of holomorphic sections is spanned by the monomials z m such that m ∈ Zd satisfies (8.39) and mj ≥ 0 for all j. The number of such monomials is precisely N (α − δ), by the definition of the partition function N (·). This proves (8.36). Remark 8.14. If the complement of W in CPd has complex codimension greater than or equal to 2, the holomorphic monomial z m extends to all of Cd , by Hartog’s theorem (see, e.g., [GH]). Hence, mj ≥ 0 for all j. However, if the level set Φ−1 (α) does not meet the coordinate plane {zj = 0}, the complement of W contains this plane and, hence, has codimension one. The remainder of this section is devoted to the proof of Lemma 8.13. The monomial z m is holomorphic on W if and only if, for each j, the exponent mj ≥ 0 once there exists z ∈ W with zj = 0. Because W = GC · Z, it is to Penough |zj |2 αj = check this criterion for z in Z. Moreover, since z ∈ Z is equivalent to 12 α + h, there exists z ∈ Z with zj = 0 if and only if there is s ∈ ∆ with sj = 0, where X ∆ = ∆α+h = {s ∈ Rd+ | sj αj = α + h}. For any m ∈ Rd , let us define the corresponding lit set Lm ⊂ ∂∆ as

(8.40)

Lm = {s ∈ ∆α+h | ∃ i such that si = 0 and mi < 0}.

Then for m ∈ Zd , the monomial z m is holomorphic on W if and only if the lit set Lm is empty. We need to prove that for each m ∈ Zd that satisfies (8.39), mj < 0 for some j implies that the lit set is non-empty. The lit set has an intuitively clear meaning, which justifies its name, in the case that m belongs to the affine space spanned by ∆, X A(α + h) = {x ∈ Rd | xj αj = α + h}. Such an m is outside ∆ if and only if mj < 0 for some j. Suppose that this is the case. Now, standing at m, let us point a flashlight at ∆. Because ∆ is a solid polytope that does not let light through, the jth facet is lit if and only if light rays from m reach that facet from the outside, without having to pass through the interior of the polytope. This exactly means that mj < 0. Therefore, the set of points that are lit by the flashlight is precisely Lm . This is the reason for the name “lit set”.

154

8.

[Q,R] = 0

? ∆ ? m

Figure 8.3. Lit set ? when m is not coplanar with ∆ Remark 8.15. The notion of a visibility set is sometimes used in Riemannian geometry: Lm is the set of points of ∆ that are visible to a spectator standing at m. If we stand outside a convex polytope and point a flashlight at the polytope, the lit set is non-empty and contractible. Therefore, if m belongs to A(α + h) but is not in ∆, the lit set Lm is non-empty and contractible. For certain m’s which are not in the affine space spanned by ∆, these properties are not always satisfied: the following example shows that, if m lies outside the affine space spanned by ∆, the lit set might not be contractible: Example 8.16. Take L(x, y) = x + y, α = 0, α + h = , and m = (m1 , m2 ) with both m1 and m2 negative. Then ∆ is an interval and Lm is its boundary, which is not contractible. See Figure 8.3. P Let us now restrict our attention to m ∈ Zd such that mj α # j = α − δ. Such an m is not in the affine plane A(α + h). However, the lit set Lm is still non-empty and contractible. Proposition 8.17. Suppose that h is a positive linear combination of αj ’s and Pr Pr P # d that δ = mi α # i = α − δ. j=1 αj = − j=1 αj . Let m ∈ Z be such that Suppose also that mj < 0 for some j. Then the lit set Lm is non-empty and contractible. Proof. Let h=

X

hj α # j

with hj > 0 for all j. We may assume, by shrinking h if necessary, that hj < 1 for all j. Let ( mj + 1 − hj 1 ≤ j ≤ r, m0j = mj + h j r < j ≤ d. Because mj ∈ Z and 0 < hj < 1, we have mj < 0 ⇐⇒ m0j < 0

(8.41)

# for each j. Also, because α# j = −αj for 1 ≤ j ≤ r and αj = αj for r < j ≤ d, we have d d r d X X X X # # m0j α# = m α + α + hj αj = (α − δ) + δ + h = α + h. j j j j j=1

j=1

j=1

j=1

ˇ 7. THE CECH COMPLEX

155

Thus m0 is in A(α+h). By (8.41) and the hypothesis, m0j < 0 for some j. Hence, m0 is outside the polytope ∆α+h . As pointed out above, these properties of m0 imply that the lit set Lm0 is non-empty and contractible. Because the lit set depends only on the collection of j’s for which mj < 0, it follows from (8.41) that Lm = Lm0 . Hence, Lm is non-empty and contractible. Proof of Lemma 8.13. Suppose that the monomial z m , for m ∈ Zd , is holomorphic on W . As we have seen, this means that the lit set Lm is empty. Suppose in addition that m satisfies (8.39). Then, by Proposition 8.17, mj ≥ 0 for all j. ˇ 7. The Cech complex It remains to prove the second part of (8.30), i.e.,
goal ˇ >0 W/GC , OL dim H (8.42) = 0, α−δ

where W = GC · Z and Z = Φ−1 (α + h), assuming that h is a positive linear combination of αj ’s. We will calculate this sheaf cohomology from a “good cover” of W/GC . We do not need to look very far: in Section 5 of Chapter 5 we obtained the following explicit description of W : [ W = (8.43) WI , I ∈ Fα+h , I

where

WI = CI × (C× )drI ,

and where Fα+h denotes the collection of subsets I ⊆ {1, . . . , d} such that there exists z ∈ (Φ# )−1 (α + h) for which zi = 0 exactly if i ∈ I. We consider the resulting covering of the reduced space, W = {W I | I ∈ Fα+h },

where W I = WI /GC . Consider the holomorphic line bundle Lβ = W ×GC Cβ for any β ∈ Z∗G ; we will soon specialize to β = α − δ. Claim 8.18. For each I ∈ Fα+h , the Dolbeault cohomology of W I twisted by the line bundle Lβ |W I vanishes in all positive degrees. Proof. By the results of Section 5.4 of Chapter 5, each of the
sets WI /GC is biholomorphically equivalent to a space of the form Ck × (C× )n−k /Γ, where Γ is a finite abelian group acting linearly on Cn . The restriction of the line bundle Lβ to this open set is
Ck × (C× )n−k ×Γ C

for some linear action of Γ on C. It follows that the Dolbeault cohomology in question can be identified with Γ-invariant
Dolbeault cohomology with coefficients in the trivial line bundle Ck × (C× )n−k × C, where the Γ-action on the fiber is perhaps non-trivial. An averaging argument shows that this cohomology is the same as the Dolbeault cohomology of Ck × (C× )n−k with trivial coefficients. By the ∂-Poincar´e lemma (see, e.g., [GH]), the latter is acyclic. Since WI ∩ WJ = WI∩J , the cover W has the property that any intersection of sets in the cover is acyclic with respect to the Dolbeault cohomology, i.e., satisfies H >0 (V, OLβ ) = 0. By Leray’s theorem (see, e.g., [GH]), this implies that the sheaf

156

8.

[Q,R] = 0

cohomology groups of OLβ over W/GC are equal to the cohomology groups of the ˇ Cech complex for the cover W:
Cˇ W, OLβ .

A k-cochain c in this complex is a map which assigns to every (k + 1)-tuple of multi-indices I0 , . . . , Ik in Fα+h a holomorphic section of Lβ over the intersection W I0 ∩ . . . ∩ W Ik . This intersection is equal to W I for I = I0 ∩ . . . ∩ Ik , and the line bundle over this intersection is L β |W I = W I × G C C β . Recall that WI = CI × (C× )drI . Let us argue as in Section 6: holomorphic sections of Lβ over WI can be viewed as holomorphic functions on WI whose Laurent expansion has the form X f= am z m , where

(8.44)

mj ≥ 0 for all j ∈ I and

X

mi α # i = β.

Denote by Om the sheaf of functions on Cd which are multiples of the monomial z . One may think of the space OLβ (WI ) as the direct sum of the spaces Om (WI ), P over all m ∈ Zd such that mi α # i = β. (Strictly speaking, this is only true for a completion of Oβ (WI ) unless the direct sum is finite.) Motivated by this observation, we now concentrate on one particular value of m. The sheaf Om can be described in the following way. The complex torus (C× )d acts on all ingredients involved: on Cd , on W/GC , and on Lβ . These actions induce an action on the sheaf of holomorphic sections. The monomial f (z) = z m transforms as m

(8.45)

md 1 f (λz) = λm 1 . . . λd f (z),

λ ∈ (C× )d .

Because the torus element λ ∈ (C× )d acts on functions by sending a function f (z) to the function f (λ−1 z), the functions which transform according to (8.45) are those functions on which the torus acts with weight −m. ˇ The (C× )d -action commutes with the Cech differential (8.46)

(Dc)J0 ,... ,Jk+1 =

k+1 X

(−1)l cJ0 ,... ,Jˆl ,... ,Jk+1 .

l=0

Therefore, this action descends to the cohomology. Moreover, for each m, the cochains on which the torus acts with weight −m form a subcomplex, and its cohomology is the subspace of the sheaf cohomology consisting of those classes on which the torus (C× )d acts with weight −m. Because W/GC is compact, the cohomology is finite-dimensional. Therefore, we have a genuine decomposition M X ˇ ˇ O Lβ ) ∼ H(W, Om ), m ∈ Zd , mi α # (8.47) H(W, = i =β m

for the covering

W = {WI | I ∈ Fα+h }.

8. THE HIGHER COHOMOLOGY

157

(In other words, all but a finite number of terms on the right of (8.47) vanish.) Therefore, we need to prove that X ˇ >0 (W, Om ) goal (8.48) H = 0 for all m ∈ Zd such that mi α # i = α − δ. ˇ The k-cochains in the complex which gives H(W, Om ) are M (8.49) Om (WI0 ∩ . . . ∩ WIk ). I0 ,... ,Ik ∈Fα+h

Recall that WI0 ∩ . . . ∩ WIk = WI for I = I0 ∩ . . . ∩ Ik . The following lemma follows immediately from the fact that WI = CI × (C× )drI : Lemma 8.19. If mj ≥ 0 for all j ∈ I, the space Om (WI ) is one-dimensional and spanned by z1m1 . . . zdmd . Otherwise, Om (WI ) = {0}. ˇ Hence, the complex (8.49) giving H(W, Om ) is isomorphic to the complex ( M C if mi ≥ 0 for all i ∈ I0 ∩ . . . ∩ Ik , Ωkm = (8.50) 0 otherwise. I ,... ,I ∈F 0

The differential D in k−1 for c ∈ Ωm ,

k

Ω?m

α+h

ˇ comes from the standard formula for the Cech differential:

(Dc)I0 ,... ,Ik =

k X

(−1)j cI0 ,... ,Iˆj ,... ,Ik .

j=0

It remains to show that the higher cohomology H >0 (Ωkm , D) vanishes when m ∈ Zd P and mi α # i = α − δ. 8. The higher cohomology

Our goal now is to prove that (8.51)

ˇ >0 (Ω?m , D) = 0 for all m ∈ Zd H

such that

X

mi α # i = α − δ,

where Ω?m is the complex (8.50). We will show that the cohomology of this complex is essentially equal to the relative cohomology of the pair (∆, Lm ), where ∆ is the polytope X ∆ = ∆α+h = {s ∈ Rd+ | s i α# i = α + h} and Lm is the “lit set”

Lm = {s ∈ ∆ | ∃ i such that si = 0 and mi < 0}.

Because both ∆ and Lm are contractible, this cohomology will be zero. Recall that the elements of Fα+h encode the faces of the polytope ∆:

Fα+h = F∆ = {I ⊆ {1, . . . , d} | ∃ s ∈ ∆ such that si = 0 ⇐⇒ i ∈ I}.

The covering of W by the sets WI , for I ∈ F∆ , corresponds to the covering of ∆ by the sets UI = {s ∈ ∆α+h | si 6= 0 for all i 6∈ I} (8.52) = the star of the Ith face of ∆, for I ∈ F∆ . Namely, WI consists of exactly those elements of W which are sent to UI by the Td -moment map W → W/GC ∼ = Z/G → ∆α ,

158

8.

[Q,R] = 0

the last arrow maps [z1 , . . . , zd ] to 12 (|z1 |2 , . . . , |zd |2 ). More explicitly, UI is the union of the interiors of faces whose closures contain the Ith face of ∆. We consider the covering of ∆ by the open sets UI . Each UI is a non-empty contractible open set, and UI∩J = UI ∩ UJ for all I and J. Therefore, the sets UI form a good covering for ∆, in the usual sense: the ˇ Cech cohomology of this covering, with constant complex coefficients, is equal to the singular cohomology of ∆. See [BT1, Theorem 15.8]. Because the intersections ˇ UI0 ∩ . . . ∩ UIk are never empty, the Cech complex for this covering is M k (8.53) C = C. I0 ,... ,Ik ∈Fα+h

The complex C ? coincides with the complex Ω?m (see (8.50)) if mj ≥ 0 for all j. Hence, in this case, the cohomology groups of Ω?m are identical to the cohomology groups of the (contractible) space ∆α : ( C k = 0, k ? k ? k H (Ωm ) = H (C ) = H (∆) = 0 k > 0.

Therefore, H >0 (Ω?m ) = 0, if mj ≥ 0 for all j, as required. It remains to prove that H >0 (Ω?m ) = 0 for all m that satisfy X (8.54) m ∈ Zd , mi α # and ∃ j such that mj < 0. i = α − δ, It will be convenient to express the complex Ω?m as a quotient: Ωkm = C k /Qkm , where C k is given by (8.53) and ( M 0 (8.55) Qkm = C I ,... ,I ∈F 0

k

if mi ≥ 0 for all i ∈ I0 ∩ . . . ∩ Ik , otherwise .

It is enough to show that (8.56)

goal

H >0 (Q?m ) = 0

ˇ k (Ω? ) = 0 for k > 0 follows for all m that satisfy (8.54). (The conclusion that H m then from the long exact sequence in cohomology induced by the short exact sequence of complexes 0 → Q?m → C ? → Ω?m → 0.) Our proof will be complete once we show that (8.57)

H k (Q?m ) = H k (Lm ),

because the lit set Lm is contractible (by Proposition 8.17). Consider the covering Vm of the lit set Lm by the sets VI = U I ∩ L m ,

I ∈ F∆ .

Claim 8.20. The covering Vm is a good covering of the lit set Lm . Proof. Recall that UI is the star of the Ith face of ∆. If the Ith face of ∆ is not lit, the entire star is not lit, and VI is empty. If the Ith face of ∆ is lit, then VI is equal to its star in Lm , which is contractible. Also, just as for the UI ’s, V I0 ∩ . . . ∩ V Ik = V I

for I = I0 ∩ . . . ∩ Ik .

It follows that Vm is a good covering, as required.

9. SINGULAR [Q,R]=0 FOR NON-SYMPLECTIC HAMILTONIAN G-MANIFOLDS

159

Remark 8.21. Vm is an indexed covering of Lm ; the set VI can be equal to the ˇ set VJ even if I 6= J. This does not cause any trouble; the Cech theory still goes through for indexed coverings. Claim 8.22. VI 6= ∅ if and only if there exists j ∈ I such that mj < 0. Proof. Suppose j ∈ I and mj < 0. Choose a point s in the jth open facet of ∆. Then s ∈ UI and s is lit. Conversely, suppose that s ∈ UI is lit. The condition that s ∈ UI implies that sl > 0 for all l 6∈ I. The condition that s be lit implies that there exists j such that sj = 0 and mj < 0. Combining these two observations, we conclude that j must be in I. ˇ Claim 8.23. The Cech complex for the covering Vm , with constant complex coefficients, coincides with the complex Q?m . ˇ Proof. The Cech complex is ˇk

C (Vm ; C) =

M

I0 ,... ,Ik ∈Fα+h

The complex

Q?m

Qkm = I0

(

C 0

if VI0 ∩ . . . ∩ VIk 6= ∅, otherwise.

(see (8.55)) can be rewritten as ( M C if ∃ i ∈ I0 ∩ . . . ∩ Ik such that mi < 0, 0 otherwise . ,... ,I ∈F k

α+h

Applying Claim 8.22 to I = I0 ∩ . . . ∩ Ik and VI = VI0 ∩ . . . ∩ VIk , we see that Cˇ k (Vm , C) = Qkm for all m and k. Now we are in a position to complete our proof: H k (Q?m )

Claim 8.23

=

H k (Vm , C)

Claim 8.20

=

ˇ k (Lm , C) H

Proposition 8.17

ˇ k (point). H

=

Hence, H >0 (Qm ) = 0, as required. 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds Let G be a torus and let (M, ω, Φ, J) be a quantizable Hamiltonian G-manifold equipped with an equivariant stable complex structure J. Assume that M is compact and M G is discrete. We stress that we do not require J to be compatible with ω nor ω to be non-degenerate. We have shown that [Q,R]=0 holds at all regular values α of Φ. When α is singular, there is difficulty even in stating this result: if ω is symplectic and J is compatible, for “singular [Q,R]=0” to hold we must desingularize by shifting, passing to α + h which is inside the moment map image Φ(M ). In the degenerate case, the moment image is meaningless. However, in the degenerate case there are two polytopes which can be used to replace Φ(M ). When stated in terms of these polytopes, the [Q,R]=0 theorem continues to hold. Alternatively, [Q,R]=0 holds with a particular desingularization which is different from the Meinrenken-Sjamaar “shift desingularization”. Whereas

160

8.

[Q,R] = 0

there might remain singular values α ∈ Z∗G to which the above results do not apply; they do establish versions of “singular [Q,R]=0” for most values α without assuming non-degeneracy. Consider the non-polarized moment cones (8.58)

Cp = Φ(p) +

d X

R+ αj,p ,

j=1

We set (8.59)

∆1 =

\

p ∈ M G.

Cp

p

and

∆2 = conv{Φ(p) | p ∈ M G }.

(8.60)

The set ∆2 , being the convex hull of a finite set of points, is always a compact convex polytope. The set ∆1 is a finite intersection of convex polyhedral cones and can be non-compact. However, if ∆1 is compact, ∆1 ⊆ ∆2 . (We leave the proof of this fact to the reader as an exercise.) Recall that if ω is symplectic and J comes from an almost complex structure compatible with ω, then ∆1 = ∆2 = Φ(M ). (See Section 2 of Chapter 2.) If ω is not symplectic, then often ∆1 6= ∆2 : Example 8.24. Let S 1 act on S 2 ⊂ C × R by rotations with standard moment map. Consider the stable complex structure coming from T S2 ⊕ R ∼ = T (C × R)|S 2 = S 2 × C × R ⊂ S 2 × C2 . Both isotropy weights are equal to +1. Hence, ∆1 is an infinite ray and ∆2 is an interval. On the other hand, if we equip S 2 with the stable complex structure which is the opposite from the standard complex structure, ∆1 is empty (being the intersection of two disjoint rays pointing in opposite directions) and ∆2 is an interval. A close examination of the proof of Theorem 8.3 yields the following, more general, result. Let (M, ω, Φ, J) be a quantizable Hamiltonian G-manifold equipped with an equivariant stable complex structure. Assume that M is compact and M G is discrete. Consider the moment cones (8.58), and, for each “polarizing vector” η (see Proposition 4.18), consider the “polarized moment cones” Cp#

= Φ(p) +

d X j=1

Let α ∈

Z∗G

R + α# j,p ,

p ∈ M G.

be a weight for G. Let α + h be a nearby regular value for Φ.

Proposition 8.25. Assume that there exists a polarization such that α + th ∈ interior(Cp ) if α ∈ Cp# , for all 0 < t ≤ 1 and for each p ∈ M G . Then (8.61)

Q(M, ω, Φ, J)α = Q(Mα+h , ωα , Jα+h ).

In particular, Q(M, ω, Φ, J)α = Q(Mα+h , ωα , Jα+h ) = 0

when there exists a polarization such that α 6∈ Cp# for all p ∈ M G .

9. SINGULAR [Q,R]=0 FOR NON-SYMPLECTIC HAMILTONIAN G-MANIFOLDS

∆1

161

∆2

Figure 8.4. Generalized moment polytopes for a Hirzebruch surface

Figure 8.5. CP3 with a 2-torus action Proposition 8.25 implies that quantization commutes with reduction for values of α that are outside ∆2 or inside ∆1 : Proposition 8.26. Let G be a torus and (M, ω, Φ, J) a quantizable Hamiltonian G-manifold equipped with an equivariant stable complex structure. Assume that ∗ M is compact and M G is discrete. Let α ∈ ZG be a weight for G and let h ∈ g∗ be such that α + th is a regular value of Φ for all 0 < t ≤ 1. If α 6∈ ∆2 , (8.62)

Q(M, ω, Φ, J)α = Q(Mα+h , ωα , Jα+h ) = 0.

If α + th ∈ interior(∆1 ) for all 0 < t ≤ 1, (8.63)

Q(M, ω, Φ, J)α = Q(Mα+h , ωα , Jα+h ).

Proof of Proposition 8.26. Suppose that α 6∈ ∆2 . Because ∆2 is convex, there exists a hyperplane that strictly separates α from ∆2 . Equivalently, there exists η ∈ g such that α(η) < Φη (p) for all p ∈ M G . Choose this η as a polarizing vector. Then α 6∈ Cp# for all p ∈ M G , and, by Proposition 8.25, (8.62) holds. Suppose that α+th ∈ interior(∆1 ) for all 0 < t ≤ 1. Then α+th ∈ interior(Cp ) for all 0 ≤ t ≤ 1 and for all p ∈ M G , regardless of whether on not α ∈ Cp# and, in particular, regardless of the polarization. By Proposition 8.25, (8.63) holds. Example 8.27. Let M be the K¨ ahler Hirzebruch surface corresponding to the first polytope in Figure 8.4. There exists a pre-symplectic form whose moment image is the second, “twisted” polygon in Figure 8.4. (See [KT1].) With this structure, ∆1 and ∆2 are as shown in the figure. For α in the interior of the bottom triangle of the “twisted polygon”, Mα is a point and dim Q(Mα ) = 1; for α in the interior of the top triangle, Mα is a point with orientation −1 and dim Q(Mα ) = −1. For α ∈ Z2 in the closure of the bottom triangle, Q(M )α = 1; for α ∈ Z2 in the interior of the top triangle or in the interior of its top edge, Q(M )α = −1; for all other α’s, Q(M )α = 0. (See [KT1].)

162

8.

[Q,R] = 0

In this example we see that Q(M )α is equal to Q(Mα0 ) when the latter corresponds to the moment map Φ0 for which the horizontal edges are shifted slightly outwards, the vertical edge is shifted slightly to the left, and the diagonal edge is shifted slightly to the right. This is a special case of the following result, which is proved exactly in the same way as Theorem 8.1 in Section 3. Proposition 8.28. Let G be a torus and let L be a a pre-quantization line bundle over a Hamiltonian G-manifold (M, ω, Φ, J) equipped with an equivariant stable complex structure. Assume that M is compact and M G is discrete. Fix α ∈ Z∗G . If α is a regular value for Φ, then [Q, R] = 0 at α by Theorem 8.1. Suppose that α is singular for Φ. Let K be the anti-canonical line bundle for J and let ΦK be the corresponding moment map. For a small enough r > 0, there exists a small (possibly zero) h0 ∈ g∗ such that α is a regular value for Φ0 := Φ + rΦK − h0 . We desingularize the reduced space by passing to (Mα0 , ωα0 ), where Mα0 = Z 0 /G for Z 0 = (Φ0 )−1 (α) and ω 0 = curvature L0α for L0α = (L|0Z ⊗ C−α ) /G. With this desingularization, we have [Q, R] = 0: Q(M, ω, Φ, J)α = dim Q(Mα0 , ωα0 ).

Example 8.29. Consider CP3 equipped with its standard K¨ ahler structure and the (S 1 × S 1 )-action with moment map shown in Figure 8.5. The middle point α is a singular value for Φ, and, moreover, for Φ + rΦK for all r ≥ 0. We can still apply Proposition 8.28, with h0 6= 0. Alternatively, [Q,R]=0 at α for any “shift desingularization”, by Proposition 8.26, because α is in the interior of the square ∆1 . 10. Overview of the literature In this section we give a brief (and admittedly incomplete) survey of publications related to the [Q,R]=0 conjecture. The reader interested in more detailed accounts should consult [Sj2] (for the results prior to 1995) and [Ve5]. Special cases of the [Q,R]=0 conjecture were proved in [GS3] and [GS4]. For example, in [GS3] this conjecture was verified for K¨ ahler manifolds, and in [GS4] it was shown that a “stable” version of this conjecture is true in the sense that it holds if one replaces the symplectic manifold (M, ω) by the symplectic manifold (M, nω) with n sufficiently large. The papers [GS3] and [GS4] appeared in the early 1980’s, and then there was a ten year hiatus during which there were very few further developments (the only development we are aware of being the article [DET]). In 1992, however, some work of Witten on two-dimensional quantum field theory opened a new avenue of approach to this conjecture: Witten was concerned with computing the cohomology ring structure of the moduli space of vector bundles over a Riemann surface; and, to do so, invented a computational technique in equivariant de Rham theory which has become known as “non-abelian localization”. (See [Wi2].) This was refined further by Jeffrey and Kirwan in the paper [JK1] (see also [JK2, JK3]) in which they showed how non-abelian localization is related to the usual abelian localization of Atiyah-Bott-Berline-Vergne. (For a cobordism version of their result, see [GGK1].) Then it was observed, simultaneously, by a number of persons (among others, Meinrenken, Jeffrey, and Kirwan) that the Jeffrey-Kirwan

10. OVERVIEW OF THE LITERATURE

163

version of non-abelian localization enables one to give a very simple heuristic proof of [Q,R]=0. (For an account of this “proof” see [Guil1].) In the spring of 1995, Meinrenken [Men1] succeeded in making this heuristic argument rigorous, and, at about the same time, Vergne [Ve4] (see also [Ve1, Ve3]) gave an alternative proof which was also, to some extent, based on non-abelian localization. Next, in the fall of 1995, a completely different proof, using Lerman’s theory of symplectic cutting, was discovered, independently, by Duistermaat-Guillemin, Meinrenken and Wu (see [DGMW] and, for the symplectic-cutting ideas, [Ler1]). In the spring of 1996, Meinrenken adapted the symplectic cutting methods of this paper to the non-abelian case and proved in [Men2] the non-abelian analogue of (8.1). Finally, in the fall of 1996, he and Sjamaar showed, in [MeSj] (see also [Sj1]), that the results of [Men2] were true for singular reduced spaces. In some sense the Meinrenken–Sjamaar theorem is end of the story as far as the [Q,R]=0 conjecture is concerned. However, some further interesting developments have occurred subsequently. In [TZ] Tian and Zhang gave a third completely new proof of the conjecture using analytic Morse theory a ` la Witten, and we started developing in [GGK1] a cobordism approach to the problem, of which we gave a fuller account in this book. Also it was discovered that [Q,R]=0 could be formulated in the setting of deformation quantization, and a proof of it in this setting was given by Fedosov [Fe]. Finally, quite recently, Meinrenken and Woodward [MW1] have given a simple analytic proof of the Verlinde factorization theorem which can be interpreted as a version of [Q,R]=0 for loop groups acting on infinite dimensional symplectic manifolds. Essentially topological proofs of the abelian case of [Q,R]=0 were obtained in [CKT, Met3, Par2]. For instance, in [Par2] the [Q,R]=0 theorem is proved for regular values of abstract moment maps for torus actions (and for regular and singular values for compact connected Lie groups acting on symplectic manifolds). We also refer the reader to [Brav4] for a non-compact version of the [Q,R]=0 theorem and to [Te] for a proof of [Q,R]=0 for smooth compact polarized varieties.

Part 3

Appendices

APPENDIX A

Signs and normalization conventions In this appendix we describe our conventions regarding signs and factors of 2π. We discuss the representation of G on C ∞ (M ) when M is a G-manifold, the integral weight lattice Z∗G for a torus G, connections and curvature, and integral equivariant cohomology. 1. The representation of G on C∞ (M) Consider an action τ of a group G on a manifold M . Thus for each g ∈ G we have a diffeomorphism τg : M → M , and τgh = τg τh . The action τ gives rise to a representation of G on C ∞ (M ) according to the rule
∗ (A.1) T : g 7→ τg−1 or, more explicitly,

(T (g)f ) (x) = f τg−1 x




for f ∈ C ∞ (M ) and x ∈ M . This assignment ensures that T (gh) = T (g)T (h) so that T is indeed a representation. If we attempt, instead, to make the simpler definition where g ∈ G acts on C ∞ (M ) by the pull-back map τg∗ : C ∞ (M ) → C ∞ (M ), we would get an anti∗ representation of G on C ∞ (M ), for (τg τh ) = τh∗ τg∗ . At the infinitesimal level, this becomes an issue of signs. Namely, to each element ξ of g one associates a one-parameter group t 7→ τ(exp tξ) of diffeomorphisms of M whose infinitesimal generator is denoted by ξM . At the infinitesimal level, the fact that g 7→ τg∗ is an anti-representation is reflected by the identity (A.2)

[ξM , ηM ] = −[ξ, η]M ,

which says that the map g → Vect(M ),

ξ 7→ ξM ,

is an anti-isomorphism of Lie algebras. If the representation of G on C ∞ (M ) is defined correctly, as in (A.1), the corresponding infinitesimal representation becomes ξ 7→ −LξM , which is a homomorphism of Lie algebras. In this monograph G will be, for the most part, an abelian Lie group. As a consequence, the sign problem, formally speaking, does not arise here, for both the right- and left-hand sides of (A.2) are zero. However, if one wants to be careful about these distinctions, one has to be careful about sign conventions in simpler situations where these distinctions are frequently ignored. For instance, an example which we often encounter is a linear representation of the torus on Cd and the resulting representation of G on C ∞ (Cd ): 167

168

A. SIGNS AND NORMALIZATION CONVENTIONS

Example A.1. Let G be a torus, and let τ be the representation of G on Cd defined by   (A.3) τ(exp ξ) z = e−iα1 (ξ) z1 , . . . , e−iαd (ξ) zd ,

the αi ’s being weights of G (see Section 2 below). From τ we get a representation of G on C ∞ (Cd ), and if we define this representation correctly (as in (A.1)), the coordinate functions z1 , . . . , zd transform by the formula (A.4)

zk 7→ eiαk (ξ) zk ,

i.e., with weights having opposite signs to the weights of the representation (A.3). Thus, at first glance, they will look as if they are transforming according to the wrong sign! We will have to exercise great care not to confuse the action of G on Cd defined by (A.3) with the representation of G on C ∞ (Cd ) defined by (A.4). For further discussion of linear representations of G on Cd , see Section 4 of Chapter 4. 2. The integral weight lattice Another problem which we often face concerns factors of 2π. Throughout most of this book G is a torus, i.e., a Lie group isomorphic to (S 1 )k for some k. Hence the exponential map, exp : g → G, is a group epimorphism. The group lattice of G is the lattice ZG = ker exp in g. The weight lattice Z∗G parametrizes the characters of G, i.e., the homomorphisms G → S 1 . Such a homomorphism is determined by its differential at the identity element e ∈ G. This differential is a linear map from the tangent space Te G = g to the tangent space T1 S 1 , hence it is an element of g∗ ⊗ Lie(S 1 ). To realize Z∗G as a lattice in the dual space g∗ = Hom(g, R), we need to make a single choice: we need to choose an isomorphism ∼ R. (A.5) Lie(S 1 ) = Equivalently, we need to specify an exponential map (A.6)

exp : R → S 1 .

The weight lattice Z∗G then becomes a lattice in g∗ which is dual to the group lattice ZG ⊂ g in the sense that Z∗G = {α ∈ g∗ | hα, ξi ∈ ZS 1 for all ξ ∈ ZG },

where ZS 1 ⊂ R is the kernel of the exponential map (A.6). We choose the isomorphism (A.5) to be

∂ ←→ 1, ∂θ where θ mod 2π is a coordinate on S 1 , so that the exponential map (A.6) becomes

(A.7)

We then have

Lie(S 1 ) = R via

exp : θ 7→ eiθ . ZS 1 = 2πZ,

and Z∗G = {α ∈ g∗ | hα, ξi ∈ 2πZ for all ξ ∈ ZG }.

A weight α ∈ Z∗G determines a homomorphism G → S 1 by exp(ξ) 7→ eiα(ξ) . (Note that, if ξ ∈ ker exp, then α(ξ) ∈ 2πZ.)

3. CONNECTION AND CURVATURE FOR PRINCIPAL TORUS BUNDLES

169

In particular, for G = S 1 , we have g = g∗ = R, the group lattice is ZG = 2πZ, and the weight lattice is Z∗G = Z. A moment map for a circle action τ becomes a real valued function Φ : M → R satisfying dΦ = ι(ξM )ω, where ξM is the vector d τ (eiθ ). The weight m ∈ Z parametrizes the homomorphism λ 7→ λm of S 1 . field dθ Our normalization convention simplifies many formulas by eliminating factors of 2π. Another advantage of this convention is that it leads to a notion of curvature which becomes the standard Riemannian geometry curvature when applied to the unit circle bundle of a two-dimensional Riemannian manifold. See the next section for details. Remark A.2. Another possible√convention would have been to choose the exponential map (A.6) to be x 7→ e2π −1x . The lattice Z∗G would have become the dual lattice to ZG ⊂ g in the standard sense: pairings between elements of ZG and Z∗G would have been integers. The curvature of a circle bundle would have becomes a real two-form which represents the first Chern class of the bundle. 3. Connection and curvature for principal torus bundles Let π : P → M be a principal K-bundle, where K is a torus with Lie algebra k. (More generally, we may allow P to be an orbi-bundle, meaning that K acts on P locally freely and M is the orbifold P/K. See Corollary B.31.) Let ζP , for ζ ∈ k, be the vector fields on P that generate the principal action. A connection form is a k-valued one-form Θ on the total space P which is K-invariant and whose restriction to the orbits is the tautological (Maurer-Cartan) form (A.8)

Θ(ζP ) = ζ

for all ζ ∈ k.

A connection form always exists; see Corollary B.37. Lemma A.3. There exists a two-form F on M such that (A.9)

π ∗ F = −dΘ.

Proof. We need to show that the two-form dΘ is basic, i.e., is K-invariant and is horizontal. It is invariant because Θ is invariant. It is horizontal because ι(ζP )dΘ = −dι(ζP )Θ = 0. Here, the first equality uses Cartan’s formula LζP = dι(ζP ) + ι(ζP )d and the fact that Θ is invariant, and the second equality uses the fact that ι(ζP )Θ ≡ ζ is constant. Remark A.4. Here is a fancier way to show that the exterior derivative dΘ is basic. The function Φζ = Θ(ζP ) ≡ ζ is a moment map for −dΘ (see Example 2.8). Because dΘ is a k-valued two-form, this moment map takes values in k∗ ⊗ k, and we have just shown that it takes the constant value identity ∈ k∗ ⊗ k ∼ = Hom(k, k). By pre-symplectic reduction (Theorem 5.1), dΘ descends to a (k-valued) closed two-form on the quotient. Definition A.5. The curvature of the connection Θ is the k-valued two-form F on M satisfying (A.9).

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A. SIGNS AND NORMALIZATION CONVENTIONS

Let us now specialize to K = S 1 and identify its Lie algebra k with R using (A.7). Then a connection one-form is a real-valued one-form Θ such that Θ(∂/∂θ) = 1, where ∂/∂θ generates the principal circle action, and the curvature is a realvalued two-form. If P is the unit circle bundle of a two-dimensional Riemannian manifold and Θ is the Levi-Civita connection, then our convention gives the same curvature as in the usual Riemannian geometry theory. For example, for M = S 2 with its standard metric, the unit circle bundle has curvature equal to the standard area form. This justifies our sign and normalization conventions. Remark A.6. Many authors define curvature by π ∗ F = dΘ. With this sign convention (which is opposite from ours), the curvature gives the negative of “infinitesimal holonomy” (see below). Also, it is common to consider these Θ and F as taking values in Lie(S 1 ) = iR and then to pass to the real-valued two-form i F . This two-form is the same as we get would from our definition of curω = 2π vature, but with the convention of Remark A.2. For the standard Riemannian two-sphere, this two-form ω is equal to 1/2π times the standard area form. Finally, let us recall how to interpret the curvature F as infinitesimal holonomy. For any two horizontal vector fields u ˜ and v˜, (A.10)

−dΘ(˜ u, v˜) = − (˜ uΘ(˜ v ) − v˜Θ(˜ u) − Θ([˜ u, v˜])) = Θ([˜ u, v˜]),

where the first equality holds for all one-forms and vector fields, and the second equality follows from the horizontality of u ˜ and v˜, which exactly means that Θ(˜ u) ≡ Θ(˜ v ) ≡ 0. Let Xt and Yt denote the flows on P generated by u ˜ and v˜. Lemma A.7. The one-parameter family of maps of P given by the diffeomorphisms Y−√t X−√t Y√t X√t , for t ∈ R, is differentiable at t = 0, and d Y √ X √ Y√ X√t = [˜ (A.11) u, v˜]. dt t=0 − t − t t

Now, let u and v be vector fields with constant coefficients with respect to some local coordinate system. Let u ˜ and horizontal lifts √ v˜ be √their √ √ to P . The holonomy over the parallelogram with sides tu, tv, − tu, and − tv (taken in this order) exactly sends p to Y−√t X−√t Y√t X√t p. By (A.11), (A.10), and (A.9), we can interpret F (u, v) as the “holonomy along the infinitesimal parallelogram generated by u and v”.

Proof of Lemma A.7. Restrict attention to a coordinate neighborhood of a point p ∈ P . The definition of the Lie bracket of vector fields gives d2 (A.12) (Ys Xt p − Xs Yt p). [˜ u, v˜](p) = dt ds t=s=0

This is equal to (A.11), because they are both equal to

(A.13)

h˜ v 0 (p), u ˜(p)i − h˜ u0 (p), v˜(p)i ,

where u ˜0 and v˜0 are the differentials of u ˜ and v˜ and
∗ h , i : Tp P ⊗ T p P × T p P → T p P

is the natural pairing. One gets (A.13) by substituting in (A.11) and (A.12) the Taylor expansions 1 Xt (q) = q + t˜ u(q) + t2 h˜ u0 (q), u ˜(q)i + O(t3 ) 2

4. CURVATURE AND CHERN CLASSES

171

and

u ˜(q(t)) = u ˜(q(0)) + t h˜ u0 (q(0)), q 0 (0)i + O(t2 ) as well as those for Yt (q) and v˜(q(t)) as many times as needed. 4. Curvature and Chern classes Let π : P → M be a principal K-bundle, for K a torus. Let F be its curvature with respect to some connection one-form Θ.

Lemma A.8. The cohomology class of the curvature F is independent of the connection one-form Θ. Proof. Let Θ and Θ0 be any two connection one-forms. The definition of a connection form implies that the difference Θ0 −Θ is basic, so there exists a k-valued one-form α on M such that Θ0 − Θ = π ∗ α. Then π ∗ (F 0 − F ) = dΘ − dΘ0 = −π ∗ dα, so F 0 = F − dα is in the same de Rham cohomology class as F . Definition A.9. The curvature class of π : P → M is the de Rham cohomology class [F ]. Lemma A.10. The curvature class [F ] belongs to the image of the natural homomorphism (A.14)

H 2 (M ; ZK ) → H 2 (M ; k).

Proof. The short exact sequence 0 → ZK → k → K → 0 gives a long ˇ exact sequence of Cech cohomology groups, from which we get an isomorphism ˇ 1 (M ; K) ∼ ˇ 2 (M ; ZK ). The transition maps of the principal bundle P → M H =H ˇ 1 (M ; K). Let cˇ denote the corresponding element of represent an element of H 2 ˇ (M ; ZK ). Then cˇ maps to [F ] under (A.14). (See [BT1].) H When K = S 1 , so that k = R and ZK = 2πZ, the Chern class of the circle 1 bundle P → M is 2π cˇ ∈ H 2 (M ; Z). Its image under the natural homomorphism (A.15)

i : H 2 (M ; Z) → H 2 (M ; R)

is 1/2π times the curvature class. The Chern class of a complex line bundle is defined as the Chern class of its unit circle bundle with respect to any fiberwise Hermitian metric. In particular, we may consider the associated line bundle L = P × K Cα

where P is a principal K (orbi-)bundle and Cα denotes the K-action on C by weight α ∈ Z∗K .

Lemma A.11. The curvature class of the principal bundle, c(P ) ∈ H 2 (M ; k), and the Chern class of the associated line bundle, c1 (L) ∈ H 2 (M ; Z), are related by 1 i(c1 (L)) = hα, c(P )i . 2π Proof. The unit circle bundle in L is P ×K S 1 where K acts on S 1 with weight α. A (k-valued) connection one-form Θ on P induces a (real valued) connection oneform on this circle bundle given by hα, Θi + dθ,

where θ mod 2π is a coordinate on S 1 . The lemma then follows from the definitions of curvature and Chern classes.

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A. SIGNS AND NORMALIZATION CONVENTIONS

5. Equivariant curvature; integral equivariant cohomology Let a Lie group G act on a principal U(1)-bundle π : P → M by bundle automorphisms, generated by vector fields ξP ,

ξ ∈ g.

Let Θ be a G-invariant connection one-form on P , considered as a real-valued oneform as explained earlier. Such a connection always exists if G acts properly; see Corollary B.38. Its equivariant curvature is the formal difference ω − Φ,

where ω is a two-form on M such that π ∗ ω = −dΘ and where Φ : M → g∗ is such that π ∗ Φξ = Θ(ξP ) for all ξ ∈ g. The equivariant curvature class of P is defined to be the equivariant cohomology ∗ class cG (P ) = [ω − Φ] ∈ HG (M ). It is independent of the choice of connection. We leave the details to the reader, and refer to Definition 2.15 and Appendix C for the relevant definitions. We have the following notion of integrality in equivariant cohomology in the ∗ Cartan model. When G is a torus, HG (point; R) = H ∗ (BG; R) are the polynomial ∗ ∗ (point; R) consists of the functions on g, and the image of HG (point; Z) → HG polynomial functions on g whose values on the group lattice ZG are integers. If [ω − Φ] is the equivariant curvature class of a G-equivariant circle bundle P → M , 1 [ω − Φ] is integral. then 2π

APPENDIX B

Proper actions of Lie groups The goal of this appendix is to give a brief introduction to the theory of proper group actions on manifolds. The first sections are devoted to the basic aspects of the theory, culminating in the slice theorem and its applications. In the last sections we treat more advanced topics such as the Mostow-Palais embedding theorem and rigidity of compact group actions. We encourage the reader to also consult the book [DK] by Duistermaat and Kolk. 1. Basic definitions 1.1. Actions. An action of a group G on a set M is a collection of maps ρa : M → M with a ∈ G such that ρab = ρa ρb and ρa = id. For the sake of brevity, to denote the action, we will also write a instead of ρa and a · m instead of ρa (m). Example B.1. If M is a linear vector space and ρa are linear transformations, this is a linear representation of G. In this book, G is always a Lie group, i.e., G is a manifold and a group, and the map G × G → G sending (a, b) → ab−1 is smooth. The set M is always a smooth (C ∞ ) manifold, and the action is smooth, meaning that the associated map (B.1)

G × M → M,

(a, m) 7→ a · m,

is smooth. We will usually, but not always, consider compact Lie groups. Recall that every compact Lie group is isomorphic to a closed subset of the orthogonal group O(N ) for some N . (This is a consequence of the Peter–Weyl theorem; see [Bre2, Chap. 0, Thm. 5.1] or Theorem B.52 of this appendix.) For instance, a torus G = (S 1 )k is the subgroup of O(2k) acting on R2 × . . . × R2 (k times) by independently rotating each factor. For some applications, however, the compactness requirement is too strong and unnecessary and can be replaced by the following condition. Definition B.2. An action of G on M is proper if the action map (B.2)

G × M → M × M,

(a, m) 7→ (a · m, m),

is proper. Recall that a map is called proper if the preimage of any compact set is compact. Also recall that a proper map between locally compact Hausdorff spaces is closed. For many applications proper group actions are as suitable as actions of compact groups. In this appendix we will work with proper group actions whenever possible. Let us list a few facts about proper group actions, leaving the proofs to the reader as a simple exercise. 173

174

B. PROPER ACTIONS OF LIE GROUPS

Lemma B.3. Let a Lie group G act on a manifold M . 1. If G is compact, the action is proper. 2. If G is not compact but M is compact, the action is not proper. 3. If the action is proper, its restriction to any closed subgroup H ⊆ G is a proper H-action on M , and its restriction to any invariant subset U of M is a proper G-action on U . 4. If the action is proper, the evaluation map evm : G → M,

(B.3)

a 7→ a · m,

is proper, for each m ∈ M . 5. A G-action on M is proper if and only if for every two compact subsets K 1 and K2 of M , the subset (B.4) of G is compact.

{a ∈ G | aK1 ∩ K2 6= ∅}

Remark B.4. If the map (a, m) 7→ a · m from G × M to M is proper, the action is proper. However, this condition is not necessary. For instance, the action of G on itself by left translations is always proper, whereas the map (a, b) 7→ a · b is proper if and only if G is compact. A G-action is locally proper if every point is contained in a G-invariant open set on which the action is proper. A G-action (on a locally compact space) is locally proper if and only if every point is contained in an open set U such that the set {a ∈ G | gU ∩ U 6= ∅} has compact closure in G. A G-space which satisfies this latter property is called by Palais [Pa3] a Cartan G-space. The following example, which we learned from D. Bar-Natan, shows that properness of an action is not a local condition, i.e., there exists an action which is locally proper but not proper: Example B.5. Consider an action of R (i.e., a flow) on R2 having the following properties: the trajectories are the curves {x = b} for |b| ≥ 1 and γc = {y = 1/(1 − x2 ) + c} for c ∈ R, and under the action a point traverses its trajectory with constant speed. (See Figure B.1.) This action is not proper: let K1 be a small neighborhood of (−1, 0) and let K2 be a small neighborhood of (1, 0). Then the trajectory γc meets both K1 and K2 for all sufficiently small c. The distance from K1 to K2 along γc becomes arbitrarily large as c gets arbitrarily small. Hence, the subset (B.4) of R is unbounded. On the other hand, the restriction of the action to each of the open subsets x < 1 and x > −1 is proper. (Exercise: what is the quotient space for this action?) 1.2. Orbits and stabilizers. Let a Lie group G act on a smooth manifold M. The stabilizer of a point m ∈ M is

Gm = {a ∈ G | a · m = m}.

Sometimes, the stabilizer of m is denoted by Stab(m). Being a closed subgroup of the Lie group G, the stabilizer Gm is a Lie group. (See [Ad, Theorem 2.27].) If the action is proper, the stabilizer Gm is compact. The action is said to be free if Gm = {e} for all m ∈ M . The action is locally free if all stabilizers are discrete.

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Figure B.1. Non-proper, locally proper flow The action is said to be effective, or faithful, if the homomorphism G → Diff(M ) T that defines the action is one-to-one or, equivalently, if m∈M Gm = {e}. The orbit of a point m ∈ M is The evaluation map,

G · m = {a · m | a ∈ G} ⊆ M.

evm : a 7→ a · m, induces a bijection from the quotient G/Gm to the orbit G · m. The manifold M decomposes into a disjoint union of orbits. The quotient M/G is the set of orbits, with the quotient topology. Example B.6. Let S 1 act on the unit sphere, S 2 ⊂ R3 , by rotations around the z-axis. Then the quotient S 2 /S 1 is the closed interval [−1, 1]. Example B.7. Let the unitary group, U(n), act on its Lie algebra, u(n), the anti-Hermitian matrices, by conjugation. The quotient is the set of all unordered ntuples, α1 , . . . , αn ∈ iR (namely, the eigenvalues). A diagonal matrix with distinct eigenvalues has a stabilizer Gm = the diagonal Unitary matrices = (S 1 )n . Thus its orbit is U(n)/(S 1 )n . This orbit is naturally isomorphic to the space of flags in Cn , i.e., decompositions Cn = L1 ⊕ . . . ⊕ Ln , where Li ⊂ Cn are mutually orthogonal complex lines (namely, Li is the eigenspace for αi ). Exercise. Let the complex torus (C× )n act on Cn by coordinate-wise multiplication: (λ1 , . . . , λn ) : (z1 , . . . , zn ) 7→ (λ1 z1 , . . . , λn zn ). Note that this action is not proper. What are the orbits? What are the stabilizers? Describe the quotient Cn /(C× )n . Describe the orbits, the stabilizers, and the quotient for the action of the compact torus (S 1 )n ⊂ (C× )n that is given by the same formula. Proposition B.8. If the action is proper, every orbit is a closed subset of M , and the orbit space M/G is Hausdorff. Proof. The assertion that the orbits are closed follows from the fact that for a proper action the evaluation map (B.3) is proper, hence closed. Suppose that the orbits of p and q cannot be separated in M/G. Let Un and Vn be balls in M of radius 1/n around p and q, respectively (for an arbitrary metric).

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Then G · Un intersects G · Vn , and hence there exist pn ∈ Un , qn ∈ Vn , and an ∈ G such that qn = an · pn . The sequence (qn , pn ) = (an · pn , pn ) is in the image of the map (B.2) and approaches (q, p). Because the map (B.2) is proper, its image is closed. Therefore, the image must contain the limit (q, p). Hence, the orbits of p and q coincide. 1.3. Preliminaries on Lie groups. Here we recall basic facts about Lie groups and Lie algebras. Let G be a Lie group. Its Lie algebra, g = Te G, is identified with the space of all left-invariant vector fields on G. The Lie bracket of two vector fields, which is given by [X, Y ]f = XY f − Y Xf for f ∈ C ∞ (M ), gives rise to the Lie algebra structure [ , ] : g × g → g. If G is a group of matrices, the Lie bracket is simply the commutator of matrices: [A, B] = AB − BA. The exponential map exp : g → G

is characterized by the fact that, for each ξ ∈ g, the curve t 7→ exp(tξ) is a group d homomorphism from (R, +) to G, and dt |t=0 exp(tξ) = ξ ∈ Te G. If G is a group of 2 matrices, the exponential map is given by exp(A) = I + A + A2! + . . . . If H ⊆ G is any closed subgroup, H is itself a Lie group and its Lie algebra is h = {ξ ∈ g | exp(tξ) ∈ H for all t ∈ R}

(see [Ad]). Example B.9. A matrix A is anti-Hermitian if and only if exp(tA) is unitary for all t. Lemma B.10 (Existence of Haar measure). Suppose G is compact. Then there exists a probability measure on G which is invariant under both left and right translations. V Proof. Let ve be any non-zero element in dim G g∗ . Let v the unique volume form on G which is invariant under left translations and whose value at the unit element is ve . Then v is also right invariant: indeed, define w = Rb∗ v, where Rb Vdim G ∗ is right translation by b ∈ G. Because g is one-dimensional, there exists λ such that we = λve . Because La and Rb commute and v is left invariant, so is w, and so w = λv everywhere. Because G is compact, these R R R Rwe can integrate R volume forms. We obtain G w = λ G v and also G w = G Rb∗ v = G v. Hence, λ = 1. Integration of v (with respect to the orientation defined by v) defines a left-invariant Borel measure with the required properties. 1.4. Averaging with respect to a compact group action. The Haar measure allows us to employ averaging arguments with respect to compact group actions. We give three applications: Corollary B.11 (Invariant inner products). Let a compact Lie group G act linearly on a vector space. Then there exists an invariant inner product h·, ·i.

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177

0

Indeed, start from any inner product h·, ·i , and take its G-average: Z hu, vi := ha · u, a · vi0 da. G

Corollary B.12 (Invariant Riemannian metrics). Let a compact Lie group G act smoothly on a manifold M . Then there exists a G-invariant Riemannian metric g(·, ·). Indeed, start from any Riemannian metric g 0 , and take its G-average: Z g(u, v) := g(a∗ u, a∗ v)da, G

for u, v ∈ Tm M .

Corollary B.13 (The complex of invariant forms). Let a compact Lie group G act smoothly on a manifold M . For any differential form µ0 , its average Z (B.5) µ := (a∗ µ0 )da G

is a G-invariant differential form, which is in the same cohomology class as µ0 if G is connected. Also, every G-invariant exact differential form has a G-invariant primitive: if ω = dµ0 for some µ0 and ω is G-invariant, then (B.5) is G-invariant and satisfies dµ = ω. Therefore, if G is connected, the G-invariant differential forms form a subcomplex of the de Rham complex with the same cohomology. Remark B.14. Some of the above results extend to proper actions of noncompact groups; see Section 3.2. However, not all do. For example, the oneform dx on R is exact and is invariant under translations but it does not have an invariant primitive. Hence, for a proper action of a non-compact group, the complex of invariant forms might fail to compute the cohomology of M . In contrast, the complex of basic forms does compute the cohomology of M/G; see Corollary B.36. 1.5. Generating vector fields. An action of a Lie group G on a manifold M induces a linear map from the Lie algebra of G to the space of vector fields g → Vect(M ),

ξ 7→ ξM ,

by

d (exp(tξ) · m) , dt t=0 where exp : g → G is the exponential mapping. (When G = Tn , we identify g with Rn , and the exponential mapping becomes Rn → Rn /2πZn . See Appendix A.) The differential of the evaluation map ξ M |m =

evm : G → M,

a 7→ a · m,

is the map d(evm )|e : g 7→ Tm M,

ξ 7→ ξM |m .

The map g → Vect(M ) sending ξ to ξM is an anti-Lie homomorphism: [ξ, η]M = −[ξM , ηM ] for all ξ, η ∈ g.

See Appendix A. Lemma B.15. The Lie algebra of the stabilizer Gm is gm = {ξ ∈ g | ξM |m = 0}.

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Proof. Since Gm is a Lie subgroup of G, gm = {ξ ∈ g | exp(tξ) ∈ Gm for all t ∈ R}.

If ξ ∈ gm , then exp(tξ) ∈ Gm for all t and exp(tξ) · m ≡ m is a constant curve. Thus its tangent is zero: ξM |m = 0. Suppose now that ξM |m = 0. To prove that ξ ∈ gm , it is enough to show that exp(tξ) ∈ Gm for all t. Hence, it is sufficient to prove that exp(tξ) · m is a constant curve, which can be seen as follows: d d = ((exp(sξ))(exp(tξ)) · m) = (exp(sξ))∗ (ξM |m ) = 0 (exp(tξ) · m) dt dt t=s t=0 by the definition of ξM |m and by the definition of the push-forward. We call gm the infinitesimal stabilizer of m. 1.6. Infinitesimal actions. Let g be a Lie algebra. By definition, a g-action on a manifold M is an antiLie-homomorphism g → Vect(M ), ξ 7→ ξM . Every action of a Lie group G induces an action of its Lie algebra g. If G is connected, the G-action is determined by the g-action, and if G is simply connected, every complete g-action (i.e., an action such that the vector fields ξM are complete) integrates to a G-action. Many aspects of group actions have infinitesimal counterparts. A g-action preserves a differential form β if all the Lie derivatives LξM β, for ξ ∈ g, vanish. A map f : M → M 0 between g-manifolds is equivariant if f∗ ξM = ξM 0 for all ξ ∈ g. A moment map for a g-action on M is an equivariant map Φ : M → g∗ such that dΦξ = ι(ξM )ω for all ξ ∈ g. A g-action is Hamiltonian if it admits a moment map. The moment map Φ is g-invariant if and only if LηM Φξ = −Φ[η,ξ] for all ξ, η ∈ g. If G is connected, these properties of the g-action are equivalent to the corresponding properties of the G-action: a form β is g-invariant if and only if it is G-invariant, a function f : M → M 0 is g-equivariant if and only if it is G-equivariant, a g-moment map is the same as a G-moment map. 2. The slice theorem 2.1. Principal bundles. Definition B.16. A principal G-bundle over a manifold M is a manifold P , a free right G-action on P , and a map π : P → M whose level sets are exactly the orbits of G, such that every point in M has a neighborhood U and a diffeomorphism π −1 (U ) → U × G

that sends the fiber π −1 (m) to the fiber {m} × G and that is equivariant with respect to the G-action which, on U × G, is given by multiplication on the right. Example B.17. Let E → M be a vector bundle of rank k. Its frame bundle is the principal GL(k) bundle whose fiber over m ∈ M is the set of linear isomorphisms g : Rk → Em , (equivalently, the set of bases of Em ), with A ∈ GL(k) acting by g 7→ g ◦ A.

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Proposition B.18. Let G be a Lie group and H a closed subgroup. Then the quotient G/H is a manifold, its tangent space at eH is g/h, and the quotient map G → G/H is a principal H-bundle. Proof. By Proposition B.8, the quotient G/H is Hausdorff. This quotient is second countable because G is. We will show that every coset in G has a neighborhood which is equivariantly diffeomorphic to U × H, where U is a subset of some Rk and where H acts on U × H by multiplication on the right. This shows both that G/H is a manifold (the U -coordinates provide an atlas) and that G → G/H is a principal bundle map. Let N be a complementary subspace to h in g. Define a map The differential

Ψ: N × H → G

by

(u, a) 7→ exp(u) · a.

dΨ|(0,e) : N ⊕ h → g is the identity map. By H-equivariance, dΨ|(0,a) is a bijection for all a ∈ H. By continuity, there exists a neighborhood U of 0 in N such that dΨ|(u,a) is a bijection for all u ∈ U . By the inverse function theorem, Ψ: U × H → G

is a local diffeomorphism. A local diffeomorphism is a diffeomorphism onto its image if and only if it is one-to-one. Let us prove that if U is small enough, then Ψ|U ×H is one-to-one. Otherwise, there would be a sequence of pairs (un , an ) 6= (vn , bn ) in N × H such that un , vn → 0 and such that exp(un ) · an = exp(vn ) · bn . Replacing an by an b−1 n and bn by e, we would get a sequence (un , an ) 6= (vn , e) with un , vn → 0 and with exp(un ) · an = exp(vn ) approaching e. This would contradict the fact that Ψ is one to one on a neighborhood of (0, e). Consequently, Ψ is an equivariant diffeomorphism from U × H to a neighborhood of H in G. Lemma B.19. The map (B.6)

G/Gm → M,

aGm 7→ a · m,

is a one-to-one immersion. If the action is proper, the orbit G · m is an embedded submanifold in M , and the map (B.6) is a diffeomorphism between the quotient G/Gm and the orbit G · m. Proof. Clearly, the map (B.6) is one-to-one. The differential at e of the map a 7→ a · m is ξ 7→ ξM |m . Its kernel is gm , by Lemma B.15. Combined with Proposition B.18, this shows that (B.6) is an immersion at the basepoint eGm . By equivariance, it is an immersion everywhere. If the action is proper, the map (B.6) is proper as a map to G · m. This follows from part 4 of Lemma B.3. Being a proper one-to-one immersion, the map (B.6) is an embedding. Lemma B.20. The tangent space to the orbit G · m is Tm (G · m) = {ξM | ξ ∈ g}.

If this space is all of Tm G, the orbit of m contains a neighborhood of m. Proof. The first assertion follows from Lemma B.19. The second follows from the inverse function theorem (the evaluation map a 7→ a · m, being a submersion, is open).

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2.2. Associated bundles. A rank k vector bundle over a manifold M is a manifold E, a map π : E → M , and the structure of a k-dimensional vector space on each fiber π −1 (m), such that every point in M has a neighborhood U and a vector space V and a diffeomorphism π −1 (U ) → U × V

that sends the fiber π −1 (m) to the fiber {m} × V and that is a linear isomorphism on each fiber. Let P → M be a principal H-bundle, and let H act linearly on a vector space W . The associated bundle is P ×H W := (P × W )/H,

where the action of H on P × W by which we quotient is a : (p, w) 7→ (pa−1 , a · w).

In what follows we denote by [p, w] the equivalence class of (p, w) ∈ P × W in P ×H W , so that [pa, w] = [p, aw] for all a ∈ H. The projection P → M induces the map P ×H W → M which sends [p, w] to π(p). Lemma B.21. This construction yields a vector bundle over M . Proof. Let U ⊂ M be an open set whose preimage in P is U × H. Then its preimage in P ×H W is (U × H) ×H W = U × W . Remark B.22. Every vector bundle can be obtained in this way: if E → M is a real vector bundle of rank k and P is its frame bundle (see Example B.17), then E = P ×GL(k) Rk . Similarly, a vector bundle with a fiberwise inner product is associated to its orthogonal frame bundle (whose structure group is O(k)), a complex Hermitian vector bundle is associated to its unitary frame bundle (whose structure group is U(k)), etc. 2.3. The slice theorem. Let a Lie group G and smoothly and properly on a manifold M . As above, denote by Gm the stabilizer of a point m ∈ M . For all a ∈ Gm , the differential of a : M → M sends Tm M → Tm M . These maps form a linear representation (called the isotropy action) of Gm on the vector space Tm M . The tangent space to the orbit, Tm (G · m), is a Gm invariant subspace of Tm M . Since the action is proper, the stabilizer Gm is compact. Hence, there exists a Gm invariant decomposition, (B.7)

Tm M = Tm (G · m) ⊕ W,

where W is the normal to the orbit. For instance, we can take W to be the orthogonal complement of Tm (G·m) with respect to any Gm invariant inner product on Tm M . (See Corollary B.11.) Remark B.23. The normal to the orbit is the quotient Tm M/Tm (G · m), by definition. It comes equipped with the linear Gm action induced by the isotropy action on Tm M . By (B.7), we can identify W with this normal. Since Gm acts on W , we can form the associated bundle G ×Gm W . Let D be a small disc in W around the origin with respect to some Gm invariant metric. Theorem B.24 (The Slice Theorem). There exists a G-equivariant diffeomorphism from the disc bundle G ×Gm D onto a neighborhood of the orbit G · m in M , whose restriction to the zero section G ×Gm {0} = G/Gm is the map (B.6).

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181

The slice theorem is also called the equivariant tubular neighborhood theorem, or the normal form theorem for proper group actions. Its proof for smooth actions of compact Lie groups is due to Koszul [Kosz]. For proper actions, the slice theorem, as well as various extensions to proper actions of results on compact group actions, is due to Palais [Pa3]. Remark B.25. The disc bundle G ×Gm D is equivariantly diffeomorphic to the entire vector bundle G ×Gm W . Thus one can also state the slice theorem as that a neighborhood of G · m is equivariantly diffeomorphic to the entire vector bundle G × Gm W . In the special case that m is a fixed point, the Slice Theorem becomes Theorem B.26 (The Local Linearization Theorem). Let a compact Lie group G act on a manifold M and let m ∈ M G be a fixed point. Then there exists a G-equivariant diffeomorphism from a neighborhood of the origin in T m M onto a neighborhood of m in M . Proof. Let U be an invariant neighborhood of m and let f : U → Tm M be any smooth map whose differential at m is the identity map on Tm M . Its R average F : U → Tm M , which is defined by F (u) = g∗ f (g −1 u)dg where dg is Haar measure, is smooth, G-equivariant, and satisfies dF |m = identity. By the implicit function theorem we can invert F on a neighborhood of m to obtain a diffeomorphism as required. Remark B.27. As an alternative proof, one may just notice that the exponential map for an invariant metric is a diffeomorphism with the required property. A shortcoming of this argument is that it does not readily generalize to actions preserving an extra structure or actions on infinite-dimensional manifolds (cf. Remark B.29). In Section 5 we outline a different approach to the linearization problem, based on rigidity of compact group actions; see Remark B.66. Proof of the Slice Theorem. Because the action is proper, the stabilizer Gm is compact. By Theorem B.26, there exists a Gm -equivariant diffeomorphism, E : (a neighborhood of 0 in Tm M ) → (a neighborhood of m in M ).

Fix any Gm -invariant metric on the subspace W of (B.7), and let D ⊂ W be a ball small enough to be contained in the domain of E. The map (B.8)

ψ : G ×Gm D → M,

[a, v] 7→ a · E(v),

is well defined and is a local diffeomorphism at the point [e, 0]. By G-equivariance, ψ is a local diffeomorphism at all points of the form [a, 0]. It remains to show that Ψ is one-to-one if D is sufficiently small. Assuming the contrary, suppose we have elements un , vn → 0 in W and an , bn ∈ G such that [an , un ] 6= [bn , vn ] and such that an · E(un ) = bn · E(vn ).

Without loss of generality, we may assume that bn = e. (Otherwise, we act by b−1 n .) Then an · E(un ) = E(vn ) → m. This does not imply that an → e. Note however that under the map G × M → M × M the sequence (an , E(un )) goes to the converging sequence (E(vn ), E(un )). Since the action is proper, there is a

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converging subsequence anj → a∞ and we obtain a contradiction with the fact that ψ is a local diffeomorphism at [a∞ , 0]. Remark B.28. It is worth noticing that the properness of the action is used twice in the above proof. The first time, it is needed to guarantee that the stabilizer Gm is compact. This is sufficient to show that G ×Gm D gives a local model of the (local) action in a neighborhood of m. The second time, it is used to show that G ×Gm D is a model of the action on a neighborhood of the entire orbit of m. The size of D has to be adjusted during this second step. The set D embeds in M by u 7→ ψ([e, u]) in the notation of (B.8). Its image is a Gm -invariant submanifold of M that is transversal to the orbit through m and which intersects each G-orbit near G·m in a Gm -orbit. Such a submanifold is called a slice for the action at m. We note that the slice theorem remains valid in the complex category: if M is a complex manifold and GC is a complex Lie group acting holomorphically and properly, then a neighborhood of each orbit is equivariantly biholomorphic to G× Gm D. The same proof goes through. One uses the fact that the exponential map is holomorphic (see [Sr]) to deduce that G/Gm is a complex manifold and to define the complex structure on G ×Gm D. In the local linearization theorem, one chooses the diffeomorphism f between a neighborhood of m in M and a neighborhood of 0 in Tm M to be holomorphic; its average F is then also holomorphic. Remark B.29. As stated, the slice theorem holds for smooth actions of compact groups on Banach manifolds. Indeed, let us first observe that the local linearization theorem holds for such actions and no modifications in the proof are required. Furthermore, to state the slice theorem, we only need to find an invariant decomposition (B.7). (Then for compact groups the proof will go through word-for-word.) To construct the decomposition (B.7) we first note that some, not necessarily invariant, complement to Tm (G · m) in Tm M exists since Tm (G · m) is finite-dimensional; see [Rudi]. Such complements are in a one-to-one correspondence with bounded projectors π : Tm M → Tm (G · m). Averaging a projector by the Gm -action, we obtain an invariant complement W . (One can easily verify that even though a projector must satisfy a non-linear equation π 2 = π, by averaging a projector we still obtain a projector.) The infinite-dimensional version of the slice theorem has applications in, for example, investigations of relative equilibria of infinite-dimensional simple mechanical systems; see, e.g., [Mat]. 3. Corollaries of the slice theorem Throughout this section we consider a Lie group G acting on a manifold M and we assume that G is compact, or, more generally, that the action is proper. 3.1. Locally free quotients. The most important corollary is almost a restatement of the slice theorem: Corollary B.30. If the G-action on M is free, the quotient M/G is a manifold, and M is a principal G-bundle over M/G. Proof. By the slice theorem, each orbit has a neighborhood U and an isomorphism ψ : G × D → U where D is a disc in Rs for some s. The projection

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183

to Rs provides local coordinates on M/G; the isomorphism ψ gives rise to a local trivialization for the principal bundle. To follow the convention that a principal action is a right action, we can let a ∈ G act as m 7→ a−1 · m when M is viewed as a principal bundle. Similarly, one can prove the following Corollary B.31. If the action is locally free, the quotient M/G is an orbifold, and M is a principal G orbi-bundle over M/G. Proof. By the slice theorem, each orbit has a neighborhood U and an isomorphism ψ : G ×Γ D → U where Γ is a finite group acting linearly on a disc D in Rs . The finite quotient map D → D/Γ ∼ = U/G provides an orbi-chart on U/G; the product G × D with the Γ-action is a trivialization of the principal G orbi-bundle over this orbi-chart. Moreover, we have Corollary B.32. Let M be a complex manifold and GC a complex Lie group acting holomorphically, properly, and locally freely. Then the quotient M/GC is a complex orbifold. Proof. This follows from the slice theorem in the complex category. 3.2. Averaging and partitions of unity. In Section 1.4 we took averages with respect to a compact group action. To apply averaging arguments to proper actions of non-compact groups, we will use invariant partitions of unity: Corollary B.33. For every covering of M by G-invariant open sets there exists a G-invariant partition of unity subordinate to the covering. This result is trickier to prove than it might appear. It relies on the following crucial observation: Lemma B.34. Let B be an invariant subset of M whose quotient B/G is compact. Then B is closed in M . Proof. Since M/G is Hausdorff (by Proposition B.8), the compact subset B/G is closed. Because the quotient map M → M/G is continuous, the preimage B of the closed set B/G is closed. Proof of Corollary B.33. Let us use the name tube for an open invariant subset of M which is equivariantly diffeomorphic to G ×H D where H ⊂ G is compact and D is an open subset of a vector space on which H acts. By the slice theorem, every invariant open set is a union of tubes. Suppose that we are given a covering of M by invariant open sets. Because M is locally compact and its topology has a countable basis, there exists a countable collection of (not necessarily invariant) open subsets Wn of M and subsets Wn00 ⊂ Wn0 ⊂ Wn which satisfy the following conditions. Each Wn is contained in a tube and is contained in some element of the given covering. The closure of Wn00 in M is a compact set contained in Wn0 , and the closure of Wn0 in M is a compact set contained in Wn . The sets Wn00 cover M . Let Un = G · Wn , Un0 = G · Wn0 , and Un00 = G · Wn00 . Let Bn0 = G · Wn0 and Bn00 = G · Wn00 , where Wn0 and Wn00 are the closures in M of Wn0 and Wn00 . Then Un00 ⊂ Bn00 ⊂ Un0 ⊂ Bn0 ⊂ Un . All these sets are invariant. The sets Un00 , Un0 , Un are

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open, and the sets Bn00 and Bn0 have compact quotients. Each Un is contained in a 00 tube and in an element given covering, and the S of the S sets 0Un cover M . 00 0 Let Vn = Un r k
Corollary B.36 (The complex of basic forms). A differential form α on a Gmanifold is called horizontal if ι(ξM )α = 0 for all ξ ∈ g. The form α is called basic if it is invariant and horizontal. Basic differential forms form a complex whose ˇ cohomology is the same as the (Cech, or singular) cohomology of the quotient M/G.

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185

We note that in the special case that G acts on M freely, or with finite stabilizers, the basic forms on M are precisely the forms on the manifold, or orbifold, M/G. We also note that it is crucial that G be compact or that the action be proper. When this requirement is not met, the quotient can have “wild” topology and the basic complex does not compute its cohomology. The result stated in Corollary B.36, which is due to Koszul [Kosz], is similar to the de Rham theorem and can be proved in the same way: Sketch of the proof of Corollary B.36. The complex of sheaves of singular cochains on M/G (with real coefficients) is a fine resolution of the sheaf R of locally constant functions on M/G. Basic forms on G-invariant open subsets of M form a complex of sheaves on M/G which is locally acyclic. Indeed, by using the fact that G is compact (or that the action is proper) and adapting the proof of the Poincar´e lemma, one can show that the basic cohomology of a neighborhood of an orbit is the same as of the orbit itself, i.e., zero in positive degrees. This sheaf is also fine because it admits partitions of unity. Thus basic forms on M provide another fine resolution of the locally constant sheaf on M/G. Since the cohomology ˇ of both resolutions are equal to the Cech cohomology of the locally constant sheaf on M/G, they are equal to each other. Corollary B.37 (Invariant connections for G-quotients). Let a Lie group G act properly and locally freely on a manifold M . Then there exists a connection one-form for the principal G-bundle M → M/G. Proof. We need to find a g-valued one form Θ which is G-invariant and whose restriction to the orbits is the tautological (Maurer-Cartan) form (B.9)

Θ(ξM ) = ξ

for all ξ ∈ g.

The slice theorem implies that M is covered by sets of the form G ×H D, where H ⊂ G is a finite subgroup. The right invariant Maurer Cartan form on G naturally extends to a form on G × D (which is zero on D) and descends to G ×H D. Therefore, each orbit has an invariant neighborhood on which there exists a connection one-form. With an invariant partition of unity, these can be patched into a connection form defined on all of M . Corollary B.38 (Invariant connections for G-equivariant bundles). Let a Lie group G act properly on a manifold M and let E → M be a G-equivariant vector bundle, or a G-equivariant principal bundle (with some structure group K). Then E admits a G-invariant connection. Proof. By the slice theorem, M is covered by sets of the form U = G ×H D where H ⊆ G is compact. Let ϕ : D → U be the natural inclusion map. The pullback ϕ∗ E is an H-equivariant bundle over D. Because D equivariantly retracts to a point, this bundle is trivial. The trivial connection on this bundle is H-basic and naturally extends to a connection on G ×H D. Therefore, each orbit has a neighborhood over which there exists a G-invariant connection. With an invariant partition of unity, these can be patched into a Ginvariant connection over all of M .

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3.3. Fixed point sets. In this section we continue our discussion of the corollaries of the slice theorem assuming that a Lie group G acts properly on a manifold M. Corollary B.39. For any non-trivial subgroup H ⊂ G, its fixed point set M H = {m ∈ M | a · m = m for all a ∈ H} is a disjoint union of closed submanifolds of M . If M is orientable and G is connected, these submanifolds have codimension two or more. We recall that a connected subset F of M is a submanifold if it has the following property. For each point in F , there exists a neighborhood U in M and a diffeomorphism of U with an open subset Ω of some vector space which carries U ∩ F to the intersection of Ω with a linear subspace. Proof. By continuity, the fixed point set of H coincides with that of the closure of H. Therefore, we may assume that the group H is closed. The set M H is closed because the action is continuous. Let F be a connected component of M H . Let m be a point in F . By Theorem B.26 (local linearization) applied to the action of H, there exists a neighborhood U of m in M and an H-equivariant diffeomorphism of U with an open subset Ω of the vector space W = Tm M . This diffeomorphism carries U ∩ M H to Ω ∩ W H , where W H is the linear subspace consisting of those vectors that are fixed by H. Suppose that M is orientable. Pick an orientation. Since G is connected, it acts by orientation preserving transformations. Hence, the elements of H preserve some orientation on the vector space W = Tm M . The quotient W/W H is then a vector space with a non-trivial orientation preserving action of a compact Lie group. As such, it must have dimension at least two. This dimension is the codimension of F. Corollary B.40. For any vector ξ in the Lie algebra g, the zero set of the corresponding vector field, M ξ = {m ∈ M | ξM |m = 0}, is a disjoint union of closed submanifolds of M . Proof. We have M ξ = M H , where M H is the fixed point set of H = {exp(tξ) | t ∈ R}, the one-parameter subgroup of G generated by ξ. Now it suffices to apply Corollary B.39. The set MH = {m ∈ M | Stab(m) = H}

is an open subset of M H . From this observation and Corollary B.39 we immediately deduce the following result. Corollary B.41. The set MH is a disjoint union of submanifolds of M . The closure of each component of MH is a component of the fixed point set M H . If M is orientable and G is connected, these submanifolds have codimension two or more.

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187

3.4. Effectiveness. Corollary B.42 (Effective ⇒ Locally Effective). Suppose that M is connected and the G-action is effective (faithful). Let U ⊂ M be any invariant open subset. Then the G-action on U is also effective. Corollary B.42 does not generally hold when the action is not proper: Example B.43. Let G = R act on M = R2 by t : [x, y] 7→ [x + tρ(y), y] where ρ(y) = 0 for y ≤ 0 and ρ(y) = e−1/y for y > 0. This action is effective on R2 , but trivial on the lower half plane. Proof of Corollary B.42. We need to prove that the group K = {a ∈ G | a · u = u for all u ∈ U }

is trivial. This is a closed subgroup of G, and its fixed point set M K contains the open set U . In particular, any connected component of M K that contains a point of U contains an open subset of M . Let N be such a component. By Corollary B.39, N is a closed submanifold of M . Since N contains an open set, its dimension must be the same as that of M . Since a submanifold of M of the same dimension as M is an open subset, N is both closed and open in M . Because M is connected, N is all of M . Hence, all the points in M are fixed by K. Since the action on M is effective, K is trivial. Corollary B.44. Suppose that G is an abelian Lie group, M is connected, and the action is effective (faithful). Let G ×Gm W be a local model for a neighborhood of some orbit in M . Then the linear representation of Gm on W is effective. Proof. Let a ∈ Gm be such that a · w = w for all w ∈ W . Then for all [b, w] ∈ G ×Gm W , a · [b, w] = [ab, w] = [ba, w] = [b, aw] = [b, w].

By Corollary B.42, the action of G on the model is effective. Hence, a = e. Corollary B.44 does not generally hold for non-abelian groups. For example, let G = SO(3) act on M = S 2 by rotations. This action is effective. Because the action is transitive, the whole space consists of a single local model, in which the slice is trivial: M = SO(3)/S 1 = SO(3) ×S 1 {0}. The S 1 -action on the slice W = {0} is not effective. 3.5. Orbit types. The orbit type of a point m ∈ M is defined to be the conjugacy class of its stabilizer Gm in G. The conjugacy class is constant along an orbit, because the stabilizer of a · m is aGm a−1 . For any closed subgroup H ⊂ G, we set and, as before,

M(H) = {m ∈ M | Gm is conjugate to H}

MH = {m ∈ M | Gm = H}. We have two decompositions of M : G G M= M(H) and M = MH . (H)

H

If G is abelian, these two are the same, because M(H) = MH . Another useful decomposition of M is its decomposition into the connected components of the sets

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M(H) . This decomposition, called the orbit type stratification, is finer than the decomposition of M into the sets M(H) . The connected components of the M(H) ’s are called the orbit type strata of M . We will also consider the infinitesimal orbit type strata, determined from the conjugacy classes of the infinitesimal stabilizers. We will now employ the slice theorem to derive results on the orbit type decomposition. Corollary B.45. Each set M(H) is a submanifold of M . Proof. It is enough to prove that the orbit of each point with stabilizer H has a neighborhood whose intersection with M(H) is a submanifold. By the slice theorem, we can assume, without loss of generality, that M = G ×H W . It suffices to show that M(H) = G ×H W H , because this is a vector sub-bundle and hence a submanifold. To verify this equality, first notice that [a, w] ∈ M(H) if and only if [e, w] ∈ M(H) . Then the stabilizer of [e, w], being contained in H, is conjugate to H if and only if it is equal to H. Corollary B.46. The number of orbit types is locally finite. More explicitly, every point in M has a neighborhood that intersects only a finite number of the sets M(H) . Consequently, if M is compact, only a finite number of orbit types occur. Proof. We apply induction on the dimension of M . It is sufficient to show that in the model G ×H W , with a compact H, the number of orbit types is finite. The orbit type of [a, 0] is (H). The orbit type of [a, w] with w 6= 0 is the same as that of [e, w/kwk] (with respect to some H-invariant norm on W ). This is equal to the G-conjugacy class of the stabilizer of w/kwk under the H-action on the unit sphere in W . By the induction hypothesis, the H action on the unit sphere in W has only a finite number of orbit types. Since any two conjugate subgroups of H are also conjugate as subgroups of G, the corollary follows. Corollary B.47. Suppose that M is connected. Then there exists a closed subgroup K ⊂ G such that the set M(K) is open and dense in M . The class (K) in Corollary B.47 is called the principal orbit type for the action. Proof of Corollary B.47. Again, assume that we have established this for manifolds of dimension less than dim M . We first prove the corollary for the model G ×H W . By the induction hypothesis, the H-action on the unit sphere S ⊂ W has a principal orbit type (K). In other words, S(K) is open and dense in S. We have (G ×H W )(K) ⊃ G ×H (R+ · S(K) ). Here, the symbol (K) on the right-hand side stands for a conjugacy class in H and on the left it means a conjugacy class in G. The set on the left, being a submanifold that contains an open dense subset, must itself be an open dense subset. Now suppose that G acts on a manifold M . The previous paragraph shows that a neighborhood of every orbit contains a principal orbit type. Connectedness of M then forces these orbit types to be the same. Indeed, if U and U 0 are two open sets with non-empty intersection, M(K) ∩ U is open and dense in U , and M(K 0 ) ∩ U 0 is open and dense in U , then M(K) and M(K 0 ) must intersect, and hence must be equal.

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189

Corollary B.48. Suppose that G is abelian, M is connected, and the action is effective. Then the set of points where the action is free (i.e., the set M{e} = {m ∈ M |a · m = m ⇒ a = e}) is open and dense in M . If M is orientable and G is connected, M{e} is connected. Proof. By Corollary B.47, there exists a principal orbit type, (H), for which M(H) is open and dense. Since G is abelian, M(H) = MH , so H acts trivially on M(H) . Since M(H) is dense, H acts trivially on M . Since the action is effective, H = {e}. Suppose that M is orientable and G is connected. The complement of M{e} is the union of the fixed point manifolds M H for subgroups H of M . By Corollaries B.46 and B.39, this complement is a locally finite union of closed submanifolds of codimension two or more. This implies that M{e} is connected. Corollary B.49. Let G be a torus and H ⊆ G a closed subgroup. Let {Ni } be the connected components of MH . Assume that each connected component of M H is orientable. Then the closure of each Ni is a connected component of M H , of which Ni is an open subset. In particular, this closure is a smooth G-invariant submanifold. Note that M H contains the closure of MH but might be strictly bigger than this closure. For instance, let S 1 act on C by rotations and let H ⊂ S 1 be a non-trivial cyclic subgroup. Then M H is the origin, but MH is empty. Proof of Corollary B.49. Because MH is contained in M H , and because the components of M H are closed and disjoint from each other, each Ni is entirely contained in some component N of M H . We will show that the intersection N ∩MH is connected, and is therefore exactly equal to Ni . Also, we will show that N ∩ MH is dense in N , and hence N is the closure of Ni . By Corollary B.39, N is a manifold with a G/H-action. Since N contains Ni (on which the G-stabilizer is exactly H), the G/H action on N is effective. The intersection N ∩ MH is precisely the principal orbit type stratum. Because G/H is connected and N is orientable, we can apply Corollary B.48 and deduce that N ∩ MH is connected, open and dense in N . 4. The Mostow–Palais embedding theorem In this section we prove that a compact G-manifold can be equivariantly embedded into a finite-dimensional linear G-representation. This result (with weaker assumptions than compactness; see Remark B.51) is due to Mostow [Most] and Palais [Pa1]. It is the equivariant analogue of the classical fact that every manifold can be embedded into a Euclidean space. The main idea is that the evaluation map embeds M into the dual space C ∞ (M )∗ . We find a sufficiently large finite-dimensional subspace V of C ∞ (M ) such that the evaluation map M → V ∗ is an embedding. This idea is also employed in complex geometry; for example, in the Kodaira embedding theorem one uses the evaluation map to embed a complex manifold into P(V ∗ ), where V is the space of holomorphic sections of a sufficiently positive line bundle.

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Theorem B.50 (The Mostow–Palais embedding theorem; compact case). Let a compact Lie group G act on a compact manifold M . Then there exists an equivariant embedding of M into a linear representation of G on a finite-dimensional vector space V . Remark B.51. Mostow and Palais did not require M to be compact. Instead, they imposed a weaker condition; namely, that G acts on M with finitely many distinct orbit types. This condition is necessary, because a linear action of a compact group only has finitely many distinct orbit types. Palais also relaxes the compactness assumption on G: it is enough to assume that G is a group of matrices and that the action is proper and has finitely many distinct orbit types. See [Pa3, Theorem 4.4.3]. Moreover, the theorem also holds when M is a separable finitedimensional metric space. It can be proved by patching together local embeddings; see [Most] for more details. However, for this one first has to construct an equivariant embedding of an orbit, which already is not entirely trivial. The existence of an embedding for a neighborhood of a fixed point follows immediately from the slice theorem. When M is compact and the action is smooth, as we assume in Theorem B.50, the theorem has a simple proof, also due to Mostow [Most, Section 7], which is given below. Alongside Theorem B.50, we will prove Theorem B.52. Every compact Lie group is isomorphic to a closed subgroup of the orthogonal group O(N ), for some N . Before proceeding with the proof, we recall that the G-action on M induces a linear G-representation on C ∞ (M ), where g ∈ G acts by sending f ∈ C ∞ (M ) to the function (gf )(x) = f (g −1 x). Denote by V (f ) the span in C ∞ (M ) of the orbit G · f . The function f is said to be a representation function if V (f ) is finite-dimensional. Lemma B.53. Representation functions are C 1 -dense in C ∞ (M ). Proof. First note that representation functions on G, with respect to the left or right action of G on itself, are uniformly dense in C ∞ (G). This follows from the Peter–Weyl theorem (see, e.g., [Ad]). Namely, the matrix coefficients of a finite-dimensional representation T of G span the finite-dimensional representation T ∗ ⊗ T in C ∞ (G). Thus, every linear combination of matrix coefficients is a representation function on G. On the other hand, by the Peter–Weyl theorem, such linear combinations are uniformly dense in C ∞ (G). To prove the lemma, we will show that the “convolution” of a function f on M and a representation function u on G, is a representation function on M . More specifically, for u ∈ C ∞ (G) and f ∈ C ∞ (M ), we set Z fu (x) = u(g)f (g −1 x) dg G

where dg is the Haar measure normalized so that G has unit measure. We claim that fu is a representation function whenever u is a representation function. Indeed, for every h ∈ G we have (B.10)

hfu = fhu ,

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191

where hu(g) := u(h−1 g). This can be seen from the following calculation: Z −1 u(g)f (g −1 h−1 x) dg (hfu )(x) = fu (h x) = G Z = u(h−1 g)f (g −1 x) dg = fhu (x). G

For a fixed f , the mapping u 7→ fu is linear and, as (B.10) shows, G-equivariant. Hence, V (fu ) is the image of V (u) and, as a consequence, V (fu ) is finite-dimensional if V (u) is finite-dimensional. To finish the proof of the lemma, observe that every function f can be arbitrarily close C 1 -approximated by functions of the form fv with v ∈ C ∞ (G). (It suffices to take as v a bump function on G which is supported near e ∈ G and has unit integral.) Uniformly approximating v by representation functions u, we obtain a C 1 -approximation of f by representation functions fu . Proof of Theorems B.50 and B.52. The evaluation maps δx : f 7→ f (x), for x ∈ M , give rise to an equivariant injection x 7→ δx of M into the dual space to C ∞ (M ). For every f ∈ C ∞ (M ), the space V (f ) is naturally a G-representation, and the evaluation map gives rise to an equivariant mapping M → V (f )∗ . Recall that every manifold can be smoothly embedded in Rm for some m, and 1 a C deformation of an embedding remains an embedding. Thus, by Lemma B.53, there exists an embedding M → Rm whose components f1 , . . . , fm are representation functions. Therefore, we obtain an equivariant evaluation map from M to the direct sum V = V (f1 )∗ ⊕ · · · ⊕ V (fm )∗ . This map is an embedding because its composition with a suitable linear mapping V → Rm is the original embedding (f1 , . . . , fm ). If the G action on M is effective, so is the G action on V . Hence, G embeds in GL(N ), where N = dim V . We can make the action orthogonal by taking any inner product on V and averaging with respect to the G-action. 5. Rigidity of compact group actions 5.1. Rigidity under deformations. In this section, we will now show that compact group actions on compact manifolds are rigid under deformations. Clearly, an action can be deformed by conjugating it by a one-parameter family of diffeomorphisms. Such a deformation is considered trivial. Rigidity means that all deformations are trivial. A family of G-actions ρt , for t ∈ [0, 1], on M is called smooth if the map [0, 1] × G × M → M given by (t, a, m) 7→ ρt (a)(m) is smooth. A family of diffeomorphisms φt : M → M is called smooth if the map [0, 1] × M → M given by (t, m) 7→ φt (m) is smooth. Proposition B.54 ([PS]). Let ρt , for t ∈ [0, 1], be a smooth family of actions of a compact Lie group G on a compact manifold M . Then there exists a smooth family of diffeomorphisms φt : M → M , with φ0 = id, which intertwines ρ0 with ρt , i.e., such that ρt (a) ◦ φt = φt ◦ ρ0 (a) for all t ∈ [0, 1] and a ∈ G. Proof. Consider the product manifold [0, 1] × M with the G-action that is equal to ρt on {t} × M . Take the vector field ∂/∂t on [0, 1] × M . Its average with

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respect to the G-action,

Z

(B.11)

G

a∗ (∂/∂t)da,

(where da is the Haar probability measure on G) is a G-invariant vector field on [0, 1] × M whose second component is ∂/∂t. This vector field integrates to a family of diffeomorphisms φt which have the desired property. Remark B.55. If M is equipped with extra structure that is preserved by the G-action, then the trivialization φt can often be chosen to respect this structure. For instance, if M is a manifold equipped with a symplectic form, a Riemannian metric, a Poisson structure, or, in fact, any tensor field, then if the action diffeomorphisms ρt (a) preserve this structure, then so do the trivializing diffeomorphisms φt . Similarly, let M be equipped with a foliation, a contact structure, or, more generally, any distribution (sub-bundle of T M ). If ρt (a) preserve this structure, then so do φt . Also, if a connected Lie group H acts on M and each ρt commutes with the H-action, so does each φt . Indeed, because the vector field ∂/∂t preserves the structure, and because the G-actions preserves the structure, the vector field a∗ (∂/∂t) also preserves the structure, for each a ∈ G. This implies that the average (B.11) preserves the structure. This method works for those structures which have the property that if two vector fields preserve the structure then so do their convex combinations. If G is connected, it is enough to assume that the set of vector fields which preserve the structure has convex connected components. Remark B.56. Compactness of M is needed to ensure that the vector field (B.11) be integrable. Proposition B.54 is false for non-compact G-manifolds, even if G is compact. See [Pa2]. 5.2. Rigidity of linear representations. Let G be a compact Lie group and M a compact manifold. Rigidity asserts that given any G-action on M , all nearby G-actions are equivalent to it. This is slightly stronger than Proposition B.54, because a priori it is not clear that nearby actions can always be connected by a family of actions. Let us begin with rigidity for linear representations. Proposition B.57 (Rigidity of representations). Let G be a compact Lie group and V a finite-dimensional vector space. Let ρ0 be a linear representation of G in V . Then for every representation ρ which is sufficiently C 0 -close to ρ0 there exists an automorphism A of V which intertwines ρ0 and ρ, i.e., such that ρ0 = A ◦ ρ ◦ A−1 . Moreover, A can be chosen to depend smoothly on ρ and ρ0 . Our proof adapts Palais’ trick from [Pa2] to the (simpler) case of linear actions. by

Proof. For each representation ρ of G in V , define a linear map A : V → V A(x) =

For any h ∈ G, A(ρ(h)x) =

Z

Z

ρ0 (g −1 )ρ(g)(x)dg,

ρ0 (g −1 )ρ(g)ρ(h)(x)dg = G

x ∈ V.

G

Z

ρ0 (g −1 )ρ(gh)(x)dg. G

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193

The change of variable g 7→ gh−1 turns the integral into Z Z ρ0 (hg −1 )ρ(g)dg = ρ0 (h) ρ0 (g −1 )ρ(g)dg = ρ0 (h)A(x). G

G

This shows that A ◦ ρ = ρ0 ◦ A. When A is invertible, the proof is finished. Because A depends continuously on ρ and is equal to the identity map when ρ = ρ0 , the map A is invertible if ρ is sufficiently close to ρ0 . Remark B.58. The assumption that V is finite-dimensional can be significantly relaxed. The proposition holds for a broad class of infinite-dimensional representations. For example, as is clear from the proof, the proposition is true for unitary representations in a Hilbert space. 5.3. Rigidity of group actions. Let us equip the space of smooth actions of G on M with the C 1 -topology. Namely, the space of all actions is embedded into the space of mappings G × M → M with the C 1 -topology. Recall that convergence in this topology means uniform convergence of the mappings and of their first derivatives. The following rigidity theorem is due to Palais. Theorem B.59 (Rigidity of compact groups actions; [Pa2]). Let ρ be an action of a compact group G on a compact manifold M . For every G-action ρ 1 on M which is sufficiently C 1 -close to ρ, there exists a diffeomorphism φ : M → M which conjugates the actions: ρ1 = φρφ−1 . In addition, φ can be chosen to depend smoothly on ρ1 . The original proof, in [Pa2], is based on the fact that G-actions embed in linear G-representations (Theorem B.50). Our proof, which is almost the same as Palais’, uses this fact together with the rigidity of linear representations (B.57). Proof of Theorem B.59. Let us apply Theorem B.50 to both ρ and ρ1 . This results in two representations of G, say ρ¯ and ρ¯1 , on vector spaces V and V1 , respectively, and equivariant embeddings M → V for ρ and M → V1 for ρ1 . A ∼ = closer inspection of the proof shows that we can find a linear isomorphism V1 → V 1 after which the embeddings become C -close and ρ¯ and ρ¯1 become close. Thus, from now on we will assume that V = V1 . By Proposition B.57, there exists a linear mapping V → V , close to the identity, which sends ρ¯1 to ρ¯. Thus we may assume that the representations are equal and the embedding ψ1 for ρ1 is still C 1 close to the embedding ψ for ρ. The image of ψ1 lies in a small tubular neighborhood of the image of ψ, which we identify with M . Let us fix a G-invariant metric on V . The composition φ of ψ1 with the orthogonal projection from the tubular neighborhood to M is clearly G-equivariant: (M, ρ1 ) → (M, ρ). Since ψ1 is C 1 -close to ψ this composition is a diffeomorphism. This construction is smooth in ρ1 . 5.4. Rigidity and group cohomology. There is a general principal by which rigidity on the infinitesimal level is equivalent to the vanishing of certain cohomology. Namely, rigidity means that a relevant moduli space is discrete, and the tangent space to the moduli space is interpreted as cohomology. For example, given a complex manifold V , the space of deformations of its complex structure is H 1 (V, Θ)

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where Θ is the sheaf of holomorphic vector fields; see [KS]. For group actions, the relevant cohomology is group cohomology. In this section we will recall the definition of the first cohomology of a group, prove that for a compact group the first cohomology vanishes, and connect this fact with the rigidity of representations of compact groups. Our exposition is essentially self-contained. The reader interested in a general introduction to group cohomology for topological and Lie groups may consult, for example, [Guic]. Let G be a Lie group and ρ a representation of G on a vector space V . A one-cocycle on G with coefficients in ρ is a function c : G → V such that (B.12)

ρ(g1 )c(g2 ) − c(g1 g2 ) + c(g1 ) = 0

for all g1 and g2 in G. The space of all one-cocycles is denoted by Z 1 (G; ρ). A cocycle c is said to be exact if there exists an element v ∈ V , called the primitive of c, such that c(g) = ρ(g)v − v. We say that c is the coboundary of v. The space of all exact cocycles is denoted by B 1 (G; ρ). The first cohomology of G with coefficients in ρ is the quotient Z 1 (G; ρ) . H 1 (G; ρ) = 1 B (G; ρ) In what follows, ρ will always be a continuous representation in a topological vector space. In this case, all cocycles are assumed to be continuous. Lemma B.60. If G is compact and V is locally convex, H 1 (G; ρ) = 0. Remark B.61. The lemma is a particular case of a well-known fact that H k (G; ρ) = 0 for k > 0, when G is compact and V is locally convex; see [Guic]. Proof. Let c be a cocycle. A direct calculation shows that Z (B.13) v= ρ(g)−1 c(g) dg, G

is a primitive of c. The integral exists because V is locally convex [Rudi]. Let us return to group actions, and consider rigidity in the context of group cohomology. We begin with linear representations.

“Proof” of rigidity of linear G-representations. Consider the space R = Hom(G, GL(V )) of all representations of G in a vector space V . Suppose that we know that R is a smooth manifold. The group GL(V ) acts on R by conjugation. Two representations are isomorphic if and only if they are in the same GL(V ) orbit. Rigidity of linear G-representations asserts that the orbits are open. Fix an element ρ0 ∈ R, and let G act on gl(V ) by the map ρ0 : G → GL(V ) followed by the conjugation action of GL(V ) on gl(V ). We claim that the tangent space to R at ρ0 is the space of cocycles, Z 1 (G; gl(V )). Let us write an infinitesimal deformation of ρ0 as ρ = (I + ξ)ρ0 where ξ : G → gl(V ). The homomorphism condition ρ(gh) = ρ(g)ρ(h) becomes the cocycle condition (B.12) on ξ, ξ(gh) = ξ(g) + ρ0 (g)ξ(h)ρ0 (g)−1 . Each ζ ∈ g gives rise to a trivial infinitesimal deformation by ρ = (I + ζ)ρ0 (I + ζ)−1 = (I + ξ)ρ0 ,

where ξ = ζ − ρ0 (g)ζρ0 (g)−1 . These ξ’s are precisely the coboundaries.

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195

We have shown that the tangent space to R at ρ0 is Z 1 (G; gl(V )) and the tangent space to the orbit GL(V ) · ρ0 is B 1 (G; gl(V )). By Lemma B.60, these two spaces are equal. This establishes the infinitesimal rigidity result. To proceed, we would like to deduce that the orbit in R is open from the fact that its tangent space is the whole tangent space to R. If we knew that R is a manifold, this fact would follow from the inverse function theorem. Remark B.62. The reason that the above “proof” is not a complete proof is that the fact that R is a manifold is not obvious. Remark B.63. The above identification of tangent spaces interprets the first group cohomology H 1 (G, gl(V )) as the space of infinitesimal deformations of ρ modulo trivial deformations, i.e., those induced from infinitesimal isomorphisms of V. Smooth G-actions on a manifold M can be treated by the same reasoning as above: “Proof” of rigidity of G-actions. A G-action on a manifold M is given by a smooth homomorphism G → Diff(M ). The group Diff(M ) acts on the space of actions by conjugation; two actions are isomorphic if and only if they are in the same Diff(M )-orbit. Denote by R the space of all G-actions. As above, the tangent space to R at ρ0 can be identified with the space of one-cocycles Z 1 (G, Vect(M )), and the tangent to the orbit can be identified with the one-coboundaries B 1 (G, Vect(M )). By Lemma B.60, these two spaces are equal. Assuming that R is a Fr´echet manifold, we may use the Nash–Moser inverse functions theorem [Ham] and deduce that the Diff(M )-orbits are open. Remark B.64. As for linear representations, it is not easy to prove directly, without first establishing some form of rigidity, that R is a Fr´echet manifold. Remark B.65. In the above reasoning, one can replace GL(V ) or Diff(M ) by any Lie group K. The tangent space to Hom(G, K) at ρ is identified with Z 1 (G; k), and the tangent space to the K-orbit, where K acts by conjugation, is identified with B 1 (G; k). Here, G acts on the Lie algebra k through the homomorphism ρ followed by the K-action on k by conjugation. Hence, the tangent space to the moduli space Hom(G, K)/K at [ρ] is identified with the group cohomology H 1 (G; k). This reasoning was given by Goldman in [Go], in a different context: The moduli space of flat connections on a principal G-bundle can be identified with Hom(π, G)/G where π is the fundamental group of the base manifold. The tangent space to the moduli space is then identified with H 1 (π; g). This identification allows one to define a natural symplectic form on Hom(π, G)/G when the base manifold is a surface [Go] or a K¨ ahler manifold [Ka1]. To deduce rigidity under deformations from infinitesimal rigidity one does not need the inverse function theorem. Instead, one may use the “homotopy method”. Following [Gin4, Section 6.2], let us apply this method to derive Proposition B.54 from its infinitesimal version. Second proof of Proposition B.54. For every g ∈ G, consider the time dependent vector field ct (g) ∈ Vect(M ) given by d ct (g)(x) = ρt (g)∗ ρt (g)−1 (x). dt

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A routine calculation shows that ct : G → Vect(M ) is a one-cocycle on G. (This calculation simply expresses the fact that the tangent space to the space of actions at ρ is Z 1 (G, Vect(M )).) Since G is compact, by Lemma B.60, the cocycle ct is exact. Moreover, by (B.13), its primitive v t ∈ Vect(M ) can be chosen smooth in t. The required family φt is then just the time-dependent flow of −v t . Remark B.66. A similar method can be applied to prove the local linearization theorem for an action of a compact group. See [Gin4], where the local linearization of Poisson actions is also considered.

APPENDIX C

Equivariant cohomology 1. The definition and basic properties of equivariant cohomology 1.1. The topological definition of equivariant cohomology. Let G be a compact Lie group. There exists a numerable1 principal G-bundle EG → BG whose total space EG is contractible. The equivariant cohomology of a G-manifold M is defined to be the (say, singular) cohomology of the Borel construction EG × G M : (C.1)

∗ HG (M ) = H ∗ (EG ×G M ).

It is well defined because EG ×G M is unique up to homotopy equivalence. q If G acts freely, HG (M ) = H q (M/G) for all q, since EG ×G M is a fiber bundle over M/G with a contractible fiber EG. If G does not act freely, this is no longer true. For instance, the equivariant cohomology of a point is equal to the cohomology of the classifying space, BG = EG/G, which is usually infinite–dimensional. For more details, see [Hus] and [Bor2, Dol1]. Example C.1. For G = S 1 , one can take EG = S ∞ , interpreted as the direct limit of odd–dimensional spheres EGk = S 2k+1 ⊂ Ck+1 with respect to the natural inclusions, and BG = CP∞ = lim CPk . For G = U(n), we obtain EG as the direct −→

limit of the Stiefel manifolds of unitary n frames in Ck . For a general G, one can take the Stiefel manifold with the G-action induced by a faithful representation G → U(n). In all these cases, EG ×G M is a direct limit of finite–dimensional manifolds, EGk ×G M , with respect to natural inclusions. For every degree q, we have (C.2)

q HG (M ) = H q (EGk ×G M )

for all sufficiently large k. This is established by a routine argument from the following two facts: • BGk+1 is obtained from BGk by attaching cells of dimension ≥ q if k is sufficiently large. • for each k there exists an open neighborhood of EGk in EG which equivariantly strongly deformation retracts to EGk . The second assertion can be seen by embedding EG in the Stiefel manifold of n-frames in a Hilbert space. In particular, for G = S 1 , ∗ HG (point) = H ∗ (BG) 1 Numerable means that there exists a partition of unity subordinate to an open covering {Ui } such that the bundle is trivial over each Ui .

197

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is the space of polynomials in one variable, R[u], with degree u = 2. When G ∼ = (S 1 )r is a torus, we can take EG = (S ∞ )r and H ∗ (BG) = R[u1 , . . . , ur ]; see [BT1, §18]. 1.2. The Cartan model. Let G be a compact connected Lie group acting smoothly on a compact smooth manifold M . Then, over R, the equivariant cohomology of M is equal to the cohomology of a differential complex, an equivariant version of the de Rham complex called the Cartan model, which we will now describe. The equivariant differential forms of degree q are, by definition, the elements of the space M ΩqG (M ) = (C.3) (S i (g∗ ) ⊗ Ωj (M ))G , 2i+j=q

i.e., G-equivariant polynomial functions α : g → Ω∗ (M ) on the Lie algebra g taking values in the space of differential forms on the manifold. Notice the grading: if α = P ⊗ ω, where P is a real valued homogeneous polynomial on g and ω is a real valued differential form on M , then deg(α) = 2 deg(P ) + deg(ω). Equivariance means that for all ξ ∈ g and b ∈ G, α(Ad(b)ξ) = b∗ α(ξ).

In particular, setting b = exp(tξ) and taking the derivative at t = 0, we see that 0 = LξM α(ξ).

(C.4)

Hence, the differential form α(ξ) is invariant under the action of the one-parameter group generated by ξ. In general, however, the forms α(ξ) are not invariant under the action of the whole group G if G is not abelian. 1 Example C.2. Let us identify Lie(S R when G = S 1 . Then an equiP ) with j variant differential form is a finite sum αj u where u is a formal variable and αj are invariant differential forms.

The equivariant exterior derivative, dG : ΩqG (M ) → Ωq+1 G (M ), is defined by (dG α)(ξ) = d(α(ξ)) + ι(ξM )α(ξ). Lemma C.3. d2G = 0 Proof. (d2G α)(ξ) = dι(ξM )α(ξ) + ι(ξM )dα(ξ) = LξM α(ξ) = 0 by (C.4). Exercise. Prove that ker dG /im dG = S ∗ (g∗ )G when M = {point}. The cohomology of the differential complex (Ω∗G (M ), dG ) turns out to be the same as the equivariant cohomology of M with real coefficients: Theorem C.4 (Equivariant de Rham Theorem). Let G be a compact connected Lie group and let M be a G-manifold. Then (C.5)

∗ HG (M ; R) = ker dG /im dG .

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199

In particular, the cohomology of the classifying space BG is the ring of Adinvariant polynomials on the Lie algebra (C.6)

∗ RG := HG (point) = H ∗ (BG) = S ∗ (g∗ )G .

In reality this fact is usually established as the first step of the proof of the equivariant de Rham theorem, not as a consequence of the theorem. The equivariant de Rham theorem, as well as the notion of the Cartan model, are due to H. Cartan [Car1, Car2]. We will outline a proof of the equivariant de Rham theorem in Section 3. The Cartan model readily generalizes to a pure algebraic setting, giving the cohomology of the so-called G-differential complexes. The reader interested in this construction and in more details on the Cartan model may consult, for example, [Car1, Car2, BV1, DKV, Gin5, GS8] and [GLS, Appendix B]. For a topological treatment, see [Hs, AB2] and references therein. 1.3. Basic properties of equivariant cohomology and examples. Proposition C.5 (Module structure). 1. A G-equivariant map f : M → N induces pullback maps on differential forms ∗ ∗ and cohomology, f ∗ : Ω∗G (N ) → Ω∗G (M ) and f ∗ : HG (N ) → HG (M ). 2. In particular, the map M → {point} gives rise to an RG -module structure q ∗ on Ω∗G (M ) and on HG (M ), so that ⊕q ΩqG (M ) and ⊕q HG (M ) are graded RG -modules. 3. For any G-equivariant homotopy, ft : M → N , t ∈ [0, 1], the pullback maps ∗ ∗ (M ), are the same for all t. (N ) → HG on cohomology, ft : HG Exercise. Prove these properties in the Cartan (differential) model (C.5) (with f and ft smooth) and in the Borel (topological) model (C.1). Example C.6 (Trivial action). Assume that the G-action on M is trivial. ∗ Then HG (M ) is canonically isomorphic to H ∗ (M ) ⊗ RG as an algebra over RG . This follows directly from the definitions, either in the Cartan model or in the ∗ topological setting. Equivalently, HG (M ) is the space of G-invariant polynomials ∗ on g with values in H (M ). Example C.7 (Zeroth equivariant cohomology). The space Ω0G (M ) is comprised of smooth invariant functions f : M → R. Since dG f = df , the closed 00 (M ) = Rk , where k is the forms are locally constant invariant functions. Hence, HG number of connected components of M/G. Example C.8 (First equivariant cohomology). The space Ω1G (M ) consists of invariant one-forms α on M . Since (dG α)(ξ) = dα + ι(ξM )α, we have dG α = 0 if and only if α is closed and horizontal. The exact equivariant one-forms are df , where f is invariant. Hence, 1 HG (M ) =

closed basic forms = H 1 (M/G), exact basic forms

by Corollary B.36. Example C.9 (Equivariant two-forms). Equivariant two-forms are sums ω+Φ, where ω is an invariant two-form and Φ ∈ g∗ ⊗ Ω0 (M ) is invariant. Equivalently, Φ is an equivariant smooth function from M to g∗ . Denote Φξ = hΦ, ξi. The exterior

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derivative, dG (ω + Φ)(ξ) = dω + ι(ξM )ω + dΦξ , is zero if and only if ω is closed and for all ξ ∈ g (C.7)

ι(ξM )ω = −dΦξ .

Example C.10 (Second equivariant cohomology). In this example we use the notion of Hamiltonian assignments introduced in Appendix E. By Proposition E.38, for a torus action we have a short exact sequence   Hamiltonian 2 → 0. 0 → H 2 (M/G) → HG (M ) → assignments Here the Hamiltonian assignment of the class [ω+Φ] associates the element A(X) = ΦH (X) of h∗ to an orbit type stratum X of M with stabilizer H = GX . Example C.11 (Vector spaces). If G acts linearly on a vector space V , two equivariant differential forms are in the same cohomology class if and only if their restrictions to the origin are equal. This follows immediately from the fact that the origin is an equivariant deformation retract of the vector space. In what follows we will need a more explicit description of the coefficient ring RG = H ∗ (BG) = (S ∗ g∗ )G . Proposition C.12 (Chevalley’s Theorem). The restriction to a maximal commutative subalgebra t ⊂ g gives rise to an isomorphism of algebras (S ∗ g∗ )G → (S ∗ t∗ )W ,

where W is the Weyl group. Since t meets every orbit of the adjoint action, the only non-obvious part of this proposition is that the restriction map is onto. This is a well-known (but not entirely trivial) result from the theory of Lie groups and their representations. We refer the reader to [Di, Theorem 7.3.5] and [Va, Theorem 4.9.2] for the proof. Here, instead, we give a simple proof of this fact for G = U(n). Example C.13. Let G = U(n) and let us, for the sake of simplicity, consider complex valued polynomials. By definition, (S ∗ g∗ )G is the algebra of polynomials on u(n) invariant under conjugation. Let us take as t the space of diagonal matrices in u(n). We will regard the diagonal entries as coordinates (x1 , . . . , xn ) on t. Then W is Sn , the group of permutations, and (S ∗ t∗ )W is just the algebra of symmetric polynomials in x1 , . . . , xn . As is well known, such polynomials can be expressed as polynomials f of elementary symmetric functions σ1 , . . . , σn . It is clear that f can be extended to a continuous G-invariant function on u(n) by simply replacing the symmetric functions in x1 , . . . , xn by the symmetric functions in the eigenvalues. Denote these functions by σk again. To show that this extension is a polynomial, it suffices to prove that all σk are polynomials, but this follows immediately from the fact that σk (A) = trace ∧k A for any matrix A.

Remark C.14. As we have pointed out above, elements of Ω∗G (M ) can be thought of as G-invariant polynomial functions on g with values in Ω∗ (M ). By replacing the ring of polynomials on g by some other class of functions, we may still obtain complex, and hence cohomology spaces. These spaces are often used as repositories for equivariant cohomology classes that are not polynomial on g. This

2. REDUCTION AND COHOMOLOGY

201

is the case, for instance, with the equivariant Todd class Td G ; see Section 2 of Appendix I. The classes of “functions” usually considered are • formal power series on g; • the germs at 0 of analytic functions on g; • smooth functions on g, or their germs at 0; • distributions on g. For formal power series, the resulting cohomology can also be described in pure topological terms, using the inverse limit construction in the setting of Example C.1. For other classes of functions, one works directly with the Cartan model, with polynomials replaced by a suitable class A of “functions”. We will refer to the resulting cohomology as the equivariant cohomology over A. Under suitable hypotheses (see, e.g., [DKV, Corollaries 64 and 104]), the equivariant cohomology over A is the extension of the standard equivariant cohomology, i.e., is equal to ∗ ∗ HG (M ) ⊗RG A. A majority of results presented below for HG (M ) extend with appropriate modifications to cohomology over other classes of “functions”. The reader interested in the precise definitions and results should consult [DKV]. 2. Reduction and cohomology The projection EG×G M → M/G induces the pull-back algebra homomorphism (C.8)

∗ H ∗ (M/G) → HG (M ).

If the G-action is free, this map is an isomorphism. The same is true for cohomology with real coefficients if G acts locally freely, i.e., with finite stabilizers. This can be established in the topological model using, for example, a spectral sequence argument, or it can be proved directly in the Cartan model. Note that, in general, the map (C.8) need not be onto (e.g., take M to be a point), nor one-to-one (see Example C.18 below). The spectral sequence argument goes as follows. Consider the projection map ρ : EG ×G M → M/G. Its “fiber” over x ∈ M/G is EG/Gx , the classifying space BGx of the stabilizer Gx of x. The Leray spectral sequence of ρ (see [Bor2] or [Hs, Section III.1]) with real coefficients collapses in the E2 -term because Gx is finite, and hence H ∗>0 (BGx ; R) = 0. Therefore, H ∗ (EG ×G M ; R) = H ∗ (M/G; R). (See, e.g., [Gin3, Lemma 5.3] for more details.) The direct proof, due to Cartan, is somewhat involved. (See [Car1, Car2], also [DKV] (Theorem 17 of Part I with the ring of polynomials on g∗ replaced by smooth functions), [GLS, Appendix B], and [GS8].) The special case of HS2 1 is particularly instructive: Lemma C.15. Let G = S 1 act on M locally freely. Let π : M → M/S 1 be the quotient map. Define π ∗ : H 2 (M/S 1 ) → HS21 (M ) by [α] 7→ [π ∗ α]. Then π ∗ is well defined, one-to-one, and onto. Remark C.16. The quotient M/S 1 is an orbifold. Locally, an orbifold is the quotient of Rn by a finite group, and a differential form is given by an invariant differential form on Rn . Just like for manifolds, the cohomology (over R) of an orbifold is ker d/image d on differential forms. In the statement of the lemma, [α] means the cohomology class represented by the differential form α on M/S 1 . A differential form β on M descends to a differential form on the orbifold M/G if and only if β is basic, i.e., invariant and horizontal (ι(ξM )β = 0 for all ξ ∈ g). See Corollary B.36.

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Proof of Lemma C.15. The pullback π ∗ α is a closed and basic differential form on M , hence it is an equivariantly closed equivariant differential form (which happened to be independent of the formal variable u). The pullback homomorphism π ∗ is well defined on cohomology, because the pullback of any d-exact form is dS 1 -exact: π ∗ dβ = dS 1 π ∗ β. Let us show that π ∗ is one-to-one in cohomology. Suppose that π ∗ ω is dS 1 exact. This means that π ∗ ω = dS 1 β = dβ + uι(ξM )β for some invariant one-form β. Since there is no variable u on the left-hand side, β is horizontal: ι(ξM )β = 0. Since β is horizontal and invariant, it is basic. Let b be the one-form on M/S 1 whose pullback to M is β. Then π ∗ ω = d(π ∗ b), and ω = db is exact. Let us show that π ∗ is onto. Let ω + uΦ be an arbitrary equivariantly closed equivariant two-form on M . Then ω is an invariant two-form, Φ is an invariant function, and dS 1 (ω + uΦ) = dω + u(dΦ + ι(ξM )ω) = 0. If Φ ≡ 0, this implies that ω is closed and basic. Hence, ω is the pullback of a closed two-form on M/S 1 . If Φ 6≡ 0, we subtract an equivariantly exact two-form to obtain a new form with Φnew ≡ 0. Namely, let γ be a connection one-form on M , i.e., a one-form satisfying (1) ι(ξM )γ ≡ 1, and (2) LξM γ ≡ 0.

Then

dS 1 (Φγ) = d(Φγ) + uι(ξM )(Φγ) = ω 0 + uΦ for some ω 0 . To complete the proof it suffices to subtract this from ω + uΦ. Example C.17. When the G-action is locally free, but not free, the map ∗ π ∗ : H ∗ (M/G; Z) → HG (M ; Z) may fail to be an isomorphism. For example, let 1 S act on the unit sphere S 3 ⊆ C2 by a · (z, w) = (az, a2 w). Then H 2 (S 3 /S 1 ; Z) pulls back to the subgroup of index two in HS2 1 (S 3 ; Z) ∼ = Z, as one can check by a Mayer-Vietoris argument. Alternatively, consider a non-effective action of G = S 1 on M which factors into the double cover G → S 1 followed by a free action of S 1 2 on M . Then H ∗ (M/G; Z) is a subgroup of index two in HG (M ; Z). This follows from the Leray spectral sequence for the fiber bundle EG ×G M → M/G whose fiber is BZ2 = RP∞ . Example C.18. As we have pointed out above, in general, the pull-back map ∗ H ∗ (M/G) → HG (M ) need not be one-to-one. The first degree in which this phenomenon can occur is three. Let us illustrate this by an example. Let M be the unit sphere S 4 in R × C2 and G = S 1 with its standard diagonal action on C2 and trivial action on R. Using the Mayer–Vietoris exact sequence, 3 it is easy to see that HG (M ) = 0. (In fact, M is equivariantly formal and hence ∗ ∗ HG (M ) = H (M )⊗RG ; see Section 4 below. This, for example, can be shown using the results of Appendix G. Indeed, the height function on S 4 is a non-degenerate abstract moment map for the action. Now formality follows from Theorem G.9. On the other hand, M/G is homeomorphic to S 3 and thus H 3 (M/G) = R. Therefore, the map

3. ADDITIVITY AND LOCALIZATION

is not a monomorphism.

203

3 R = H 3 (M/G) → HG (M ) = 0

3. Additivity and localization 3.1. Additivity techniques. An important feature of (co)homology is that information about the cohomology of a space can be recovered from the cohomology of its smaller pieces. This property of cohomology allows one, under favorable conditions, to determine the cohomology of the space, or at least to obtain some valuable information about it. One way to do this is by using the long exact sequence of a pair. Another way to state the same principle is to say that cohomology is additive, where additivity is understood in a very broad sense. More formally, this is expressed by the Mayer–Vietoris exact sequence. These two exact sequences, which are essentially equivalent to each other (see [Sw]), are among the most frequently used and conceptually significant properties of (co)homology. The long exact sequence of a pair and the Mayer–Vietoris exact sequence also exist for equivariant cohomology and play a role similar to that of their ordinary versions. Namely, as follows immediately from the topological description of equivariant cohomology, there is a long exact sequence associated with a pair of G-spaces. To be more precise, let Z be a G-invariant subset of a G-space M . (For example, ∗ (M, Z) = H ∗ (EG ×G M is a G-manifold with boundary and Z = ∂M .) Set HG M, EG ×G Z). Then the long exact sequence for the ordinary cohomology of the pair (EG ×G M, EG ×G Z) turns into the exact sequence ∗ ∗ ∗ . . . → HG (M, Z) → HG (M ) → HG (Z) → . . . .

In the same vein, let U and V be G-invariant subsets of M such that M = int U ∪ int V . Then we have the exact sequence ∗ ∗ ∗ ∗ . . . → HG (M ) → HG (U ) ⊕ HG (V ) → HG (U ∩ V ) → . . . ,

called the Mayer–Vietoris exact sequence for equivariant cohomology. Remark C.19. For the Mayer–Vietoris exact sequence to hold, it is enough to assume that U , V , and U ∩ V are, respectively, equivariant deformation retracts of U 0 , V 0 , and U 0 ∩ V 0 , where U 0 and V 0 are open and M = U 0 ∪ V 0 . As does the long exact sequence for pairs of spaces, the Mayer–Vietoris exact sequence for equivariant cohomology follows directly from its ordinary, nonequivariant, version. (See, e.g., [Mas3, p. 58] or [Sw].) Alternatively, when U and V are open it can be proved from the Cartan model in the same way as the ordinary Mayer–Vietoris exact sequence is proved for de Rham cohomology. (See, e.g., [BT1].) Finally, the Mayer–Vietoris exact sequence extends to pairs of subsets (triads) in a natural way [Sw]. 3.2. An application: the equivariant de Rham theorem. As an application of the additivity techniques, we outline the proof of Theorem C.4, which asserts that the cohomology of the Borel construction EG ×G M , over R, is isomorphic to the cohomology of the Cartan complex, ker dG /im dG . Assume first that M is a point. Then we need to prove (C.6), which is done in Section 5.2.

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More generally, let M be an orbit, i.e., a homogeneous space G/K, where K is a closed subgroup of G. The space EG ×G M is then homotopy equivalent to ∗ BK, and hence HG (M ) = H ∗ (BK). On the other hand, a somewhat non-trivial algebraic calculation (see, e.g., [DKV, pp. 53–57] or [GLS, Appendix B]) shows that the cohomology in the Cartan model is (S ∗ k∗ )K . (For example, if M = G, the complex Ω∗G (G) is simply the Weil complex Λ∗ g∗ ⊗ S ∗ g∗ which is known to be acyclic.) Now, using (C.6) with K in place of G, we obtain the equivariant de Rham theorem for M . (The reader interested in more recent results about the equivariant cohomology of homogeneous spaces should consult [BT3].) Passing to the general case, we adapt the proof of the de Rham theorem for ordinary cohomology given in [BT1]. Assume, for the sake of simplicity, that M is compact. Then we can cover M by a finite number of invariant open sets such that each non-empty multiple intersection is (smoothly) equivariantly retractable to an orbit. The equivariant de Rham theorem holds for all intersections of open sets from the cover since the equivariant cohomology is an invariant of smooth equivariant homotopy, both in the topological model and in the Cartan model (see Proposition C.5). Now the proof is finished in the same way as for the ordinary de Rham theorem in [BT1], but using the five–lemma and the Mayer–Vietoris sequences for the equivariant cohomology in the topological model and in the Cartan model. 3.3. Borel’s Localization. A key feature of equivariant cohomology which has no analog for ordinary cohomology is that a substantial part of the structure of ∗ HG (M ) can be recovered, when G is abelian, from the fixed point set. One of the incarnations of this general principle is the following, slightly weakened, version of a theorem due to Borel, [Bor2]. Theorem C.20 (Borel’s localization theorem). Let M be a compact manifold, possibly with boundary, acted upon by a torus G. Then the restriction homomorphism ∗ ∗ HG (M ) → HG (M G )

is an isomorphism modulo torsion over the ring RG of Ad-invariant polynomials on g. Moreover, if the G-action on ∂M has no fixed points, the restriction homomorphism ∗ ∗ HG (M, ∂M ) → HG (M G )

is also an isomorphism modulo RG -torsion.

∗ Since HG (M G ) = H ∗ (M G ) ⊗ RG , this, in particular, implies that

(C.9)

∗ rkRG HG (M ) = dim H ∗ (M G ).

Note that a torsion RG -module may be quite large since RG is the ring of polynomials. For example, if G = S 1 acts freely on M , the entire space HS∗ 1 (M ) = H ∗ (M/S 1 ) is a torsion RG -module. Theorem C.20 is a special case of the following result, in which we take either Z = ∂M or Z = ∅. Theorem C.21. Let a torus G act on a compact manifold M , possibly with boundary, and let Z ⊂ M be a subset which is itself a compact manifold. Then the kernel and cokernel of the restriction homomorphism ∗ ∗ HG (M, Z) → HG (M G , Z G )

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205

are RG -torsion modules. In other words, the restriction homomorphism is an isomorphism modulo RG -torsion. Let us outline a simple proof of Theorem C.21 using the Mayer–Vietoris exact sequence as in [AB2]. This method also provides some information on the torsion ∗ of HG (M ). The reader interested in more details should consult [AB2]. ∗ Proof. First observe that HG (M ) is an RG -torsion module when the G-action has no fixed points on M . To see this, let us cover M by a finite number of Ginvariant open sets of the form G ×K B, where K is a subgroup of G, and the slice B is diffeomorphic to a subset of a vector space with a linear K-action. The existence of such a cover follows from the slice theorem. (See Appendix B and, in particular, Theorem B.24.) ∗ ∗ Each HG (G×K B) is a torsion RG -module. Indeed, HG (G×K B) = H ∗ (EG×G ∗ ∗ G ×K B) = H (EG ×K B) = HK (B), where for the last equality we take EK to be EG with the action restricted to K. Since the action of RG on this space factors through the restriction RG → RK , every Ad-invariant polynomial on g ∗ that vanishes on k annihilates HG (G ×K B). Any G-invariant open subset of G ×K B is again of the form G ×K B 0 , where the slice B 0 is obtained as the intersection of the slice B with the open subset. Therefore, the equivariant cohomology of the subset is again a torsion RG -module. ∗ By the Mayer–Vietoris exact sequence, HG (M ) is a torsion RG -module. Indeed, let V be a union of a finite number of sets of the form G ×K B, and let V 0 be an additional such set. The Mayer–Vietoris exact sequence for V and V 0 implies that ∗−1 ∗ ∗ ∗ HG (V ∩ V 0 ) → HG (V ∪ V 0 ) → HG (V ) ⊕ HG (V 0 )

is exact. The above reasoning shows that V 0 and V ∩ V 0 are torsion RG -modules. ∗ (V ) is a torsion RG -module. By exactArguing inductively, let us assume that HG 0 ∗ ness, the middle term, HG (V ∪ V ), is also a torsion RG -module. Let us proceed by induction in the number of components of the fixed point set of M . We have just proved the theorem if this number is zero. Pick a connected component F of the fixed point set M G , let U be a small neighborhood of F , equivariantly contractible to F , and let V = M rU . The Mayer–Vietoris exact sequences for M = V ∪ U and M G = V G t F read ∗ ∗ ∗ ∗ (∂U) −−−−→ . . . . . . −−−−→ HG (M ) −−−−→ HG (V ) ⊕ HG (U ) −−−−→ HG       y y y

∗ ∗ ∗ . . . −−−−→ HG (M G ) −−−−→ HG (V G ) ⊕ HG (F ) −−−−→

0

−−−−→ . . .

Suppose that the theorem is true for manifolds whose fixed point set has fewer connected components than M G does. Then all vertical arrows but the first one are isomorphisms modulo RG -torsion (i.e., the kernels and cokernels of these homomorphisms are torsion modules). The five-lemma implies that the first arrow is also an isomorphism modulo torsion. ∗ For the relative equivariant cohomology, HG (M, Z), the result follows immediately from the long exact sequence for the pair (M, Z). 4. Formality In this section we introduce the notion of formality and prove some properties of formal G-manifolds, to be used later in Appendix G.

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4.1. Definitions of formality. Throughout this section we consider only cohomology with real coefficients. As above, we denote H ∗ (BG) by RG .

∗ Definition C.22. The pair (M, Z) is formal if HG (M, Z) is isomorphic to H (M, Z) ⊗ RG as an RG -module: ∗

(C.10)

∗ HG (M, Z) ' H ∗ (M, Z) ⊗ RG

Thus, the RG -module action appears “trivial” to

over RG .

∗ structure on HG (M, Z) does not see the G-action; ∗ it. In particular, HG (M, Z) has no RG -torsion.

the

Lemma C.23. Let M be a G-manifold, let U and V be invariant open subsets of M such that M = U ∪ V , or such that U and V satisfy the hypothesis of Remark C.19. Let Z = U ∩ V . Suppose that the pairs (U, Z) and (V, Z) are formal. Then so is the pair (M, Z). Proof. Consider the following segment of the Mayer–Vietoris exact sequence ∗−1 ∗ ∗ ∗ ∗ HG (Z, Z) → HG (M, Z) → HG (U, Z) ⊕ HG (V, Z) → HG (Z, Z).

∗ Since HG (Z, Z) = 0,

∗ ∗ ∗ HG (M, Z) = HG (U, Z) ⊕ HG (V, Z).

If the right-hand side is formal, so is the left. The Serre spectral sequence of the fibration EG ×G M → BG converges to its E∞ -term, which is the graded RG algebra associated with a certain filtration of ∗ ∗ the cohomology HG (M, Z). As an RG -module, E∞ is equal to HG (M, Z). The ∗ E2 -term of this spectral sequence is H (M ) ⊗ RG . Similarly, for a pair (M, Z) of ∗ G-spaces, there is a spectral sequence converging to HG (M, Z) whose E2 -term is ∗ H (M, Z) ⊗ RG . Recall that the En+1 -term of a spectral sequence is the cohomology of the En -term with respect to some differential dn . By definition, a spectral sequence collapses in the E2 -term if the differentials dn vanish for all n ≥ 2. Lemma C.24. The pair (M, Z) is formal if and only if the spectral sequence ∗ (M, Z) = dim H ∗ (M, Z). collapses at the E2 -term or, equivalently, if rkRG HG Proof. If the spectral sequence collapses at the E2 -term, we obviously have ∗ E∞ = E2 . Because E∞ = HG (M, Z) and E2 = H ∗ (M, Z)⊗RG as RG modules, this ∗ implies formality. Formality clearly implies that rkRG HG (M, Z) = dim H ∗ (M, Z). Let us now assume that the spectral sequence does not collapse at the E2 term. Let n be such that the differentials d2 , . . . , dn−1 are all zero and dn 6= 0. Then En = . . . = E2 = H ∗ (M, Z) ⊗ RG . In particular, En is a free RG -module. We claim that rkEn+1 < rkEn . Denote by QG the field of rational functions on g, i.e., the field of fractions of RG . The differential dn induces a differential on En ⊗RG QG , which is again non-zero because En is a free RG -module. It follows that dimQG H ∗ (En ⊗RG QG ) < dimQG En ⊗RG QG .

By the universal coefficients formula,

H ∗ (En ⊗RG QG ) = H ∗ (En ) ⊗RG QG = En+1 ⊗RG QG .

By definition, rkEi = dimQG Ei ⊗RG QG . Therefore, rkEn+1 < rkEn .

4. FORMALITY

Finally,

207

∗ rkHG (M, Z) = rkE∞ ≤ rkEn+1 < rkEn = dim H ∗ (M, Z).

Among the most interesting examples of formal G-manifolds are compact symplectic manifolds with Hamiltonian G-actions. (In fact, such manifolds are equivariantly perfect, which is a geometric property that implies formality, [Kir].) We will see other examples of formal pairs in Appendix G. The following result gives a useful criterion for formality: Proposition C.25. Let G be a compact Lie group. A compact G-manifold M ∗ is formal if and only if HG (M ) is torsion–free as an RG -module. When G is a torus, this fact is an easy consequence of Borel’s localization theorem (see Section 4.2). We defer the proof of the general case to Section 5.3. To check formality, it suffices to only consider torus actions: Proposition C.26. Let G be a compact Lie group and T its maximal torus. A compact G-manifold M is formal if and only if M is formal as a T-manifold. We will prove this result in Section 5.3. 4.2. Formality for torus actions. Combining Lemma C.24 with Borel’s localization theorem, we obtain the following result: Corollary C.27. Let G be a torus. A pair of G-manifolds (M, Z) is formal if and only if one of the following conditions holds: • dim H ∗ (M, Z) = dim H ∗ (M G , Z G ); ∗ (M, Z) has no RG -torsion; • HG ∗ ∗ (M G , Z G ) = H ∗ (M G , Z G ) ⊗ RG is a (M, Z) → HG • the restriction j ∗ : HG monomorphism. In general, the geometrical meaning of formality is not completely clear. Let us prove, however, a simple geometric consequence of formality, [GGK3]. Recall that an orbit type stratum in M is called minimal if it is closed. Proposition C.28. Let G be a torus. On a compact formal G-manifold M , every minimal stratum is a connected component of the fixed point set M G . Proof. Let X be a minimal stratum and let H be the connected component of the identity in Gx for x ∈ X. Assume H 6= G. Then the equivariant Thom class τ of the normal bundle to X (see, e.g., Section 7) is a non-zero torsion element ∗ (M ). In fact, τ is annihilated by the image of H ∗ (B(G/H)) → H ∗ (BG). in HG Alternatively, j ∗ τ = 0, because X ∩ M G = ∅. For example, when M is compact symplectic with G acting Hamiltonianly, M is formal and Proposition C.28 applies. In this case, however, to show that a minimal stratum X consists of fixed points, it suffices to observe that X is a compact symplectic manifold and H acts Hamiltonianly on X, so H must have fixed points on X. Remark C.29. The condition stated in Proposition C.28 is not sufficient for formality. For instance, let G be a torus and let M be a formal G-manifold. Identify a neighborhood of a free orbit with G×D where D is a disc. Attach M to G×N , for an interesting compact manifold N , by taking the product of G with the connected

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sum of D and N . This gives a non-formal manifold with the same minimal strata as in M . The following result will be used in Appendix G. Lemma C.30. Let G be a torus and let (M, Z) be a pair of G-spaces. Suppose that the pair is formal, and suppose that Z G = ∅. Then the restriction map is onto.

∗ ∗ HG (M ) → HG (Z)

Proof. The long exact sequence of the pair (M, Z) gives rise to the sequence (C.11)

∗+1 ∗ ∗ HG (M ) → HG (Z) → HG (M, Z),

which is exact in the middle term. Since Z G = ∅, by Borel’s localization theorem, ∗ ∗ HG (Z) is an RG -torsion module. Therefore, its image in HG (M, Z) is also a torsion ∗ ∗ module. By formality, HG (M, Z) is torsion free, and hence the image of HG (Z) is zero. Since (C.11) is exact and the second map in (C.11) is zero, the first map is onto. 5. The relation between H∗G and H∗T 5.1. The splitting principle. Let G be a compact Lie group, T a maximal torus in G, and P → B a principal G-bundle with a compact base B. In this subsection we prove a general result, known as the splitting principle, relating H ∗ (B) and H ∗ (P/T). Denote by π : P/T → B the natural projection. Note that W , the Weyl group of G, acts on P/T in a fiberwise way, and hence on H ∗ (P/T). Theorem C.31 (The splitting principle). The induced homomorphism π ∗ : H ∗ (B) → H ∗ (P/T)

is a monomorphism whose image is exactly H ∗ (P/T)W . Remark C.32. The assertion that π ∗ is a monomorphism (and its image is in H (P/T)W ) is relatively straightforward (see Fact 1 below) and can be proved by entirely elementary means. We will refer to this assertion, which is sufficient for many applications, as the easy part of the splitting principle. The statement that the image is exactly H ∗ (P/T)W is more delicate. Its proof relies heavily on Borel’s calculation of H ∗ (G/T). It will also be clear from the proof that H ∗ (P/T) = H ∗ (B) ⊗ H ∗ (G/T) as ∗ H (B)-modules. The cohomology of the “flag manifold” G/T will be described in the proof of Theorem C.31. ∗

Remark C.33. The references to the original proofs of this theorem and other results of this section can be found in [Bor1]. (See also [DKV] for the approach using equivariant differential forms.) Example C.34. Let G = U(r). Consider the vector bundle E over B associated with P , i.e., P is the unitary frame bundle in E. Then the pull-back π ∗ E over P/T splits into the direct sum of r complex line bundles L1 ⊕ . . . ⊕ Lr . (Hence, the name of the theorem.) Therefore, the total Chern class c(π ∗ E) is the product of Chern classes: c(π ∗ E) = c(L1 ) · . . . · c(Lr ). Combined with the fact that π ∗ is a monomorphism in cohomology, this often allows one to assume in cohomology calculations that a given vector bundle is the direct sum of line bundles. (See, e.g.,

∗ 5. THE RELATION BETWEEN H∗ G AND HT

209

Appendix I where we use the splitting principle to define the Todd classes.) Note also that this version of the splitting principle holds for complex K-equivariant vector bundles over K-manifolds. The formulation of the precise statement and its proof are left to the reader as an exercise. Proof of Theorem C.31. The proof is based on the following two facts. Fact 1. Let π : Q → B be a fiber bundle with fiber F such that the restriction H ∗ (Q) → H ∗ (F ) is an epimorphism. Then H ∗ (Q) = H ∗ (B) ⊗ H ∗ (F ) as an H ∗ (B)-module. This observation, sometimes called the Leray–Hirsch theorem, is an immediate consequence of the Serre spectral sequence of π. Alternatively, it can be proved by using the Mayer–Vietoris exact sequence for the restriction of π to some nice cover of B (see, e.g., [Hus, Chapter 17]). Note that the easy part (monomorphism) of the splitting principle readily follows from this fact. Fact 2. H ∗ (G/T) = S ∗ t∗ /(S ∗>0 t∗ )W , where t is the Lie algebra of T. More explicitly, consider the complex vector bundle over G/T associated with the principal T-bundle G → G/T and the standard representation of T ∼ = (S 1 )r on Cr . This vector bundle decomposes as the sum of complex line bundles L1 , . . . , Lr . Denote by t1 , . . . , tr the first Chern classes of these line bundles. Then H ∗ (G/T) = R[t1 , . . . , tr ]/(Rdeg>0 [t1 , . . . , tr ])W , where Rdeg>0 [t1 , . . . , tr ] stands for space of polynomials of positive degree. This fact is proved by a careful analysis of the Serre spectral sequence of the fiber bundle BT → BG whose fiber is G/T; see [Bor1]. To prove the theorem, it suffices to note that the vector bundle associated with the principal T-bundle P → P/T splits into the sum of line bundles whose restrictions to the fiber are exactly the bundles L1 , . . . , Lr . Thus H ∗ (P/T) → H ∗ (G/T) is an epimorphism, and hence H ∗ (P/T) = H ∗ (B) ⊗ H ∗ (G/T). It follows that π ∗ is a monomorphism and, since H ∗>0 (G/T)W = 0, the image of π ∗ is exactly H ∗ (P/T)W . 5.2. The coefficient ring: H∗ (BG) = (S∗ (g∗ ))G . As an immediate application of the splitting principle we obtain the identification (C.6), i.e., H ∗ (BG) = (S ∗ (g∗ ))G , which is the first step in the proof of the equivariant de Rham theorem (Theorem C.4). Indeed, let B = BG and P = EG. Then we have a natural identification EG/T = BT, and π ∗ : H ∗ (BG) → H ∗ (BT)W = (S ∗ t∗ )W is an isomorphism by the splitting principle. Finally, by Proposition C.12, (S ∗ t∗ )W is naturally isomorphic to (S ∗ g∗ )G . Alternatively, (C.6) can be described more geometrically using the Chern–Weil construction. Then the proof that the resulting map is an isomorphism uses only the easy part of the splitting principle. Indeed, the Chern–Weil construction (see, e.g., [BT2, KN]) gives rise to an algebra homomorphism I : (S ∗ g∗ )G → H ∗ (BG).

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Let us show that this homomorphism is in fact an isomorphism. It is not hard to see that the following diagram is commutative: I

(S ∗ g∗ )G −−−−→ H ∗ (BG)     π∗ y y

(S ∗ t∗ )W −−−−→ H ∗ (BT)W Here the horizontal arrows are given by the Chern–Weil construction. The first vertical arrow is the restriction map from Proposition C.12 and π ∗ is the splitting homomorphism. It is clear that the bottom horizontal arrow is an isomorphism. By the easy part of the splitting principle, π ∗ is a monomorphism and by Proposition C.12 the first vertical arrow is an isomorphism. It readily follows from the diagram that I is an isomorphism. Finally, the identification (C.6) can be proved directly by analyzing the spectral sequence of the fibration EG → BG. Namely, one can show that H ∗ (G) has generators which are transgressive and that their differentials generate (S ∗ g∗ )G . 5.3. H∗G (M) and H∗T (M). Let, as above, G be a compact Lie group and T ⊂ G a maximal torus. In this section we recall how the cohomology groups ∗ (M ) and HT∗ (M ), of a G-manifold M , are related to each other. HG ∗ The inclusion T → G gives rise to a homomorphism HG (M ) → HT∗ (M ) induced by the mapping (EG × M )/T → (EG × M )/G, where on the left-hand side we can take EG as ET. Note that, as before, the Weyl group W of G acts on HT∗ (M ). For example, when M is a point we have ∗ (pt) = H ∗ (BG) = (S ∗ t∗ )W ,→ S ∗ t∗ as discussed HT∗ (pt) = H ∗ (BT) = S ∗ t∗ and HG in Section 5.2. Theorem C.35. Let M be a compact G-manifold. ∗ (M ) ∼ (1) HG = HT∗ (M )W as algebras. ∗ ∗ ∼ (2) HT (M ) = HG (M ) ⊗RG RT as RT -modules.

Proof. The first assertion is just the statement of Theorem C.31 for the principle G-bundle P = EG × M → (EG × M )/G = B. Strictly speaking, Theorem C.31 does not apply to this bundle because the base B is not compact or even finite–dimensional. However, B becomes compact when EG is replaced by its finite–dimensional approximation EGk as in Example C.1. Moreover, this substitute has no effect on the qth cohomology if k is large enough. Now the first assertion does follow from Theorem C.31. To prove the second assertion, consider the commutative diagram ρT

EG ×T M −−−−→ EG/T   πy ρG

BT   y

EG ×G M −−−−→ EG/G BG in which the horizontal arrows are given by the projections to the first component and the vertical arrows are associated bundles with fiber G/T. The structure of the RT -module on HT∗ (M ) is defined by ρ∗T and the structure of the RG -module on ∗ HG (M ) comes from ρ∗G . Therefore, the RT -linear homomorphism ∗ ϕ : HG (M ) ⊗RG RT → HT∗ (M )

6. EQUIVARIANT VECTOR BUNDLES

211

which sends v ⊗u to (ρ∗T u)(π ∗ v) is well defined. Denote by u1 , . . . , ur the generators of H ∗ (BT) which are the first Chern classes of the corresponding line bundles. Then ρ∗T ui restricts to the class ti on the fiber G/T of π for all i = 1, . . . , r. It follows from the proof of Theorem C.31 that ϕ is an isomorphism of graded vector spaces. As a consequence, ϕ is also an isomorphism of RT -modules. Remark C.36. The second assertion of Theorem C.35 does not follow directly from the Serre spectral sequence for the fibration π because this spectral sequence might, in principle, lose some information about the multiplication on HT∗ (M ) and, in particular, about the RT -module structure on HT∗ (M ). A different topological proof of the second assertion can be found in [Hs, Section III.1] and an analytical proof in [DKV, page 160]. ∗ Theorem C.35 is very useful to reduce the analysis of HG (M ) to the analysis of which is usually simpler. Let us illustrate this by applying Theorem C.35 to prove Propositions C.25 and C.26. More examples will follow in Section 8.

HT∗ (M )

Proof of Proposition C.26. Assume that M is G-formal. Then, by defini∗ tion, HG (M ) is the free RG -module H ∗ (M ) ⊗ RG . By the second part of Theorem C.35, HT∗ (M ) = (H ∗ (M ) ⊗ RG ) ⊗RG RT = H ∗ (M ) ⊗ RT . Hence, M is T-formal. Conversely, assume that M is T-formal. Then HT∗ (M ) = H ∗ (M ) ⊗ RT and the Leray spectral sequence collapses. The W -action on HT∗ (M ) induces a W -action on H ∗ (M ) ⊗ RT which is trivial on the first term and equal to the standard action of W on RT on the second. Hence, by the first part of Theorem C.35, and M is G-formal.

∗ ∗ HG (M ) = H ∗ (M ) ⊗ RW T = H (M ) ⊗ RG ,

Proof of Proposition C.25. Assume that M is formal. Then, by defini∗ tion, HG (M ) is a free RG -module and, in particular, a torsion–free RG -module. ∗ Conversely, assume that HG (M ) is a torsion–free RG -module. By Proposition C.26 it suffices to show that M is T-formal, where T is a maximal torus in G. This, in turn, is equivalent to that HT∗ (M ) is torsion–free as an RT -module by Corollary C.27. Recall that RT is a free RG -module; see [Va, Theorem 4.15.28]. Applying the second part of Theorem C.35, we conclude that HT∗ (M ) is torsion–free as an RG ∗ module, since it is a tensor product of the torsion–free RG -module HG (M ) and the free RG -module RT . Let us show that HT∗ (M ) is torsion–free as an RT -module. Let u ∈ HT∗ (M ) and f ∈ R QT be such that f 6= 0 and f · u = 0. Our goal is to show that u = 0. Set F = γ∈W γ(f ). Clearly, F ∈ RG = RW T and F 6= 0, since RT is the ring of polynomials on t. Furthermore, F · u = 0 and hence u is a torsion element in HT∗ (M ) over RG . As we have shown, HT∗ (M ) is torsion–free over RG . Thus u = 0 as required. 6. Equivariant vector bundles and characteristic classes In this section we introduce equivariant vector bundles and characteristic classes and “classify” equivariant complex line bundles. We refer the reader to, e.g., [Hus, MiSt], for the construction of ordinary (non-equivariant) characteristic

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classes. Throughout this section, G is a compact Lie group and M is a connected G-manifold. 6.1. Equivariant vector bundles. Definition C.37. A G-equivariant vector bundle over M is a vector bundle E together with a lift of the G-action to E by fiberwise linear transformations. Example C.38. Let V be a linear representation of G. The vector bundle E = M × V with the diagonal G-action is a G-equivariant vector bundle. It is a trivial G-equivariant vector bundle if the representation of G on V is trivial. When V is a non-trivial representation, E is trivial as an ordinary vector bundle, but not as an equivariant vector bundle. Example C.39. The tangent bundle T M and any tensor bundle associated with T M , e.g., T ∗ M , are naturally G-equivariant vector bundles over M . Example C.40. The pre-quantization line bundle L for a Hamiltonian G-action on M is a G-equivariant complex line bundle over M . (See Section 2 of Chapter 6.) 6.2. Equivariant characteristic classes. A G-equivariant vector bundle E ˜ on EG ×G M whose pull-back to EG × M is on M gives rise to a vector bundle E EG × E. The equivariant characteristic classes of E are, by definition, the charac˜ These characteristic classes lie in H ∗ (M ). This construction teristic classes of E. G applies to essentially any characteristic class. In particular, one has the equivariant Pontrjagin classes pG k (E) when E is any equivariant real vector bundle, the equivariant Euler class eG (E) when E is an oriented equivariant real vector bundle, and the equivariant Chern classes cG k (E) and the Todd class when E is an equivariant complex vector bundle. A slightly different construction of equivariant Chern classes, based on the splitting principle, is outlined in Section 2 of Appendix I. As follows from the definition, equivariant characteristic classes have many of the properties of ordinary characteristic classes, such as the product formula. Example C.41. Assume that the G-action on M is trivial. Let L be a Gequivariant complex line bundle over M . (Note that the action of G on L may ∗ (M ) = H ∗ (M ) ⊗ RG , as pointed out in Example C.6. still be non-trivial.) Then HG Recall that the curvature class of a line bundle is essentially equal to the cohomology class of the curvature form on the base; this class differs from the first Chern class by a factor, see Appendix A. The equivariant curvature class of L is ∗ [wG ](L) = [w](L) − α ∈ HG (M ) = H ∗ (M ) ⊗ RG ,

where [ω](L) is the curvature class of L and α ∈ (g∗ )G is the weight of the Grepresentation on a fiber of L, and the equivariant Chern class of L is 1 α. 2π (See Section 2 of Appendix I and, in particular, Lemma I.3 for further details on this example.) cG 1 (L) = c1 (L) −

Example C.42. Let L be a G-equivariant complex line bundle over M . Let p : P → M be the unit circle bundle in L with respect to some G-invariant fiberwise Hermitian metric. Pick a G-invariant connection form Θ on P . Then cG 1 (L) =

6. EQUIVARIANT VECTOR BUNDLES

213

1 2π [ω

2 − Φ] in HG (M ), where ω is the curvature form on M , i.e., p∗ ω = −dΘ and the moment map is defined by the condition π ∗ Φ = Θ(ξP ). In other words, 1 ∗ (C.12) p (ω − Φ) = −dG Θ. 2π (For more details on these examples, see Section 2.4 of Chapter 6 and Appendix A.) This definition of cG 1 is a simple example of the equivariant Chern–Weil construction which can also be carried out to define other equivariant characteristic classes; see [BT2, BV1, BV2] and [BGV].

Example C.43. For a G-manifold M , the equivariant Euler class eG (M ) and the equivariant Pontrjagin classes pG k (M ) are, by definition, the equivariant Euler and Pontrjagin classes of T M . Similarly, if M carries a G-invariant almost complex structure or a G-equivariant stable complex structure, one has the equivariant Chern classes cG k (M ). Example C.44. The normal bundle V to the fixed point set M G is a Gequivariant vector bundle. Thus the equivariant characteristic classes of V are ∗ defined. Since the G-action on M G is trivial, these classes belong to HG (M ) ⊗ RG , where, as above, RG is the space of G-invariant polynomials on g. The Euler class eG (V) will play a particularly important role in Section 7. Let us assume that G is a torus and obtain an expression for the leading term of this class as a polynomial on g. To this end, we will restrict our attention to one connected component F of M G . Its normal bundle VF can be turned into a complex 1 G-equivariant vector bundle. Indeed, with the L choose a subcircle S ⊆ G that acts same fixed points as G. Let VF = Vm be the splitting such that √ S 1 acts on Vm with stabilizer {e2πik/m | 0 ≤ k < m}. Define the multiplication by −1 to be Jv = e2πi/4m · v

for v ∈ Vm . Let α1 , . . . , αk , be the weights of the G-representation on a fiber of VF . Then (C.13)

eG (VF ) = (−1)k

k Y

αi + . . .

i=1

where the dots stand for lower degree terms. This formula follows from Example C.41 and the splitting principle and is used in Section 7. Since F is a component of M G , all weights αi are non-zero and, as a consequence, so is the leading term. In particular, eG (VF ) is a polynomial of degree codim F/2. Note in conclusion that whereas the weights αi depend on the choice of complex structure, their product depends only on the choice of orientation. Indeed, as in the non-equivariant case, the Euler class is defined for an oriented vector bundle. Also note that to determine the leading term of eG (V) it suffices to consider the representation of G on a single fiber. Remark C.45. In the definition of equivariant characteristic classes we used the fact that a G-equivariant vector bundle E over M gives rise to a vector bundle ˜ over EG ×G M . This construction leads to the question of whether or not the E ˜ is a bijection between the sets of isomorphism classes of correspondence E 7→ E equivariant vector bundles on M and ordinary vector bundles over EG×G M , which we denote by VectG (M ) and Vect(EG ×G M ), respectively. This turns out to be a

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very subtle question even when M = pt. For example, recall that VectG (pt) is the set of isomorphism classes of representations of G and Vect(EG×G pt) = Vect(BG). When G is a torus, VectG (pt) → Vect(BG) is a bijection. Furthermore, when G is connected, this map induces an isomorphism of the corresponding Grothendieck groups. If G is not connected, the map VectG (pt) → Vect(BG) may fail to be a bijection. The reader interested in the references to these and other related results should consult the survey [Ol] and also [JMcCO]. The situation considerably simplifies (at least for continuous G-actions) for complex line bundles or more generally for vector bundles with structural group U(1)n . Namely, in this case there is a one-to-one correspondence between (continuous) G-equivariant bundles and bundles over EG ×G M ; see [HY]. 6.3. Equivariant complex line bundles. In this section we classify Gequivariant line bundles. Recall that, as is well known, ordinary complex line bundles are classified by the first Chern class: Proposition C.46. The first Chern class establishes a one-to-one correspondence between the collection of isomorphism classes of complex line bundles on M and H 2 (M ; Z). Proof. Fix a cover Ui of M by contractible open sets. A complex line bundle ˇ L → M trivializes over each Ui . Its transition functions form a Cech one-cocycle × ϕij : Ui ∩ Uj → C . Two cocycles, ϕij and ψij , correspond to isomorphic line bundles if and only if they are in the same cohomology class: ϕij = fi ψij fj−1 ˇ 1 (M ; C× ). This for some fj : Uj → C× . Hence, line bundles are classified by H 2 group is isomorphic to H (M ; 2πZ). This follows from the long exact sequence in exp cohomology coming from the short exact sequence of sheaves 0 → 2πZ → C → C× → 1, combined with the fact that the sheaf of smooth C-valued functions is flabby. Dividing by 2π, we conclude that the line bundles L are classified by the resulting elements c1 (L) ∈ H 2 (M ; Z). For more details we refer the reader to, e.g., [Ki4, Kost]. Alternatively, the proposition is a consequence of the identification K(Z, 2) = CP∞ = B U(1). This correspondence between line bundles and integral cohomology classes was used, for example, in Section 2.2 of Chapter 6 to determine which closed two-forms admit a pre-quantization. Our goal now is to extend this result to equivariant line bundles. This result is of interest for us for two reasons: it gives a necessary and sufficient condition for a Hamiltonian G-manifold to be pre-quantizable (Theorem 6.7), and it is also used in the classification of equivariant SpinC -structures (Section 2 of Appendix D). Theorem C.47. Let G be a compact Lie group and M a connected G-manifold. The equivariant first Chern class cG 1 gives rise to a one-to-one correspondence between equivalence classes of G-equivariant complex line bundles over M and ele2 ments of HG (M ; Z). Remark C.48. For locally finite CW-complexes with continuous G-actions, this result is proved by Hattori and Yoshida, [HY]. For smooth actions of connected groups, Theorem C.47 is due to Riera, [Ri]. The proof given below is different from either of these proofs.

6. EQUIVARIANT VECTOR BUNDLES

215

Remark C.49. Let Γ be the set of isomorphism classes of G-equivariant complex line bundles over M . It is easy to see that Γ is a group with multiplication 2 given by tensor product of line bundles, and that cG 1 : Γ → HG (M ; Z) is a group homomorphism. According to Theorem C.47, this is in fact an isomorphism. Proof of Theorem C.47. Consider a finite–dimensional approximation EGk → EGk /G = BGk of the universal bundle EG → BG, as in Example C.1, with the following properties: • EGk is a smooth manifold with a free G-action; • EGk is simply connected and H 2 (EGk ; Z) = 0; • the inclusion j : M ×G EGk ,→ EG ×G M induces an isomorphism

2 j ∗ : HG (M ; Z) = H 2 (EG ×G M ; Z) → H 2 (EGk ×G M ; Z).

The actual construction of such an approximation is immaterial for the argument below and we only need the fact that this approximation exists. See Example C.1 for a construction of such an approximation. Recall that to every G-equivariant line bundle F over M one can associate an ordinary line bundle L over EGk ×G M as follows. Let π : EGk × M → M be the natural projection. The pull-back π ∗ F is G-equivariant with respect to the diagonal G-action on EGk × M . Since the diagonal action is free, π ∗ F descends to a line bundle L = F/G over EGk ×G M . In other words, L is characterized by the condition ˜ = π ∗ F, L

(C.14)

˜ is the pull-back of L to EGk × M . where L Lemma C.50. (1) j ∗ cG 1 (F) = c1 (L). (2) The correspondence F 7→ L gives rise to a bijection between G-equivariant complex line bundles F over M and ordinary complex line bundles L over EGk ×G M , up to isomorphism. The theorem follows from the lemma combined with Proposition C.46 and the fact that j ∗ is an isomorphism in degree two. Proof Of The Lemma. Let us prove that j ∗ cG 1 (F) = c1 (L). First note that replacing EGk by EG in the definition of L, we obtain a line bundle L0 over EG ×G M for a G-equivariant line bundle F. (This is exactly the construction described in Section 6.2.) The diagram j

L −−−−→ EGk ×G M ,→EG ×G M ←−−−− L0 x x    

˜ ∗ F −−−−→ EGk × M ,→ EG × M L=π     πy y F

−−−−→

M

=

M

←−−−− F

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0 is commutative, and, hence, L = j ∗ L0 . By definition, cG 1 (F) = c1 (L ) and, therefore, ∗ 0 ∗ 0 j ∗ cG 1 (F) = j c1 (L ) = c1 (j L ) = c1 (L).

Let us prove that the correspondence F 7→ L is surjective. Let L be a complex ˜ the pull-back of L to EGk ×M . line bundle over EGk ×G M . As above, denote by L ˜ By (C.14), we need to find a G-equivariant line bundle F over M such that π ∗ F = L. 2 Since H (EGk ; Z) = 0, every line bundle over EGk is trivial. In particular, the ˜ to every fiber of the projection π is trivial. As a consequence, L ˜ restriction of L admits a fiberwise flat connection, i.e., a connection whose restriction to every fiber of π is flat. Indeed, such a connection exists locally in M , i.e., over sets EGk × U , ˜ EG ×U is trivial. Using a where U are contractible open subsets of M , because L| k partition of unity in M subordinate to a locally finite cover of M by such open sets U , we can patch these connections together into a fiberwise flat connection. Furthermore, the fiberwise connection can be made G-invariant by averaging over the diagonal G-action on EGk ×M . More precisely, let P be the U(1)-principal ˜ and Θ0 the connection form on P for a fiberwise flat connection. Then, bundle for L since the G-action on EGk × M preserves the fibers Rof π, the connections g ∗ Θ0 are also fiberwise flat and so is the connection Θ = G g ∗ Θ0 dg; cf. Remark C.51. (Note that this argument uses the fact that U(1) is commutative.) Furthermore, π1 (EGk ) = 0 implies that the connection Θ is π-fiberwise trivial, i.e, has trivial holonomy, or, in other words, is globally flat. Now let F be the complex line bundle over M whose fiber over x ∈ M is the ˜ over π −1 (x) with respect to Θ. Using the trivialization set of flat sections of L ˜ over EGk × U , it is easy to show that F is really a line bundle. Clearly, π ∗ F = L. Furthermore, F is G-equivariant, since Θ is G-invariant. This completes the proof of surjectivity. ˜ = π∗ F Let us prove that F 7→ L is injective. Assume that L is trivial. Then L admits a G-invariant globally flat connection. Denote by p : P → EGk × M the U(1)-principal bundle and by Θ the corresponding flat G-invariant connection form on P . This connection, however, need not project to M , for Θ may not agree with the connection Θπ along the fibers of π, which π ∗ F acquires as a pull-back bundle. We will modify Θ to obtain a connection that does project to a globally flat connection on M . Note that Θπ is G-invariant and that Θ−Θπ = p∗ α, where α is a closed G-invariant one-form defined along the π-fibers. Since the fibers are simply connected, α is in fact exact on every fiber. Hence, on every fiber, there exists a function f such that df = α. This function is defined up to a constant. Let us then fix a section {q} × M , where q ∈ EGk , and choose the function f on every fiber so that f (q) = 0. These functions fit together to form a smooth function, denoted again by f , on EGk × M , such that dπ f = α, where dπ is Rthe de Rham differential along the fibers of π. Finally, let f¯ be the average f¯ = G g ∗ f dg. Then f¯ is Ginvariant and dπ f¯ = α, for α is also G-invariant. The connection Θ0 = Θ0 − df¯ is G-invariant, globally flat on EGk × M , and such that Θ0 = Θπ on the fibers of π. It remains to show that Θ0 projects to a globally flat connection on M . (Note that the projection connection will automatically be G-invariant if it exists.) For a path γ in M connecting points x and y, consider its arbitrary lift γ˜ to EGk × M , connecting points x˜ and y˜. We have natural identifications (π ∗ F)x˜ = Fx and (π ∗ F)y˜ = Fy . For the projection connection, the parallel transport ϕ from Fx to Fy along γ is then defined as the parallel transport ϕ˜ from (π ∗ F)x˜ to (π ∗ F)y˜ along γ˜.

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Since the connection on EGk × M is globally flat, ϕ˜ is independent of the lift γ˜ as long as x˜ and y˜ are fixed. Furthermore, since Θ0 = Θπ , it follows that ϕ is actually independent of x ˜ ∈ π −1 (x) and y˜ ∈ π −1 (y) and of the path connecting x and y. This establishes injectivity and completes the proof of Lemma C.50 and Theorem C.47. Remark C.51. Let L be a G-equivariant line bundle over M which admits a globally flat connection, i.e., a connection with trivial holonomy. Then L also admits a G-invariant connection which is locally flat, i.e., has zero curvature. The proof is by averaging. Let p : P → M be the U(1)-principal bundle for L and Θ0 a connection form on P with trivial holonomy. For every g ∈ G, the connection form g ∗ Θ0 has zero curvature. Thus g ∗ Θ0 = Θ0 + p∗ αg , where αg is a closed one-form on M . It follows that the G-invariant connection Z Z Θ= g ∗ Θ0 dg = Θ0 + p∗ αg dg G

G

also has zero curvature. (Here the Haar measure dg is normalized so that G has unit volume.) In general, Θ need not be globally flat. However, this is always the case when G is connected. Indeed, the form αg depends smoothly on g and its cohomology class is integral. Hence, for a connected group G, the form αg is exact for all g. As a consequence of the last remark, the proof that F 7→ L is injective can be considerably simplified when G is connected: it suffices first to pull-back a globally flat connection on π ∗ F to a globally flat connection on F by any section M → EGk × M . The resulting pull-back connection can then be made G-invariant, while still kept globally flat, by averaging over G. 7. The Atiyah–Bott–Berline–Vergne localization formula In this section we will prove a localization formula expressing the integral of a closed equivariant form over a G-manifold through integrals over the fixed point set. This formula was obtained in [BV1, BV2] and [AB2]. There are different proofs of the localization formula known today. For example, it can be proved using cobordism techniques. Here, however, we mainly follow the topological approach of Atiyah and Bott, [AB2]. Throughout this section we assume that G is a connected compact Lie group. 7.1. The localization formula. Let β be a compactly supported equivariant differential form of degree q on an oriented G-manifold M . Denote by βk the component of β in (Ωk (M ) ⊗ S ∗ (g∗ ))G . For example, β0 is a smooth function on M taking values in polynomials on g. More generally, interpreting βk as a polynomial valued k-form, we set Z Z β= βn ∈ R G , M

M

where n = dim M . Note that, by (C.3), this integral may be non-zero even when deg β 6= dim M in contrast with ordinary integration of differential forms. It is not hard to see that the equivariant analogue of Stokes’ formula holds: Z Z dG β = β. M

∂M

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Assume now that ∂M = ∅ and, for the sake of simplicity, that M is compact. Then, by Stokes’ formula, the integral over M is a well–defined RG -linear map Z ∗ (C.15) : HG (M ) → RG . M

Note that this map is not an algebra homomorphism.

Example C.52. An equivariant characteristic number of M is the integral over M of an equivariant characteristic class. Note that this “number” is an element of RG . R It is worth noticing that, since the range of M is torsion–free, the integration ∗ map (C.15) kills the torsion part of HG (M ). In a similar fashion, integration gives rise to a QG -linear homomorphism Z ∗ (C.16) : HG (M ) ⊗ QG → QG , M

where QG is the field of G-invariant rational functions on g. The localization formula expresses the integral over M through the integrals over the components of M G , when G is torus. This formula, which does not have a non-equivariant analogue, is yet another incarnation of the general principle that for an abelian G many properties of the equivariant cohomology can be recovered from the fixed point set. (See Section 3.) To state the formula, denote by VF the normal bundle to a connected component F of M G . Fix an orientation of M and a fiberwise orientation of VF ; these determine an orientation of F . By Example C.44 (in particular, (C.13)), the Euler class eG (VF ) is invertible in the algebra of H ∗ (F )-valued rational functions on g, when G is a torus. In contrast, the non-equivariant Euler class is never invertible. Theorem C.53 (A-B-B-V localization theorem, I). Let G be a torus acting on ∗ (M ), compact manifold M . For any class u ∈ HG Z Z X u|F (C.17) , u= G (V ) e F F M F

where u|F is the restriction of u to F .

This formula takes a particularly simple form when the fixed points are isolated: Z X u(p) Q (C.18) , u = (−2π)n αi,p M G p∈M

1 2

where n = dim M and where αi,p are the isotropy weights for the G-action on Tp M (with the normalization convention of Appendix A). Remark C.54. Note that every term on the right-hand side of (C.17) is, in general, an element of QG , while their sum is an element of RG . ∗ The localization theorem holds as stated when u is an element of HG (M )⊗QG . Moreover, one can further extend the polynomial ring. For example, the formula remains correct when in the definition of equivariant cohomology the polynomial ring on g is replaced by formal power series or analytic functions with values in Ω∗ (M ). (See Remark C.14.) Similar formulas hold, of course, for closed equivariant differential forms (rather than cohomology classes) with an appropriate differential form replacing the cohomology class eG (V).

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Example C.55. Let G be a circle and β an equivariantly closed equivariant form. Identifying g with R, we can write β = β1 + . . . + βdim M where βk is a k-form on M with values in the polynomials in an independent variable t. Furthermore, the weights αi then take the form αi = wi t, where wi ∈ Z. Thus (C.18) becomes n X  Z β0 (p)(t) 2π Q . β= − t wi,p M G p∈M

Remark C.56. Recall that equivariant differential forms on M can be thought of as G-invariant polynomials on g with values in Ω∗ (M ). For every ξ ∈ g we have the evaluation homomorphism Ω∗G (M ) → Ω∗ (M ) which sends an equivariant differential form α to an ordinary differential form α(ξ). This homomorphism, in general, does not commute with the differential (if ξ 6= 0) and, hence, does not induce a homomorphism in cohomology. Moreover, the form α(ξ) may not even be closed when α is exact. R However, it is easy to see that M (dG β)(ξ) = 0 for any form β when M is compact. Hence, for a closed form α, we have Z  Z [α] (ξ) = α(ξ). M

M

R For this reason the value of this integral is sometimes written as M [α](ξ); see, e.g., Remark I.6. The evaluation at ξ = 0 gives rise to the forgetful algebra homomorphism ∗ HG (M ) → H ∗ (M ) which is induced by the restriction to the fiber of the fibration EG ×G M → BG. ∗ Recall from Example C.6 that when the G-action on M is trivial, HG (M ) is ∗ canonically identified with H (M ) ⊗ RG . In this case the evaluation at every ξ is ∗ a well–defined homomorphism HG (M ) → H ∗ (M ). In general, when the action is not trivial, the evaluation at ξ 6= 0 as a map in cohomology is not well defined. With this remark in mind we are now in a position to state a slightly more general version of Theorem C.53, which is due to Berline and Vergne, [BV1]. This theorem can be proved similarly to Theorem C.53.

Theorem C.57 (A-B-B-V localization theorem, II). Let α be a closed equivariant differential form on M . For every ξ ∈ g, Z XZ α|F (ξ) α(ξ) = (C.19) , G M F e (VF )(ξ) F

where F runs through the connected components of M ξ = {ξM = 0} and eG (VF ) is an equivariant differential form representing the equivariant Euler class of V F . 7.2. The proof of the localization theorem. Following [AB2], let us first recall some properties of the push-forward homomorphism in ordinary cohomology. Let f: N →M

be a continuous map of compact oriented manifolds of dimensions n and m, respectively. The push-forward f∗ : H ∗ (N ) → H ∗−n+m (M )

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is defined as the composition (C.20)

D −1

f∗

D

M f∗ : H ∗ (N ) →N Hn−∗ (N ) → Hn−∗ (M ) → H ∗+m−n (M ),

where the first and the last arrows are the Poincar´e duality homomorphisms and the middle arrow is the induced map in homology. Since we work with cohomology with real coefficients, f∗ v ∈ H ∗ (M ), for v ∈ H ∗ (N ), is the unique cohomology class such that Z Z (C.21) (f∗ v)x = v(f ∗ x) M

N

for all x ∈ H ∗ (M ). Equivalently, fR∗ is dual to f ∗ with respect to the natural inner product on cohomology: hx, yi = xy.

Example C.58. When M = pt, the push-forward H ∗ (N ) → R is given by the integral over N . Furthermore, if f is a fibration, f∗ is the integration over the fibers of f . Let us now list the properties of the push-forward homomorphism which are important for what follows. Proposition C.59 (The properties of push-forward). (1) The push-forward is functorial: (f g)∗ = f∗ g∗ . (2) The push-forward of f : N → M is linear over H ∗ (M ). In other words, f∗ (vf ∗ (u)) = f∗ (v)u for any u ∈ H ∗ (M ) and v ∈ H ∗ (N ). (3) Let j : N ,→ M be an embedding. Denote by E → N a tubular neighborhood of N and let k = codim N . Then j∗ factors into the following sequence of homomorphisms: Th

H ∗−k (N ) → Hc∗ (E) → Hc∗ (M ) = H ∗ (M ), where the first arrow Th is the Thom isomorphism (see [BT1]). (4) In the setting of the previous statement, j ∗ j∗ (v) = v e(VN ), where VN is the normal bundle to N in M . Let us give some hints on the proofs of these facts. The first property is an immediate consequence of the definition (C.20). The characterization of f∗ v given by (C.21) implies the second property. The third property follows from the fact that the image in H ∗ (M ) of the Thom class Th(1) of VN is Poincar´e dual to N and that Th(v) = Th(1)π ∗ v, where π is the projection E → N . (Note also that the third assertion is equivalent to that j∗ can be factored as Th

H ∗−k (N ) → H ∗ (E, ErN ) → H ∗ (M, M rN ) → H ∗ (M ), where the second arrow is the excision homomorphism and the third one is induced by the inclusion.) The fourth assertion is a consequence of the third one and the definition of the Euler class: j ∗ Th(1) = e(VN ). We leave detailed proofs to the reader as an exercise. Alternatively, the proofs can be found, for example, in [Dol2, Chapter 8]. Remark C.60. As a side remark, let us point out the following useful application of the push-forward. Let N and M be compact manifolds of equal dimensions and f : N → M a continuous mapping of non-zero degree. Then

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f ∗ : H ∗ (M ) → H ∗ (N ) is a monomorphism. Indeed, by the second property of the push-forward, f∗ f ∗ u = f∗ (1)u = (deg f )u. Thus, (deg f )−1 f∗ is a left inverse of f ∗ . For example, let N and M be closed surfaces of genera k and l, respectively. Then a non-zero degree map N → M exists if and only if k ≥ l. (If N or M is not orientable, one should use the push-forward in cohomology with Z2 -coefficients.) The push-forward is also defined in the equivariant setting. Namely, a Gequivariant mapping f : N → M of compact oriented G-manifolds gives rise to the push-forward RG -linear homomorphism ∗−n+m ∗ f ∗ : HG (N ) → HG (M ).

This homomorphism has the properties 1–4 of Proposition C.59, with all the cohomology spaces and classes replaced by their equivariant counterparts. To define the equivariant push-forward, the ordinary push-forward construction cannot be applied to the map EG ×G N → EG ×G M , for these spaces are infinite– dimensional and the Poincar´e duality homomorphism is not defined. However, the Poincar´e duality is defined when the space EG is replaced by its finite–dimensional approximation EGk as in Example C.1. Thus, for every q we have the pushforward homomorphism H q (EGk ×G N ) → H q−n+m (EGk ×G N ). When k is large enough these cohomology spaces are equal to the equivariant cohomology of N and M , respectively, and the push-forward is independent of k. The resulting homomorphism is called the push-forward in the equivariant cohomology. The equivariant version of Proposition C.59 is now relatively straightforward to verify. R Example C.61. Let π : N → pt be the constant map. Then π∗ = N . Furthermore, π∗ can be understood as the integration over the fibers of the fibration EG ×G N → BG. It is also clear that the push-forward extends to a QG -linear homomorphism ∗ ∗ HG (N ) ⊗ QG → HG (M ) ⊗ QG . Thus, to prove the localization theorem, we only need to show that  F ∗  X (j ) u π∗M u = π∗F (C.22) , eG (VF ) F

M

F

where π : M → pt and π : F → pt are constant maps and j F : F ,→ M is the natural inclusion. Consider the embedding j : M G ,→ M. Lemma C.62. The push-forward ∗ ∗ j ∗ : HG (M G ) ⊗ QG → HG (M ) ⊗ QG P F ∗ is an isomorphism, and its inverse is u 7→ (j ) u/eG (VF ). In particular, for ∗ every u ∈ HG (M ), X (j F )∗ u (C.23) j∗ ≡ u mod RG -torsion. eG (VF ) F

Remark C.63. This lemma does not have a non-equivariant analogue because the Euler class in the ordinary cohomology is never invertible.

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Assuming the lemma, let us prove the localization theorem in the form (C.22). P G G Note that π∗M j∗ = π∗M and π∗M = F π∗F . Then, by the lemma, π∗M u = π∗M j∗

X (j F )∗ u eG (VF ) F

F ∗ G X (j ) u = π∗M G e (VF ) F   X (j F )∗ u π∗F . = eG (VF ) F

Proof of Lemma C.62. Since, by Borel’s localization, j ∗ is an isomorphism modulo torsion, to prove (C.23), it suffices to show that X (j F )∗ u = j ∗ u. j ∗ j∗ eG (VF ) F

Furthermore, by the the fourth property of the push-forward, (j F )∗ j∗F v = v eG (VF ). Thus,  F ∗  X (j F )∗ u X (j ) u F ∗ F j ∗ j∗ = (j ) j ∗ eG (VF ) eG (VF ) F

F

X (j F )∗ u eG (VF ) = eG (VF ) F

= j ∗ u.

This proves (C.23). P F ∗ By (C.23), the homomorphism u 7→ (j ) u/eG (VF ) is a right inverse of ∗ G ∗ j∗ : HG (M ) ⊗ QG → HG (M ) ⊗ QG. By Borel’s localization, the QG -vector spaces ∗ ∗ HG (M G ) ⊗ QG and HG (M ) ⊗ QG have equal dimensions. It follows that j∗ is an isomorphism. Remark C.64. The reader interested in a refinement of this argument, giving ∗ some information on the torsion of HG (M ), should consult, e.g., [AB2, Section 5]. 8. Applications of the Atiyah–Bott–Berline–Vergne localization formula 8.1. The Duistermaat–Heckman formula and the relations among characteristic numbers. The localization theorem has numerous applications. One of them is a short proof of the Duistermaat–Heckman formula (see Section 6 of Chapter 4) originally obtained in [DH1] by a different method. In its simplest form, when the fixed points are isolated, the Duistermaat–Heckman formula reads Z X ehΦ(p),ξi 1 Q (C.24) ehΦ,ξi ω n = (2π)n hαi,p , ξi n! M G p∈M

for any equivariantly closed equivariant two-form ω − Φ and any vector ξ ∈ g such that {ξM = 0} = M G , where αi,p are the isotropy weights at p. The Duistermaat– Heckman formula (C.24) follows from the integral localization theorem by applying (C.18) to u = e−(ω−Φ) and plugging in ξ ∈ g. The condition {ξM = 0} = M G

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guarantees that the denominators on the right-hand side are non-zero. (For more details see [BV1, BV2, AB2].) Another interesting class of results is obtained by applying the localization formula to the characteristic classes of M . This yields relations among the equivariant characteristic numbers of M (i.e., the integrals of the characteristic classes) and the mixed equivariant characteristic classes of the fixed point data (M G , V). For instance, by applying the localization formula to u = 1, we already obtain a non-trivial relation which is particularly simple when the fixed points are isolated: X 1 Q = 0. αi,p G p∈M

In particular, if G is the circle group this turns into the relation X 1 Q (C.25) =0 wi,p G p∈M

on the integers ωi,p ∈ Z, in the notation of Example C.55.

Remark C.65. The relation (C.25) can be used to give a simple proof of a theorem of McDuff, [McD2], that a symplectic S 1 -action on a compact symplectic four-manifold is Hamiltonian if and only if it has fixed points. (See [Gin3] for a detailed argument.) Remark C.66. There are other relations among the isotropy weights αi,p . For example, for an almost complex S 1 -manifold with isolated fixed points, X Y wi,p = 0 p∈M G

as shown in [Hat2] by using the localization theorem in K-theory. It would be interesting to find explicitly all independent relations among weights αi,p that hold for all (e.g., almost-complex) manifolds of a given dimension (cf. [GZ2]). 8.2. The equivariant Poincar´ e duality. Since the space EG ×G M is infinite dimensional, it is not clear how to extend to this space the Poincar´e duality theorem relating homology and cohomology in complementary dimensions. There is however a different version of the Poincar´e duality which does extend to equivari∗ ant cohomology HG with real coefficients in a nearly trivial way: according to this version of the Poincar´e duality (in its non-equivariant form), the standard pairing on real cohomology given by the integration over the manifold is non-degenerate. ∗ In a similar fashion, the equivariant cohomology HG ⊗ QG carries a non-degenerate QG -bilinear paring. (Here, as above, QG is the field of fractions for RG .) Let G be a torus acting on a compact oriented manifold M . Recall from Section 7 (see, in particular, (C.15) and (C.16)) that the integration over M , or equivalently by the map M → pt, gives rise to a QG -linear homomorphism Rthe push-forward ∗ ∗ (M ) ⊗ QG , we set : H (M ) ⊗ Q → QG . For u and v in HG G G M Z hu, viM = uv ∈ QG . M

∗ (M ) ⊗ QG over QG . This is a QG -bilinear pairing on the vector space HG

Proposition C.67 (Equivariant Poincar´e duality). The pairing h, iM is nondegenerate.

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∗ Remark C.68. Recall that HG (M ) ⊗ QG is a finite–dimensional vector space over QG . Thus, the non-degeneracy condition is understood in the standard sense as for finite–dimensional vector spaces. More explicitly this means that for every ∗ ∗ QG -bilinear map φ : HG (M ) ⊗ QG → QG there exists a unique v ∈ HG (M ) ⊗ QG such that φ(x) = hv, xi for all x. In particular, u = 0 if and only if hu, xi = 0 for all x. ∗ Proof of Proposition C.67. We have HG (M G ) ⊗ QG = H ∗ (M G ) ⊗ QG , G because the G-action on M is trivial. Hence, for M G , the theorem follows from the ordinary Poincar´e duality. We will derive the equivariant Poincar´e duality for M from that for M G . Let ∗ φ : HG (M ) ⊗ QG → QG be QG -linear. Consider the QG -linear map ∗ φ ◦ j ∗ : HG (M G ) ⊗ QG → QG ,

where, as in the previous section,

j : MG → M

is the inclusion map. Since the equivariant Poincar´e duality holds for M G , there exists a cohomology ∗ (M G ) ⊗ QG such that class v ∈ HG φ ◦ j∗ (x) = hv, xiM G

∗ (M G ) ⊗ QG . Then for all x ∈ HG



 hv, xiM G = j ∗ (j ∗ )−1 v, x M G = (j ∗ )−1 v, j∗ x M .

(Note that j ∗ is an algebra isomorphism by the Borel localization theorem.) In other words, for y = j∗ x, (C.26)

φ(y) = hu, yiM , where u = (j ∗ )−1 v.

Since j∗ is an isomorphism (over QG ), the identity (C.26) holds for all y. The ∗ uniqueness is a consequence of the fact that HG (M ) ⊗ QG is finite–dimensional over QG .

∗ The pairing h, iM is, of course, also defined on HG (M ) and is RG -bilinear on ∗ this space. This pairing ignores the torsion of HG (M ).2 It is not clear if an RG ∗ linear map φ : HG (M ) → RG can always be represented as the pairing with some ∗ u ∈ HG (M ).

8.3. Generalizations to non-abelian groups. Let G be a compact Lie group which is not necessarily abelian. In this case, the fixed point set M G can ∗ carry rather little information about HG (M ), and Borel’s localization theorem and its consequences, taken literally, do not hold for G-actions. Example C.69. Let M = G/T, where T is a maximal torus. Then M G = ∅. ∗ However, HG (M ) = HT∗ (pt) = H ∗ (BT) = RT . This is a torsion–free module over W RG = RT , where W is the Weyl group. (In fact, RT is a free module over RG ; see, ∗ e.g., [Va, Theorem 4.15.28].) Thus, in this example HG (M G ) gives no information ∗ on HG (M ). 2 We emphasize that here, as almost everywhere in this appendix, the torsion is understood over RG , not over Z.

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However, Borel’s localization theorem and the Atiyah–Bott–Berline–Vergne localization extend to G-actions when M G is replaced by a larger set. Let M max be the set of points in M whose stabilizers have maximal rank: M max = {x ∈ M | rkGx = rkG}. Clearly, M max is invariant under the G-action. Note that M max may fail to be a smooth submanifold of M . Theorem C.70. Let G be a compact group acting on a compact manifold M . Borel’s localization theorem (Theorem C.20) holds for M with M G replaced by M max . If, in addition, M max is smooth and M and M max are orientable, the Poincar´e duality (Proposition C.67) also holds for M and the Atiyah–Bott–Berline– Vergne localization formula (C.17) remains valid with M G replaced by M max . Remark C.71. The first assertion of this theorem is a very particular case of [Hs, Theorems III.1 and III.10 ]. It is also worthwhile to notice that Theorem C.70 is not what is usually referred to as non-abelian localization in symplectic geometry. Proof of Theorem C.70. Let T ⊂ G be a maximal torus. Because M max contains the fixed point set M T , the restriction of HT∗ (M ) to M T factors as the composition HT∗ (M ) → HT∗ (M max ) → HT∗ (M T ). It follows from Borel’s localization (over T) that HT∗ (M ) → HT∗ (M max ) is an isomorphism modulo RT -torsion. ∗ with (HT∗ )W and RG with By the first part of Theorem C.35, let us identify HG ∗ W (M ) is exactly RT . Furthermore, observe that the RG -torsion submodule of HG ∗ equal to the intersection of the RT -torsion submodule of HT (M ) with HT∗ (M )W . It follows that ∗ ∗ HG (M ) = HT∗ (M )W → HT∗ (M max )W = HG (M max )

is a monomorphism modulo RG -torsion. Let us show that this map is in fact an epimorphism modulo RG -torsion. Assume that u is not in the image of this map. We need to show that F · u is in the image for some F ∈ RG . Since as we have pointed out above HT∗ (M ) → HT∗ (M max ) is an isomorphism modulo torsion, f · u is in the image of HT∗ (M ) for some f ∈ RT . Because this image is an RT -submodule, the element Y F · u, where F = γ(f ) ∈ RG , γ∈W

is also in the image, i.e., F · u is the image of some x ∈ HT∗ (M ). P Note that F · u is W -invariant. Hence, F · u is the image of the cohomology class γ∈W γ(x)/|W | ∗ which is W -invariant and hence an element of HG (M ). ∗ W ∗ The identifications HG = (HT ) and, more generally, the W -action commute with push-forwards. Poincar´e duality for G follows from Proposition C.67 and its proof. We leave the details to the reader as an exercise. The Atiyah–Bott–Berline–Vergne localization formula is a formal consequence of the properties of push-forward, Borel’s localization theorem, and the fact that the Euler class of the normal bundle to the fixed point set is invertible over QG . Push–forwards for G-actions have the same properties as for torus actions. Borel’s localization theorem has just been extended to G-actions with M max taken in place of M G .

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∗ It remains to prove that eG (VM max ) is invertible in HG (M ) ⊗ RG . Let F be a max connected component of M . Then, over the field of rational functions,

(C.27)

eG (VF ) = eT (VF ) = eT (VF )|F T

under the sequence of identifications ∼ =

∼ =

∗ HG (F ) ⊗ QG → (HT∗ (F ) ⊗ QT )W → (HT∗ (F T ) ⊗ QT )W .

(Note that W can act non-trivially on both terms in HT∗ (F T ) ⊗ QT .) Furthermore, eT (VF )|F T is still given by (C.13), where the product is over all T-irreducible components of VF |F T . Since none of these components is a trivial representation, the leading term is non-zero and eT (VF )|F T is invertible as a rational function. By (C.27), the class eG (VF ) is also invertible. This completes the proof of the Theorem C.70. Remark C.72. It would be interesting to see if, or under what conditions, ∗ HG (M max ) is a torsion–free RG -module. 9. Equivariant homology In this section we give a geometrical interpretation of the equivariant homology H∗G (M ) = H∗ (EG ×G M ). Although the main result of this section holds for homology with any coefficients, for the sake of simplicity we restrict our attention to the coefficient group Z. Recall that the homology of a CW-complex X can be defined by the following construction. Consider oriented manifolds Σ with singularities in codimension two or greater. (See [RS] and references therein for piecewise-linear versions of these definitions.) We assume Σ to be compact but possibly with boundary ∂Σ. Note that ∂Σ is again an oriented manifold with singularities in codimension two or greater. Let Ck (X) be a free group with generators f : Σ → X for all Σ and f , with the standard convention about the orientations. It is clear that C∗ (X) is a complex with differential ∂f = f |∂Σ . The homology of this complex is equal to the ordinary singular homology H∗ (X; Z). Now let M be a CW-complex with an action of a compact group G. Consider the complex C∗G (M ) defined as above, but now with Σ carrying a free G-action and f being equivariant. Remark C.73. Similar maps of principal G-bundles into a symplectic G-manifold have been used to define pseudo-holomorphic curves and Gromov–Witten invariants for symplectic quotients. See [GaSa, Mu1, CGS]. Theorem C.74. The homology of the complex C∗G (M ) in degree q is naturally G isomorphic to the equivariant homology Hq−n (M ; Z), where n = dim G. Proof. Consider the homomorphism of complexes Φ : C∗G (EG × M ) → C∗−n (EG ×G M )

arising from taking the quotient by G. Namely, let Σ be equipped with a free G-action and F : Σ → EG × M be G-equivariant. Then F descends to the map f : Σ/G → EG×G M , where Σ/G is again a manifold with singularities in codimension two or greater. By setting Φ(F ) = f we obtain a homomorphism of complexes. The homomorphism Φ is in fact an isomorphism. To define its inverse, denote by ρ the principal G-bundle EG × M → EG ×G M . Let f : Σ → EG ×G M

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˜ be the total space of the pull-back be an element of C∗−n (EG ×G M ) and let Σ ∗ ˜ G-principal bundle f ρ : Σ → Σ. It is clear that f lifts to a G-equivariant map ˜ → EG × M , where Σ ˜ is a manifold with singularities in codimension two or F: Σ greater. By the construction, Φ(F ) = f . As a consequence, the homology of C∗G (EG × M ) is naturally isomorphic to G H∗−n (M ; Z). It remains to show that C∗G (EG×M ) has the same homology as C∗G (M ). Every map F ∈ C∗G (EG × M ) has the form (κ, φ) where φ ∈ C∗G (M ) and κ : Σ → EG is the lift of a classifying map Σ/G → BG. Since the G-action on Σ determines the classifying map up to a homotopy, it also determines κ up to a G-equivariant homotopy. This shows that the projection F 7→ φ induces an isomorphism in the homology of these complexes. (We leave the details to the reader as an exercise.) Remark C.75. A similar construction applies to maps of genuine manifolds (without singularities) and gives a geometrical interpretation of equivariant bordisms of M . Namely, the ring of equivariant bordisms of M , understood as ordinary bordisms of EG ×G M , can be described via equivariant mappings to M of manifolds with free G-actions. Remark C.76. It would be interesting to understand what “homology theory” results from the above construction when free actions are replaced by locally free actions. (For example, a regular level of a moment map for a torus action gives an element of such a homology theory.) Remark C.77. For every n, there exists an obvious non-degenerate pairing n HG (M ; R) × HnG (M ; R) → R. However, it is not clear how H∗G (M ) fits into the picture of equivariant Poincar´e duality in the context of Section 8.2.

APPENDIX D

Stable complex and Spinc -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spinc structure. Specifically, we use these structures to define quantization. In this appendix we review the definitions and basic properties of these structures. We refer the reader to [ABS, Du, Fr] and [LM, Appendix D] for alternative introductions to Spinc -structures. The reader may also consult the early works [ASi, BH, Bot2]. In this chapter, all G-actions are assumed to be proper. We introduce two equivalence relations, that we call bundle equivalence and homotopy, both for stable complex structures and for Spinc structures. In the literature, a stable complex structure is usually taken up to bundle equivalence, and a Spinc -structure is usually taken up to homotopy. This choice leads to problems which we avoid by keeping track of both equivalence relations for both structures. (See Section 3.) Finally, we note that the notion of a Spinc structure is essentially equivalent to the notion of a “quantum line bundle” [Ve4] and, when the underlying manifold is symplectic, to the notion of an Mpc (“metaplecticc ”) structure [RR]. 1. Stable complex structures In this section we discuss stable complex structures and their equivalences. 1.1. Definitions. Definition D.1. A stable complex structure on a real vector bundle E is a fiberwise complex structure on the Whitney sum E ⊕ Rk for some k, where Rk denotes the trivial bundle with fiber Rk . A stable complex structure on a manifold is a stable complex structure on its tangent bundle. Suppose that a Lie group G acts on E by bundle automorphisms. For instance, if E = T M is the tangent bundle of a G-manifold M , we take the natural induced action unless stated otherwise. An equivariant stable complex structure on E is a stable complex structure such that G acts on E ⊕ Rk by complex bundle automorphisms. Here, Rk is equipped with the trivial G-action. Example D.2. An almost complex structure on M is a fiberwise complex structure on the tangent bundle T M , i.e., an automorphism of real vector bundles J : T M → T M such that J 2 = identity. This is a special case of a stable complex structure. However, not every stable complex structure arises in this way. For instance, S 5 admits a stable complex structure via T S 5 ⊕ R = S 5 × R6 = S 5 × C3 , whereas an almost complex manifold must be even dimensional. Example D.3. A complex manifold is almost complex, and hence stable complex. (Let us recall why: holomorphic coordinates identify each tangent space 229

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with Cn ; holomorphic transition functions transform these Cn ’s by complex linear transformations, by the Cauchy-Riemann equations.) 1.2. Equivalence relations. One usually works with an equivalence class of stable complex structures. We have two notions of equivalence: bundle equivalence and homotopy. For the sake of brevity, we only give the definitions in the presence of a group action. Let E0 and E1 be G-equivariant vector bundles over a G-manifold M . Let J0 and J1 be stable complex structures, given by G-invariant fiberwise complex structures on E0 ⊕ Rk and E1 ⊕ Rl . Definition D.4. The structures J0 and J1 are bundle equivalent if there exist a and b such that E0 ⊕ Rk ⊕ Ca and E1 ⊕ Rl ⊕ Cb are isomorphic as G-equivariant complex vector bundles, where Ca and Cb are equipped with trivial G-actions. Bundle equivalence of stable complex structures is sufficient for most purposes and is sometimes referred to as equivalence. (In our papers [CKT] and [GGK2], a “stable complex structure” meant a bundle equivalence class.) For stable complex structures on the same vector bundle we have a second notion of equivalence: Definition D.5. Suppose that E0 = E1 = E. The structures J0 and J1 are homotopic if there exist a and b such that k +2a = l +2b and such that the resulting complex structures on the vector bundle E⊕Rm , where m = k+2a = l+2b, obtained from its identifications with (E ⊕ Rk ) ⊕ Ca and with (E ⊕ Rl ) ⊕ Cb , are homotopic through a family of G-invariant fiberwise complex structures. Stable complex structures that arise from geometric constructions, such as restricting to a boundary or reducing with respect to a group action are usually defined up to homotopy. See Section 1.3. The quantization of a stable complex manifold (M, J) equipped with a complex line bundle L is invariant under both equivalence relations: homotopy and bundle equivalence. The fact that homotopic J’s give the same quantization is a direct consequence of the Fredholm homotopy invariance of the index. This fact is used to show that the quantization of a reduced space is well defined; see Example D.13. On the other hand, to prove that bundle-equivalent stable complex structures have the same quantization, one must invoke the index theorem. See Section 3 of this appendix or Section 7 of Chapter 6 for more details. Proposition D.6. Homotopic stable complex structures are bundle equivalent. Proof. It is enough to show that if Jt is a smooth family of invariant fiberwise complex structures on a G-equivariant real vector bundle E, then (E, J0 ) and (E, J1 ) are isomorphic as G-equivariant complex vector bundles. ˜ of E to M × We view Jt as a fiberwise complex structure on the pull-back E ˜ [0, 1]. This turns E into a G-equivariant complex vector bundle over M × [0, 1] which restricts to (E, J0 ) and (E, J1 ) on the components of the boundary. Fix a ˜ The vector field on E ˜ obtained as the horizontal G-invariant connection on E. lift of the vector field ∂/∂t on M × [0, 1] integrates to a family of isomorphisms (E, J0 ) → (E, Jt ), t ∈ [0, 1]. Remark D.7. The same argument applies to reductions of the structure group to any closed subgroup, not necessarily GL(n, C) ⊆ GL(2n, R). Bundle equivalent stable complex structures need not be homotopic:

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√ √ Example D.8. Consider M = C with the complex structures −1 and − −1. As stable complex structures, they are bundle equivalent: the tangent bundle T C with either of these structures is isomorphic to the trivial complex line bundle M × C. However, these structures are not homotopic; for instance, they induce opposite orientations. (See Section 1.4.) Moreover, bundle equivalent stable complex structures need not be homotopic even if they induce the same orientation. See Example D.25. Notice that in Definition D.4 the equivalence relation is taken with the group action. It would make no sense to first consider a (non-equivariant) bundle equivalence class and then let a group act. The reason is that a group action on a manifold (or on a vector bundle) induces a group action on the set of stable complex structures but not on the set of their bundle equivalence classes or on a particular class. There is an analogy here with the notion of an equivariant vector bundle over a G-manifold; a lift of a G-action to a bundle is not natural and might not exist. √ Example D.9. On the manifold M = C, the stable complex structures −1 √ and − −1 are bundle equivalent. However, if we let G = S 1 act by rotations, these structures become√non-equivalent. (The isotropy weight at the √ origin is equal to 1 for the structure −1 and is equal to −1 for the structure − −1. See Proposition D.17.) Unless stated otherwise, in the presence of a G-action all stable complex structures are assumed to be invariant and all homotopies or bundle equivalences are assumed to be equivariant. 1.3. Geometric constructions. In this section we discuss geometric constructions which gives rise to homotopy classes of stable complex structures: the almost complex structure compatible with a symplectic form, the reduction of a stable complex structure, the restriction to a boundary, and the restriction to a fixed point set and to its normal bundle. The first important source of homotopy classes of stable complex structures is symplectic manifolds: Definition D.10. An almost complex structure J : T M → T M is compatible with a symplectic form ω if hu, vi := ω(u, Jv) defines a Riemannian metric. Remark D.11. It is not hard to see that, if J is an almost complex structure, a (real valued) two-form ω that satisfies any one of the following conditions satisfies them all: 1. hu, vi := ω(u, Jv) is symmetric; 2. ω(Ju, Jv) = ω(u, v) for all u, v; 3. ω is a differential form of type (1, 1) with respect to J. Compatibility means, in addition, that hu, vi is positive definite. It implies that ω is non-degenerate. Example D.12. On a symplectic G-manifold (M, ω) there exists an equivariant almost complex structure J compatible with ω, unique up to homotopy. See [Ste, Section 41], [Wei1, Lecture 2], or [McDSa, Sections 2.5 and 4.1]. Sketch of proof. Given an arbitrary Riemannian metric on M , define an 1 operator A by the condition hu, vi = ω(u, Av). Then J = A(−A2 )− 2 is an almost

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complex structure compatible with ω. In this way, we obtain a continuous map h , i 7→ J from the space of Riemannian metrics onto the space of almost complex structures compatible with ω. The former is non-empty and convex, and hence connected. As a consequence, the latter is non-empty and connected, which proves the claim. In the presence of a G-action preserving the symplectic structure, we use the same argument, but starting with the space of G-invariant metrics. Stable complex structures arise when we consider reduction: Example D.13. Let Ψ : M → g∗ be an (abstract) moment map. An invariant almost complex structure J on M generally does not induce an almost complex structure on the reduced space Mred = Ψ−1 (0)/G, unless Ψ is a moment map for an genuine symplectic form ω and J is compatible with ω. (See [CKT] for conditions under which J descends to Mred , when G = S 1 .) However, J does induce a stable complex structure on Mred , unique up to homotopy. See Section 2.3 of Chapter 5 for details. Moreover, homotopic or bundle equivalent structures on M induce, respectively, homotopic, or bundle equivalent, structures on Mred. Finally, because quantization is defined for a homotopy class (or, more generally, for a bundle equivalence class) of stable complex structures, the quantization of the reduced space is well defined. Another important reason to consider stable complex structures is that the notion of a cobordism of such structures is particularly simple. This notion relies on the crucial observation that a stable complex structure on a manifold induces one on its boundary: Proposition D.14. Let M be a G-manifold with boundary ∂M . An equivariant stable complex structure on M induces one on ∂M , which is canonical up to homotopy. Homotopic or bundle equivalent, equivariant stable complex structures on M induce, respectively, homotopic or bundle equivalent, equivariant stable complex structures on ∂M . Proof. We have a short exact sequence of vector bundles over the boundary, (D.1)

0 → T (∂M ) → T M |∂M → N (∂M ) → 0,

where T (∂M ) is the tangent bundle to the boundary and where N (∂M ) = (T M |∂M ) /T (∂M )

is the normal bundle to the boundary. The normal bundle to the boundary is a one-dimensional real vector bundle, oriented by choosing the “outward” direction to be positive. This determines an isomorphism with the trivial vector bundle, N (∂M ) ∼ = ∂M × R, and the isomorphism is unique up to homotopy. The sequence (D.1) splits, and the splitting is unique up to homotopy. (A short exact sequence of equivariant vector bundles 0 → A → B → C → 0 always splits, and any two splittings are homotopic: the space of all splittings can be (noncanonically) identified with the space of sections of the vector bundle HomG (C, A), and this space is connected.) We obtain an isomorphism T M |∂M = R ⊕ T (∂M ),

which is canonical up to homotopy. The lemma follows.

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Remark D.15. One can also define cobordisms of almost complex structures on manifolds. The structure required on a cobording (2n + 1)-dimensional manifold is then a reduction (“tangent” to the boundary) of the structure group of the tangent bundle to GL(n, C). As usual, all manifolds are assumed to be oriented. In the non-equivariant case, the resulting cobordism ring is in fact equal to the cobordism ring of stable complex structures, as follows, e.g., from the results of [Bak, Gin2, Mor]. However, stable complex cobordisms are considerably more tracktable and for many purposes more natural objects than their almost complex counterparts. In this book we need to consider fixed point sets and their normal bundles. Let an abelian Lie group G act properly on a manifold M . Let F be a connected component of the fixed point set M H for some closed subgroup H of G. Recall that F is a closed submanifold of M , on which G acts (non-effectively, with H acting trivially), and the normal bundle N F = (T M |F )/T F is a G-equivariant real vector bundle over F . Proposition D.16. An equivariant stable complex structure on M induces an equivariant stable complex structure on F and a fiberwise complex structure on the normal bundle N F such that G acts on N F by complex bundle automorphisms. Homotopic, or bundle equivalent, structures on M induce homotopic, or bundle equivalent, structures on F and N F . Corollary D.17. Let G be a torus and let M be a stable complex G-manifold. At a fixed point for the G-action, the non-zero isotropy weights are well defined, even if the stable complex structure is defined only up to equivalence. Proof of Corollary D.17. Let F ⊆ M G be a connected component of the fixed point set. The non-zero isotropy weights at a point p ∈ F carry exactly the same information as the G-action on the complex vector space Np F . Proof of Proposition D.16. Let J be a fiberwise complex structure on T M ⊕ Rk . Then T M |F ⊕ Rk is a G-equivariant complex vector bundle over F . The sub-bundle T F ⊕ Rk is G-invariant and complex, because it consists precisely of those vectors that are fixed by H. The normal bundle N F = T M |F /T F is naturally isomorphic to the quotient (T M |F ⊕ Rk )/(T F ⊕ Rk ), making it into a complex vector bundle. Because an equivariant homotopy or bundle equivalence of complex structures must preserve the sub-bundle of H-fixed vectors, the effects of these equivalences on the structures on F and on N F are as stated. In particular, let π: E → F be a G-equivariant real vector bundle, and suppose that there exists a subgroup H ⊆ G whose fixed point set E H is precisely the zero section F . An equivariant stable complex structure J on the total space of E gives rise to an equivariant stable complex structure JF on F and a fiberwise complex structure Jf on E. The converse is also true: Proposition D.18. An equivariant stable complex structure JF on F and an invariant fiberwise complex structure Jf on E determine an equivariant stable complex structure J on the total space E, defined up to homotopy, which, in turn, induces the structures JF and Jf . If Jf and JF are defined up to homotopy, or bundle equivalence, so is J.

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Proof. We have a short exact sequence of bundles over E, 0 → π ∗ E → T E → T F → 0.

We have a complex structure on π ∗ E coming from Jf , and a stable complex structure on T F coming from JF . A splitting of the sequence gives an isomorphism TE ∼ = π ∗ E ⊕ T F, unique up to homotopy. We set J = Jf ⊕ JF .

Propositions D.16 and D.18 immediately imply the following result, which is used to specify the right-hand side of the stable complex “Linearization Theorem” in Chapter 4. Proposition D.19. Let J be an equivariant stable complex structure on M and let F be a connected component of the set M H of fixed points for some subgroup H of G. Then there is a unique up to homotopy stable complex structure J F on the total space N F which induces the same stable complex structure on F and fiberwise complex structure on N F as those induced by J. Homotopic or bundle equivalent, structures on M induce homotopic, or bundle equivalent, structures on N F . 1.4. Orientations and Chern numbers. A stable complex structure induces an orientation, obtained as the “difference” of the complex orientation on E ⊕ Rk and the standard orientation on Rk . Homotopic stable complex structures determine the same orientation. However, bundle equivalent stable complex structures need not induce the same orientation. So, for instance, a manifold equipped with a stable complex structure up to bundle equivalence is orientable, but is not naturally oriented. √ √ For instance, the complex structures −1 and − −1 on C induce opposite orientations although they are bundle equivalent (cf. Example D.8). More generally, Lemma D.20. Every stable complex structure J is bundle equivalent to a stable complex structure J 0 which induces the opposite orientation. √ Proof. Let J be a fiberwise complex structure on E ⊕Rk . Let J 0 = J ⊕− −1 √ on E ⊕ Rk ⊕ C. Then J 0 is isomorphic to J ⊕ −1 (via conjugation of the last factor), but J and J 0 induce opposite orientations. An oriented bundle equivalence between stable complex structures J0 and J1 defined on E ⊕Rk and on E ⊕Rl is an isomorphism of G-equivariant complex vector bundles E ⊕ Rk ⊕ Ca and E ⊕ Rl ⊕ Cb which is fiberwise orientation preserving. (This is well defined even if E is not a priori oriented.) If J0 and J1 induce opposite orientations on E, an oriented bundle equivalence between them does not exist. If they induce the same orientation, an oriented bundle equivalence between them is the same thing as a bundle equivalence. In particular, if J0 and J1 are homotopic, there exists an oriented bundle equivalence between them. From Lemma D.20 we conclude that an oriented bundle equivalence class carries exactly the same information as a bundle equivalence class plus an auxiliary orientation. Bundle equivalent (equivariant) stable complex structures have the same (equivariant) Chern classes. To integrate these classes and obtain characteristic numbers, one also needs an orientation of the manifold. We specialize to E = T M , so that an orientation of M is the same thing as a fiberwise orientation of E. It is then

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convenient to fix an orientation O of M as an additional structure. This is the approach often taken in complex cobordism theory. (See, e.g., [Sto, Rudy], and [May].) Example D.21. In Example 5.5 of Chapter 5 we described non-standard stable complex structures on CPn whose Chern classes are different from those of any almost complex structure. We deduce that these stable complex structures are not even bundle equivalent to almost complex structures. Let M be an oriented manifold. Let J denote the set of stable complex structures on M . Let J0 denote the set of stable complex structures that induce the given orientation on M . Then the natural map J0 /oriented bundle equivalence → J /bundle equivalence

is a bijection, as follows from the above discussion. The natural map (D.2)

J0 /homotopy → J /bundle equivalence

is well defined and onto, but is not always one-to-one. See Example D.25. From now on we will consider oriented manifolds, equipped with compatible stable complex structures, up to oriented bundle equivalence. 1.5. Oriented complex cobordisms. We now define cobordisms of oriented stable complex G-manifolds. In this context, stable complex structures are taken up to bundle equivalence. Definition D.22 (Oriented complex cobordism). Let M0 and M1 be oriented stable complex G-manifolds. An oriented complex cobordism between these manifolds is an oriented stable complex G-manifold with boundary W and a diffeoF morphism of ∂W with −M0 M1 which transports the stable complex structure on ∂W to structures on M0 and M1 that are bundle equivalent to the given ones. (The minus sign indicates that the diffeomorphism reverses the orientation of M0 and preserves the orientation of M1 .) Cobordant oriented stable complex manifolds have the same characteristic numbers. This follows immediately from Stokes’s theorem. The converse is also true: if two oriented stable complex manifolds have the same characteristic numbers, then there exists an oriented complex cobordism between them, by a theorem of Milnor and Novikov. See [MiSt, Sto]. 1.6. Relations with other definitions of a stable complex structure. In the literature one encounters several notions of stable complex structures. For example, weak complex structures from [BH] are bundle equivalence classes, whereas weakly complex structures from [CF1, CF2] are homotopy classes. Stable complex structures are sometimes also referred to as stable almost complex structures [May, Rudy], weakly almost complex structures [BH], or U -structures. Cobordisms of stable complex structures are called complex cobordisms, or unitary cobordisms. In algebraic topology, a stable complex structure is usually defined on the stable normal bundle to M (see, e.g., [Sto] and [Rudy]). We, as, e.g., in [May], define stable complex structures on G-manifolds to be on stable tangent bundles. In the non-equivariant setting, the two approaches are equivalent. This is no longer true in the equivariant case. (See [May], p. 337 for details.) An equivariant stable complex

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structure on M gives rise to an equivariant complex structure on the stable normal bundle to M , but the converse is not true. The key feature of Definition D.1 is that the group action on the “stabilizing” part of the stable tangent bundle is required to be trivial. Without this requirement, the notions of equivariant stable complex structures on the tangent and stable normal bundles would be essentially equivalent. However, some of the properties of equivariant stable complex structures would be lost; for instance, Proposition D.16 and Corollary D.17 would not hold for “normal” equivariant stable complex structures, and many applications, such as in [GGK2, Section 4], would not be true for manifolds with stable complex structures on the normal bundle. 1.7. Stable complex structures on spheres. In conclusion we give some examples of classification results for stable complex structures on manifolds. In the non-equivariant case these classification questions are handled using homotopy theory and, in particular, obstruction theory (see, e.g., [Hu]). Below we outline the solutions in the simplest case, where the manifold is a sphere. We restrict our attention to stable complex structures compatible with a fixed orientation, as explained in Section 1.4. Let us first recall some general facts and set notation to be used later on. Let Vn be the space of complex structures on R2n compatible with a fixed orientation. It is easy to see that Vn = SO(2n)/ U(n). The stable homotopy groups of Vn are as follows (see [Mas1]). (Here, q > 0 and n is large enough so that 2n − 1 > q.)   Z for q ≡ 2 mod 4; πq (Vn ) = Z2 for q ≡ 0, 7 mod 8;   0 otherwise. The natural inclusion Vn → Vn+1 induces an isomorphism of these stable homotopy groups. As usual, let B SO(2n) and B U(n) denote the classifying spaces for SO(2n) and U(n). The inclusion U(n) → SO(2n) induces a map B U(n) → B SO(2n) which is a fiber bundle with fiber Vn . The inclusion of Vn into B U(n) as a fiber induces a map πq (Vn ) → πq (B U(n)). Denote the image of this map by Γq . Here, as above, we assume that n is large enough for a fixed q so that Γq is independent of n. Proposition D.23. Stable complex structures on S q compatible with a fixed orientation are classified by the elements of πq (Vn ) up to homotopy and by the elements of Γq up to bundle equivalence. The projection πq (Vn ) → Γq is the natural map between equivalence classes (cf. Proposition D.6). Proof. For any q, the tangent bundle T S q is stably trivial. In fact, T S q ⊕R = S × Rq+1 . It follows that homotopy classes of stable complex complex structures on S q , compatible with a given orientation, are in a one-to-one correspondence with πq (Vn ), where n is large enough. Isomorphism classes of real oriented 2n-dimensional vector bundles over M are classified by homotopy classes of maps M → B SO(2n), [Hus]. Likewise, complex n-dimensional vector bundles are classified by the homotopy classes of maps M → B U(n). Forgetting the complex structure on a vector bundle corresponds to the map B U(n) → B SO(2n). Fiber bundle equivalence classes of complex structures on an oriented real vector bundle are described by the homotopy classes of lifts of M → B SO(2n) to M → B U(n). Specializing this to the case of stable complex q

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structures on S q we note that the classifying map S q → B SO(2n) is contractible. In other words, this map represents 0 ∈ πq (B SO(2n)). As follows from the long exact sequence . . . → πq (Vn ) → πq (B U(n)) → πq (B SO(2n)) → . . . of the homotopy groups, the lifts of a contractible map are classified by the image Γq of πq (Vn ). We leave the proof of the last assertion to the reader as an exercise. Example D.24 (Stable complex structures on S 2 ). Stable complex structures on S 2 , compatible with a fixed orientation, are classified by Z up to either homotopy or bundle equivalence, and these two equivalence relations are identical on S 2 . Indeed, for q = 2, the natural map Z = π2 (Vn ) → Γ2 is an isomorphism. Example D.25 (Stable complex structures on S 7 and S 8 ). The first sphere for which the two classifications are inequivalent is S 7 . Applying Proposition D.23 to this sphere, we see that up to homotopy the stable complex structures on S 7 are classified by π7 (Vn ) = Z2 . Up to bundle equivalence, stable complex structures on S 7 are classified by Γ7 which is zero, for π7 (B U(n)) = π6 (U(n)) = 0. Similarly, the two classifications are different for S 8 , where π8 (Vn ) = Z2 and Γ8 = 0. (Indeed, Γ8 is the image of Z2 in π8 (B U(n)) = π7 (U(n)) = Z.) Remark D.26. Calculations similar to Examples D.24 and D.25 also show that the two classifications are equivalent for all S q with q ≤ 6. Combining this with elementary obstruction theory (see, e.g., [Hu]), one can show that the two classifications of non-equivariant stable complex structures on M (compatible with a fixed orientation) are equivalent when dim M ≤ 6. Classification problems for stable complex structures become more subtle in the equivariant setting. Example D.27 (Equivariant stable complex structures on S 2 ). Consider the standard action of G = S 1 on S 2 by rotations about the z-axis. Then any two G-equivariant stable complex structures on S 2 , compatible with a fixed orientation, are equivariantly homotopy equivalent and hence also bundle equivalent. Proof. Let J be a G-equivariant complex structure on T S 2 ⊕ Rk compatible with the orientation. If p is the north or the south pole, we have an equivariant decomposition of the stable tangent space as Tp S 2 ⊕ Rk , where Rk is fixed by G, and both components are J-complex subspaces. Since J is compatible with the orientation, J is equivariantly homotopic to a stable complex structure which has some standard form near the poles. Hence, without loss of generality, we can assume that J has such a form near the poles. Let us fix a trivialization of T S 2 ⊕ Rk along the arc C of a meridian connecting the two poles. Then J is completely determined by its values along C. Now it is easy to see that equivariant homotopy classes of stable complex structures on S 2 are in a one-to-one correspondence with homotopy classes of the maps of C to V n with fixed end-points. This set is essentially π1 (Vn ) = 0, which implies the assertion of Example D.27.

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2. Spinc -structures 2.1. The definition of Spinc -structures. Recall that the group Spin(n) is the connected double covering of the group SO(n). (In fact, π1 (SO(n)) = Z2 , so Spin(n) is the universal covering of SO(n), when n > 2.) Let q : Spin(n) → SO(n)

denote the covering map, and let  denote the non-trivial element in the kernel of this map. Then Spinc (n) = Spin(n) ×Z2 U(1) is the quotient of Spin(n) × U(1) by the two element subgroup generated by (, −1). In other words, elements of Spinc (n) are equivalence classes [s, c], where s ∈ Spin(n), c ∈ U(1), and [s, c] = [s, −c]. The projections to the two factors give rise to two natural homomorphisms, π : Spinc (n) → SO(n),

(D.3) and

[s, c] 7→ q(s),

det : Spinc → U(1),

(D.4)

[s, c] 7→ c2 ,

which give rise to short exact sequences

π

1 → U(1) → Spinc (n) → SO(n) → 1

and

det

1 → Spin(n) → Spinc (n) → U(1) → 1. Let E → M be a real vector bundle of rank n. On the conceptual level, a Spinc -structure on E is an “extension” of its structural group to Spinc (n). More precisely, let GL(E) denote the frame bundle of E, that is, the principal GL(n)bundle whose fiber over p ∈ M is the set of bases of the vector space Ep . The group π Spinc (n) acts on GL(E) through the composition Spinc (n) → SO(n) ,→ GL(n). Definition D.28. A Spinc -structure on E is a principal Spinc (n)-bundle c

P →M

together with a Spin (n)-equivariant bundle map c

p : P → GL(E).

A Spin structure on a manifold M is a Spinc structure on its tangent bundle E = T M. Although a Spinc -structure is more than just a principal bundle P , we will often refer to a Spinc -structure (P, p) as simply P . The map p determines an isomorphism of vector bundles, ∼ E, (D.5) P × π Rn =

and vice versa: an isomorphism (D.5) determines an equivariant map p : P → GL(E). The standard metric and orientation on Rn transport to E through (D.5), such that the bundle SO(E) of oriented orthogonal frames is precisely the image of the map p : P → GL(E). In particular, a vector bundle that admits a Spin c structure is orientable. In the literature, a Spinc structure is commonly defined for a vector bundle which is a priori equipped with a fiberwise metric and orientation. We will consider this notion under a slightly different name:

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Definition D.29. Let E → M be a vector bundle. If E is equipped with a fiberwise orientation, an oriented Spinc -structure on E is a Spinc structure which induces the given orientation. If E is equipped with a fiberwise metric, a metric Spinc -structure on E is a Spinc structure which induces the given metric. Hence, Spinc structures as are usually referred to in the literature are what we call oriented metric Spinc structures. One may also discard the metric and orientation completely; this is achieved through the notion of an MLc (“metalinearc ”) structure, which we discuss in Section 2.6. This notion is superior to others in that it requires the least a priori structure. Our definition of a Spinc structure is an intermediate notion, which, on one hand, avoids an a priori choice of metric and orientation, but does give rise to these structures. One advantage of this notion is purely psychological, in that it allows us to keep the common name “Spinc ”. In Sections 2.4, 2.5, and 2.6, we will see that these different notions of Spinc structures are essentially equivalent. In practice, one may work with any one of these notions. The determinant line bundle associated with the Spinc -structure is the complex line bundle Ldet = P ×det C over M , associated through the homomorphism (D.4). We may think of P as a circle bundle over SO(E), with p : P → SO(E) being the projection map.

Lemma D.30. The associated line bundle, P ×U(1) C → SO(E), is a square root of the pullback to SO(E) of the determinant line bundle Ldet . We leave the proof as an exercise to the reader. Because the subgroup U(1) is the center of Spinc (n), its right action on P commutes with the entire principal Spinc (n) action. Therefore, P is a Spinc (n)equivariant circle bundle over SO(E). Conversely, a Spinc (n) equivariant circle bundle over SO(E) is a Spinc structure on E if K := ker π ∼ = U(1) acts by the principal action. Proposition D.31. An oriented vector bundle E over M admits a Spinc structure if and only if its second Stiefel-Whitney class w2 (E) is integral, i.e., lies in the image of the homomorphism ρ : H 2 (M ; Z) → H 2 (M ; Z2 ). We refer the reader to [FF] or [Fr] for the proof. The definitions of the StiefelWhitney class w2 (E) and the Chern class c1 (L) can be found in [MiSt]. One may attempt to define an equivalence of Spinc structures (P, p) and (P 0 , p0 ) to be a principal bundle map F : P → P 0 which respects the maps to GL(E). Spinc structures that are equivalent in this sense must induce the same metric and orientation. In fact, this will be our notion of equivalence of metric Spinc structures in Section 2.4. We have formulated our definition of a Spinc structure in such a way as to avoid fixing a metric and an orientation, and we will work with notions of equivalence which allow different metrics or orientations. In the next sections we introduce two such notions, which we call bundle equivalence and homotopy of Spinc structures. In the literature, it is often unclear what equivalence relation is taken, and this results in seemingly contradictory classification results

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for Spinc -structures. We outline the homotopy classification of Spinc structures in Section 2.7. Finally, we note that the quantization of a Spinc structure (see Section 3 of this appendix or Section 7 of Chapter 6) is invariant under both equivalence relations: homotopy and bundle equivalence. The fact that homotopic Spinc structures give the same quantization is a direct consequence of the Fredholm homotopy invariance of the index. On the other hand, to prove that bundle-equivalent Spinc structures have the same quantization one must invoke the index theorem. 2.2. Bundle equivalence of Spinc -structures. The simplest equivalence relation between Spinc -structures is the equivalence of principal bundles: Definition D.32. Let (P, p) and (P 0 , p0 ) be Spinc -structures on vector bundles E and, respectively, E 0 over M . These structures are bundle equivalent if P and P 0 are equivalent as Spinc (n)-principal bundles. Remark D.33. An isomorphism of principal bundles F : P → P 0 gives rise to a unique isomorphism of principal bundles f : GL(E) → GL(E 0 ) such that the following diagram commutes:

(D.6)

Indeed, f is determined by p

P   py

F

−−−−→ f

P0   p0 y

GL(E) −−−−→ GL(E 0 ). F

p0

GL(E) ∼ = GL(E 0 ). = P ×Spinc (n) GL(n) → P 0 ×Spinc (n) GL(n) ∼

Similarly, if two Spinc -structures are bundle equivalent, the underlying vector bundles E and E 0 are isomorphic, via E∼ = P ×Spinc (n) Rn → P 0 ×Spinc Rn ∼ = E0.

Note that we do not need E and E 0 to be the same vector bundle, and that we impose no restriction on the isomorphism f in (D.6). We will later encounter stricter notions of equivalence, which apply to the case E = E 0 , and in which we insist that f be equal to, or homotopic to, the identity map.

Remark D.34. A Spinc (n)-principal bundle can also be described in terms of transition functions in the usual way. Fix a cover of M by contractible open sets Ui . A Spinc -principal bundle P is given by a collection of functions ϕij : Uij = Ui ∩ Uj → Spinc (n) such that the cocycle condition holds, i.e., ϕij ϕjk ϕki = 1. Then the vector bundle E is determined by the GL+ (n)-valued cocycle πϕij . Two cocycles ϕij and ψij give rise to bundle equivalent Spinc -structures on E if and only if these cocycles differ by a coboundary: ψij = fi ϕij fj−1 for some collection of functions fi : Ui → Spinc (n). The determinant line bundle of P is given by the U(1)-cocycle det ϕij . Remark D.35. Recall that every Spinc (n)-bundle over a manifold is a pullback of the universal Spinc (n)-bundle, E Spinc (n) → B Spinc (n), through a map to the classifying space B Spinc (n), and this map is unique up to homotopy. Two Spinc structures are bundle equivalent exactly if the corresponding maps to the classifying

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space are homotopic. This notion of homotopy is different (in fact, weaker) than the notion of homotopy that will be introduced in the next section. Remark D.36. The real (or, equivalently, rational) characteristic classes of Spinc (n)-structures are described by the cohomology ring H ∗ (B Spinc (n)). For n even, which is the case of most interest for us, this ring (over R) is the polynomial ring with generators c1 (of degree two), e of degree n = 2m, and p1 , . . . , pm with deg pj = 4j; see, e.g., [FF]. The class c1 corresponds to the first Chern class of Ldet , the class e corresponds to the Euler class of E, and the classes pj are the Pontrjagin classes. As a consequence, for Spinc -structures on an even–dimensional manifold, all characteristic classes but c1 are determined by the topology of M and hence are independent of the Spinc -structure. The notion of bundle equivalence is most natural in the topological context. The characteristic classes of (P, p) are entirely determined by P and thus are invariants of bundle equivalence. 2.3. Homotopy of Spinc -structures. For Spinc structures over a fixed vector bundle E one has the following equivalence relation, which is finer than bundle equivalence: Definition D.37. Spinc structures (P, p) and (P 0 , p0 ) over a vector bundle E are homotopic if there exists a principal bundle isomorphism F : P → P0

and a smooth family of Spinc -equivariant maps pt : P → GL(E),

for t ∈ [0, 1], such that p0 = p and p1 = p0 ◦ F : F

P −−−−→   0 yp∼p ◦F

GL(E)

P0   0 yp

GL(E).

We have the following useful characterization of homotopy: Proposition D.38. Let E be a vector bundle. Two Spinc structures (P, p) and (P 0 , p0 ) over E are homotopic if and only if there exists a principal bundle isomorphism F : P → P 0 such that the induced automorphism f : GL(E) → GL(E) is homotopic to the identity through bundle automorphisms: P   py

F

−−−−→ f ∼identity

P0   p0 y

GL(E) −−−−−−−→ GL(E). Moreover, it is enough to assume that f is homotopic to the identity through SO(n)equivariant maps. Proof. Let F : P → P 0 be an isomorphism of principal bundles. Let f be the unique automorphism of GL(E) such that f ◦ p = p0 ◦ F , as in (D.6).

242

D. STABLE COMPLEX AND SPINc -STRUCTURES

Suppose that there exists a family of Spinc -equivariant maps pt : P → GL(E) such that p0 = p and p1 = p0 ◦ F . Let ft : GL(E) → GL(E) be the unique automorphism such that the following diagram commutes:

(D.7)

P   pt y

F

−−−−→ ft

P0   p0 y

GL(E) −−−−→ GL(E).

(See Remark D.33.) Then, by uniqueness, f0 = f and f1 = identity. On the other hand, suppose that there exists a family of SO(n)-equivariant maps ft of GL(E) such that f0 = f and f1 = identity. Let pt = ft−1 ◦ p0 ◦ F . Then pt : P → GL(E) is Spinc (n)-equivariant, p0 = p, and p1 = p0 ◦ F . Clearly, if two Spinc structures on E are homotopic, they are also bundle equivalent. Finally, we note that the definitions of Spinc structures and their equivalences naturally extend to equivariant structures in the presence of a proper G-action. Various claims that we have made regarding these objects have equivariant analogues. In fact, the same proofs work, with the word “equivariant” or “invariant” inserted wherever appropriate. Note that some of these proofs rely on the existence of an invariant fiberwise metric on a G-equivariant vector bundle E, or the existence of an equivariant connection on a G-equivariant principal bundle. Properness of the action is needed to guarantee that such metrics or connections do exist. See Section 3.2 of Appendix B for how to “average” with respect to a proper group action. 2.4. Metric Spinc structures. Let E → M be a vector bundle. Recall that a Spinc structure on E is a principal Spinc bundle P → M together with a Spinc (n) equivariant bundle map p : P → GL(E). If E is a priori equipped with a fiberwise orientation or metric, recall that an oriented, resp., metric Spinc structure is one which induces the given orientation, resp., the given metric. Let E be a vector bundle with a fiberwise metric and let O(E) be its orthonormal frame bundle. An equivalence of metric Spinc structures (P, p) and (P 0 , p0 ) over E is an isomorphism F : P → P 0 which lifts the identity map on O(E): P   py

F

−−−−→ identity

P0   p0 y

O(E) −−−−−→ O(E)

Spinc structures up to homotopy are the same as metric Spinc structures up to equivalence: Proposition D.39. Let E → M be a vector bundle with a fiberwise metric. Then: (1) Every Spinc -structure is homotopic to a metric Spinc -structure. (2) Two metric Spinc -structures are equivalent if and only if they are homotopic. Because homotopy preserves orientation, and so does equivalence of metric Spinc structures, it is enough to prove, for any pre-chosen orientation and metric on E,

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(1’) Every oriented Spinc -structure is homotopic to an oriented metric Spinc structure. (2’) Two oriented metric Spinc -structures are equivalent if and only if they are homotopic. Proof of (1’): Let GL+ (E) denote the oriented frame bundle and SO(E) the oriented orthogonal frame bundle of E. Let (P, p0 ) be an oriented Spinc structure and SO(E 0 ) the oriented orthonormal frame bundle for the induced metric. Metrics on E exactly correspond to sections of GL+ (E)/ SO(n). Because this bundle has contractible fibers, there exists a smooth family of bundle maps ϕ+

GL+ (E)/ SO(n) −−−−→ GL+ (E)/ SO(n)     y y

M M such that ϕ0 =identity and ϕ1 sends the section SO(E 0 )/ SO(n) to the section SO(E)/ SO(n). Considering GL+ (E) as a principal SO(n)-bundle over GL+ (E)/ SO(n), the family ϕt lifts to a family of SO(n)-equivariant maps, GL+ (E)   y

ft

−−−−→ ϕt

GL+ (E)   y

GL+ (E)/ SO(n) −−−−→ GL+ (E)/ SO(n)

such that f0 = identity and f1 sends SO(E 0 ) to SO(E). We then have maps P   p0 y

f1

P   py

SO(E 0 ) −−−−→ SO(E)

where p = f1 ◦ p0 . Then (P, p) is an oriented metric Spinc structure. The structures (P, p) and (P, p0 ) are homotopic by Proposition D.38. Proof of (2’): Let (P, p) and (P 0 , p0 ) be oriented metric Spinc structures. Suppose that there exists an isomorphism F : P → P 0 of principal bundles, inducing an isomorphism f : GL+ (E) → GL+ (E), and a family ft : GL+ (E) → GL+ (E) of bundle isomorphisms such that f1 = f and f0 = identity. These descend to maps ϕt : GL+ (E)/ SO(n) → GL+ (E)/ SO(n) such that ϕ0 = identity and ϕ1 sends SO(E)/ SO(n) to itself. Because the bundle GL+ (E)/ SO(n) → M has contractible fibers, the homotopy ϕt can be deformed, through homotopies ϕt,s with the same endpoints, to a homotopy which sends SO(E)/ SO(n) to itself for all t. The lifting of ϕt to ft extends to a family of SO(n)-equivariant maps GL+ (E)   y

ft,s

−−−−→ ϕt,s

GL+ (E)   y

GL+ (E)/ SO(n) −−−−→ GL+ (E)/ SO(n).

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such that f0,s = identity and f1,s = f for all s. The map ft,1 sends SO(E) to itself for all t. Considering P and P 0 as Spinc (n)-equivariant U(1)-bundles over SO(E), the family ft,1 lifts to a family of automorphisms, P   py

F

t −−−− →

ft,1

P0   p0 y

SO(E) −−−−→ SO(E).

such that F1 = F . The automorphism F0 : P → P 0 descends to the identity map on SO(E). This gives the required equivalence of metric Spinc structures. 2.5. Spinc structures and Pinc structures. A Pinc structure is a “Spinc structure without an orientation”. To define it, consider the group Pinc (n) = Pin(n) ×Z2 U(1),

where Pin(n) is the non-trivial double covering of O(n). A Pinc -structure on a vector bundle E is a principal Pinc (n)-bundle P together with a Pinc (n)-equivariant map p : P → GL(E), where Pinc (n) acts on GL(E) through the homomorphisms Pinc (n) → O(n) ,→ GL(n). A Pinc structure induces a fiberwise metric on E (but not a fiberwise orientation). The group Pinc (n) contains two connected components. The connected component that contains the identity element is Spinc (n). Let D denote the other connected component of Pinc (n). We have the following properties: (1) If a, b ∈ D, then ab ∈ Spinc (n). (2) Spinc (n) acts on D by multiplication from the right, freely and transitively. (3) Spinc (n) acts on D by multiplication from the left, freely and transitively. These properties imply that we have an isomorphism of spaces with left and right Spinc (n)-actions ∼ =

D ×Spinc (n) D → Spinc (n)

(D.8) given by

[a, b] 7→ ab. Consider the map π : D → GL(n) obtained as the composition π :

D ,→ Pinc (n) → O(n) ,→ GL(n).

For each Spinc structure (P, p), we obtain another Spinc structure (P 0 , p0 ) by setting P 0 = P ×Spinc (n) D and p0 ([u, a]) = p(u)π(a). (Note that π(a) ∈ GL(n) acts on p(u) ∈ GL(E) from the right.) This defines an orientation reversing involution τ : (P, p) 7→ (P 0 , p0 )

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on the set of homotopy classes of Spinc structures. (The fact that τ is an involution follows from (D.8).) Let us now consider a vector bundle E equipped with a fiberwise orientation O. Then we have a natural bijection between the three sets: (1) Pinc structures on E; (2) Spinc structures on E which are compatible with O; (3) Spinc structures on E which are incompatible with O.

Indeed, given a Pinc structure (Q, q), we get Spinc structures (P, p) and (P 0 , p0 ) by P = π −1 (GL+ (E)), P 0 = π −1 (GL− (E)), p = q|P , and p0 = q|P 0 , where GL− (E) is the bundle of disoriented frames. Conversely, given a Spinc structure (P, p) (either compatible or incompatible with O) we get a Pinc structure (Q, q) with Q = P ×Spinc (n) Pinc (n). Finally, the involution τ interchanges oriented Spinc structures and disoriented Spinc structures. Therefore, on a vector bundle E (without a pre-chosen orientation), a Spinc structure contains the same information as a Pinc structure together with an orientation. 2.6. Pinc structures and MLc structures. To eliminate metrics and orientations altogether, we may work with the “metalinearc ” group MLc (n) = ML(n) ×Z2 U(1),

where ML(n) is the non-trivial double covering of GL(n). The group MLc (n) contains the group Pinc (n) as a maximal compact subgroup. An MLc -structure on a vector bundle E is a principal MLc (n)-bundle P˜ , together with an MLc -equivariant map p˜: P˜ → GL(E), where MLc (n) acts on the principal GL(n)-bundle GL(E) through the homomorphism π : MLc (n) → GL(n). The pair (P˜ , p˜) plays the role of a “Spinc -structure without the metric or orientation”. When E = T M , and after fixing an orientation, the concept of an MLc structure is equivalent to Duflo and Vergne’s notion of a “quantum line bundle” on M ; see [Ve4]. To a Pinc structure (P, p) we associate an MLc structure (P˜ , p˜) by P˜ = P ×Spinc (n) MLc (n) and p˜([u, a]) = p(u) · π(a). Conversely, from an MLc structure (P˜ , p˜) and a metric on E we get a Pinc structure by taking P to be the preimage of O(E) in P˜ . Therefore, on a vector bundle E, a Pinc structure contains the same information as an MLc structure together with a fiberwise metric. An equivalence of MLc structures (P˜ , p˜) and (P˜ 0 , p˜0 ) is an isomorphism F˜ : P˜ → 0 ˜ P which lifts the identity map on GL(E):

(D.9)

P˜   p˜y

F˜

−−−−→ identity

P˜ 0   p˜0 y

GL(E) −−−−−→ GL(E)

On a vector bundle with a fiberwise metric, it is easy to see that MLc structures up to equivalence are the same as metric Pinc structures up to equivalence. Combining this with the results of the previous sections yields the following result:

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Proposition D.40. Let E → M be an oriented vector bundle. There exists a natural one to one correspondence between the following sets of structures: (1) Oriented Spinc structures on E, up to homotopy; (2) Pinc structures on E, up to homotopy; (3) MLc structures on E, up to equivalence. Hence, the homotopy classification of oriented Spinc structures is the same as the classification of each of the structures (1)–(3). See Proposition D.43 below. Finally, on a symplectic vector bundle, an MLc structure is the same as a Mpc (metaplecticc ) structure, as we now explain. The metaplectic group Mp(2n) is the non-trivial double covering of the (real) symplectic group Sp(2n). The metaplectic c group is Mpc (2n) = Mp(2n) ×Z2 U(1).

A metaplecticc structure on a symplectic vector bundle E of rank 2n is an extension of its structure group from the symplectic group Sp(2n) to the metaplecticc group Mpc (2n). A symplectic structure on a vector bundle E → M determines a reduction of structure group of any MLc structure to an Mpc structure: Let Sp(E) denote the bundle of symplectic frames (whose fiber over m ∈ M is the set of linear symplectomorphisms of Em with R2n ). If (P˜ , p˜) is an MLc structure, the preimage of Sp(E) in P˜ is a principal bundle with structure group Mpc and provides us with a metaplecticc structure on E. 2.7. Classification of Spinc -structures, and the distinguishing line bundle. We will now classify the Spinc -structures on a vector bundle E, up to homotopy. Recall that Spinc (n) = Spin(n) ×Z2 U(1) and there is a natural projection π : Spinc (n) → SO(n). We identify K = ker π with U(1) by [1, c] 7→ c. Note that Spinc (n) ×K U(1) = Spinc (n).

Given a complex Hermitian line bundle L over M , we can “twist” (P, p) by L and obtain another Spinc -structure, (P 0 , p0 ), by P 0 = P ×K U(L),

where U(L) ⊂ L is the unit circle bundle. The projection p0 : P 0 → GL(E) and the principal Spinc (n)-action are induced from (P, p). The fact that these are well defined follows from the fact that K = ker π and that K is the center of Spinc (n). Note that (P 0 , p0 ) induces the same metric and orientation as (P, p). Lemma D.41. Let (P, p) be a Spinc structure. Let (P 0 , p0 ) and (P 00 , p00 ) be the results of twisting (P, p) by line bundles L0 and L00 . If the line bundles L0 and L00 are equivalent, then the Spinc structures (P 0 , p0 ) and (P 00 , p00 ) are homotopic. Proof. An isomorphism L0 → L00 naturally extends to an isomorphism P ×K U(L ) → P ×K U(L00 ) which respects the maps p0 and p00 . 0

Because line bundles up to equivalence are classified by H 2 (M ; Z), we get an action of H 2 (M ; Z) on the set of homotopy classes of Spinc -structures: Definition D.42. Let δ ∈ H 2 (M ; Z), We define the action of δ on (P, p) to give the Spinc structure with P 0 = P ×K U(L), where L is a complex Hermitian line bundle with c1 (L) = δ.

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In exactly the same way we can twist a Pinc structure or an MLc structure. For each of these structures, we replace the principal bundle P by the bundle P ×K U(L) where K ∼ = U(1) is in the center of the structure group Pinc (n) or MLc (n). Proposition D.43. Let E → M be an oriented vector bundle which admits a Spinc -structure. (1) The action of H 2 (M ; Z) on the set of homotopy classes of oriented Spinc structures on E is effective and transitive. In particular, this set is (noncanonically) in one-to-one correspondence with the cohomology group H 2 (M ; Z). (2) Let (P, p) be an oriented Spinc -structure on E with determinant line bundle Ldet , and let L0det be the determinant line bundle of the Spinc -structure obtained by the action of u ∈ H 2 (M ; Z) on (P, p). Then c1 (L0det ) = c1 (Ldet ) + 2u.

(3) A line bundle L is the determinant line bundle for some Spinc -structure on E if and only if c1 (L) = w2 (E) mod 2. Remark D.44. This proposition shows that the determinant line bundle Ldet does not determine the homotopy class of a Spinc -structure on E uniquely. Namely, Ldet determines a Spinc -structure up to elements of the kernel of the homomorphism H 2 (M ; Z) → H 2 (M ; Z) induced by multiplication by 2.

The proof of Proposition D.43 follows from a construction which is reverse, in a certain sense, to the H 2 (M ; Z) action of Definition D.42. Namely, we will give a recipe which associates to any two oriented Spinc -structures (P, p) and (P 0 , p0 ) on E a “distinguishing” complex line bundle L over M , such that twisting (P, p) by L will give (P 0 , p0 ). The Chern class δ = c1 (L) ∈ H 2 (M ; Z) will only depend on the homotopy classes of the Spinc -structures. By Proposition D.39, we may restrict our attention to oriented metric Spinc structures, and will need to show that equivalent structures give isomorphic line bundles. Recall that P and P 0 can be viewed as principal K-bundles over SO(E) = P/K = P 0 /K, where K = ker π ∼ = U(1). Since K commutes with all elements of Spinc (n), we can view P → SO(E) and P 0 → SO(E) as Spinc (n)-equivariant K-bundles. Denote by F and F0 the associated equivariant complex line bundles. Then Spinc (n) also acts on (F)∗ ⊗F0 so that the action of K is trivial. (Here we view K as a subgroup of Spinc (n) and not as the structural group.) Thus the Spinc (n)action on (F)∗ ⊗ F0 factors through an SO(n)-action. It follows that this line bundle descends to (i.e., is a pull-back of) a line bundle Lδ over SO(E)/ SO(n) = M . We set δ to be the first Chern class of the resulting line bundle on M . Definition D.45. The line bundle Lδ = Lδ (P, P 0 ) constructed above is called the distinguishing line bundle of the Spinc -structures (P, p) and (P 0 , p0 ). The cohomology class δ(P, P 0 ) = c1 (L) is called the distinguishing cohomology class. . Remark D.46. If we replace (P, p) and (P 0 , p0 ) by metric Spinc structures that are homotopic to them, the resulting distinguishing line bundle is isomorphic to Lδ . By Proposition D.39, we get a line bundle Lδ , defined canonically up to equivalence of complex line bundles, for every two Spinc structures (P, p) and (P 0 , p0 ), which only depends on their homotopy class.

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Remark D.47. We have Lδ (P 0 , P ) ∼ = Lδ (P, P 0 )∗ and (D.10)

Lδ (P, P 00 ) ∼ = Lδ (P, P 0 ) ⊗ Lδ (P 0 , P 00 )

for Spinc -structures P , P 0 , and P 00 . Likewise, δ(P 0 , P ) = −δ(P, P 0 ) and δ(P, P 00 ) = δ(P, P 0 )+δ(P 0 , P 00 ). We emphasize that δ(P, P 0 ) and, up to isomorphism, Lδ (P, P 0 ) are invariants of homotopy but not of bundle equivalence of Spinc -structures. However, 2δ(P, P 0 ) and Lδ (P, P 0 )⊗2 are invariants of bundle equivalence. The class δ depends only on the equivalence class of the metric Spinc structures (P, p) and (P 0 , p0 ). Proof of Proposition D.43. To prove the first assertion, it is enough to observe that the distinguishing line bundle for the Spinc -structures P and P 0 = P ×K U(L) is Lδ = L. The second assertion follows immediately from the fact that the composition det U(1) ∼ = K ,→ Spinc (n) → U(1) is the map c 7→ c2 . The third statement follows from Proposition D.31. Remark D.48. In practice we will need to compare Spinc structures which may induce opposite orientations: on a vector bundle without a pre-chosen orientation, two Spinc structures are distinguished by (1) the distinguishing line bundle Lδ ; (2) whether or not they induce the same orientation. This follows immediately from the results of Section 2.5. Note that the distinguishing line bundle can be defined even if (P, p) and (P 0 , p0 ) induce opposite orientations. For instance, we may define it by passing to the associated Pinc structures and noting that the above construction of the distinguishing line bundle works word-for-word for a pair of Pinc structures, with Spinc (n) replaced by Pinc (n) and SO(E) replaced by O(E) everywhere. Equivalently, we may replace (P, p) by τ (P, p) (see Section 2.5) and then carry out the above construction of Lδ . All the above remains valid in the presence of a proper G-action. In particular, we can twist a G-equivariant Spinc structure by a G-equivariant line bundle, and, for any two G-equivariant Spinc structures P and P 0 , the distinuishing line bundle Lδ (P, P 0 ) is a G-equivariant line bundle. 3. Spinc -structures and stable complex structures In this section we associate a Spinc structure (P, p) to any stable complex structure J, and we compare the Spinc structures coming from different stable complex structures. Remark D.49. In the literature, one usually takes stable complex structures up to bundle equivalence and Spinc structures up to homotopy. With this approach, the map J 7→ (P, p) is not well defined: if a stable complex structure is only defined up to bundle equivalence, so is the resulting Spinc -structure, and one does not get a well defined homotopy class. One of the reasons that we keep track of both equivalence relations is to avoid this problem.

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3.1. The canonical Spinc -structure on a complex vector bundle. In the context of this book, the main sources of Spinc -structures are complex and stable complex structures. The anti-canonical line bundle of a complex vector bundle E is the top wedge K ∗ = Λm C E, where m = rkC E. (The symbol K is usually reserved for the canonical line bundle; the two are dual to each other.) Similarly, if E is equipped with a stable complex structure, its anti-canonical line bundle is the anti-canonical line bundle of the complex vector bundle (E ⊕ Rk , J) through which the stable complex structure on E is defined. The isomorphism class of the anti-canonical line bundle only depends on the bundle equivalence class of the stable complex structure. Proposition D.50. A complex vector bundle E and complex line bundle L over M determine a Spinc -structure on E uniquely up to homotopy. The determinant line bundle of this Spinc -structure is isomorphic to K ∗ ⊗ L⊗2 where K ∗ is the anticanonical line bundle of E. Homotopic (resp., bundle equivalent) complex structures on E give rise to homotopic (resp., bundle equivalent) Spinc structures. Proof. We first construct an inclusion of U(n) into Spinc (2n). We start with the standard inclusion U(n) ,→ SO(2n) and lift it to an inclusion U(n) ,→ Spin(2n) for the connected double covering U(n) of U(n). Likewise, the homomorphism det : U(n) → U(1) lifts to a homomorphism U(n) → U(1) ∼ = U(1). These homomorphisms give rise to an inclusion U(n) → Spin(2n) × U(1) which descends to the required inclusion U(n) ,→ Spin(2n) ×Z2 U(1) = Spinc (2n). Set P = U(E) ×U(n) Spinc (2n) with its natural projection p to SO(E) = U(E) ×U(n) SO(2n), where U(E) is the principal U(n)-bundle of unitary frames in E with respect to some Hermitian metric, and where the projection p arises from the homomorphism π : Spinc (2n) → SO(2n). Then (P, p) is a Spinc -structure on E, and, up to homotopy, it is independent of the choice of the Hermitian metric. We get a Spinc -structure with the required determinant line bundle by twisting P by L as in Definition D.42. Suppose now that we have a family Jt of complex structures on E, parametrized by t ∈ [0, 1]. This makes the product E × [0, 1] into a complex vector bundle over M × [0, 1]. The previous construction gives rise to a Spinc structure (P˜ , p˜) on E × [0, 1], whose restriction to E × {t} is the Spinc structure associated with Jt . The trivial connection on the fibration GL+ (E) × [0, 1] → [0, 1] lifts to a connection on P˜ → [0, 1] whose parallel transport provides us with a homotopy (see Definition D.37) between the Spinc structures associated with J0 and J1 . An equivalence of complex line bundles E0 → E1 gives an equivalence of principal U(n) bundles U(E0 ) → U(E1 ) and, further, an equivalence of the associated Spinc bundles. Combining Proposition D.50 with the results of Section 2.7, we get the following result. Lemma D.51. Any two complex structures J, J 0 on a real vector bundle E determine a “distinguishing” complex line bundle Lδ = Lδ (J, J 0 ) uniquely up to isomorphism. We have K 0 = K ⊗ L⊗2 δ

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where K 0 and K are the anti-canonical line bundles associated to J and J 0 . If J 00 is a third complex structure, Lδ (J, J 0 ) ⊗ Lδ (J 0 , J 00 ) = Lδ (J, J 00 ).

Applying this to the bundles E ⊕ Rk , we get a similar result for stable complex structures: Lemma D.52. Any two stable complex structures J, J 0 on a real vector bundle E determine a “distinguishing” complex line bundle Lδ = Lδ (J, J 0 ) uniquely up to isomorphism. We have K 0 = K ⊗ L⊗2 δ

where K 0 and K are the anti-canonical line bundles associated to J and J 0 . For any line bundle L, the Spinc structure associated to J 0 and L is homotopic to the Spinc structure associated to J and L ⊗ Lδ . If J 00 is a third stable complex structure, Lδ (J, J 0 ) ⊗ Lδ (J 0 , J 00 ) = Lδ (J, J 00 ).

We will work out in detail one important special case: Example D.53. Consider Cd = Cr ×Cd−r with the standard complex structure J = Jr ⊕ Jd−r and the non-standard complex structure J # = (−Jr ) ⊕ Jd−r , where Jr and Jd−r are, respectively, the standard complex structures on Cr and Cd−r . Let (S 1 )d act by rotating the coordinates. Consider the spinc structures associated Vr r C . to J and J # . Their (equivariant) distinguishing line bundle is the top wedge Note that J and J # induce the same orientation if and only if r is even but that their distinguishing line bundle is always defined by Remark D.48.

Remark D.54. Every line bundle over Cd is trivial. The example is meaningful when we consider all structures as G-equivariant with G = (S 1 )d . Proof of Example D.53. It is enough to consider the case r = d = 1; the general case follows by flipping the complex structure in stages, one coordinate at a time. Because orientation is flipped, we must work with Pinc , not Spinc structures. Groups. We identify U(1) = SO(2) and denote   cos θ − sin θ eiθ = . sin θ cos θ

An element of O(2) = SO(2) o Z2 can be written uniquely as either eiθ or eiθ ( 01 10 ). In multiplying two such elements, note that eiθ1 ( 01 10 )eiθ2 = ei(θ1 −θ2 ) ( 01 10 ). iθ

iθ

We write an element of Pin(2) formally as either e 2 or e 2 ( 01 10 ) with θ ∈ R/4πZ. The homomorphism iθ 2

iθ

iθ 2

sends e to e and e The inclusion map

q : Pin(2) → O(2) ( 01 10 )

to eiθ ( 01 10 ).

U(1) ,→ Pinc (2) := Pin(2) ×Z2 U(1)

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is iθ

iθ

eiθ 7→ [e 2 , e 2 ].

Frame bundles. Consider V = R2 = C with the standard circle action and with the distinguished element e = 1. Its orthogonal frame bundle is O(V ) = {(w, v1 , v2 ) | w ∈ V, v1 , v2 ∈ V, kv1 k = kv2 k = 1, v2 ⊥ v1 }.

The principal O(2) action is B : (v1 , v2 ) 7→ (v1 , v2 )B for B ∈ O(2). The circle action on V lifts to the left circle action on O(V ) given by λ · (w, v1 , v2 ) = (λw, λv1 , λv2 ). Fix a √ complex structure J on V . Concretely, we will consider either J1 = or J2 = − −1. The unitary frame bundle is

√ −1

U(V ) = {(w, u) | w ∈ V, u ∈ V, kuk = 1}.

The principal U(1) action is eiθ : (w, u) 7→ (w, eJθ u)

where eJθ u = cos θu + sin θJu. The left circle action is Note that if λ = eiα , then

λ · (w, u) = (λw, λu).

λu = e±Jα u √ √ according to whether J = −1 or J = − −1. c The associated Pin (2) bundle over V is

(D.11)

P = {[w, u, A, a] | (w, u) ∈ U(V ), [A, a] ∈ Pinc (2)} with (D.12)

iθ

iθ

[w, eJθ u, A, a] = [w, u, e 2 A, e 2 a].

The map to O(V ) sends [w, u, A, a] to (w, v1 , v2 ) by (v1 , v2 ) = (u, Ju)q(A). The left circle action, λ · [w, u, A, a] = [λw, λu, A, a], can, by (D.11) and (D.12), be √ 1 1 written√as λ · [w, u, A, a] = [λw, u, λ± 2 A, λ± 2 a], according to whether J = −1 or J = − −1. Line bundles. We now take the associated line bundle over O(V ). An element of L = P ×U(1) C is written as [w, u, A, a, z]. However, since {u = e} is a trivialization of U(V ) and Pinc (2) ×U(1) C = Pin(2) ×Z2 C, we can set u = e and a = 1, and get that L = {[w, A, z] | w ∈ V, [A, z] ∈ Pin(2) ×Z2 C}. The map to O(V ) sends [w, A, z] to (w, v1 , v2 ), where (v1 , v2 ) = (e, Je)q(A). The left circle action is 1

1

λ · [w, A, z] = [λw, λ± 2 A, λ± 2 z], √ √ according to whether√J = −1 or J = √ − −1. Let L1 and L2 denote the bundles obtained from J1 = −1 and J2 = − −1.

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We need to show that the distinguishing line bundle is V × C with the circle action λ(w, ζ) = (λw, λζ). Let π ∗ Lδ denote its pullback to O(V ). We define the required isomorphism L2 ⊗ π ∗ Lδ → L 1 by πi [w, A, z ⊗ ζ] 7→ [w, ( 01 10 )e 4 A, zζ]. The fact that this respects the maps to O(V ) follows easily from the facts that (e, J2 e) = (e, −J1 e) and   πi πi 0 q(A). q ( 01 10 )e 4 A = ( 01 10 )e 2 q(A) = 10 −1 Equivariance with respect to the left circleaction follows easily from the facts that πi 1 1 1 πi 1 ( 01 10 )e 4 λ− 2 A = λ 2 ( 01 10 )e 4 A and λ− 2 z (λζ) = λ 2 (zζ).

3.2. Destabilization of Spinc -structures. By Proposition D.50, every almost complex manifold equipped with a complex line bundle has a canonical Spinc structure. Likewise, a stable complex structure on M and a complex line bundle give rise to a “stable” Spinc -structure, i.e., a Spinc -structure on the stable tangent bundle. It turns out, however, that Spinc -structures naturally destabilize: every “stable” Spinc -structure induces a genuine Spinc -structure. Proposition D.55 (Cannas da Silva, [CKT, Lemma 2.4]). Let E be a real ndimensional vector bundle. Every Spinc -structure on the Whitney sum E ⊕ Rk canonically induces a Spinc -structure on E with the same determinant line bundle. Homotopic Spinc -structures on E ⊕ Rk induce homotopic Spinc -structures on E.

Proof. Let P 0 be a principal Spinc (n + k) bundle for a Spinc structure on E ⊕ Rk and let p0 : P 0 → SO(E ⊕ Rk ) be the corresponding map of principal bundles. Recall that this map makes P 0 into a Spinc (n + k) equivariant U(1) bundle over SO(E ⊕ Rk ). The inclusion E ,→ E ⊕ Rk induces an SO(n)-invariant inclusion SO(E) ,→ SO(E ⊕ Rk ). Let P denote the preimage of SO(E) in P 0 and let p = p0 |P . Then we have a pull-back diagram −−−−→

P   py

P0   p0 y

SO(E) −−−−→ SO(E ⊕ Rk ) which is equivariant with respect to the natural homomorphisms Spinc (n) −−−−→ Spinc (n + k)     y y SO(n) c

−−−−→ SO(n + k).

The pair (P, p), which is a Spin (n) equivariant principal U(1) bundle over SO(E), provides the required Spinc structure on E. Destabilization of MLc structures is defined in a similar way. It is easy to see that equivalent MLc structures on E ⊕ Rk destabilizer to equivalent MLc structures on E. Because MLc structures up to equivalence are the same as oriented Spinc structures up to homotopy (see Proposition D.40), homotopic Spinc structures on E ⊕ Rk destabilize to homotopic Spinc structures on E.

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Remark D.56. This construction gives a one-to-one correspondence between the sets of homotopy classes of Spinc -structures on E ⊕ Rk and on E. (The inverse comes from the natural inclusion Spinc (n) → Spinc (n + k).) Spinc -structures are analogous, in some sense, to O(n)-structures on real vector bundles (i.e., fiberwise Euclidean metrics). Proposition D.55 is well known to also hold for SO(n) structures or O(n) structures; the proof is similar to the one given above. Furthermore, the determinant complex line bundle Ldet is the Spinc analogue of the determinant real line bundle D = O(E) ×det R. From this point of view, the last assertion of Proposition D.43, which gives a necessary and sufficient condition for the existence of a Spinc -structure with a given determinant bundle, is similar to the condition w1 (E) = w1 (D) for E to have an O(n)-structure with real determinant line bundle D. (In particular, E is orientable if and only if w1 (E) = 0.) Recall in this connection that w2 (E) = 0 is the necessary and sufficient condition for E to admit a Spin-structure. 3.3. Stable complex structures and Spinc -structures; the shift formula. We recall some definitions from Section 1. A stable complex structure on a vector bundle E is a complex structure on a Whitney sum E ⊕ Rk . Two notions of equivalence for stable complex structures are of interest for us. Stable complex structures J0 and J1 on vector bundles E0 and, respectively, E1 are said to be bundle equivalent if the complex vector bundles E0 ⊕ Rk0 and E1 ⊕ Rk1 on which J0 and J1 are actually defined become isomorphic after addition of some numbers of trivial complex vector bundles to each of them. On the other hand, stable complex structures J0 and J1 on the same vector bundle E are said to be homotopic if the complex structures on E ⊕ Rk0 and E ⊕ Rk1 become homotopic after addition of some numbers of trivial complex vector bundles. Combining Propositions D.50 and D.55, we obtain Proposition D.57. A stable complex vector bundle E and complex line bundle L over M determine a Spinc -structure on E uniquely up to homotopy. The determinant line bundle of this Spinc -structure is isomorphic to Ldet = K ∗ ⊗ L⊗2 , where K ∗ is the anti-canonical line bundle of the stable complex structure. Homotopic stable complex structures determine homotopic Spinc -structures on E. In particular, every stable complex structure on M and complex line bundle over M determine a Spinc -structure on M , up to homotopy. Furthermore, we also have the following “Spinc Shift Formula”: Proposition D.58. Let J0 and J1 be stable complex structures on E and L0 and L1 be complex line bundles over M . Denote by Lδ the distinguishing line bundle for the Pinc -structures determined by (J0 , L0 ) and (J1 , L1 ) (cf. Remark D.48). Then the pairs (J0 , L0 ⊗ Lδ ) and (J1 , L1 ) give rise to homotopic Pinc -structure on E. If J0 and J1 induce the same orientation, we get homotopic Spinc structures on E. Remark D.59. It follows from Propositions D.50 and D.58 that ⊗2 ⊗2 ∗ ∗ L⊗2 0 ⊗ K 0 ⊗ Lδ = L1 ⊗ K 1 ,

where K∗0 and K∗1 are the anti-canonical bundles of J0 and J1 . In particular, if ∗ L0 = L1 , we have L⊗2 δ = K0 ⊗ K1 . Note also that, as is clear from the proof given below, Lδ is completely determined by J0 and J1 when L0 = L1 . Proof of Proposition D.58. By adding, if necessary, a trivial bundle, we may ensure that all complex structures are defined on the same vector bundle, which

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we denote again by E. Denote by Pi , i = 0, 1, the Spinc -structures determined by (Ji , Li ) and by P 0 the Spinc -structure arising from (J0 , L0 ⊗ Lδ ). Then Lδ (P0 , P 0 ) = Lδ = Lδ (P0 , P1 ).

By the cocycle equation (D.10) of Remark D.46, Lδ (P1 , P 0 ) = Lδ (P0 , P1 )∗ ⊗ Lδ (P0 , P 0 ) = L∗δ ⊗ Lδ is trivial. We finish the proof by the classification of Section 2.7. 3.4. Equivariant Spinc -structures and reduction. On a G-equivariant vector bundle E → M , we define a G-equivariant Spinc -structure by requiring P → M to be G-equivariant principal Spinc (n)-bundle and p : P → GL+ (E) to be a G × Spinc (n)-equivariant map. The definitions and results of Sections 3.1–3.3 extend immediately to the equivariant case. (In Proposition D.43, one needs to use Theorem C.47 to identify the group of G-equivariant complex line bundles with 2 HG (M ; Z).) Since all constructions are canonical, Propositions D.50, D.55, D.57, and D.58 hold literally for equivariant Spinc -structures. In same vein, these results extend to Spinc -structures on orbifolds. The reduction procedure for Spinc -structures is very similar to reduction of stable complex structures (see Section 2.3 of Chapter 5). As in Section 2 of Chapter 5, let G be a torus, M a G-manifold, and Ψ : M → g∗ an abstract moment map. For a regular value α ∈ g∗ , the reduced space is the orbifold Mα = Ψ−1 (α)/G. Proposition D.60 (Reduction of Spinc -structures). A G-equivariant Spinc structure (P, p) on M gives rise to a Spinc -structure (Pα , pα ) on the reduced space Mα , unique up to homotopy. Homotopic Spinc -structures reduce to homotopic Spinc -structures on Mα . Proof. Let Z = Ψ−1 (α). Recall that TZ M decomposes into TZ M = π ∗ (T Mα ) ⊕ g ⊕ g∗ as an equivariant vector bundle, where π : Z → Mα = Z/G is the natural projection, and that this decomposition is canonical up to homotopy. (See equation (5.6).) The quotient P/G is an (orbifold) Spinc -structure on the vector bundle T Mα ⊕ g ⊕ g∗ over Mα . Applying the destabilization procedure of Proposition D.55, we obtain the reduced Spinc -structure on the reduced space Mα . The second assertion follows easily from the definitions. Now let J be a G-equivariant stable complex structure on M and P the associated Spinc -structure. Reducing J as in Section 2.3 of Chapter 5, we obtain a stable complex structure Jα on Mα , and reducing P , we obtain the reduced Spinc -structure Pα on Mα . Proposition D.61. Pα is the Spinc -structure associated with Jα . We leave the proof as an exercise to the reader. 3.5. Quantization. In this book, we use a Spinc -structure to make sense of “quantization”. Let us recall how this is done. A Spinc -structure (P, p) on a manifold X, together with the choice of a connection on the bundle P → X, give rise to an elliptic operator D acting on sections of certain vector bundles over X. (See below.) We define the quantization to be the

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index (kernel minus cokernel) of this operator. We will see below that this index is well defined. Let us recall in greater detail how to get an elliptic operator from a Spinc structure (P, p) and a connection on P . (See [Bot2, LM, Du, Fr].) We assume that the underlying manifold X is 2n-dimensional, so that P is a principal Spinc (2n) bundle and p gives a vector bundle isomorphism TX ∼ = P ×Spinc (2n) R2n . The group Spinc (2n) has two famous complex linear representations, called the spinor representations, and denoted M+ and M− , and a Spinc (2n)-equivariant linear map

(D.13)

R2n ⊗ M+ → M− .

Let S+ and S− denote the bundles associated to P with fibers M+ and M− : S+ = P ×Spinc (2n) M+

and

S− = P ×Spinc (2n) M− .

From (D.13) we get a bundle map (D.14)

T X ⊗ S + → S− .

Choose a connection on P . This induces a connection on S+ . Covariant differentiation defines a map (D.15)

∇ : sections of S+ → sections of T ∗ X ⊗ S+ .

The metric allows us to identify T ∗ X = T X. Then we can compose the maps (D.14) and (D.15) and get the Dirac Spinc operator D : sections of S+ → sections of S− .

The above definition of the Dirac Spinc operator depends on the choice of a connection on the principal bundle P → X. However, the space of such connections is connected. This implies that different choices of a connection give rise to homotopic Dirac operators. By the Fredholm homotopy invariance of the index, these operators have the same index. Hence, the index associated to a Spinc structure is well defined and does not depend on the choice of connection. Homotopic Spinc -structures also give rise to homotopic Dirac operators. Therefore, the index associated to a Spinc -structure only depends on the homotopy class of the Spinc -structure. One applications of this is in showing that the quantization of a reduced space is well defined. See Example D.13. This argument fails to apply to Spinc structures that are merely bundle equivalent. However, in this case we can invole the Atiyah–Singer index theorem, [ASi]. By this theorem, the index of the Spinc Dirac operator is determined by the characteristic classes of the Spinc structure. Thus we conclude that the index is also an invariant of bundle equivalence. Explicitly, the Atiyah Singer index theorem gives Z ˆ M ), dim Q(M ) = ec1 (Ldet ) A(T M

where we integrate with respect to the orientation induced by the Spinc structure. ˆ M ) is independent of the Spinc structure and that ec1 (Ldet ) only Notice that A(T depends on the determinant line bundle. So Q(M ) is defined for any Pinc structure and orientation. If we keep the Pinc structure but flip the orientation, this has the effect of flipping the sign of Q(M ). Concretely, Q(M ) is the index (kernel minus

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co-kernel) of an elliptic operator, and flipping the orientation has the same effect as switching the kernel and co-kernel. In Section 3.1 we have associated a Spinc structure to a pair (J, L) consisting of a stable complex structure J and a line bundle L on a manifold M . We define the quantization Q(M, J, L) to be the index associated to this Spinc structure. By the results of Sections 3.5 and 2.6, one can also associate a Dirac operator to a Pinc structure and orientation, or to an MLc structure and orientation. The operator itself depends on certain auxiliary choices, but its index is independent of these choices. This index is, by definition, the quantization of the given orientation and Pinc , or MLc , structure. We have also defined a distinguishing line bundle Lδ = Lδ (J2 , J1 ) associated to any two stable complex structures J1 and J2 , such that the Spinc structure associated to J1 and L is homotopic to the Spinc structure associated to J2 and L ⊗ Lδ . Therefore, Q(M, J2 , L ⊗ Lδ ) = ±Q(M, J1 , L), where the sign is according to whether or not J2 and J1 induce the same orientation. When ω is a symplectic form on M , one usually defines the quantization Q(M, ω) to be the index of the Dirac Spinc operator D associated a the prequantization line bundle L → M and an almost complex structure J compatible with ω. (See Section 3.) However, there is a slightly different definition which has some advantages: one may take the index of D to define the quantization of (M, ω) where ω is half the curvature of the determinant bundle associated to the Spin c structure. (This leads to a different notion of whether (M, ω) is at all quantizable.) In this appendix we do not work with two-forms; we refer the reader to Section 7.3 of Chapter 6 for a further discussion of this notion of Spinc -quantization.

APPENDIX E

Assignments and abstract moment maps In this appendix we answer the questions of existence of a (proper) abstract moment map for a given torus action and existence of a compatible two-form for a given abstract moment map. This is done using the notion of an assignment, which is a combinatorial counterpart of an abstract moment map. We largely follow [GGK3]. 1. Existence of abstract moment maps We recall that an abstract moment map on a G-manifold M is a G-equivariant map Ψ : M → g∗ such that for each subgroup H of G the composition ΨH : M → g∗ → h∗ is locally constant on the H fixed point set M H . Every manifold with a G-action admits an abstract moment map: the zero map. This map is never proper unless the manifold is compact. In this section we answer the question of when a G-manifold admits a proper (in fact, polarized) abstract moment map. A necessary condition for an action to admit a proper abstract moment map Ψ is that each component of the fixed point set be compact. (Recall that Ψ is constant on each such component.) Is this condition sufficient? Moreover, does a (proper) abstract moment map exist with prescribed values at the fixed points? 1.1. Existence of abstract moment maps for circle actions. Answers to the above questions take a particularly simple and attractive form if G is a circle, when abstract moment maps are simply G-invariant functions that are constant on the connected components of the fixed point set. Theorem E.1. Let G be a circle acting on M , and let ψ : M G → R be a locally constant function. (1) There exists an abstract moment map Ψ : M → R with Ψ|M G = ψ. (2) Assume that ψ is proper and bounded from below. Then Ψ can be chosen to be proper and bounded from below. In other words, if G is the circle group, we can prescribe the values of an abstract moment map on the connected components of M G completely arbitrarily. If M G is compact, the condition of the second assertion is satisfied automatically, and every locally constant function on M G extends to a proper abstract moment map. Proof of Theorem E.1. The theorem follows from the following two facts, applied to X = M G and f = ψ. (1) Let X ⊂ M be a closed submanifold and f : X → R a smooth function. Then there exists a smooth function F : M → R such that F |X = f . Moreover, 257

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if f is proper and bounded from below, F can be chosen to be proper and bounded from below too. (2) Let F : M → R be proper and bounded from below. Then the average F of F by a compact group action is also proper and bounded from below. Let us prove the first fact. Fix a tubular neighborhood U of X in M , and let π : U → X be a smooth projection which extends to a proper map from the closure U to X. Let ρ, 1 − ρ be a smooth partition of unity subordinate to the covering of M by the two open sets U and M rX. Pick a smooth function ϕ : M → R which is proper and bounded from below (see, e.g., [GP, Chapter 1, Section 8]). Then F = ρf + (1 − ρ)ϕ has the desired properties. −1 To prove the second fact, notice that the pre-image F ([−a, a]) is contained in the set G · F −1 ([−a, a]), which is the image of the compact set G × F −1 ([−a, a]) under the continuous action mapping G × M → M . In Theorem E.1, it is not true that if ψ is just proper (but not bounded), then Ψ can be chosen to be proper. In general, a proper map on a closed submanifold X of M might not extend to a proper map on M . For instance, the function f (0, y) = y on the y-axis does not extend to a continuous proper function F from R2 to R. A similar counterexample involving abstract moment maps is given below. Example E.2. Let M be obtained by the following plumbing construction: M = Z × S 2 × D2 / ∼,

where S 2 is the unit sphere {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} and D2 is the disc {(u, v) ∈ R2 | u2 + v 2 < 2 }, and the equivalence relation ∼ is defined by p p (n, x, y, 1 − x2 − y 2 , u, v) ∼ (n + 1, u, v, − 1 − u2 − v 2 , x, y)

for all n. We equip M with the diagonal circle action where

eiθ · [n, x, y, z, u, v] = [n, x0 , y 0 , z, u0 , v 0 ],

    x u cos θ − sin θ x0 u 0 . = y v sin θ cos θ y0 v0 The function ψ([n, 0, 0, 1, 0, 0]) = n is proper and locally constant on the fixed point set, but it does not extend to a proper function Ψ : M → R. Note that the function ψ extends to a (non-proper) Hamiltonian moment map, for a closed two-form whose pullback to each {n} × S 2 × {0} is non-negative and has total area equal to 2π. Also note that M does admit some proper abstract moment map: take the G-average of any function M → R which is proper and bounded from below. 

1.2. Assignments. Let us now investigate more carefully the question of existence of an abstract moment map for an action of a torus whose dimension is greater than one. Theorem E.1 is no longer true in this case. Example E.3. Let Ψ = (Ψ1 , Ψ2 ) be an abstract moment map on M = S 2 × S 2 with G = S 1 × S 1 acting by rotating each of the two factors. Then Ψ must send the four fixed points to the corners of a rectangle in R2 whose sides are parallel to the axes. Thus the values Ψ(M G ) cannot be assigned arbitrarily. Moreover, abstract moment maps might not even separate the components of the fixed point set:

1. EXISTENCE OF ABSTRACT MOMENT MAPS

259

Example E.4. Let S 4 be the unit sphere in C × C × R, and let G = S 1 × S 1 act on it by rotating each of the first two factors. There are two fixed points: the North Pole and the South Pole. The set of points fixed by the first S 1 is connected and contains both poles. The same is true for the set of points fixed by the second S 1 . Consequently, any abstract moment map on S 4 must have the same value at the poles. These examples stress the role of the orbit type strata other than the components of the fixed point set. For a stratum X, we denote by gX the infinitesimal stabilizer of any of its points. Suppose that Ψ : M → g∗ is an abstract moment map. Then for each infinitesimal orbit type stratum X in M , the map Ψ followed by the projection g∗ → g∗X gives an element A(X) of g∗X . An important observation is that the existence question for an abstract moment map is equivalent to the existence question for such an assignment, X 7→ A(X). We make this precise in Theorems E.16 and E.19, which rely on the definition and example below. Note that the infinitesimal orbit type strata form a poset (partially ordered set) with X  Y if and only if X is contained in the closure of Y . Definition E.5. An assignment is a function A that associates to each infinitesimal orbit type stratum X in M an element A(X) of g∗X and that satisfies the following compatibility condition: if X  Y , then A(Y ) is the image of A(X) under the restriction map g∗X → g∗Y . The linear space of all assignments on M is denoted by A(M ). In Appendix F we identify the space of assignments as the degree zero piece in a cohomology theory, and provide natural generalizations of this notion.

Example E.6. Let Ψ : M → g∗ be an abstract moment map. Then A(X) = Ψ (X) is an assignment. The assignment A and the moment map Ψ are said to be associated with each other. If the abstract moment map is exact, i.e., Ψ arises from a one-form µ so that Ψξ = µ(ξM ), the corresponding assignment is zero. gX

Example E.7. When G is the circle group, an assignment simply associates a real number to each component of the fixed point set. Thus, in this case, A(M ) = G (g∗ )π0 (M ) . Example E.8. Consider the action of the two-dimensional torus G = S 1 × S 1 on M = CP2 given by (t1 , t2 )[z0 : z1 : z2 ] = [z0 : t1 z1 : t2 z2 ]. This action has three fixed points, and every assignment A is uniquely determined by its value at the fixed points. There are, however, three relations between the values A|M G ∈ (g∗ )3 , coming from the strata with one-dimensional stabilizers. As a result, A(M ) is three-dimensional. Geometrically, the assignment values at the fixed points are the vertices of a triangle in R2 with two equal sides parallel to the coordinate axes. Example E.9. Let M1 be a G1 -manifold and M2 be a G2 -manifold. Then A(M1 × M2 ) = A(M1 ) ⊕ A(M2 ) for the G1 × G2 -action on M1 × M2 . Remark E.10. Replacing in Definition E.5 the infinitesimal orbit type stratification by the orbit type stratification notion leads to the same class of assignments A(M ). Namely, a function that associates to each orbit type stratum X an element of g∗X and that satisfies the compatibility condition of Definition E.5 is in fact constant on each infinitesimal orbit type stratum, and, hence, is an assignment.

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Example E.11. Let M be a K¨ ahler toric manifold (see Section 5 of Chapter 5, or [Fu] or [Au]) with moment map Ψ : M → g∗ . The image Ψ(M ) is a convex polytope [At3, GS2]. A convex polytope is stratified, with the strata being its open faces of various dimensions. The orbit type strata in M are exactly the preimages in M of the open faces in Ψ(M ), [De]. As a consequence, the poset of (infinitesimal) orbit type strata of M is isomorphic to the poset of faces of Ψ(M ). Moreover, the stabilizers can be read from the faces and vice versa: the affine plane spanned by the face Ψ(X) is a shift of the annihilator in g∗ of the Lie algebra gX . The shifts are exactly given by the assignment A(X) = ΨgX (X): affine span(Ψ(X)) = preimage of A(X) under g∗ → g∗X .

The polytope Ψ(M ), and hence the assignment A, determines the manifold, the G-action, and the symplectic form up to an equivariant symplectomorphism, [De], and the equivariant K¨ ahler structure on the strata, [Guil1]. More generally, Example E.12. Let M be an n-dimensional complex manifold and let G be an n-dimensional torus that acts on M . Suppose that each point of M with stabilizer H ⊆ G has a neighborhood which is biholomorphic to a neighborhood of the origin in Cn with an action of H of the following form. The H-action is obtained as the composition of an isomorphism H → (S 1 )k with the (S 1 )k -action on Ck × Cn−k which is standard on the first factor and trivial on the second. (For instance, this is the case if M is a toric manifold.) It follows that for any stratum X, the natural map X M ⊕πY g∗X −→ g∗Y {Y |XY, dim gY =1}

is a linear isomorphism. Therefore, an assignment is determined by its values on the strata Y with dim gY = 1, and these values can be prescribed arbitrarily. Hence, for such M , M A(M ) = g∗Y ∼ = R#{Y |dim gY =1} . Y

Remark E.13. In Example E.11 we saw that the assignment of a symplectic toric manifold M determines its moment polytope Ψ(M ). Similarly, for a toric manifold with a closed invariant two-form which may have degeneracies, the assignment determines its twisted polytope in the sense of [KT1]. An obvious, but important, fact is

Lemma E.14. Let Ψ0 and Ψ1 be abstract moment maps which have the same assignment, A. Then (1 − ρ)Ψ0 + ρΨ1 is also an abstract moment map with assignment A, for any invariant smooth function ρ. Proof. For any H ⊆ G, on every component X of M H ,

H (1 − ρ)ΨH 0 + ρΨ1 = (1 − ρ)A(X) + ρA(X) ≡ A(X)

is constant on X.

Remark E.15. The definition of an assignment can be extended to actions of a non-commutative group G. An assignment can then be defined as a function x 7→ A(x) ∈ g∗x on M such that the following conditions hold:

1. EXISTENCE OF ABSTRACT MOMENT MAPS

• • In the rise to

261

A(g · x) = Ad∗g A(x) for all x ∈ M and g ∈ G. Ah is locally constant on the set M h of points x with h ⊆ gx . non-commutative case, as in Example E.6, an abstract moment map gives an assignment.

1.3. Existence of abstract moment maps for torus actions. The relation between abstract moment maps and assignments is expressed in the following theorem, which asserts that every assignment is associated with some abstract moment map. Let G be a torus. Theorem E.16. Let M be a manifold with a G action. Let A : X 7→ A(X) be an assignment. Then there exists an abstract moment map Ψ : M → g∗ which is associated with A, i.e., such that ΨgX (X) = A(X) in g∗X for every orbit type stratum X. Proof. Let m be a point in M , let h be the infinitesimal stabilizer of m, and let A(m) ∈ h∗ be the element assigned to the orbit type stratum containing m. Let Ψm ∈ g∗ be any element whose projection to h∗ is A(m). Pick an open neighborhood Um of the orbit G · m which equivariantly retracts to the orbit. The constant function Ψm is an abstract moment map on Um whose assignment is A|Um . Choose an invariant partition of unity {ρj } subordinate to the covering of M by the open subsets Um , with Pthe support of ρj contained in the open set Umj . The convex combination Ψ = ρj Ψmj is an abstract moment map. This follows from Lemma E.14, applied to open subsets of the manifold. On a non-compact manifold, it is sometimes required that an abstract moment map be proper. (See [Ka3, GGK2, GGK3].) In fact, we often need a component of Ψ to be proper and bounded from below. Let η be an element of the Lie algebra g. Recall that a map to g∗ is η-polarized if its η-component is proper and bounded from below. Definition E.17. An assignment A is η-polarized if its η-component on M η , (E.1)

Aη : M η → R,

is proper and bounded from below. Notice that the function (E.1) is well defined and locally constant, by the definition of an assignment. Remark E.18. Let H be the closure in G of the one-parameter subgroup generated by η. Then the zero set M η of the vector field ηM is equal to the fixed point set M H of H in M . On this set, the H-component of an assignment is a well–defined function AH : M H → h∗ . An assignment A is η-polarized if and only if this function is η-polarized. For a generic η, we have H = G. For such an η, an assignment is η-polarized if its restriction to the fixed point set, which is an ordinary function A : M G → g∗ , is η-polarized. Theorem E.19. Let M be a manifold with a G action. For every η-polarized assignment X 7→ A(X) on M there exists an η-polarized abstract moment map Ψ : M → g∗ whose assignment is A.

Corollary E.20. Assume that M G is compact. Then every assignment extends to a proper abstract moment map.

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As a consequence, if M G is compact, there always exists a proper abstract moment map (e.g., one which extends the zero assignment). In particular, if the fixed point set is empty, there always exists a proper abstract moment map. Proof of Theorem E.19. By assumption, the function Aη : M η → R is proper and bounded from below. Since M η is closed, Aη extends to a function ϕ : M → R that is proper and bounded from below. (See the proof of Theorem E.1.) For each m ∈ M , let Ψm ∈ g∗ be an element whose projection to g∗m is A(m). We choose Ψm ∈ g∗ to meet the following additional requirement: Ψηm = hΨm , ηi = ϕ(m). If m ∈ M η , this condition is automatically satisfied, and if m 6∈ M η , this choice is possible because η 6∈ gm . Let Um be a tubular neighborhood of the orbit through m which equivariantly retracts to the orbit and on which the function ϕ differs from the value ϕ(m) by less than 1. Then the constant function Ψm is an abstract moment map on Um with assignment A and whose η-component is bounded from below by ϕ − 1. Choose an invariant partition of unity {ρj } subordinate to the covering of M by the open subsets Um , with the P support of ρj contained in the open set Umj . Then the convex combination Ψ = ρj Ψmj is an abstract moment map; this follows from Lemma E.14, applied to open subsets of the manifold. Moreover, since the η-component of each Ψm is bounded from below by ϕ − 1, the same holds for Ψ. Since Ψη ≥ ϕ − 1, and ϕ − 1 is proper and bounded from below, Ψη is proper and bounded from below. In general, a G-manifold M may admit no proper abstract moment maps, even when every connected component of the fixed point set M G is compact (and so a proper locally constant map ψ : M G → g∗ does exist). The obstruction lies in the compatibility condition; the manifold M might not admit a proper assignment. We will now construct an example of such a G-manifold. Example E.21. Let G = S 1 × S 1 act on the four–dimensional sphere S 4 as in Example E.4. Recall that the fixed points are the North and South Poles and that any abstract moment map on S 4 must take the same value at these points. Fix some small  > 0, and let D 4 be the -ball in C × C with the G-action that rotates each of the two factors. Take the trivial disc bundle over S 4 , N

= S 4 × D4 = {(z, z 0 , x, w, w0 ) | |z|2 + |z 0 |2 + x2 = 1 and |w|2 + |w0 |2 < 2 } ⊂ C2 × R × C 2

with the diagonal action of G. Since the neighborhood of each of the two fixed points in N is equivariantly diffeomorphic to D 4 × D4 , we can plumb an infinite sequence of such N ’s. More explicitly, take M = N × Z/ ∼ where the equivalence relation ∼ is

(E.2)

(z, z 0 , x, w, w0 , n) ∼ (w, w0 , −x, z, z 0 , n + 1)

for all x > 0 and n ∈ N, whenever both |z|2 + |z 0 |2 and |w|2 + |w0 |2 are less than . Then M is a G-manifold. The gluing map (E.2) reverses the orientation; however, we can get an orientation on M by flipping the orientation of every other copy of N . An abstract moment map on M must take a constant value on the infinite sequence of fixed points; such a map cannot be proper.

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1.4. Minimal stratum assignments. Theorem E.16 can be understood as that assignments are combinatorial counterparts of abstract moment maps. The amount of information needed to determine an assignment can be further reduced by taking full advantage of the compatibility condition, as follows. Recall that the (infinitesimal) orbit type strata in M are partially ordered; X  Y if and only if X is contained in the closure of Y . The strata that are minimal under this ordering are exactly those that are closed subsets of M . The closure of any orbit type stratum in M is a smooth submanifold which contains a minimal stratum. Every component of the fixed point set, M G , is a minimal stratum. However, there can exist minimal strata outside the fixed point set M G . Whether or not such strata exist is related to the formality property discussed in Section 4 of Appendix C. Definition E.22. A minimal stratum assignment is an assignment of an element A(X) ∈ g∗X to each minimal stratum X, where gX is the infinitesimal stabilizer of x ∈ X, such that the following compatibility condition is satisfied: if two minimal strata X1 and X2 are such that X1  Y and X2  Y for some stratum Y , then the restrictions to gY of A(X1 ) and of A(X2 ) are the same: A(X1 )gY = A(X2 )gY . Notice that this condition holds automatically for the zero assignment. The following theorem follows immediately from the definitions. Theorem E.23. The restriction of any assignment to the minimal strata is a minimal stratum assignment. Conversely, any minimal stratum assignment extends to a unique assignment. Hence, every minimal stratum assignment is associated with an abstract moment map. Remark E.24. It appears that in Theorem E.23 the minimal stratum assignment cannot be replaced by a function defined only on the fixed point set. Namely, we expect there to exist a G-manifold M with isolated fixed points and a function ψ : M G → g∗ which does not extend to an assignment (hence does not extend to an abstract moment map) but which satisfies the following compatibility condition: if x, y ∈ M G belong to the same connected component of M H , then ψ H (x) = ψ H (y). Remark E.25. In Remark E.15 we proposed a definition of assignments for an action of not necessarily abelian Lie group. It appears to be an interesting and feasible problem to check whether or not the results of this section generalize to such actions. 2. Exact moment maps We have already shown that the natural forgetful homomorphism from the space of abstract moment maps on a G-manifold M to the space A(M ) of assignments on M is onto (Theorem E.16). In this section we study the kernel of this epimorphism. Recall that an abstract moment map Ψ is said to be exact if there exists a G-invariant one-form µ with Ψξ = µ(ξM ) for all ξ ∈ g (see Example 3.7). The assignment associated with such a map is zero. The following result, which is proved later in this section, shows that the converse is also true. Theorem E.26. An abstract moment map whose assignment is identically zero is exact. More explicitly, suppose that Ψ : M → g∗ is an abstract moment map such that for each subgroup H ⊂ G, the function ΨH : M → h∗ vanishes on the H-fixed

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point set M H . Then there exists an invariant one-form µ such that Ψξ = µ(ξM ) for all ξ ∈ g. Combining Theorems E.16 and E.26 we obtain Corollary E.27. The sequence      exact   abstract  0→ moment → moment → A(M ) → 0     maps maps

is exact.

The proof of Theorem E.26 relies on the following key result, which we will prove in Section 4. Theorem E.28. Let G be a torus acting linearly on Rm , and let Ψ be an abstract moment map on a neighborhood of the origin, vanishing at the origin. Then there exists a G-invariant one-form µ on a neighborhood of the origin such that µ(ξM ) = Ψξ for all ξ ∈ g. We will also need a parametric version of this theorem. Corollary E.29. Let G be a torus acting linearly on the fibers of a vector bundle V → Y , and let Ψ be an abstract moment map on a neighborhood of the zero section, vanishing on the zero section. Then there exists a smooth family µ of G-invariant one-forms on the fibers of V, such that µ(ξM ) = Ψξ near the zero section. Proof of Corollary E.29. By using a partition of unity on Y , the corollary can be reduced to the case where Y is a linear space and V = Rm × Y . This case follows immediately from Theorem E.28 when Rm is replaced by Rm × Y with the trivial G-action on the second factor. Assuming Theorem E.28, let us prove a preliminary result, which is a local version of Theorem E.26 that will be used in the next section, and deduce Theorem E.26 from it. Proposition E.30. Let G be a torus acting on a manifold M , and let Ψ : M → g∗ be an abstract moment map. Let p be a point in M and H = Gp its stabilizer. Suppose that ΨH (p) = 0. Then there exists an open G-invariant neighborhood V of p in M and a G-invariant one-form µ on V such that (E.3)

µ(ξM ) = Ψξ

on V for all ξ ∈ g. Proof. Let us first examine the case where the action is locally free near p. Fix a basis ξ1 , . . . , ξn in g. Then the vector fields (ξi )M form a basis in the tangent space to the orbit at every point of a G-invariant neighborhood V of the orbit through p. By setting (E.4)

µ((ξi )M ) = Ψξi ,

we thus obtain a form defined along the orbits in V . We extend it to a differential form µ on V by taking its composition with an orthogonal projection to the orbit with respect to a G-invariant metric. It is easy to see that µ satisfies the condition µ(ξM ) = Ψξ .

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265

Let us now prove the proposition in the general case. Pick a closed subgroup K ⊂ G whose Lie algebra k is complementary to h in g. A small G-invariant neighborhood V of the orbit Y through p can be identified, by the slice theorem, with a neighborhood of the zero section in the normal bundle π : V → Y to Y in M , with the action induced by that on M . We can apply Corollary E.29 to the linear H-action on the fibers of V, equipped with the abstract moment map ΨH induced from M . As a result, we get a smooth family µ of one-forms on the fibers of V, such that µ(ξM ) = Ψξ for all ξ ∈ h. The K-orbits form a foliation which is transverse to the fibration π. We extend µ to a one-form on a whole neighborhood of Y by making µ vanish on the vectors tangent to the K orbits. The resulting form is a G-invariant form µH on V such that µH (ξM ) = Ψξ for all ξ ∈ h, and µH (ξM ) = 0 for all ξ ∈ k. The K-action on V is locally free. Let µK be the form defined as above by (E.4) and extended to V so that it vanishes on the vectors tangent to the fibers of π. Then µK (ξM ) = Ψξ for all ξ ∈ k. Since the vector fields ξM for ξ ∈ h are tangent to the fibers of π, we also have µK (ξM ) = 0 for all ξ ∈ h. The form µ = µH + µK has the desired property, that µ(ξM ) = Ψξ for all ξ ∈ g. Proof of Theorem E.26. By Proposition E.30, there exists an open covering of M by invariant sets Uα , and on each Uα there exists an invariant one-form µα such that Ψξ = µα (ξM ) for all ξ ∈ g. Let ρj be a partition of unity P subordinate to this covering, with ρj supported in Uαj for each j. Define µ = ρj µαj . Then Ψξ = µ(ξM ) on M for all ξ ∈ g. 3. Hamiltonian moment maps As we have already seen (Example 3.6), every moment map which is associated with a closed invariant two-form is an abstract moment map. We will examine now the question of which abstract moment maps arise in this way. Recall that such abstract moment maps are called Hamiltonian. Thus we fix an abstract moment map Ψ, and we look for a closed two-form ω with ι(ξM )ω = dΨξ . Note that such an ω would necessarily be G-invariant. Our first observation is an immediate consequence of the fact that every exact moment map, Ψξ = µ(ξM ), is automatically Hamiltonian with ω = −dµ. Thus Theorem E.26 implies Corollary E.31. Let Ψ : M → g∗ be an abstract moment map with zero assignment. Then Ψ is associated with an exact two-form. In particular, Ψ is Hamiltonian. 3.1. Local existence of two-forms. Our next result shows that there are no local obstructions to the existence of ω, if G is abelian. Quite surprisingly, a similar local existence result fails to hold for non-abelian compact Lie groups, [Brad]. Corollary E.32 (Local existence of two-forms). Let G be a torus acting on a manifold M , and let Ψ : M → g∗ be an abstract moment map. For every p ∈ M , Ψ is associated with an exact two-form ω on some open G-invariant neighborhood V of p. In particular, Ψ is Hamiltonian on a neighborhood of p. Proof. Consider the new abstract moment map Ψ − Ψ(p). By Proposition E.30, there exists an invariant neighborhood V of p in M and a G-invariant oneform µ on V such that µ(ξM ) = Ψξ −Ψξ (p) on V . Let ω = −dµ; then ι(ξM )ω = dΨξ on V .

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The following semi-local result is also of interest. Corollary E.33. On a manifold with a unique minimal stratum, X, every abstract moment map is Hamiltonian. Proof. Pick an element γ ∈ g∗ whose restriction to gX is equal to ΨgX (X), and apply Theorem E.26 to the abstract moment map Ψ − γ. Corollary E.34. In every G-manifold, there exists an invariant neighborhood of the fixed point set, M G , on which every abstract moment map is Hamiltonian. Proof. Take a tubular neighborhood of M G which retracts to M G , and apply Corollary E.33 to each of its connected components. Remark E.35. It is well known that moment maps Ψ associated with symplectic forms satisfy a certain non-degeneracy condition. For example, for circle actions the Hessian d2 Ψ must be non-degenerate on the normal bundle to the fixed point set. In Appendix G, we state explicitly a necessary and sufficient condition for Ψ to be associated with a symplectic form locally, near M G . Furthermore, we will prove that abstract moment maps satisfying this non-degeneracy condition globally have many properties of moment maps on symplectic manifolds. These include the convexity theorem ([At3] and [GS2]) and formality ([Kir], see also Appendices G and C). 3.2. Global existence of two-forms. Let us now turn to the problem of global existence for ω. The following example shows that not every abstract moment map is Hamiltonian. Example E.36. Let S 1 act on CP2 by λ · [z0 : z1 : z2 ] = [z0 : λz1 : λ2 z2 ]. There are three fixed points: [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. Denote by a, b, c their respective images by an abstract moment map. If the abstract moment map is associated with a closed two form, ω, then it is an easy consequence of Stokes’ theorem that the differences, b−a and c−b are, respectively, equal (up to a common factor) to the integrals of ω on the 2-spheres [∗ : ∗ : 0] and [0 : ∗ : ∗] in CP2 . Since these lie in the same cohomology class, the values a, b, c must then be equidistant: a − b = b − c. However, an abstract moment map can take arbitrary values a, b, c at the three fixed points, by Theorem E.1. 2 (M ) are represented by Recall that the equivariant cohomology classes in HG the differences ω − Ψ where Ψ is a Hamiltonian moment map and ω is a compatible two-form; see Example 3.6. The forgetful mapping which sends ω − Ψ to the assignment A corresponding to Ψ gives rise to a homomorphism 2 ρ : HG (M ) → A(M ).

Theorem E.37. An abstract moment map Ψ is Hamiltonian if and only if A ∈ im ρ, where A is the assignment of Ψ. Proof. It is clear by definition that A ∈ im ρ if Ψ is Hamiltonian. Conversely, assume that A ∈ im ρ. Then there exists a G-equivariant equivariantly closed two-form ω − Φ such that the assignment of Φ is also A. The difference F = Ψ−Φ is an abstract moment map with the zero assignment. By Theorem E.26, F is exact and therefore Hamiltonian (Corollary E.31). Thus Ψ is Hamiltonian as the sum of two Hamiltonian abstract moment maps, F and Φ.

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267

The space of Hamiltonian assignments, i.e., assignments associated with Hamiltonian abstract moment maps, is the quotient of the space of all Hamiltonian abstract moment maps by the space of exact abstract moment maps. This follows from Corollary E.27. These three spaces fit together to form a part of a commutative square of exact sequences which summarizes some of our results. Proposition E.38. The following diagram is commutative and all of its rows and columns are exact:

0 −−−−→

0 −−−−→

0   y

basic exact 2-forms   y

basic closed 2-forms   y

−−−−→

−−−−→

0 −−−−→ H 2 (M/G) −−−−→   y 0

0   y

equivariantly exact 2-forms   y

equivariantly closed 2-forms   y 2 (M ) HG   y

0

−−−−→

−−−−→

−−−−→

0   y

exact moment maps   y

Hamiltonian moment maps   y

Hamiltonian assignments   y

−−−−→ 0

−−−−→ 0

−−−−→ 0

0

Proof. The exactness of the left column is a particular case of a more general fact, that the cohomology of the basic de Rham complex of M is equal to H ∗ (M/G), if G is compact or, more generally, if the G-action is proper, even when the action is not free. (See Corollary B.36.) The middle column is exact by the definition of equivariant cohomology via the equivariant de Rham complex. (See, e.g., [AB2] and [DKV].) Exactness of the right column follows from Corollary E.27. The fact that the top two rows are exact follows directly from the definitions of the spaces involved. The commutativity of the diagram is clear. Finally, commutativity with the exactness of the columns and the top two rows implies that the bottom row is exact by simple diagram chasing. Remark E.39. Our notion of assignments has an interesting connection with a theorem of Goretsky-Kottwitz-MacPherson [GKM]. Assume that a compact oriented G-manifold M is formal and satisfies the so-called GKM condition, which we will recall below. Then the Goretsky-Kottwitz-MacPherson theorem implies that every assignment is Hamiltonian, and hence every abstract moment map is associated with a two-form. The “GKM condition” is that the fixed points are isolated, and, in addition, every orbit type stratum with stabilizer of codimension one is two-dimensional.

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Let A be an assignment. Its restriction to the fixed point set is a locally constant function, AG : M G → g∗ . Such a function can be identified with an element of 2 HG (M G ). For each subgroup H ⊂ G of codimension one, the set of H-fixed points is a disjoint union of two-spheres in M , on each of which G/H acts with exactly two fixed points; this is a consequence of the “GKM condition” and formality. The assignment compatibility condition implies that in each such a two-sphere, the images in h∗ of AG are the same at the two fixed points. The Goretsky-KottwitzMacPherson theorem asserts that this condition on AG implies that there exists an equivariantly closed equivariant two-form ω + Ψ on M , whose restriction to M G is AG . By formality, an assignment on M is uniquely determined by its restriction to M G (see Proposition C.28). Hence, A is the assignment associated with Ψ; hence, it is Hamiltonian. 3.3. Global existence of two-forms for circle actions. When G is a circle, a necessary and sufficient condition for Ψ to be associated with a closed two-form can be stated in terms of equivariant cohomology and the fixed point set. Recall G that A(M ) = (g∗ )π0 (M ) for G = S 1 , i.e., A(M ) is the set of functions from π0 (M G ) to g∗ (Example E.7). Furthermore, since the G-action on M G is trivial, we have G 2 HG (M G ) = H 2 (M G ) ⊕ (g∗ )π0 (M ) , and thus 2 A(M ) = HG (M G )/H 2 (M G ).

Theorem E.37 implies the following corollary which can also be proved directly: Corollary E.40. Let G be a circle. An abstract moment map Ψ : M → R is 2 (M ) → Hamiltonian if and only if Ψ|M G is in the image of the homomorphism HG G 2 G 2 HG (M )/H (M ) Remark E.41. Assume that the fixed points of the action are isolated. Then H 2 (M G ) = 0 and the necessary and sufficient condition of Corollary E.40 simply 2 2 (M G )). Note also that (M ) → HG turns into the condition that Ψ|M G ∈ im (HG even when G is a torus, Corollary E.40 gives a necessary condition for Ψ to be Hamiltonian. The next example shows that Corollary E.40 does not extend to tori of dimension greater than one, i.e., the necessary condition is not then sufficient. Example E.42. Let Ψ : M → g∗ be an abstract moment map that is not Hamiltonian. (For instance, we can take M = CP2 and G = S 1 as in Example E.36.) Consider the product action of G × S 1 on M × S 1 . Thus G acts on the first factor M and fixes the second factor S 1 , and S 1 acts freely on the second factor S 1 and fixes the first factor M . Since there are no fixed points, the restriction of any abstract moment map to the fixed point set is trivially contained in the image of 2 1 HG×S 1 (M × S ). ˜ a) := (Ψ(p), 0) is an abstract moment map which is not We claim that Ψ(p, Hamiltonian. ˜ is an abstract moment map, take any subgroup H ,→ G × S 1 . To see that Ψ If the composition H → G × S 1 → S 1 is not trivial, the fixed point set M H is empty, and we have nothing to check. Otherwise, H is a subgroup of G. Thus ˜ H is equal to ΨH , and its restriction to M H is locally constant because Ψ is an Ψ abstract moment map. ˜ were Hamiltonian, Ψ ˜ G = Ψ would also be Hamiltonian and so would be If Ψ the restriction of Ψ to M × {1}, contradicting the original assumption about Ψ.

4. ABSTRACT MOMENT MAPS ON LINEAR SPACES ARE EXACT

269

For tori of dimension greater than one, the condition for an abstract moment map to be Hamiltonian should involve its restriction to the “minimal strata” rather than the fixed point set. 4. Abstract moment maps on linear spaces are exact A crucial step in the proof of Theorem E.26 and Proposition E.30, on which many of our subsequent results rely, is Theorem E.28. In this section we recall this theorem and present two different proofs of it. The second proof is elementary while the first proof is more conceptual. We recall the statement of the theorem. Theorem E.28. Let G be a torus acting linearly on Rm , and let Ψ be an abstract moment map on a neighborhood of the origin, vanishing at the origin. Then there exists a G-invariant one-form µ on a neighborhood of the origin such that µ(ξM ) = Ψξ for all ξ ∈ g.

Proof of Theorem E.28. By adding, if necessary, an additional copy of R (with the trivial G-action) to Rm , we can always make m even, say, m = 2d. Fix a G-invariant complex structure on the vector space Rm . We obtain a representation of G on Cd with weights α1 , . . . , αd . The infinitesimal action of the Lie algebra g of G on Cd is then given by the vector fields   d √ X ∂ ∂ (E.5) . − z¯i αi (ξ) zi ξM = −1 ∂zi ∂ z¯i i=1 We will look for a complex-valued one-form µ such that

(E.6)

µ(ξM ) = Ψξ .

When we find such a form, we will replace it by the real form (µ + µ ¯)/2, which will still have the desired properties because ξM are real vector fields. Also, we will not require µ to be G-invariant: because Ψ is G-invariant, if µ satisfies (E.6), so will its G-average. Any one-form on Cd can be written as (E.7)

d √ X µ = − −1 fi dzi − gi d¯ zi i=1

for some smooth functions fj and gj , j = 1, . . . , d. For such a one-form, the function Ψ : Cd → g∗ defined by Ψξ = µ(ξM ), is (E.8)

d X Ψ= (zj fj + z j gj )αj . j=1

Conversely, for any function Ψ which has the form (E.8) for some smooth functions fj , gj , there exists a one-form µ such that Ψξ = µ(ξM ) for all ξ ∈ g. Namely, just take µ given by (E.7). To prove Theorem E.28, it is thus enough to prove the following result. Proposition E.43. Let Ψ be a g∗ -valued function on a neighborhood of the origin in Cd , vanishing at the origin and satisfying the second condition of an abstract moment map: for any subgroup H ⊂ G, the function ΨH : Cd → h∗ is locally constant on the set of H-fixed points.

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Then there exist smooth functions fj and gj such that Ψ is given by (E.8) on a neighborhood of the origin. Remark E.44. The converse of the proposition is also true. Namely, it is clear that any Ψ of the form (E.8) satisfies the second condition of an abstract moment map. Our first proof of Proposition E.43 relies on the polynomial version of this result: Proposition E.45. Let Ψ : Cd → g∗ be a polynomial function which vanishes at the origin and which satisfies the second condition of an abstract moment map. Then there exist polynomials fj and gj on Cd such that Ψ is given by (E.8). Proof. Since Ψ is polynomial, we can write it as a sum of monomials in a unique manner, X βk,l z k z l , Ψ= k,l

summing over k = (k1 , . . . , kd ) and l = (l1 , . . . , ld ) in Zd+ , where the coefficients βk,l are in g∗ . For every subset I ⊂ {1, . . . , d}, denote by (C× )I the subset of Cd consisting of all vectors (z1 , . . . , zd ) for which zi 6= 0 if and only if i ∈ I. In other words, Y C× (C× )I = i , i∈I

C× i

is the complement to the origin in Ci . where All the points z ∈ (C× )I have the same stabilizer GI , whose Lie algebra is \ gI = (E.9) ker αi . i∈I

Since Ψ satisfies the second condition of an abstract moment map, ΨgI is constant on (C× )I . Since, additionally, Ψ is continuous on Cd and vanishes at the origin, ΨgI vanishes on (C× )I . Let us analyze what this condition tells us about the coefficients βk,l . The P polynomial Ψξ = k,l βk,l (ξ)z k z l vanishes on (C× )I for all ξ ∈ gI if and only if for each k, l, the summand βk,l (ξ)z k z l vanishes on (C× )I for all ξ ∈ gI . Let us fix k and l, and restrict our attention to I = Ik,l = {i | ki 6= 0 or li 6= 0}.

k l

Since the monomial z z does not vanish on (C× )I , its coefficient βk,l (ξ) must vanish for all ξ ∈ gI . By (E.9), a linear functional that vanishes on gI is a linear P combination of αi , i ∈ I. Therefore, βk,l = λi,k,l αi and Ψ=

X i

i∈Ik,l

αi

X

λi,k,l z k z l .

k,l such that i∈Ik,l

Since for each i ∈ Ik,l either zi or z i factors out of the monomial z k z l , the function Ψ is of the form (E.8). Let us now deduce Proposition E.43 from its polynomial version, i.e., from Proposition E.45. To this end, we reformulate these propositions as assertions that certain sequences of homomorphisms are exact.

4. ABSTRACT MOMENT MAPS ON LINEAR SPACES ARE EXACT

271

Denote by P the ring of complex-valued polynomials in zj and z¯j , j = 1, . . . , d. Define the modules Mi , i = 1, 2, 3, over P as follows: P • M1 is the space of one-forms fi dzi + gi d¯ zi with fi and gi in P. • M2 is the tensor product P ⊗ g∗ over C. • For each subset I ⊆ {1, . . . , n}, denote by PI the ring of polynomial functions on (C× )I . The restriction homomorphism P → PI makes PI into a P-module. Set M M3 = PI ⊗ g∗I , I

where gI is the Lie algebra of the stabilizer of (C× )I , given by (E.9). Define the sequence of homomorphisms (E.10)

β

α

M1 → M 2 → M 3

by setting α : µ 7→ Ψ with Ψξ = µ(ξM ) and β to be the homomorphism M P ⊗ g∗ → PI ⊗ g∗I I

associated with the restrictions P → PI and g∗ → g∗I . In other words, the Ith component of β(Ψ) is the gI -component of Ψ restricted to (C× )I . Hence, Ψ satisfies the second condition of an abstract moment map if and only if β(Ψ) = 0, and Ψ is associated with a one-form if and only if it is in the image of α. Proposition E.45 is equivalent to the sequence (E.10) being exact. Denote by O and E, respectively, the algebras of germs of analytic, respectively, , where i = 1, 2, 3, be the smooth functions on Cd at the origin. Let Msmooth i modules defined similarly to Mi but in the category of smooth germs at the origin. are modules over E. Note that Msmooth i As before, we have a sequence of homomorphisms (E.11)

α

β

Msmooth → Msmooth → Msmooth . 1 2 3

To prove the theorem in the smooth category, it suffices to show that this sequence is exact. Note that the inclusions P → O and P → E make O and E into P-modules. The following lemma is obvious. Lemma E.46. Msmooth = Mi ⊗P E. i To finish the proof of Theorem E.28, we need to recall some facts from commutative algebra. Let B be a commutative ring and A a sub-ring of B. The ring B is said to be flat over A if for every exact sequence of A-modules M1 → M 2 → M 3 the sequence M1 ⊗ A B → M 2 ⊗ A B → M 3 ⊗ A B

is also exact. It is known that O is flat over P (see [Mal], page 45, Example 4.11) and E is flat over O (see [Mal], page 88, Corollary 1.2). This in turn implies that E is flat over P. Therefore, the exactness of (E.10) implies the exactness of (E.11). This completes our first proof of Proposition E.43.

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Second Proof of Proposition E.43. Step 1. To prove the proposition, it is enough to find functions h1 (z), . . . , hd (z) such that X Ψ(z) = (E.12) hi (z)αi i

and such that for each i (E.13)

hi (z) = 0 if zi = 0

where z = (z1 , . . . , zd ). Indeed, by Hadamard’s lemma, there then exist smooth functions fi and gi for which hi = zi fi + z i gi and Ψ takes the form (E.8). Step 2. For each subset I ⊂ {1, . . . , d}, denote by CI the set of z ∈ Cd for which zi 6= 0 implies i ∈ I. In other words, let Ci be the “ith coordinate axis” in Cd , then Y CI = Ci . i∈I

The second abstract moment map condition is equivalent to the following requirement: \ Ψξ (z) = 0 for all ξ ∈ ker αi and all z ∈ CI . i∈I

This, in turn, is equivalent to the following condition:

(E.14)

Ψ(z) ∈ span{αi }i∈I for each I and each z ∈ CI .

This condition implies that for each z there exists a collection of constants hi (z) satisfying (E.12) and (E.13). Our goal is thus to choose hi (z) smoothly in z. Plan. We will define the functions hi by induction. At each step we will assume that there exists a subset of Cd where these functions are well defined and satisfy (E.12) and (E.13), that this subset of Cd is a union of subspaces of the form CI , and that the functions hi are smooth on each subspace CI on which they are defined. Step 3. We begin the induction with a set I of one element; without loss of generality, we may assume that I = {1}. By (E.14), for z = (z1 , 0, . . . , 0), the value Ψ(z) is a multiple of α1 . We define h1 (z) = Ψ(z)/α1 and hi (z) = 0 for all i 6= 1, for all z ∈ C1 . Notice that this definition is consistent at z = 0 because Ψ(0) = 0, by assumption. Step 4. This is the beginning of the inductive step. Suppose that we have defined the functions h1 , . . . , hd on a subset of Cd which contains the subspace CJ for all J I and such that the conditions (E.12) and (E.13) are satisfied wherever these functions are defined. For z ∈ CI , let us first consider the auxiliary functions X ˜ i (z) = (E.15) h (−1)|I|−|J|−1 hi (zJ ), J I

where zJ is the projection of z to CJ . We claim that, for all z ∈ CI , ˜ i (z) = 0 whenever zi = 0; (1) h ˜ (2) hi (z) = hi (z) whenever z ∈ CJ CI .

The first assertion follows from the fact that each summand in (E.15) satisifes (E.13). To prove the second assertion, it is enough to consider the case when J is obtained from I by removing a single element. Thus, without loss of generality, we

5. FORMAL COBORDISM

273

may assume that I = J t {1}. Then, for z ∈ CJ , ˜ hi (z) = hi (zJ ) P + K J (−1)|I|−|K|−1hi (zK ) P + K J (−1)|I|−|Kt{1}|−1hi (zK∪{1} ).

Because 1 6∈ I, we have z1 = 0, so that zK = zKt{1} , and the corresponding summands above cancel pairwise. We get ˜ hi (z) = hi (zJ ) = hi (z), because z ∈ CJ . Step 5. Let us now complete the inductive step. Unfortunately, the functions ˜ hi (z) fail to satisfy (E.12) in general. Let us correct this. For each z ∈ CI , the space of solutions v = (vi )i∈I to the equation X Ψ(z) = v i αi i∈I

is a non-empty affine subspace of R , and we define the vector (hi (z))i∈I to be the ˜ i (z))i∈I to this solution space. For i 6∈ I and orthogonal projection of the vector (h z ∈ CI we define hi (z) = 0. Then hi (z) are smooth functions of z ∈ CI which satisfy (E.12). Let us show that this definition does not lead to inconsistencies. Suppose that z ∈ CI and hi (z) was already defined before. Then z ∈ CJ for some J I. Then ˜ ˜ hi (z) = hold i (z), and the induction hypothesis implies that the hi (z) satisfy (E.12) new ˜ and (E.13). Our definition then gives hi (z) = hi (z). It remains to check that (E.13) holds for hi : CI → R. If i 6∈ I, this follows from the definition of hi . If i ∈ I and zi = 0, then z ∈ CJ for some J I, and (E.13) follows from the induction hypothesis I

5. Formal cobordism of Hamiltonian spaces In this section we apply the techniques developed above to analyze the relationship between cobordisms of Hamiltonian spaces discussed in Chapter 2 and cobordisms of abstract moment maps introduced in Chapter 3. For this purpose we need to consider triples (M, [α], Ψ) where M is an oriented G-manifold, [α] is an equivariant cohomology class (see Appendix C), and Ψ is a proper abstract moment map. A cobordism between two such triples, (M0 , [α0 ], Ψ0 ) and (M1 , [α1 ], Ψ1 ), is an equivariant oriented cobordism W between M0 and M1 , equipped with a proper abstract moment map Ψ and equivariant cohomology class [α] such that Ψ|∂W = Ψ0 t Ψ1 and [α]|∂W = [α0 ] t [α1 ]. Any Hamiltonian space (M, ω, Φ) determines such a triple, with [α] = 2 [ω − Φ] ∈ HG (M ) (see Definition 2.15) and Ψ = Φ. We stress that, in general, the abstract moment map Ψ and the equivariant cohomology class [α] are completely independent of each other. A formal cobordism of Hamiltonian spaces (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ) is a cobordism of the corresponding triples (M0 , [ω0 − Φ0 ], Φ0 ) and (M1 , [ω1 − Φ1 ], Φ1 ). Clearly, any Hamiltonian cobordism gives a formal cobordism. In the reverse direction, we have the following result: Theorem E.47. Let (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ) be η-polarized Hamiltonian G-manifolds. Suppose that there exists an η-polarized formal cobordism between them. Then there exists an η-polarized Hamiltonian cobordism between them. The first step is to replace the cobording equivariant cohomology class by a cobording equivariant two-form:

274

E. ASSIGNMENTS AND ABSTRACT MOMENT MAPS

Lemma E.48. Let (W, [α], Ψ) be a formal cobordism between the Hamiltonian spaces (M0 , ω0 , Ψ0 ) and (M1 , ω1 , Ψ1 ). Then there exists an equivariant two-form ω − Φ on W such that 2 • [ω − Φ] = [α] ∈ HG (W ); • ω|∂W = ω0 t ω1 ; • Φ|∂W = Φ0 t Φ1 .

Proof. Let ω 0 −Φ0 be any representative for [α]. Then [ω 0 −Φ0 ]|Mr = [ωr −Φr ]. Let βr be a one-form on Mr such that ωr − Φr = (ω 0 − Φ0 ) |Mr + dG βr for r = 0, 1. Identify a neighborhood of the boundary piece Mr of W with the collar (0, 1] × Mr . Let ϕ : (0, 1] → [0, 1] be a smooth function such that ϕ(t) = 0 for t near 0 and ϕ(t) = 1 for t near 1. Define a one-form β on W by β := ϕ(t)βr on the collar neighborhood of Mr and β := 0 outside these collar neighborhoods. Define ω − Φ := (ω 0 − Φ0 ) + dG β.

Then [ω − Φ] = [ω 0 − Φ0 ] = [α], and

(ω − Φ) |Mr = (ω 0 − Φ0 ) |Mr + dG βr = ωr − Φr .

We can now prove the theorem. Proof of Theorem E.47. Let (W, [α], Ψ) be a formal cobordism such that [α]|Mr = [ωr − Φr ], Ψ|Mr = Φr , and Ψ : W → g∗ is η-polarized. Let ω 0 − Φ0 be an equivariant two-form on W such that [ω 0 − Φ0 ] = [α] and (ω 0 − Φ0 ) |Mr = ωr − Φr for r = 0, 1. (See Lemma E.48.) Since each Φηr : Mr → R is proper and bounded from below, there exists a η neighborhood U 0 of ∂W on the closure of which Φ0 is proper and bounded from η below. The zero set W = {ηW = 0} is compact, because the restriction Ψη : W η → R is, on the one hand, proper, and, on the other hand, constant. Therefore, there exists a neighborhood U 00 of W η whose closure is still compact. The union U = η U 0 ∪ U 00 is an open set, containing ∂W ∪ W η , such that the restriction of Φ0 to the closure of U is proper and bounded from below. Let µ be a one-form, defined on the complement of W η , with the property that η

For instance, we can take

µ(ηW ) = Ψη − Φ0 . η

µ(·) = (Ψη − Φ0 )

h·, ηW i hηW , ηW i

where h , i is a Riemannian metric. Finally, let ρ : M → [0, 1] be an invariant smooth function which is equal to one outside the open set U and is equal to zero on a smaller neighborhood of ∂W ∪ W η . The two-form on W given by and the moment map (E.16)

ω = ω 0 − d(ρµ) ξ

Φξ = Φ0 + ρµ(ξW )

provide the desired η-polarized Hamiltonian cobordism. Indeed, the function η

Φη = (1 − ρ)Φ0 + ρΨη

5. FORMAL COBORDISM

275

is proper and bounded from below, and coincides with Φ0 , hence with Φ0 t Φ1 , on ∂W . In the above proof, the formal cobordism does not come from the Hamiltonian cobordism: the cobording Hamiltonian moment map Φ is generally different from the original cobording abstract moment map Ψ. We may also ask whether a cobording triple actually comes from a Hamiltonian cobordism, or, more generally, whether a triple (M, [α], Ψ) (with or without boundary) comes from a Hamilton2 ian space (M, ω, Φ). That is, given [α] ∈ HG (M ) and an abstract moment map ∗ Ψ : M → g , we look for a two-form ω for which Ψ is a moment map and, additionally, such that [ω − Ψ] = [α]. It is easy to come up with a necessary condition: Exercise. Choose a representative α = ω 0 − Φ0 for [α]. Let H be the stabilizer of a point x ∈ M . Show that the image of α(x) := Φ0 (x) under the natural map g∗ → h∗ is independent of the choice of representative for [α]. (Hint: all representatives of [α] differ by dG β = dβ + β(ξM ).) Hence, a necessary condition is that the assignment of the abstract moment map Ψ must coincide with the assignment of the equivariant cohomology class [α]. This condition is also sufficient: Proposition E.49. Let a torus G act on a manifold M (with or without bound2 ary), and let (M, [α], Ψ) be a formal triple, with [α] ∈ HG (M ) and Ψ an abstract moment map. Suppose that the assignment for [α] is equal to the assignment of Ψ. Then there exists a Hamiltonian triple (M, ω, Φ) such that Ψ = Φ and [α] = [ω −Φ]. Proof. Let ω 0 − Φ0 be some representative for [α]. The difference Ψ − Φ0 is an abstract moment map with zero assignment. By Theorem E.26 there exists an invariant one-form β such that Ψξ − (Φ0 )ξ = β(ξM ) for all ξ ∈ g. Let ω = ω 0 − dβ and Φ = Ψ. Then Φ is a moment map for ω and [ω − Φ] = [α − dG β] = [α] as required. More generally, there always exists a Hamiltonian triple which gives the formal triple outside a neighborhood of the set where the assignments of [α] and Ψ are different. We can keep all moment maps polarized: Theorem E.50. Let M be a G-manifold (with or without boundary) and let 2 (M, [α], Ψ) be a formal triple, with [α] ∈ HG (M ) and Ψ : M → g∗ an η-proper abstract moment map. Let X be the set where the assignments of Ψ and [α] are different, and let U be a neighborhood of X. Then there exists an η-proper Hamiltonian triple (M, ω, Φ) such that [ω − Φ] = [α] and such that Φ = Ψ outside U . Proof. Let ω 0 − Φ0 be some representative of [α]. By Proposition E.49, there exists a one-form β on M rX such that Ψξ − (Φ0 )ξ = β(ξM )

on M rX, for all ξ ∈ g. Because Ψ is η-proper, X η = X ∩ M η is compact. Let U1 , U2 , and U3 be a covering of M by invariant open sets such that • U1 contains X η , and its closure in M is compact; • U2 ∩ M η = ∅, and X ⊆ U1 ∪ U2 ⊆ U ; • U3 ∩ X = ∅.

276

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Let ρ1 , ρ2 , and ρ3 be a partition of unity on M such that supp ρi ⊂ Ui . Note that the one-form β is defined on U3 . Let h·, ηM i µ(·) = hηM , ηM i

where h·, ·i is some invariant Riemannian metric. This is a one-form defined on U2 , satisfying µ(ηM ) = 1. Let Then

ω − Φ := (ω 0 − Φ0 ) + dG (ρ3 β + ρ2 (Ψη − (Φ0 )η ) µ) . [ω − Φ] = [ω 0 − Φ0 ] = [α],

and


Φξ = (Φ0 )ξ + ρ3 Ψξ − (Φ0 )ξ + ρ2 (Ψη − (Φ0 )η ) µ(ξM ). Outside U , we have ρ1 = ρ2 = 0 and ρ3 = 1, so Φ = Ψ. Everywhere, we have Φη

= (1 − ρ2 − ρ3 )(Φ0 )η + (ρ2 + ρ3 )Ψη = ρ1 (Φ0 )η + (ρ2 + ρ3 )Ψη .

Outside U1 , we have ρ1 = 0 and ρ2 + ρ3 = 1, so Φη = Ψη , and Φ is η-proper. A variant of this result holds for proper moment maps that are not necessarily polarized. For this we need to make a certain finiteness assumption. Proposition E.51. Let (W, [α], Ψ) be a proper formal triple. Let X be the set where the assignments of α and Ψ differ. Let U be a neighborhood of X. Suppose that X contains finitely many orbit type strata. Then the complement W rX admits a closed two-form ω and a corresponding proper moment map Φ such that [ω −Φ] = [α]|W rX and such that Φ|W rU = Ψ|W rU . The finiteness assumption allows us to prove Proposition E.51 by induction. The induction step is given in the following technical lemma. Lemma E.52. Let W be a G-manifold, with or without boundary. Let X ⊂ W be a minimal orbit type stratum, and let Ψ : W → g∗ be a proper abstract moment map. Let U be a neighborhood of X in W . Then there exists a proper abstract moment map Ψ0 : W rX → g∗ such that Ψ = Ψ0 outside U . Proof. Let η ∈ g be such that η ∈ gX but η 6∈ gm (equivalently, ηM |m 6= 0) for all m in a punctured neighborhood of X. We will define (Ψ0 )ξ := Ψξ + µ(ξM ) for an appropriate one-form µ. Because η ∈ gX , the function Ψη is constant on X, hence bounded on a neighborhood of X. Let V be such a neighborhood, such that, additionally, V ⊆ U and ηM 6= 0 on V rX. Let ρ : M rX → [0, ∞) be such that supp ρ ⊂ V and ρ(m) −−−−→ ∞. m→X

We will define µ(·) := ρ

h·, ηM i hηM , ηM i

5. FORMAL COBORDISM

277

for an appropriate Riemannian metric h·, ·i. Let ξ i ∈ g be such that η and the ξ i form a basis and such that ξ i 6∈ gX . Then we may consider g∗ with the norm X |α(ξ i )|. kαk = |α(η)| + i

We choose the Riemann metric h·, ·i on M such that the functions

i  ξM , η M hηM , ηM i

will be bounded on V . This is possible because of the assumptions η ∈ gX and ξ i 6∈ gX . Then X i kΨ0 k = |(Ψ0 )η | + (Ψ0 )ξ i



i  X i ξM , ηM ξ = |Ψ + ρ| + Ψ + ρ . hηM , ηM i i η

Outside V we have Ψ0 = Ψ, hence, Ψ0 is proper on M rV . On the closure of V , kΨ0 k ≥ |ρ| − sup |Ψη | . V

Therefore, on the preimage of any bounded set in g∗ , the function ρ is bounded. On any subset of V where ρ is bounded, Ψ0 differs from Ψ by a bounded function, and since Ψ is proper, so is Ψ0 . These results give partial generalizations of Theorem E.47 to moment maps that are proper, not just polarized. For instance, Theorem E.53. Suppose that there exists a proper formal cobordism (W, [α], Ψ) between the Hamiltonian spaces (M0 , ω0 , Φ0 ) and (M1 , ω1 , Φ1 ). Suppose, in addition, that the closure of every orbit type stratum in W meets the boundary ∂W . Then there exists a proper Hamiltonian cobordism between the Hamiltonian spaces. Proof. From Ψ|Mr = Φr we see that the difference Ψ − Φ0 has a zero assignment on ∂W . This and the assumption on orbit type strata guarantees that this difference has a zero assignment on all of W . By Proposition E.49 we can choose a representative ω 0 − Φ0 of [α] in such a way that Φ0 = Ψ. We get a Hamiltonian cobordism (W, ω 0 , Φ0 ) with ∂W = M0 t M1 , and such that [ω 0 − Φ0 ]|Mr = [ωr − Φr ] and Φ0 |Mr = Φr . By Lemma E.48 we can add an equivariantly exact form and get a Hamiltonian cobordism (W, ω, Φ) with (ω − Φ)|Mr = ωr − Φr as forms. Remark E.54. If the group G is the circle group and the cobordism W carries an equivariant stable complex structure, the assumption in Theorem E.53 can be relaxed: it is enough to assume that the closure of every component of the fixed point set W G meets the boundary ∂W . Namely, any non-fixed stratum which does not reach the boundary can be removed through a “Gusein-Zade surgery” which is described in [GGK2, Section 3]. We do not know whether this assumption on the fixed point set can be further removed in the presence of a stable complex structure. Alternatively, if G is the circle group and if W does not contain any two orbit type strata with stabilizers Zm and Z2m for any m, again it is enough to assume that the closure of every component of W G meets ∂W . See [GGK2, Section 5]. It

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is not clear whether this argument can be extended to tori G of dimension bigger than one.

APPENDIX F

Assignment cohomology In Appendix E we defined the space of assignments A(M ) of a G-manifold M . In the present appendix, we show that A(M ) fits as the zeroth space in a sequence of vector spaces HA∗ (M ), called the assignment cohomology. Our exposition closely follows [GGK3, Section 7]. A more general version of such a cohomology, in a purely combinatorial context, is introduced in [Bac]. 1. Construction of assignment cohomology Let M be a manifold with an action of a torus G. Denote by PM the set of its infinitesimal orbit type strata. For each stratum X, denote by gX the infinitesimal stabilizer of the points of X, and let V (X) = g∗X be the dual space. Recall that PM is a partially ordered set, a poset for brevity, with X  Y if X is contained in the closure of Y . Denote by πYX : V (X) → V (Y ) the natural projection dual to the inclusion map gY ⊆ gX when X  Y . Recall that the space of assignments is Y A(M ) = {v ∈ V (X) | πYX vX = vY for all X  Y }. X∈PM

Every abstract moment map induces an assignment. We will call elements of A(M ) moment assignments, to distinguish them from assignments with other coefficients V (X), which we introduce later. Define the assignment cohomology HA∗ (M ) to be the cohomology of the following cochain complex which we will denote by C ∗ (M ; V ). A k-cochain is a function ϕ that associates to each ordered (k + 1)-tuple X0  . . .  Xk of elements of PM an element in V (Xk ). The differential d is defined by the formula dϕ(X0 , . . . , Xk+1 )

=

k X `=0

(F.1)

c` , . . . , Xk+1 ) (−1)` ϕ(X0 , . . . , X

Xk ϕ(X0 , . . . , Xk ), +(−1)k+1 πX k+1

where, as usual, the hat over X` means that X` is omitted. Example F.1. The zeroth assignment cohomology is simply the space of assignments: HA0 (M ) = A(M ). Remark F.2 (Functoriality). Assignment cohomology is functorial with respect to equivariant maps of manifolds: Let M and N be G-manifolds and let f : M → N be a G-equivariant map. Such a map might not send a stratum in M to a stratum in N . (For example, the function 279

280

F. ASSIGNMENT COHOMOLOGY

f (z, w) = z from C2 with the diagonal circle action to C with the standard circle action sends the open dense stratum C2 r0 to the union of strata {0} t C× = C.) However, it does induce a monotone mapping of posets, f˜: PM → PN , in the following way. For each stratum X in M there exists a unique stratum Y in N with gX ⊆ gY whose closure contains f (X). (To see this, consider the infinitesimal gX -action in N . Since X is connected, f (X) is contained in a unique component of the gX -fixed point set of N . This component is a G-invariant submanifold of N . ˜ The stratum Y is the open dense stratum in this component.) We set f(X) = Y. Note that gX ⊆ gf˜(X) for all X. By definition, the pullback map on the cochain complexes sends a cochain ϕ ∈ C k (N ; V ) to the cochain f ∗ ϕ ∈ C k (M ; V ) given by f˜(Xk )

(f ∗ ϕ)(X0 , . . . , Xk ) = πXk

˜ k )). ϕ(f˜(X0 ), . . . , f(X

The map f ∗ commutes with d and thus induces a pullback map in cohomology. Theorem F.3. HAk (M ) = 0 when k ≥ dim M or k ≥ dim G. Theorem F.3 will easily follow from the following alternative construction of assignment cohomology. Let C0k (M ; V ) be the space of functions ϕ that associate to each ordered (k + 1)-tuple X0 ≺ X1 ≺ . . . ≺ Xk of distinct elements of PM an element of V (Xk ). This can be identified with the subspace of C k (M ; V ) consisting of those cochains that vanish on (X0 , . . . , Xk ) whenever Xi = Xi+1 for some i. Theorem F.4. C0∗ (M ; V ) is a subcomplex of C ∗ (M ; V ), and the inclusion map of complexes induces an isomorphism in cohomology. This result is standard. However, for the sake of completeness, we prove it below. The complex C0∗ (M ; V ) is much smaller than C ∗ (M ; V ) and is more convenient to use for explicit calculations. Its disadvantage is that this complex is not functorial with respect to mappings of posets: a map f : M → N that sends strata to strata sends a tuple X0 ≺ . . . ≺ Xk to a tuple f (X0 )  . . .  f (Xk ), but the f (Xj ) might not be distinct even if the Xj are. Proof of Theorem F.3. By Theorem F.4 it suffices to prove Theorem F.3 for the cohomology of the complex C0∗ (M ; V ). The first part of the theorem follows from the fact that if X ≺ Y , then dim X < dim Y . Therefore, the longest possible tuple X0 ≺ . . . ≺ Xk of distinct strata has k = dim M . Thus C0k (M ; V ) = 0 for k > dim M . When k = dim M , the maximal stratum Xk is the open dense stratum in M , on which V (Xk ) = 0. As a result, C0dim M (M ; V ) = 0. The second part of the theorem follows from the fact that if X ≺ Y , then dim gX > dim gY . The same argument as before shows that C0k (M ; V ) = 0 whenever k ≥ dim G. Notice that the proof of the second part of the theorem breaks down if the infinitesimal orbit type stratification is replaced by the orbit type stratification. Proof of Theorem F.4. Denote by (X0k0 , . . . , Xlkl ) the tuple (X0 , . . . , X0 , . . . , Xl , . . . , Xl ) in which each Xj occurs kj times and the strata Xj are ordered and distinct: X0 ≺ X1 ≺ . . . ≺ Xl . Denote by n(X0k0 , . . . , Xlkl ) the number of j’s such that

2. ASSIGNMENTS WITH OTHER COEFFICIENTS

281

kj > 1; call this number the fatness of the tuple. As is easy to check, the fatness of a (k + 1)-tuple is no greater than k. For each integer n ≥ 0 let Cnk (M ; V ) be the space of (k + 1)-cochains that are supported on tuples of fatness n. (This is consistent with the previous definition of C0k (M ; V ).) Then C k (M ; V ) = C0k (M ; V ) ⊕ . . . ⊕ Ckk (M ; V ) as vector spaces. Set k C>0 (M ; V ) =

k M

Cnk (M ; V ).

n=1

For every non-zero cochain ϕ in this space, there exists a unique integer n between 1 and k and a unique decomposition (F.2)

ϕ = ϕn + ϕn+1 + . . . + ϕk

such that ϕj ∈ Cj∗ (M ; V ) for all j and such that ϕn 6= 0. k+1 An easy computation shows that d(Cnk (M ; V )) ⊆ Cnk+1 (M ; V ) + Cn+1 (M ; V ) k+1 k ∗ for all n ≥ 1 and that d(C0 (M ; V )) ⊆ C0 (M ; V ). In particular, C0 (M ; V ) and ∗ C>0 (M ; V ) are subcomplexes, and the assignment cohomology splits: HA∗ (M ; V ) = HA∗0 (M ; V ) ⊕ HA∗>0 (M ; V ).

It remains to show that HA∗>0 (M ; V ) vanishes. Define a linear map L : C k (M ; V ) → C k−1 (M ; V ) by (Lϕ)(X0k0 , . . . , Xlkl ) =

l X

k +1

(−1)k0 +...+kj−1 ϕ(X0k0 , . . . , Xj j

, . . . , Xlkl ).

j=0

An explicit computation shows that (F.3)

dLϕ + Ldϕ = πϕ,

where (πϕ)(X0k0 , . . . , Xlkl ) = n(k0 , . . . , kl )ϕ(X0k0 , . . . , Xlkl ). Therefore, π : C ∗ (M ; V ) → C ∗ (M ; V ) is chain homotopic to zero and, as a con∗ sequence, induces the zero map π∗ in homology. Denote by j : C>0 (M ; V ) → ∗ ∗ ∗ C (M ; V ) the natural inclusion. It is clear that πj : C>0 (M ; V ) → C>0 (M ; V ) is an isomorphism. Thus π∗ j∗ : HA∗>0 (M ; V ) → HA∗>0 (M ; V )

is an isomorphism. Since π∗ = 0, this is possible only when HA∗>0 (M ; V ) = 0. 2. Assignments with other coefficients The definitions of assignments and assignment cohomology extend word–for– word to other systems of coefficients. A system of coefficients V on the poset P M is a function that associates a vector space V (X) to each stratum X and a linear map Y X X πYX : V (X) → V (Y ) to each pair X  Y , so that πZ πY = π Z whenever X  Y  Z X and πX = id. We define a differential complex (C ∗ (M ; V ), d) as before and denote its cohomology by HA∗ (M ; V ).

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F. ASSIGNMENT COHOMOLOGY

A morphism V1 → V2 of two systems of coefficients consists of a linear map V1 (X) → V2 (X) for each X ∈ PM , such that the squares V1 (X) −→ V2 (X) ↓ ↓ V1 (Y ) −→ V2 (Y )

commute for all X  Y . The systems of coefficients V on PM form a category. The assignment cohomology groups, HAn (M ; V ), and in particular the space of assignments A(M ; V ) are functorial in V . Remark F.5. One can think of the poset PM as a category in which there exists a single arrow Y → X whenever X  Y . A system of coefficients is a contravariant functor from PM to the category of vector spaces, and a morphism of systems of coefficients is a natural transformation. The space of assignments is the inverse limit (F.4)

A(M ; V ) = lim

←− X∈PM

V (X),

which is a functor in V .1 A system of coefficients V can be viewed as a pre-sheaf on the category P M , in the sense of [SGA4, Expose I, Definition 1.2]. (Also see [Moe, Ch. I §2 and §4].) The assignments are the global sections (F.5)

A(M ; V ) = Γ(PM ; V ),

and the assignment cohomology is equal to the cohomology of this pre-sheaf. More explicitly, the cohomology groups of the pre-sheaf are defined to be the derived functors of the global section functor (F.5) (as a functor in V ); equivalently, the cohomology groups are the derived functors of the inverse limit functor (F.4). The fact that these derived functors are the same as the cohomology of the complex C ∗ (M ; V ) is Proposition 6.1 in [Moe, ch. II]. For some applications it is beneficial to work with systems of coefficients with values in categories other than the category of vector spaces. Let us illustrate this by some examples. Example F.6. Assignments with values in the functor (F.6)

V (X) = { equivalence classes of representations of gX }.

are called isotropy assignments. A G-action gives rise to a canonical isotropy assignment defined as follows: to each X associate the isotropy representation of g X on Tp M , p ∈ X. In a similar manner, any G-equivariant vector bundle over M gives rise to an isotropy assignment. Example F.7. For a symplectic manifold with a Hamiltonian torus action, its X-ray is, roughly speaking, the direct sum of the isotropy assignment and the moment map assignment. The notion of X-rays, introduced in [To], is central to the study of Hamiltonian torus actions. See [To, Ka2, KT2, Met1, Met2]. 1 Note that the convention on the direction of morphisms commonly used to turn a poset into a category is opposite of the one employed in this paper. Alternatively, the poset of strata is sometimes given the order inverse of the one used above. However, the only essential point in the choice of directions of morphisms is that a poset should be made into a category so that V becomes a contravariant functor.

3. ASSIGNMENT COHOMOLOGY FOR PAIRS

283

A broad class of systems of coefficients can be obtained by the following construction. Consider the category of sub-algebras of g with morphisms given by inclusion maps. Let V 0 be any functor from this category to the category of vector spaces (or modules, abelian groups, etc.). Then we get a system of coefficients with V (X) = V 0 (gX ). For instance, we get moment assignments from V 0 (h) = h∗ , and we get isotropy assignments from V 0 (h) = {virtual representations of h}. Example F.8. Assume that M has a unique minimal stratum, X0 . This is the case, for instance, when M is a vector space on which G acts linearly. Then HA0 (M ; V ) = V (X0 ) and HAk (M ; V ) = 0 for all k > 0. Indeed, the operator (Qϕ)(Y1 , . . . , Yk ) = ϕ(X0 , Y1 , . . . , Yk ) satisfies dQϕ+Qdϕ = ϕ, hence it is a homotopy operator for the complex C ∗ (M ; V ). (Also see Remark F.16.) Theorem F.4 holds for assignment cohomology with many other systems of coefficients. For example, the theorem clearly holds whenever V takes values in the category of vector spaces. When Theorem F.4 applies, we also have the following variant of Theorem F.3: Proposition F.9. 1. Let V be such that V (X) = 0 for the open stratum X. Then HAk (M ; V ) = 0 when k ≥ dim M . 2. Let V be obtained as the pull-back of a functor on the sub-algebras of g which vanishes on the zero sub-algebra. Then HAk (M ; V ) = 0 when k ≥ dim G. The proof of this fact is entirely similar to the proof of Theorem F.3. 3. Assignment cohomology for pairs Let N be a subset of M which is a union of strata. Define the relative assignment cohomology HA∗ (M, N ; V ) to be the cohomology of the sub-complex C ∗ (M, N ; V ) of C ∗ (M ; V ) formed by those cochains which vanish on all (k + 1)tuples X0  . . .  Xk in which all of the strata Xj are in N . Theorem F.10. There is a long exact sequence (F.7) . . . → HA∗ (M, N ; V ) → HA∗ (M ; V ) → HA∗ (N ) → HA∗+1 (M, N ; V ) → . . . , where the connecting homomorphism is given by the standard formula. Proof. The theorem is an immediate consequence of the fact that the sequence of complexes (F.8)

0 → C ∗ (M, N ; V ) → C ∗ (M ; V ) → C ∗ (N ; V ) → 0

is exact. To prove the exactness of (F.8), note that the space C ∗ (M, N ; V ) is the kernel of the restriction map C ∗ (M ; V ) → C ∗ (N ; V ) by its definition. To see that the restriction map is onto, note that every cochain ϕ in C ∗ (N ; V ) can be extended to a cochain ϕ˜ in C ∗ (M ; V ) by declaring ϕ(X ˜ 0 , . . . , Xk ) to be zero whenever not all of the Xj ’s are in N . An alternative proof of Theorem F.10 in the case where N is open is given in Remark F.14.

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F. ASSIGNMENT COHOMOLOGY

Remark F.11. It is not clear if there is a way to define relative assignment cohomology so that the sequence (F.7) is exact in the case where N ⊆ M is Ginvariant but not a union of strata. For instance, consider a G-manifold B and let G act on M = B × [0, 1] by acting on the first factor. Let N = B × {0, 1}. Since every stratum in M meets N , it seems reasonable to set HA∗ (M, N ) = 0. Then an exact sequence (F.7) would give an isomorphism between A(M ) and A(N ). However, these spaces are not isomorphic; in fact, A(N ) is isomorphic to A(M ) ⊕ A(M ). As is with other “cohomology theories”, an exact sequence of coefficients gives rise to a long exact sequence in assignment cohomology: Theorem F.12. A short exact sequence of systems of coefficients, 0 → V1 → V2 → V3 → 0,

(F.9)

induces a long exact sequence in cohomology, (F.10)

δ

→ HAk (M, N ; V1 ) → HAk (M, N ; V2 ) → HAk (M, N ; V3 ) → HAk+1 (M, N ; V1 ) → Proof. The short exact sequence (F.9) naturally induces a short exact sequence of complexes, 0 → C ∗ (M, N ; V1 ) → C ∗ (M, N ; V2 ) → C ∗ (M, N ; V3 ) → 0,

and hence the long exact sequence (F.10) in cohomology.

Remark F.13. Suppose that N ⊆ M is an open subset which is a union of strata. Note that being open is equivalent to the following condition: (F.11)

For every pair X ≺ Y of strata in M, if X ⊆ N, then also Y ⊆ N.

Then the relative assignment cohomology is equal to the ordinary assignment cohomology with a different system of coefficients: HAn (M, N ; V ) = HAn (M ; VM/N ) where VM/N is the system of coefficients given by ( V (X) if X 6⊆ N , VM/N (X) = 0 otherwise for any stratum X in M , with the projection maps ( πYX if both X and Y are not in N , (πM/N )X Y = 0 otherwise. The compatibility condition, (F.12)

X (πM/N )YZ ◦ (πM/N )X Y = (πM/N )Z whenever X  Y  Z,

which is required for VM/N to form a system of coefficients, is satisfied if and only if N meets the requirement (F.11). Remark F.14. If N ⊆ M is a union of strata and is open, the long exact sequence for the pair (M, N ) in Theorem F.10 follows from the long exact sequence for coefficients in Theorem F.12. To see this, recall that HA n (M, N ; V ) = HAn (M ; VM/N ) as explained in Remark F.13. Furthermore, we set VN (X) = X X V (X)/VM/N (X) with (πN )X Y = πY if X and Y are both in N and (πN )Y = 0 otherwise. Then we have HA∗ (M ; VN ) = HA∗ (N ; i∗ V ), where i : N → M is the

4. EXAMPLES OF CALCULATIONS OF ASSIGNMENT COHOMOLOGY

285

inclusion map. The sequence of systems of coefficients 0 → VM/N → V → VN → 0 is exact. By Theorem F.12, this sequence gives rise to the long exact sequence which coincides with the sequence (F.7) of Theorem F.10. Remark F.15. Relative assignment cohomology is a sequence of functors V 7→ HAn (M, N ; V ) from the category CM of systems of coefficients on PM to the category of vector spaces. This sequence, together with the maps δ of (F.10), form a δ-functor. (Essentially, this means that short exact sequences in CM induce long exact sequences in cohomology, as in Theorem F.12. See [La2].) In the non-relative case N = ∅ this δ-functor is universal, i.e, the functors V 7→ HAn (M ; V ) are the derived functors of the assignment functor V 7→ A(M ; V ); see Remark F.5. This remains true in the relative case if N ⊆ M is a union of strata and is open: the relative assignment cohomology functors V 7→ HA n (M, N ; V ) are then the derived functors of the relative assignment functor V 7→ A(M, N ; V ) which associates to each V the space of assignments that vanish on N . However, for a general N , it is not clear if these functors are universal or, equivalently, whether or not V 7→ HA∗ (M, N ; V ) are the derived functors for V 7→ A(M, N ; V ). Remark F.16. The poset of strata PM does not in general satisfy the following condition which is routinely required in some sources (e.g., [Jen], [Mas2], and [Rudy]): (F.13)

For any X ∈ PM and Y ∈ PM there exists Z ∈ PM such that Z  X and Z  Y.

(The reader should keep in mind that our order convention is opposite from the standard one; see footnote 1.) This condition is met, for example, when PM has a minimal element, i.e., an X0 ∈ PM such that X0  X for all X ∈ PM . (Equivalently, X0 is a stratum which is contained in the closure of every stratum X.) With the condition (F.13), the poset P is called a directed set. Under this condition, an inverse system V of finite–dimensional vector spaces is automatically flabby (see, e.g., [Jen] and [Rudy]) and lim(k) V = 0 for all k > 0. This generalizes Example ←− F.8. 4. Examples of calculations of assignment cohomology The following simple example shows that relative assignment cohomology can be non-trivial in degrees greater than zero. Example F.17. Let M = CP2 and G be the torus T2 acting on M as in Example E.8, and N = M G . Then A(M, N ) = 0, A(N ) = (g∗ )3 is six-dimensional, and dim A(M ) = 3. Furthermore, HA∗>0 (M ) = HA∗>0 (N ) = 0. Thus (F.7) turns into the exact sequence 0 → A(M ) → A(N ) → HA1 (M, N ) → 0,

where dim HA1 (M, N ) = 3, as can also be checked by a direct calculation. Example F.18 (Assignment cohomology for toric varieties). Let M be a compact smooth K¨ ahler toric manifold of complex dimension n with moment map Ψ : M → g∗ . (See Examples E.12 and E.11.) Recall that the poset PM of orbit

286

F. ASSIGNMENT COHOMOLOGY

type strata is isomorphic to the poset of faces Ψ(X) of a simple polytope Ψ(M ), and for each stratum X, dimC X = dim Ψ(X) = n − dim g∗X .

We will work with the system of coefficients V (X) = g∗X . The zeroth assignment cohomology is the space of assignment which was computed in Example E.12. Namely, M g∗Y , HA0 (M ; V ) = Y

where the summation is over all strata Y with dim gY = 1. These strata correspond to the (n − 1)-dimensional faces of Ψ(M ). In particular, dim HA0 (M ; V ) = the number of facets of Ψ(X).

Let us prove that the higher cohomology groups vanish: HAk (M ; V ) = 0 for all k ≥ 1.

(F.14)

By Theorem F.4 it is enough to work with the complex C0∗ (M ; V ). For a closed cochain ϕ ∈ C0k (M ; V ), k ≥ 1, we will find a primitive (k − 1)-cochain ψ ∈ C0k−1 (M ; V ), i.e., a cochain ψ such that dψ = ϕ. Let X0 ≺ . . . ≺ Xk−1 be any ordered k-tuple of distinct strata. Recall that the natural map X

(F.15)

V (Xk−1 )

⊕πX k−1 k −→

M

V (Xk ),

Xk

where Xk is such that codim Xk = 1 and Xk−1  Xk , is a linear isomorphism. Therefore, to define the value ψ(X0 , . . . , Xk−1 ), which is an element of V (Xk−1 ), it is enough to specify the projections of these elements to all of the spaces V (Xk ) with Xk as above. We require these projections to be (F.16)

X

πXkk−1 ψ(X0 , . . . , Xk−1 ) = (−1)k ϕ(X0 , . . . , Xk ).

Let us show that dψ = ϕ. Set ϕ0 = ϕ − dψ. Note that ψ(X0 , . . . , Xk−1 ) = 0 when codim Xk−1 = 1. Then it follows from the definition (F.16) of ψ and the definition (F.1) of the differential that ϕ0 vanishes on all tuples X0 ≺ . . . ≺ Xk in which Xk codim Xk = 1. Again, by (F.1), dϕ0 (X0 , . . . , Xk+1 ) = πX ϕ0 (X0 , . . . , Xk ) for all k+1 tuples X0 ≺ . . . ≺ Xk ≺ Xk+1 in which codim Xk+1 = 1. Since dϕ0 = 0, and since (F.15) is a linear isomorphism, this implies that ϕ0 ≡ 0. Hence, ϕ = dψ is exact. We now give an example of a manifold which has a non-trivial (absolute, not relative) first assignment cohomology. Example F.19. Let M = S 2 × S 2 × S 2 , and let G = S 1 × S 1 act by (a, b) · (u, v, w) = (a · u, b · v, ab−1 · w)

where on the right the dot denotes the standard S 1 action on S 2 by rotations. The moment assignments can be drawn as pictures showing the moment map images of the orbit type strata (the “x-ray”). Such a picture is shown in Figure F.1. Notice that this picture is two-, not three-dimensional. This arrangement can be moved around as long as the edges are shifted but not rotated. An assignment is therefore

5. GENERALIZATIONS OF ASSIGNMENT COHOMOLOGY

287

Figure F.1. A T 2 -manifold with non-zero first assignment cohomology determined by the location of the bottom left vertex and the lengths of the three edges coming out of it. Therefore, (F.17)

dim HA0 (M ; V ) = dim A(M ; V ) = 5.

We will find the dimension of the first assignment cohomology space by using the Euler characteristic of the complex C0∗ (M ; V ) of Theorem F.4. We have (F.18) P k k dim HA0 (M ; V ) − dim HA1 (M ; V ) = k (−1) dim C0 (M ; V ) 0 = dim C0 (M ; V ) − dim C01 (M ; V )

because C0k (M ; V ) = 0 for all k ≥ 2 (see also Theorem F.3). A 0-cochain associates to each vertex an element of a two–dimensional space and to each edge an element of a one–dimensional space. Therefore, (F.19)

dim C00 (M ; V ) = 2(number of vertices) + (number of edges) = 2 · 8 + 12 = 28.

A 1-cochain associates an element of a one-dimensional space to each pair consisting of a vertex and an edge coming out of it. Therefore (F.20)

dim C01 (M ; V ) = 24.

Substituting (F.19), (F.20), and (F.17) in (F.18), we get hence

5 − dim HA1 (M ; V ) = 28 − 24, dim HA1 (M ; V ) = 1.

Remark F.20. A version of the Leray spectral sequence for cohomology similar to the one considered here is introduced in [Bac]. This spectral sequence can also be used for calculations of assignment cohomology. 5. Generalizations of assignment cohomology The results and definitions of this section can be generalized or altered in many natural ways. For instance, the assignment cohomology can be defined for an arbitrary system of coefficients with values in an abelian category on an arbitrary poset (cf. [Bac]). In particular, in a more geometrical realm, the infinitesimal orbit type stratification can be replaced by the orbit type stratification and V can then be the pull-back of a contravariant functor on subgroups of G. Furthermore, instead

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of working with functors with values in finite–dimensional vector spaces one may consider functors with values in abelian groups or graded vector spaces or rings. In fact, such functors do arise in the study of symplectic manifolds, and A(M ; V ) can be viewed as a repository containing many of the invariants of the action. One important example is the isotropy assignments of Example F.6 above. Moreover, most of the results of this section extend with obvious modifications to actions of finite or compact non-abelian groups. For example, we can take V to be the pull-back of a functor on the sub-algebras of g which is invariant under conjugations. However, the space of moment map assignments for actions of nonabelian groups (Remark E.15) does not arise as the zeroth cohomology groups of this type. In the non-abelian case, the space of moment map assignments does not seem to be associated with a functor on the poset of strata. Instead, to obtain this space one should work with the singular foliation of M given by the decomposition of M according to actual stabilizers, but not just their conjugacy classes. This renders a correct generalization of assignment cohomology to actions of non-abelian groups much less straightforward. Remark F.21. The assignment cohomology appears to be related to Bredon’s equivariant cohomology, [Bre1], and, perhaps, to Borel’s equivariant cohomology with twisted coefficients. The nature and explicit form of these relations are, however, unclear to the authors.

APPENDIX G

Non-degenerate abstract moment maps A moment map Ψ associated with a symplectic form satisfies a certain nondegeneracy condition. For example, for a circle action, the Hessian d2 Ψ is nondegenerate on the normal bundle to the fixed point set. In this appendix we will show that a condition of this type is also sufficient in order for Ψ to be locally associated with a symplectic form. We will also derive some topological consequences of this non-degeneracy requirement similar to the properties of moment maps for Hamiltonian torus actions on symplectic manifolds. Implicitly, non-degenerate abstract moment maps have already been used to obtain topological results of this type; see e.g., [LT2]. 1. Definitions and basic examples Let G be a torus acting on a manifold M . Definition G.1. An abstract moment map Ψ : M → g∗ is non-degenerate if for every vector ξ ∈ g, (1) Crit(Ψξ ) = {ξM = 0}, and (2) Ψξ : M → R is a Morse-Bott function. Remark G.2. Condition G.1 is equivalent to: ∗ (10 ) for all p ∈ M , the image dΨp (Tp M ) is equal to the annihilator g⊥ p ⊆ g of the Lie algebra gp of the stabilizer of p. Given that this condition is satisfied, Condition G.1 is equivalent to (20 ) For every subtorus H of G and every p ∈ M H the subspace of Tp M annihilated by the quadratic forms d2 Ψξp ,

(G.1)

where ξ ∈ h,

H

is the space Tp M . Remark G.3. The Hessians d2 Ψξp are well defined, since, by assumption, dΨξp = 0 when ξ ∈ h. Also notice that for any abstract moment map im dΨp ⊂ g⊥ p and H Tp M is contained in the common annihilator space of the quadratic forms (G.1). Thus Ψ is non-degenerate if and only if these inclusions are equalities. Remark G.4. Note that for an abstract moment map Ψ : M → g∗ it makes sense to assert that Ψ is non-degenerate at a point p ∈ M . The set of points where Ψ is non-degenerate is open. Example G.5. Let Ψ : M → g∗ is a non-degenerate moment map on a Gmanifold. Then for any closed subgroup H ⊆ G, the H-component ΨH : M → h∗ is a non-degenerate moment map for the H-action on M . 289

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Example G.6. An abstract moment map for a circle action, Ψ : M → R, is non-degenerate if and only if it is a Morse–Bott function and its critical points are the fixed points. In particular, this shows that the manifold M need not have even dimension. However, if the fixed points of the action are isolated (and G is a torus) dim M is necessarily even. The basic example of a non-degenerate moment map is still the one associated with a Hamiltonian action of G on a symplectic manifold. In Section 3 we will discuss the converse: given a non-degenerate abstract moment map Ψ, when is Ψ associated to a symplectic form? In particular, we will give conditions under which Ψ is always locally associated with a symplectic form if the dimension of M is even. Example G.7. Let M be a G manifold. Consider the product N × M , where G acts on N trivially. Let Ψ : N × M → g∗ be an abstract moment map. Suppose that Ψ is independent of the N component, i.e., is a pull-back of ϕ : M → g∗ . Then ϕ is an abstract moment map, and Ψ is non-degenerate if and only if ϕ is non-degenerate. This example shows that the local existence of symplectic forms cannot be generalized to a global result. This cylinder construction also gives a local model (with N being an interval) for a non-degenerate abstract moment map on an orientable odd–dimensional manifold (see Section 3.2). Remark G.8. Definition G.1 makes sense even when G is a non-commutative Lie group. However, in this case it is not clear whether Definition G.1 is sufficient to guarantee any of the topological consequences of non-degeneracy or that an abstract moment map is locally associated with a symplectic form (cf. Section 3 of Appendix E). 2. Global properties of non-degenerate abstract moment maps Many subtle properties of symplectic manifolds with Hamiltonian torus actions are, in fact, formal consequences of the non-degeneracy of their moment maps. As a result, these properties are also shared by manifolds with non-degenerate abstract moment maps. Below we state four theorems establishing such properties. Let M be a compact manifold acted upon by a torus G and let Ψ be a nondegenerate abstract moment map for this action. The first two theorems (Theorems G.9 and G.13) are due to Kirwan [Kir] in the case of Hamiltonian actions of arbitrary compact Lie groups. The proof of Theorem G.13 given below seems to be new. 2.1. Formality. Theorem G.9 (Formality). Let a torus G act on a compact manifold M , and suppose that there exists a non-degenerate abstract moment map, Ψ : M → g ∗ . Then (1) the restriction homomorphism ∗ ∗ j ∗ : HG (M ) → HG (M G ) = H ∗ (M G ) ⊗ H ∗ (BG)

is a monomorphism, ∗ (2) the G-manifold M is formal; in particular, HG (M ) ∼ = H ∗ (M ) ⊗ H ∗ (BG) as H ∗ (BG)-modules, and ∗ (3) the forgetful homomorphism HG (M ) → H ∗ (M ) is an epimorphism.

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Remark G.10. Recall that formality means that the Serre spectral sequence converging to the equivariant cohomology of M collapses at the E2 -term and H ∗ (M ) ⊗ H ∗ (BG) is the graded H ∗ (BG)-algebra associated with a certain fil∗ ∗ tration on HG (M ). In particular, the H ∗ (BG)-module structure on HG (M ) does not see the G-action; the action appears “trivial” to it. (See Section 4 of Appendix C.) Proof of Theorem G.9. Let us prove the second assertion first. The nondegeneracy of Ψ implies that for ξ ∈ g, the function f = hΨ, ξi is a Morse–Bott function on M . Thus dim H ∗ (Crit f ) ≥ dim H ∗ (M ). On the other hand, Crit f = M G for a generic ξ, and so (G.2)

dim H ∗ (M G ) ≥ dim H ∗ (M ).

Due to the non-degeneracy of Ψ, the critical points of f have even indices when these points, i.e., the fixed points of G, are isolated. Thus, in this case f is a perfect Morse function and dim H ∗ (Crit f ) = dim H ∗ (M ). We will show that this is also true in general and thus f is a perfect Morse–Bott function.) The Serre spectral sequence of the fibration EG ×G M → BG converges to ∗ HG (M ) and has E2 = H ∗ (M ) ⊗ RG , where RG = H ∗ (BG). This implies

(G.3)

∗ rkRG E2 ≥ rkRG HG (M ).

Equality holds if and only if the spectral sequence collapses, for the E2 term has no RG -torsion (cf. Section 4 of Appendix C). By equation (C.9), which is a con∗ sequence of Borel’s localization theorem, rkRG HG (M ) = dim H ∗ (M G ). (See Theorem C.20 or [Bor2] and [AB2].) Combining this fact with (G.2) and (G.3), we obtain ∗ ∗ rkRG HG (M ) = dim H ∗ (M G ) ≥ dim H ∗ (M ) = rkRG E2 ≥ rkRG HG (M ).

Therefore, all inequalities in this series are in fact equalities, and the spectral se∗ (M ) as modules over RG . This proves the second quence collapses. Thus E2 = HG assertion. ∗ In particular, the forgetful homomorphism is onto and HG (M ) is a free RG module. Again by Borel’s localization theorem, j ∗ is an isomorphism modulo RG ∗ torsion. Since HG (M ) is torsion-free, j ∗ is a monomorphism. 1 The proof of Theorem G.9 only used one, generic, component of the abstract moment map. In fact, the proof shows that if a G-manifold admits a Morse-Bott function whose critical set coincides with the fixed point set, the manifold is a formal G-space. In what follows we will need a relative version of this: Lemma G.11. Let N be a G-manifold with boundary such that the G-action on the boundary has no fixed points. Assume that there exists a function ϕ : N → R with the following properties: (1) ϕ is Morse-Bott, (2) Crit ϕ = N G , and (3) ϕ−1 (0) = ∂N . Then the pair (N, ∂N ) is formal. 1 It is important to note that in this proof we do not directly use the fact that the connected components of Crit f have even Morse indices (nor that f is G-invariant). The only properties of f we need are that Crit f = M G and that f is Morse–Bott.

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Proof. By the Serre spectral sequence, ∗ rkRG HG (N, ∂N ) ≤ rkRG E2 = dim H ∗ (N, ∂N ).

(G.4) By Morse theory, (G.5)

dim H ∗ (N, ∂N ) ≤ dim H ∗ (Crit ϕ) = dim H ∗ (N G ).

Finally, ∗ ∗ dim H ∗ (N G ) = rkHG (N G ) = rkR HG (N, ∂N )

by Borel localization. This together with (G.5) provides the inequality opposite of (G.4). Therefore, (G.4) is an equality. Thus the spectral sequence collapses, and the pair (N, ∂N ) is formal. Remark G.12. In Appendix E we discussed the existence problem for abstract moment maps making no non-degeneracy requirements. The existence problem for non-degenerate abstract moment maps is also of interest. The local existence is an immediate consequence of the local linearization theorem for compact group actions (Theorem B.26 in Appendix B). The global existence problem for non-degenerate abstract moment maps is much harder already for a compact manifold. Theorem G.9 shows that the existence of a non-degenerate abstract moment map yields a strong constraint on the equivariant cohomology of the manifold. In particular, there are compact manifolds acted on by the circle G which do not admit non-degenerate abstract moment maps. (By Example G.6, on such a manifold M there is no invariant Morse–Bott function Ψ with Crit Ψ = M G .) It would be interesting to see if formality of the manifold (as in Theorem G.9 and Remark G.10) is also sufficient for the existence of a non-degenerate abstract moment map. 2.2. Kirwan’s epimorphism and convexity. Theorem G.13 (Kirwan’s epimorphism). Let a torus G act on a compact manifold M , let Ψ : M → g∗ be a non-degenerate abstract moment map, let λ be a regular value of Ψ and Mλ = Ψ−1 (λ). Then the restriction homomorphism ∗ ∗ (Mλ ) = H ∗ (Mλ /G) HG (M ) → HG

is an epimorphism. Recall that, as in the symplectic case, for a regular value λ the G-action on M λ ∗ ∗ (Mλ /G). (Mλ ) = HG is locally free, which implies the identification HG Remark G.14. The idea of the proof is to argue inductively by performing “reduction in stages”. Namely, let ξ be a generic rational vector in g. Denote by K the circle generated by ξ in G. The theorem is not hard to prove (cf. Lemma C.30) for circle actions by applying a version of the formality theorem (Theorem G.9) for manifolds with boundary. This version of formality theorem can be proved similarly to Theorem G.9. Thus ∗ ∗ HK (M ) → HK ({Ψξ = 0}) = H ∗ ({Ψξ = 0}/K)

is an epimorphism. Moreover, by using an equivariant formality theorem one can show that ∗ ∗ ∗ HG (M ) → HG ({Ψξ = 0}) = HG/K ({Ψξ = 0}/K)

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293

is an epimorphism. Now it suffices to repeat this procedure for the G/K-action on {Ψξ = 0}/K which also admits a “non-degenerate abstract moment map”. The problem with this approach arises because the quotient is in general an orbifold, not a manifold. Therefore, to do this one would have to develop an orbifold version of the theory of non-degenerate abstract moment maps. It is easier, however, to apply Lemma G.15 below which uses equivariant Morse– Bott functions instead of non-degenerate abstract moment maps. Thus, we can carry the K-action along when restricting to {Ψξ = 0} rather than dividing by K. This is the crucial point of the proof that allows us to avoid working with orbifolds. Proof of Theorem G.13. Without loss of generality, we may assume that λ = 0 ∈ g∗ . First, let us prove the following result. Lemma G.15. Let X be a compact manifold without boundary, acted upon by a torus G. Let ϕ : X → R be a function with the following properties: (1) ϕ is Morse–Bott, (2) 0 is a regular value of ϕ, (3) Crit ϕ = X G , and (4) G acts without fixed points on {ϕ = 0}. ∗ ∗ Then the restriction map HG (X) → HG ({ϕ = 0}) is onto. Proof of the lemma. Let X+ = {ϕ ≥ 0}, X− = {ϕ ≤ 0}, and X0 = {ϕ = 0} = ∂X+ = ∂X− . By Lemma G.11, the pairs (X+ , X0 ) and (X− , X0 ) are formal. By Lemma C.23, this implies that the pair (X, X0 ) is formal. By Lemma C.30, the restriction map is onto. Let us return to the proof of the theorem. Without loss of generality, assume that λ = 0. Pick a basis ξ1 , . . . , ξr in g with the following properties: (1) for all s, 1 ≤ s ≤ r, the origin 0 ∈ Rs is a regular value of the mapping (f1 , . . . , fs ), where fi = hΨ, ξi i, i = 1, . . . , r. (2) for each ξs , the zero set of the corresponding vector field coincides with the G-fixed point set: M G = {(ξs )M = 0}. Let Ys = (f1 , . . . , fs )−1 (0) for s = 1, . . . , r and Y0 = M . Then Ys is a Gmanifold, and the restriction of Ψ to Ys is an abstract moment map, which may fail to be non-degenerate. However, for any ξ which is not in the linear span of ξ1 , . . . , ξs , the component Ψξ |Ys satisfies Conditions 1 and 2 Definition G.1. In particular, for ξ = ξs+1 , the function fs+1 |Ys : Ys → R is Morse–Bott, and its critical set is YsG . So this function satisfies Conditions 1–3 of Lemma G.15, with X = Ys and ϕ = fs+1 |Ys . Condition 4 of the lemma follows from Condition 2 because ϕ is a component of an abstract moment map. By applying this lemma to s = 0, 1, . . . , r − 1, we obtain a sequence of epimorphisms: ∗ ∗ ∗ ∗ ∗ (Y1 )  . . .  HG (Yr ) = HG (M0 ). HG (M ) = HG (Y0 )  HG

This completes the proof of the theorem. Remark G.16. Even in the symplectic case, |Ψ|2 need not be a Morse-Bott function. However, |Ψ|2 satisfies Kirwan’s condition which is weaker than being Morse–Bott but is still sufficient for Morse theory, [Kir]. As S. Tolman pointed out, it would be interesting to check whether or not the same is true for non-degenerate abstract moment maps.

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Theorem G.17 (Convexity). The image Ψ(M ) is a convex polytope in g ∗ and for every λ ∈ g∗ the preimage Mλ = Ψ−1 (λ) is connected. The local convexity is an immediate consequence of the assumption that Ψ is non-degenerate. The global convexity follows from the local convexity in the same way as in the symplectic case. (See [At3] and [GS2].) Remark G.18. We emphasize that Condition 1 of Definition G.1 is crucial for Theorem G.17 to hold. The reason is that in order for critical manifolds to have even indices, the G-action on the critical set should be trivial, forcing nearby orbits to be close to particular points of the critical manifold rather than to the entire critical manifold. For example, contact moment maps for invariant contact forms (see [Ler2] and also [Ar] for the notion of a contact Hamiltonian) may fail to be non-degenerate abstract moment maps in our sense (see Example G.29). Furthermore, Theorem G.17, as stated above, fails in general for such moment maps. (However, a different form of the convexity theorem still holds; see [Ler2] and references therein.) This also shows that although Definition G.1 appears completely adequate for working with even–dimensional manifolds, its usefulness in odd-dimension is unclear. We will return to the discussion of the odd–dimensional case in Section 3.2. Finally, by Theorem G.9, the theorems of Goretsky, Kottwitz and MacPherson, [GKM], and the Brion–Vergne theorem, [BrV], readily apply to non-degenerate moment maps. Let N be the set of points p ∈ M such that the stabilizer Gp is one¯ be the closure of N , i.e., N ¯ = N ∪ M G . As in the symplectic dimensional and let N ∗ ∗ ¯ ∗ ∗ (M G ) (N ) → H G (M G ) and HG (M ) → HG case, the images of the restrictions HG coincide: ∗ ¯ ∗ (N ) is an isomorphism. (M ) → HG Theorem G.19. The restriction HG

Simple proofs of this result, not using the Goretsky–Kottwitz–MacPherson theorem, can be found in [TW] and [BrV]. 3. Existence of non-degenerate two-forms 3.1. Local existence of symplectic forms on even dimensional manifolds. Example G.20. Let Ψ : M → g∗ be a moment map for a Hamiltonian torus action on a symplectic manifold. Then Ψ is a non-degenerate abstract moment map. The converse does not hold literally: Example G.21. Let S 2n ⊂ Cn × R be the unit sphere and consider the projection to the R component, Ψ : S 2n → R. This is a non-degenerate abstract moment map with respect to the S 1 action induced by the standard diagonal S 1 action on Cn . However, by Corollary E.40, it is not even associated with a closed two-form when 2n ≥ 4. Considering direct products as in Example G.7, it is easy to construct examples of non-degenerate abstract moment maps which do satisfy the hypotheses of Corollary E.40 and are therefore associated with closed two-forms but are not associated

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with symplectic forms. However, we note that we are not aware of such examples in which the fixed points are isolated. In the above examples, the obstruction to the existence of a symplectic form is global. However, the non-degeneracy condition for an abstract moment map is local. We will show that a non-degenerate abstract moment map is always associated with a symplectic form locally, if (and only if) it satisfies an additional condition, which we now derive. Suppose that Ψ : M → g∗ is associated with a symplectic form on a neighborhood of an orbit O = G · p. The symplectic slice to the orbit at p is

(G.6)

S = (Tp O)ω /Tp O

where (Tp O)ω is the “orthogonal complement” to Tp O in Tp M with respect to ω. The vector space S acquires from M a symplectic form and a linear symplectic G p action, where Gp is the stabilizer of p in G. Let H be the connected component of the identity in Gp . Let K ⊆ G be a complementary torus to H in G, i.e., a torus with the property that the multiplication homomorphism K ×H →G is an isomorphism. Then Γ = K ∩ Gp ∼ = Gp /H is a finite abelian group, and we have an isomorphism K ×Γ Gp ∼ (G.7) =G given by [k, a] 7→ k · a. Consider the cotangent bundle T ∗ K = k∗ × K with its standard symplectic form, K action, and moment map k∗ × K → k ∗ .

Combining this with the symplectic slice, we get an induced symplectic form, G action (through the isomorphism (G.7)), and moment map on the “model” Y := T ∗ K ×Γ S.

The local normal form for a Hamiltonian torus action asserts that a neighborhood of O in M is isomorphic to a neighborhood of the zero section in the model Y (thought of as a bundle over K/Γ ∼ = G/Gp ∼ = O). Given a non-degenerate abstract moment map, we establish the local existence of a symplectic form by more or less reversing the above argument. First note that, because (Tp O)ω = ker dΦ|p , we can express the symplectic slice (G.6) as S = ker dΦ|p /Tp O.

In this form, the slice is defined for any non-degenerate abstract moment map Φ. It is then a real vector space with a linear Gp action. A necessary condition for Φ to be associated with a symplectic form near O is for S to admit a Gp -invariant symplectic form. Because every compact subgroup of the symplectic group Sp(2n) is conjugate to a subgroup of U (n), we can then identify S with Cn such that Gp acts by unitary transformations. Therefore, a necessary condition for Φ to be associated with a symplectic form near O is

(G.8)

The slice S admits a Gp -invariant complex structure.

This condition is also sufficient:

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Theorem G.22. Let Ψ : M → g∗ be a non-degenerate abstract moment map on an even-dimensional G-manifold, let O = G · p ∼ = G/Gp be the orbit of a point p ∈ M , and let Gp ⊆ G be its stabilizer. Suppose that the slice S = ker dΦ|p /Tp O admits a Gp invariant complex structure. Then there exists an invariant neighborhood of O on which Ψ is associated with a symplectic form. The condition of Theorem G.22 is automatically satisfied in the presence of a stable complex structure: Corollary G.23. Let M be an even-dimensional G-equivariant stable complex manifold, for G a torus. Let Ψ : M → g∗ be a non-degenerate abstract moment map. Then every G-orbit O has a neighborhood on which Ψ is associated with a symplectic form. We will prove Theorem G.22 and Corollary G.23 later in this section. At this point let us just mention that, in Corollary G.23, the given stable complex structure need not be induced from an almost complex structure compatible with the symplectic form. To begin, we note that on a neighborhood of an orbit O there always exists a closed two-form for which Ψ is a moment map by Corollary E.33. We need to show that such a two-form can be chosen to be symplectic. Let us begin with the case that p is an isolated fixed point. Lemma G.24. Let M be a G-manifold, for G a torus, and let p ∈ M G be an isolated fixed point. Let Ψ : M → g∗ be an abstract moment map. Let ω be a closed two-form on a neighborhood of p for which Ψ is a moment map. Then ω is non-degenerate at p if and only if Ψ is non-degenerate at p. Proof. From the local normal form for a compact group action we may assume that M is a vector space with a linear G action and p = 0 is the only fixed point. Let V1 ⊕ . . . ⊕ V r be the isotypic decomposition of this vector space. Then there exist weights αi ∈ Z∗G such that αi 6= αj for all i 6= j and Vi is a complex vector space√on which G acts with weight αi , i.e., acts by the character G → S 1 , exp(ξ) 7→ e −1αi (ξ) , followed by the standard S 1 action by scalar multiplication. By G-invariance, it is not hard to show that 1

ω|p = ω1 ⊕ . . . ⊕ ωr

where ωi is an S -invariant two-form on Vi . Let us first assume that dimC Vi = 1 for all i. Then there exist c1 , . . . , cr such that √ X (G.9) cj dzj dz j ω|p = − −1

where zj is a complex coordinate on Vj . The action is generated by the vector fields   √ X ∂ ∂ ξM = −1 − zj . αj (ξ) zj ∂zj ∂z j This implies that X d2 Ψξ (p) = cj αj (ξ)|zj |2 .

Therefore, the abstract moment map Ψ is non-degenerate at p exactly if all the c j ’s are non-zero, which holds exactly if the two-form (G.9) is non-degenerate at p.

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297

Let us now allow the Vi to be of any dimension. Lemma G.25. Let ω be an S 1 invariant alternating two-form on a complex vector space V . Then there exist coordinates zj , for 1 ≤ j ≤ dim V , such that X ω= cj dzj ∧ dz j

for some constant cj .

Proof of the lemma. By S 1 invariance, the kernel of ω is a complex subspace of V . It is enough to bring ω to the desired form on a complementary complex subspace W , on which ω is non-degenerate. By S 1 -invariance, X ω= cij dzi ∧ dz j

where zi are complex coordinates on W and cji = −cij . There exists a onedimensional complex subspace W1 on which ω is non-degenerate. Indeed, if cij 6= 0, ∂ + ∂z∂ j ∈ W ⊗R C. we can take W1 to be generated by the real part of the vector ∂z i Because ω is S 1 invariant, the “orthogonal complement” of W1 with respect to ω is a complex vector space. By induction, we get W = W1 ⊕ . . . ⊕ Wm where each Wi is a one-dimensional complex vector space and ω|Wi ×Wj = 0 if i 6= j. The lemma follows. We can now finish the proof of Lemma G.24. Applying Lemma G.25 to each of the spaces , for 1 ≤ j ≤ dim M on V1 ⊕ . . . ⊕ Vr such that P Vi , we find coordinates zjP ω|p = cj dzj ∧ dz j and d2 Ψξ (p) = cj αj (ξ)|zj |2 . As before, ω is non-degenerate at p if and only if cj 6= 0 for all j, if and only if Ψ is non-degenerate at p. Lemma G.24 does not hold when G is disconnected. For example, the zero map is a non-degenerate abstract moment map for a Γ action if Γ is finite but the zero two-form is not non-degenerate. Moreover, in this situation there might not even exist an invariant symplectic form: Example G.26. Let Γ denote the group of transformations of R2n of the form (x1 , . . . , x2n ) 7→ (1 x1 , . . . , 2n x2n ) Q with i ∈ {−1, 1} for each i and i = 1. If 2n > 2, the only Γ-invariant two-form V2 n ∗ in (R ) is P zero. (This can be see easily by writing an arbitrary two-form in coordinates as i<j aij dxi ∧ dxj .) Our assumption that the slice S admits an invariant complex structure allows us to avoid situations as in Example G.26. We will need the following technical result:

Lemma G.27. Let a compact abelian Lie group G act on an even-dimensional vector space S. Suppose that there exists a G-invariant complex structure on S. Consider the product N × S where N is a manifold with a trivial G-action, and the point (α, 0), where α ∈ N . Let Ψ : N × S → g∗

be an abstract moment map which is non-degenerate at (α, 0). Then there exists a closed two-form on N × S for which Ψ is a moment map and whose restriction to {α} × S is symplectic at (α, 0).

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G. NON-DEGENERATE ABSTRACT MOMENT MAPS

Proof. By Corollary E.33, there exists a closed two-form σ1 on N ×S for which Ψ is a moment map. Let H ⊂ G be the component of the identity element. Let S H be the subspace of S fixed by H. Then there exists a G-invariant complementary subspace S 0 such that S = S H ⊕ S 0 , so that N × S = N × S H × S0.

It is not hard to see that the restriction of Ψ to {α} × {0} × S 0 is non-degenerate at (α, 0, 0). By Lemma G.24, the restriction of σ1 to {α} × {0} × S 0 is symplectic. Consider the (non-effective) G action on S H ∼ = Cm . Because every compact subgroup of GL(m, C) is conjugate to a subgroup of U(m), there exists a G-invariant symplectic form on S H . Let σ2 be its pull-back to N × S H × S 0 . Then for a large enough r, the restriction of ω := σ1 + rσ2 to {α} × S is symplectic at (α, 0), and Ψ is a moment map for ω. Proof of Theorem G.22. As before, let us write (G.10)

G = K × Γ Gp

with Γ finite abelian. Because K acts with a finite stabilizer at p, and by the first condition of a non-degenerate abstract moment map, the K-component of Ψ is a submersion at p, i.e., dΨK |p : Tp M → k∗ is onto. Because all points of O have the same stabilizer, this is true at all points of O. Let α = ΨK (p) ∈ k∗ . Then there exists an invariant neighborhood Z of O in (ΨK )−1 (α) which is a manifold and an equivariant diffeomorphism between a neighborhood of O in M and a neighborhood of {α} × O in k∗ × Z which carries ΨK to the projection map, which we denote by the same symbol: Ψ K : k∗ × Z → k ∗ .

A neighborhood of O in Z is equivariantly diffeomorphic to a neighborhood of G ×Gp {0} in

(G.11)

G ×Gp S,

where S = Tp Z/Tp O = ker dΨ|p /Tp O is the slice; this follows from the local normal form for a compact group action (see Appendix B). We rewrite (G.11) as K ×Γ S,

where G acts through the decomposition (G.10), the K action on itself, and the Gp action on S. Therefore, we may assume that M = k ∗ × K ×Γ S

and p = [α, e, 0]. The decomposition (G.10) gives a decomposition g∗ = k∗ ⊕ g∗p so that Ψ = Ψ K + Ψ Gp . Recall that ΨK : M → k∗ is the projection to the first coordinate. By K-invariance, ΨGp is the pull-back to M of a non-degenerate abstract moment map, which we denote by the same symbol, ΨGp : k∗ × S → g∗p .

Let ω1 be the pull-back to M of the standard symplectic form on the cotangent bundle T ∗ K = k∗ × K. Note that its Gp moment map is zero.

3. EXISTENCE OF NON-DEGENERATE TWO-FORMS

299

By Lemma G.27, there exists a closed two-form on k∗ × S for which ΨGp is a moment map and whose restriction to {α} × S is symplectic at (α, 0); let ω2 be its pull-back to M . Note that its K moment map is zero. Then ω1 + ω2 is a closed two-form for which G ΨK + ΨGp is a moment map and which is symplectic at p, and hence on an entire neighborhood of O. Proof of Corollary G.23. By Theorem G.22 it is enough to show that for each p ∈ M the slice S = ker dΦ|p /Tp O admits a Gp -invariant complex structure. As before, we decompose our torus as G = K ×Γ Gp and the moment map as Ψ = ΨK ⊕ ΨGp . Then ker dΨ|p = ker dΨK |p , and Tp M ∼ = ker dΨK |p ⊕ k∗ , where ∗ ∼ Gp acts on k trivially. Also, ker dΨ|p = S ⊕ Tp O, where Gp acts on Tp O trivially. An equivariant stable complex structure on M then gives a Gp -invariant complex structure on Tp M ⊕ Rm = S ⊕ Tp O ⊕ k∗ ⊕ Rm for some m. Because the subspace fixed by Gp is (Tp M ⊕ Rm )Gp = S Gp ⊕ Tp O ⊕ k∗ ⊕ Rm , this is a complex subspace, and the quotient (Tp M ⊕ Rm )/(Tp M ⊕ Rm )G ∼ = S/S Gp acquires a Gp -invariant complex structure. Finally, because S ∼ = S/S Gp ⊕ S Gp , Gp and because S is an even-dimensional vector space with a trivial Gp action, an arbitrary complex structure on S Gp and the Gp -invariant complex structure on S/S Gp fit together to give a Gp invariant complex structure on S. 3.2. Odd-dimensional analogues. In this section we briefly touch upon the question of the local structure of non-degenerate moment maps on odd–dimensional manifolds. Example G.28. Let ω be a closed G-invariant two-form of maximal rank on an odd–dimensional manifold M such that the field of directions ker ω (called the characteristic field) is nowhere tangent to G-orbits. Also, let Ψ be an abstract moment map associated with ω. (Similarly to the the symplectic case, Ψ always exists locally under the above non-tangency condition, but need not exists globally.) Then Ψ is non-degenerate. Example G.29. In particular we note, continuing Remark G.18, that a contact moment map is non-degenerate exactly at the points where the Reeb vector field is not tangent to the G-orbit. (For a compact contact manifold, a tangency always occurs at some points.) In fact, the converse of Example G.28, i.e., an odd–dimensional analogue of Theorem G.22, also holds locally. First note that arguing as in Section 3.1, it is not hard to prove the following result. Theorem G.30. Assume that M is odd–dimensional and orientable and let Ψ be a non-degenerate abstract moment map on M . Then locally (M, Ψ) is a cylinder over an even–dimensional manifold with abstract moment map. In other words, for every point p ∈ M , there exists a G-invariant open neighborhood U containing p such that U is equivariantly diffeomorphic to the product U0 × I, where U0 is even-dimensional and I is an interval with trivial G-action, and Ψ is independent of the I-component. Applying Theorem G.22 we conclude that, if the slice in U0 admits an invariant complex structure, then locally M is a product of a symplectic manifold and

300

G. NON-DEGENERATE ABSTRACT MOMENT MAPS

an interval, where the action is trivial on the interval and the moment map is independent of the second component. Alternatively, this can be expressed as the odd-dimensional analogue of Theorem G.22. Namely, for every point p ∈ M , there exists a G-invariant open set U containing p and a G-invariant form ω on U of maximal rank such that the action of G on (U, ω) is Hamiltonian and Ψ is the moment map associated with this action. The characteristic foliation of ω is invariant under the G-action, nowhere tangent to the G-orbits, and Ψ is constant along this foliation. All of these statements (including Theorem G.30) are equivalent to each other and can be easily derived from Theorem G.22. For example, let us prove the local existence of ω, which readily implies Theorem G.30. First, we apply Theorem G.22 to the product M × R where the G-action on R is trivial and Ψ is extended to M ×R to be independent of the second component. Then the form ω is obtained by restricting the symplectic form on a neighborhood of (p, 0) to the intersection of this neighborhood with M × {0}. It is clear that ω has maximal rank. Since the form ω is invariant, the characteristic field ker ω is G-invariant. Since Ψ is associated with ω, it is constant along characteristics. Finally, the first non-degeneracy condition implies that ker ω is nowhere tangent to G-orbits.

APPENDIX H

Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 1. The Hamiltonian cobordism ring and characteristic classes In this section we discuss the problem of calculating the Hamiltonian cobordism ring. We start this discussion with compact cobordisms of compact G-manifolds. Throughout this section all manifolds are assumed to be stable complex1 and, as usual, G is a torus. 1.1. Formulation of the problem. Recall that a compact stable complex Hamiltonian G-manifold is a stable complex G-manifold equipped with a cohomol2 ogy class u ∈ HG (W ). The Hamiltonian cobordism ring H∗G of such manifolds is very large, and probably does not have an elegant description in terms of generators and relations. Here, however, we will be more interested in the invariants of cobordism. Hence, our focus will be on the “dual” problem of finding a full collection of invariants of Hamiltonian cobordism. A satisfactory solution for this problem is unknown also, but partial information is available. Since in this book we are mainly interested in mixed characteristic numbers (see Section 2 for the definition), we will concentrate on the following questions: (A) Do mixed characteristic numbers form a full system of invariants of Hamiltonian cobordism? (B) What are the relations among the mixed characteristic numbers of cobordism classes in a given dimension? (C) Which cobordism classes of Hamiltonian G-manifolds are represented by symplectic manifolds? Are there additional constraints on the mixed characteristic numbers of symplectic manifolds? Complete answers to these questions are known only in the non-equivariant case, when the group is trivial and the moment map is zero. Let us discuss this case. 1.2. Non-equivariant Hamiltonian cobordisms. The ring of non-equivariant Hamiltonian cobordisms with zero cohomology classes is the same as the ring of (stable) complex bordisms MU ∗ , [MiSt]. As a ring, MU∗ ⊗Q is freely generated by complex projective spaces. The set of generators of MU∗ itself is also known, [MiSt]. The mixed characteristic numbers become in this case the ordinary Chern numbers. They form a a complete set of independent invariants of MU∗ and they may assume arbitrary rational values on MU∗ ⊗Q, [MiSt]. 1 A complex cobordism theory is always simpler than a real one. This is true already in the non-equivariant case; see [MiSt].

301

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H. CHARACTERISTIC NUMBERS

Turning to general non-equivariant Hamiltonian cobordisms, for simplicity, we restrict our attention to the case where the cohomology classes are integral. Then the theory of Hamiltonian cobordisms becomes identical with the theory of stable complex cobordisms equipped with integral cohomology classes of degree two. The resulting cobordism ring can be identified with the stable complex bordism ring MU∗ (CP∞ ) of CP∞ for which MU∗ is the “coefficient ring”. The ring MU∗ (CP∞ ) is calculated in [Sto]. Its free (additive) generators are the homotopy classes of the maps of the generators of MU∗ to CP∞ . As above, the mixed characteristic classes form a full system of independent invariants of integral non-equivariant Hamiltonian cobordism, [Sto], and they can assume arbitrary values over Q. This answers questions (A) and (B) in this case. One can also show that every integral non-equivariant Hamiltonian manifold is cobordant to a symplectic manifold, [Bak, Gin2, Mor].2 (For example, a stable complex manifold equipped with the zero cohomology class is still cobordant to a symplectic manifold, which necessarily has more than one connected component.) In other words, symplectic manifolds generate MU ∗ (CP∞ ). Hence, the mixed characteristic numbers form a full system of independent invariants of Hamiltonian cobordism classes of symplectic manifolds. By “dualizing” we also conclude that every collection of mixed characteristic numbers which is possible for an integral non-equivariant Hamiltonian manifold is also attainable for an integral symplectic manifold. 1.3. The equivariant cobordism ring and the Hamiltonian cobordism ring. The questions (A)–(C) become considerably more subtle in the equivariant case. As above, the key to understanding question (A) is the stable complex equivariant cobordism ring U∗G , i.e., the ring of Hamiltonian cobordisms in which equivariant cohomology classes are assumed to be zero. This ring plays the role of the coefficient ring for the theory of stable complex Hamiltonian cobordisms. To avoid possible confusion we note that there are two essentially different Gequivariant (compact) stable complex cobordism theories. One theory is the naive geometric theory of G-cobordisms defined as above with manifolds equipped with (tangential) G-equivariant stable complex structures. This theory arises in applications we are interested in and is the only one considered in this book and U∗G denotes the ring of such cobordisms. However, this theory is relatively poorly understood. 3 The second theory, introduced by tom Dieck, [tD], is a certain “extension” of the geometric theory by homotopy theoretical means. This theory is accessible by topological methods and is much better understood; see, e.g., [tD, Kr, May, Si1, Si2] and references therein. This theory is however less suitable for our goals. Although a complete proof is unavailable at the moment, there is some evidence in favor of the conjecture that the answer to question (A) is affirmative for equivariant cobordisms: equivariant Chern numbers distinguish equivariant cobordism classes. For G-manifolds with isolated fixed points this is established in Section 3. A relevant result in the homotopy theoretical setting is obtained by Sinha in [Si1]. Note that a positive answer to (A) for Tn -equivariant cobordisms for all n would imply the affirmative answer to that question for stable complex Hamiltonian cobordisms; see Section 2. 2 We emphasize again that the orientations of symplectic manifolds are not assumed to be induced by the symplectic forms. 3 For some discrete groups (e.g., Z ) the real analogues of the ring R are described in [CF3]. p

1. THE HAMILTONIAN COBORDISM RING

303

In contrast with the non-equivariant setting, the answer to question (B) is unknown already for equivariant cobordisms. In this connection we note that in every even dimension there are non-vanishing equivariant Chern classes of arbitrarily high degrees. By the localization theorem, the corresponding Chern numbers can be expressed through the (cobordism classes of) fixed point data. However, the group of these cobordism classes is infinitely generated in every dimension. Hence, there is no reason to expect that relations exist. For example, in dimension two the only non-trivial Chern classes are powers of the first Chern class and there appear to be no relations among the Chern numbers. The answer to question (C) is also unknown. In the non-equivariant case the argument is based on a complete understanding of the generators of the cobordism ring, lacking for the equivariant cobordism ring. Yet it is not unreasonable to conjecture that modulo torsion the Hamiltonian cobordism ring is generated by symplectic manifolds when G is a torus. From the symplectic geometry perspective, the main reason to consider Hamiltonian cobordisms is that mixed characteristic numbers are invariants of Hamiltonian cobordism. To summarize the above discussion, we can say that the theory of Hamiltonian cobordisms is likely to be the “minimal” theory with this property. Furthermore, once we are only concerned with such integral invariants, there is no reason to restrict one’s attention to symplectic manifolds. Moreover, as we have already seen, non-symplectic Hamiltonian manifolds arise naturally already in symplectic geometry problems. We conclude this discussion by mentioning that the theory of Hamiltonian cobordisms and its applications to symplectic geometry have been inspired by the theory of non-degenerate cobordisms to which we will turn in Section 4.

1.4. Relations between compact and non-compact cobordisms. In this book we consider proper Hamiltonian cobordisms, where non-compact manifolds are allowed but all moment maps are assumed to be proper. In particular, we then work with the equivariant forms ω −Φ rather than their cohomology classes [ω −Φ]. Furthermore, we may discard the two-forms and work with abstract moment maps (see Chapter 3). As we have already mentioned, the main problem which arises when one works with non-compact cobordisms is that compact manifolds may all turn out to be cobordant to zero in the resulting cobordism theory. This is certainly very undesirable and it is important to know that this does not happen when we pass to proper cobordisms of abstract moment maps. Thus let us compare G-equivariant compact cobordisms and proper cobordisms of abstract moment maps. Note that any compact G-manifold or cobordism can be equipped with the zero abstract moment map which is automatically proper due to compactness. In this way we obtain a map from the G-equivariant compact cobordism theory to the theory of proper cobordisms of abstract moment maps. The main result of [GGK2] is that this map is one-to-one, i.e., the compact theory is embedded into the proper one. In other words, introducing non-compact cobordisms does not result in new relations between compact G-manifolds. The linearization theorem (Theorem 4.12), comes close to a satisfactory description of the non-compact theory. Using reduction, we also conclude that fiber bundles of toric orbifolds (with possibly exotic stable complex structures) generate

304

H. CHARACTERISTIC NUMBERS

the image of the compact G-equivariant cobordism theory in the compact equivariant orbifold cobordism theory. (Also see [Mart1].) The question of which non-compact manifolds are cobordant to compact ones is still open. In other words, as of today, the image of the inclusion of the compact theory into the proper one is unknown. The problem is closely related, via the localization formula, to describing relations among the representations of G on the normal bundles to M G when M is compact. For G = S 1 , the relations are found by Gusein-Zade, [GZ2]. However, this description is rather complicated and difficult to decipher, even in the case of isolated fixed points. When G is a torus, the problem of describing the set of relations is completely open. An important ingredient of the theory of compact cobordisms which does not have an immediate extension to the non-compact case is the notion of characteristic numbers. Although one can sometimes define characteristic numbers in the non-compact case using, for example, the cutting construction, reduction, or the localization theorem, it is not clear whether the resulting invariants are useful and tangible. 2. Characteristic numbers In this section we focus on characteristic numbers for the ring of G-equivariant (stable) complex cobordisms, U∗G , and the ring of (stable) complex Hamiltonian cobordisms of G-manifolds, H∗G . All Hamiltonian G-manifolds in this sections are assumed to be integral, i.e., the corresponding equivariant cohomology classes 1 2π [ω − Φ] are assumed to be integral. The equivariant characteristic numbers cG I (W ) of a compact oriented stable complex G-manifold are defined as 

G i1 G ik ∗ cG I (W ) = (c1 ) · · · (ck ) , [W ] ∈ H (BG),

where I = (i1 , . . . , ik ) is a multi-index and cG l is the lth equivariant Chern class of W . For a stable complex Hamiltonian G-manifold W , the mixed equivariant characteristic numbers cG j,I (W ) are defined by

j  G i1 G ik ∗ cG (H.1) j,I (W ) = u · (c1 ) · · · (ck ) , [W ] ∈ H (BG),

where u is the equivariant cohomology class of degree two. Note these characi1 G ik teristic numbers can be non-zero if the degree of the product (cG or 1 ) · · · (ck ) i1 G ik is greater than dim W/2. ) uj · (cG ) · · · (c 1 k In the next section we will present some evidence supporting the conjecture that the equivariant characteristic numbers separate elements of U∗G when G is a torus. G Here we will construct a monomorphism ΣG : H∗G → U∗+2 and show that if this conjecture holds, then mixed characteristic numbers also separate stable complex Hamiltonian equivariant cobordisms. 2 By Theorem C.47, for a given integral class u ∈ HG (W ; Z), there exists a G-equivariant principle U(1)-bundle E → W . (Note that, in contrast with the non-equivariant case, this is a non-trivial fact.) Moreover, this bundle is unique up to an equivariant principle bundle isomorphism. Let W → W be the S 2 -bundle associated with E → W , where S 2 is equipped with the standard U(1)-action by rotations about the z-axis. The G-action on W inherited from E commutes with the natural right U(1) = S 1 -action. Hence, W can be viewed as a stable complex

3. CHARACTERISTIC NUMBERS AS A FULL SYSTEM OF INVARIANTS

305

(G × S 1 )-manifold. It is not hard to see that the mapping (W, u) → W gives rise G×S 1 to a linear homomorphism ΣG : H∗G → U∗+2 . Proposition H.1. The homomorphism ΣG is a monomorphism. Furthermore, the G×S 1 -characteristic numbers of W can be expressed via the mixed G-characteristic numbers of (W, u). Proof. Note that W is canonically embedded in W as the “North Pole secS1

tion”, i.e., as the connected component of W with the representation of S 1 on the normal bundle isomorphic to the standard representation of S 1 on C. (The “South Pole section” has the complex conjugate normal representation.) The class u is then the first G-equivariant Chern class of the normal bundle to W in W . Let (W0 , u0 ) and (W1 , u1 ) be oriented stable complex G-manifolds equipped with equivariant cohomology classes. Assume that W0 is G × S 1 -cobordant to S1

S1

W1 . Then W0 is G-cobordant to W1 . Moreover, the same is true for the components with a given normal representation of S 1 . Thus W0 is cobordant to W1 as an oriented stable complex G-manifold. Finally, the cobordism between the cohomology classes is given by the first G-equivariant Chern class of the normal bundle. This proves that ΣG is injective. The second assertion follows from the appropriate version of the Atiyah–Bott– Berline–Vergne localization theorem (cf. Section 7 of Appendix C). Remark H.2. As is clear from the proof, the theorem holds for any compact Lie group G. 1

Example H.3. The group U2S is the direct sum of the following summands. • The group of (stable) complex cobordisms of two-manifolds with trivial S 1 action. This group is isomorphic to Z, generated by S 2 , and the only invariant is the first Chern number (see [MiSt]). • The free abelian group generated by S 2 with the standard S 1 -action. • For every non-zero integer k 6= 1, the free abelian group generated by S 2 with the S 1 -action which is obtained from the standard one via the k-fold covering S 1 → S 1 . (The trivial action would correspond to k = 0 and the standard action would result from k = 1.) A similar description applies when G = SU(2) except that in this case one does not have components corresponding to the k-fold coverings. 3. Characteristic numbers as a full system of invariants Our goal in this section is to show that two stable complex compact G-manifolds with isolated fixed points are cobordant if and only if their equivariant characteristic numbers are equal. Here, as usual, G is a torus. In one direction the assertion is clear: by Stokes’ formula, cobordant manifolds have equal characteristic numbers. Hence we need to prove Theorem H.4. Let G be a torus and let W be a compact stable complex Gmanifold such that W G is discrete and the mixed characteristic numbers of W are equal to zero. Then W is cobordant to zero. Remark H.5. As we have mentioned above, the condition that the fixed points are isolated is probably purely technical. Conjecturally, Theorem H.4 holds without this assumption.

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Proof of Theorem H.4. The tangent spaces Tp W , for p ∈ W G , are stable complex representations of G. Moreover, these vector spaces are in fact genuine complex representations of G, as readily follows from the definition of equivariant stable complex structures (see Section 1 of Appendix D). In addition, the spaces Tp W inherit orientations from W , and these orientations may differ from the orientations induced by the complex structures. Let mV be the number of times with which a complex representation V of G occurs among the representations Tp W , for p ∈ W G . Note that mV is understood algebraically: mV = #{p ∈ W G | Tp W = V } − #{p ∈ W G | Tp W = −V },

where Tp W = −V means that the two representations are equal, but the orientations are different. Lemma H.6. mV = 0 for every complex representation V of G. Let us postpone the proof of the lemma and first finish the proof of the theorem. This can be done directly using the methods of [GZ1]. The following argument relies on the linearization theorem and the results of [GGK2]. Let us equip W with an abstract moment map which is identically zero and “polarize” by picking an arbitrary vector η ∈ g∗ such that the zero set M η = {ηM = 0} coincides with the fixed point set M G . By the linearization theorem for abstract moment maps (Theorem 4.12), we have G (H.2) Tp W W ∼ p∈W G

where the tangent spaces Tp W are equipped with the induced G-actions, orientations, and complex structures. The tangent spaces Tp W are also equipped with abstract moment maps Φp which are polarized and equal to zero at the origin. As is shown in [GGK2], the theory of compact cobordisms is embedded into the theory of proper cobordisms of abstract moment maps. Thus it suffices to show that the right-hand side of (H.2) is cobordant to zero. Note that two quadratic abstract moment maps on the same representation of G, which are polarized with respect to the same vector, are cobordant if and only if these moment maps assume equal values at the origin. (This can be easily proved in the same manner as Lemma 3.31.) Since Φp (0) = 0 for all p ∈ W G , the contribution of Tp W to the cobordism class of the right-hand side of (H.2) is entirely determined by the complex representation of G on Tp W and the orientation of Tp W and is independent of the moment map Φp . Therefore, the right-hand side of (H.2) is cobordant to zero if and only if every representation V occurs in it with zero multiplicity, i.e., mV = 0. Hence the theorem follows from Lemma H.6. Proof of Lemma H.6. A representation of G on Cn is determined by its weights α1 , . . . , αn , which are elements of g∗ . The weights, in turn, are determined by the values σ1 (α1 , . . . , αn ), . . . , σn (α1 , . . . , αn ), where σ1 , . . . , σn are the elementary symmetric functions. Since α1 , . . . , αn are linear functions on g, the symmetric function σk (α1 , . . . , αn ) is a polynomial on g of degree k. Denote by σ(p) = (σ1 (p), . . . , σn (p)) the collection of such symmetric functions for the representation of G on Tp W , p ∈ W G . Clearly, σ(p) determines the representation, but not the orientation of Tp W inherited from W . For a collection σ of n polynomials on g, let mσ be the algebraic number of representations Tp W with σ(p) = σ. (Of

3. CHARACTERISTIC NUMBERS AS A FULL SYSTEM OF INVARIANTS

307

course, we set mσ = 0 if σ does not occur as the collection of symmetric functions for any of the Tp W .) In other words, mσ = m V , where the collection of symmetric functions for V is σ. Thus our goal is to show that mσ = 0 for any σ. The restriction of the kth equivariant Chern class cG k (T W ) of W to p is equal to the kth equivariant Chern class cG (T W ) of the vector bundle Tp W → p which p k in turn is σk (p): G k cG k (T W ) |p = ck (Tp W ) = σk (p) ∈ S (g). The normal bundle to p in W is Tp W with the orientation inherited from W . The equivariant Euler class of this bundle is ±cn (Tp W ) = ±σn (p), where the sign is positive if the orientation of Tp W agrees with the complex orientation and negative otherwise. Thus using the Atiyah–Bott–Berline–Vergne localization theorem (Theorem C.53), we obtain the following expression for the equivariant Chern numbers of M :

G  in cG c1 (T W )i1 · . . . · cG I (W ) = n (T W ) , [W ] X σ1 (p)i1 · . . . · σn (p)in ± = σn (p) p∈W G X ±σ1 (p)i1 · . . . · σn (p)in −1 = p∈W G

X

=

σ

mσ σ1i1 · . . . · σnin −1 ,

where I = (i1 , . . . , in ) is a multi-index. In what follows we will always assume that in ≥ 1 to make every term on the right-hand side polynomial. By the hypothesis of the theorem, cG I (W ) = 0 for all I. To show that mσ = 0 for any σ, we will make suitable choices of multi-indices I. Let η1 , . . . , ηs be all the distinct polynomials that occur as σ1 (p) for p ∈ W G . Then s X X G mσ σ1i1 · . . . · σnin −1 cI (W ) = j=1

=

s X j=1

σ σ1 =ηj

ηji1

X

σ σ1 =ηj

mσ σ2i2 · . . . · σnin −1

= 0. Let i1 vary in the range 0, . . . , s − 1. Then (ηji1 ) is a van der Monde matrix (with polynomial entries). Hence, X mσ σ2i2 · . . . · σnin −1 = 0 σ σ1 =ηj

for all j and any multi-index (i2 , . . . , in ). Note that if σ is such that σ1 6= ηj for any j, we automatically have mσ = 0. So the last identity can also be written as X mσ σ2i2 · . . . · σnin −1 = 0 σ

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H. CHARACTERISTIC NUMBERS

for any multi-index (i2 , . . . , in ), where we sum over σ = (σ2 , . . . , σn ). Proceeding inductively, we conclude that mσ = 0 for any σ.

Remark H.7. In the proof of the theorem, to conclude that M is cobordant to zero, we used only a finite number of different characteristic classes. The exact number needed depends on the dimension and the size of the fixed point set. 4. Non-degenerate cobordisms One of the themes of this book stems from the fact that geometric quantization is an invariant of cobordism. For the major part of the book, both cobordism and geometric quantization are understood as formal objects. Cobordism is stable–complex equivariant cobordism of manifolds equipped in addition with an equivariant cohomology class of degree two. Geometric quantization is understood as a virtual representation given for example by the Riemann–Roch formula. In this section we show that, at least in the non-equivariant case, geometric quantization remains, in some sense, an invariant of cobordism when both quantization and cobordism are defined more geometrically. Let us first recall the definition of non-degenerate cobordism of symplectic manifolds given in [Gin1, Gin2].4 Let (W0 , ω0 ) and (W1 , ω1 ) be compact oriented (integral) symplectic manifolds of dimension 2n. Note that the orientations of W0 and W1 are not assumed to be induced by the symplectic structures. We say that (W0 , ω0 ) and (W1 , ω1 ) are cobordant if there exists an oriented cobordism M between W0 and W1 equipped with a closed (integral) two-form σ of maximal rank (equal to 2n), which restricts to ω0 and ω1 on W0 and, respectively, W1 . We emphasize again that this notion of cobordism is different from the ones used in other parts of this book. Remark H.8. Since σ has maximal rank on an odd–dimensional manifold, the null space ker σ is one–dimensional at every point of M . The integral curves of the line field ker σ are called the characteristics of σ. The orientation of M together with σ n give rise to the orientation of ker σ and hence to the orientations of the characteristics. For almost all x ∈ M , the characteristic through x begins and ends on ∂M . There may be, however, characteristics that do not have beginnings and/or ends on the boundary. Similarly, for a given connected component W of ∂M , either every x ∈ W is the beginning of a characteristic or its end, depending on whether R n ω is positive or negative. For almost all x ∈ W , the characteristic through x W reaches some other component of ∂M . As usual, cobordism equivalence classes form a ring. In what follows we will restrict our attention to the case where the symplectic forms and cobording twoforms are assumed to have integral cohomology classes. In this case, the cobordism ring, denoted by B∗ , has been calculated. Example H.9. Let (Σg , k) be the surface of genus g ≥ 0 with symplectic area equal to k. It is not hard to show (see [Gin1]) that (Σg1 , k1 ) + (Σg2 , k2 ) ∼ (Σg1 +g2 −1 , k1 + k2 ),

where ∼ denotes non-degenerate cobordism and k1 > 0 and k2 > 0. 4 Throughout

this section, all manifolds are assumed to be compact.

4. NON-DEGENERATE COBORDISMS

309

Furthermore, B2 is a free abelian group of rank 2. The generators are the sphere and the torus of unit area. For example, a surface of genus two is cobordant to the difference of a torus and a sphere. A Rfull system of independent invariants in dimension two is given by the total area W ω of a symplectic surface (W, ω) and the twisted Euler characteristic Z  X χ(W, ω) = sign ω χ(Wi ), Wi

where Wi are the connected components of W . An elementary proof of this fact can be found in [Gin1]. Note that these invariants cannot assume arbitrary integer values: χ(W, ω) is necessarily even.

To describe B∗ note that there is a forgetful homomorphism F from B∗ to the ring of stable complex cobordisms equipped with integral cohomology classes of second degree. This homomorphism sends the cobordism class of (W, ω) to the cobordism class of the triple (W, J, [ω]), where J is an almost complex structure on W compatible with ω. The ring of such stable complex cobordisms is naturally isomorphic to the ring of bordisms MU∗ (B U(1)), where B U(1) = CP∞ . Theorem H.10 ([Bak, Gin2, Mor]). The forgetful homomorphism F : B∗ → MU∗ (B U(1)) is an isomorphism. Recall that for any multi-index I = (i1 , . . . , ik ), the Ith mixed characteristic number of (W 2n , ω) is by definition D E cI (W ) = [ω]n−kIk ci11 . . . cikk , [W ] , (H.3)

where ci is the ith Chern class of W with respect to J, as usual, [W ] is the fundamental class of W with respect to the given orientation, and k I k= 1i1 + 2i2 + · · · + kik is the weight of I. Note that cI (W ) = 0 automatically whenever k I k> n. It follows from Theorem H.10 (see [Sto]) that the mixed characteristic numbers form a full system of independent invariants of cobordisms of symplectic manifolds. In other words, the evaluation homomorphism {cI } : B2n → Zk(n) , where k(n) is the number of multi-indices I with k I k≤ n, is one-to-one and its image has a full rank. Note, however, that this homomorphism is not onto (see Example H.9). The assertion that F is a monomorphism is an easy consequence (see [Gin2]) of an appropriate h-principle, [Grom, McD1]. Then, using the fact that the ring of stable complex cobordisms is generated over the rationals by projective spaces, it is not hard to show that F and {cI } are isomorphisms modulo torsion, [Gin2]. The proof of the fact that F is a genuine epimorphism is more involved; see [Bak, Mor]. It is worth emphasizing that the non-degenerate cobordism of symplectic manifolds considered in this section is “soft”—it does not capture any deep symplectic invariants of (W, ω) beyond those determined by J and [ω]. This is exactly the reason why the calculation of the ring B∗ can be relatively easily reduced to an algebraic topological problem. In conclusion note that a cobordism (M, σ) between (W0 , ω0 ) and (W1 , ω1 ) gives rise to a Lagrangian relation L in (W0 × W1 , ω0 ⊕ (−ω1 )): by definition, (x, y) ∈ L if and only if x and y are connected by a characteristic of σ. According to a general principle (see., e.g., [Wei2]) Lagrangian relations should, in turn, induce operators between quantization spaces. Later on we will see that the cobordism (M, σ) does induce a homomorphism from the geometric quantization of (W0 , ω0 )

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H. CHARACTERISTIC NUMBERS

to the geometric quantization of (W1 , ω1 ). However, defining this homomorphism, as well defining the geometric quantization, requires fixing an additional structure. 5. Geometric quantization 5.1. Two constructions of geometric quantization. The virtual geometric quantization of a symplectic manifold (W, ω) is, by definition, the index Q(W, ω) = ker D − coker D of a suitable elliptic operator D (see Chapter 6). When there is no Hamiltonian group action involved, the index is just a virtual vector space, i.e., it is determined by the integer equal to its virtual dimension and given by the Riemann–Roch formula. Thus, in the non-equivariant situation, the virtual geometric quantization is a rather weak invariant of a symplectic manifold (similar to the Euler characteristic or the signature in algebraic topology). However, one can associate to a given symplectic manifold geometric quantizations which are genuine, not only virtual, vector spaces. Here we consider two constructions of such spaces, both based on various versions of vanishing theorems according to which coker D = 0 if the symplectic form is “large enough”. Thus, by definition, the geometric quantization is just ker D. The dimension of the space is, of course, “correct”, i.e., equal to the index of D. An essential feature of both of these constructions is that the resulting space depends on additional data and is not canonically determined by the symplectic structure alone. Let us give some more details on these constructions. Let (W, ω) be a symplectic manifold. Fix an almost complex structure J on W , compatible with ω. As shown in Section 6 of Chapter 6, the pair (J, ω) gives rise to a first order elliptic operator D, the rolled-up Dolbeault operator. Consider now the symplectic structures kω, k = 1, 2, . . . on W , with J fixed, and let Dk be the family of corresponding elliptic operators. As shown in [GU], coker Dk = 0 if k is large enough. Thus, for such k, the quantization Q(W, kω) = ker Dk is a genuine vector space. It should be noted, however, that the required value of k may depend on J as well as on ω. The second construction is similar to the one which we just described but uses the Spinc -Dirac operator in place of D. Now the initial data is a Spinc -structure on W . For example, one can start again with an almost complex structure J compatible with ω, associate to the pair (kω, J) a Spinc -structure (see Appendix D) and consider the corresponding Spinc -Dirac operator Dk . Again, coker Dk = 0 when k is large enough, [BU], and we just have Q(W, kω) = ker Dk . (See also [Brav1].) It is worth noting that the elliptic operators used in these two constructions differ from each other (although they do have equal (leading) symbols). As a result, in spite of their similarity, the constructions give, strictly speaking, different vector spaces, which have, of course, equal dimensions. Note also that although the orientations of the manifolds do not explicitly enter either of the constructions, the manifolds should be oriented via the symplectic forms, for then the results are consistent with the Riemann–Roch formula. We refer the reader to Chapter 6 for a more detailed discussion of these quantization constructions. 5.2. Dependence on the almost complex structure. Both methods of obtaining a genuine vector space as a quantization rely on additional structures, such as an almost complex structure J compatible with ω. The quantization space Q(W, kω) depends on J. To emphasize this dependence, let us fix W and ω and

5. GEOMETRIC QUANTIZATION

311

denote the quantization space, resulting from either of the two constructions, by Qk (J). Let us now examine closer the dependence of Qk (J) on J, following [GM]. (For a completely different approach to the problem in the case of linear polarizations of Cd see [Ki4, Section 4.2] and references therein.) Let J be the space of almost complex structures compatible with ω. The collection of spaces Qk (J) can be viewed as a vector bundle over J . Literally, this is not accurate: the value of k required for the vanishing theorem to apply depends on J. However, the space J is exhausted by open subsets Uk such that the spaces Qk (J) form a vector bundle Vk over Uk , for each k. Remark H.11. Sometimes one can choose k so that Uk = J , that is, the bundle Vk is defined over the entire space J . This is the case, for example, when dim W = 2. If we could identify the quantization spaces Qk (J) with each other in some natural way, which only depends on the symplectic structure, they would fit together into one single “canonical” quantization, independent of a choice of J. Therefore, we would like to choose a flat connection on the bundle Vk which is preserved under the group of symplectomorphisms. Let us explain this more precisely. The group of symplectomorphisms acts on the space J and on each of the open sets Uk . However, this action does not lift in a canonical way to the bundles Vk . To lift a symplectomorphism to Vk , it suffices to lift it to a contactomorphism of the pre-quantization circle bundle E → W , which requires fixing a Hamiltonian. To be more specific, let H be the group of Hamiltonian symplectomorphisms, i.e., of symplectomorphisms generated by time-dependent Hamiltonians. Fix a prequantization circle bundle E → W with a connection form Θ. (See Section 2 of Chapter 6 for the definitions.) Recall that Θ is a contact form and thus the connection ker Θ is a contact structure on E. Let C be the identity connected component in the group of all contactomorphisms of ker Θ. It is easy to see that there is an epimorphisms C → H and that, in fact, C is an extension of H by S 1 . By definition, the group C acts on W through the action of H. This action canonically lifts to the original action of C on E. As a result, we obtain the induced action of C on J (and on Uk ) and its lift to Vk . One way to say that quantization is independent of J is to find a natural flat (or projectively flat) connection on Vk . By a natural connection we mean one that is C-invariant. The following result shows that for many symplectic manifolds W the bundle Vk does not admit a natural flat connection. Theorem H.12 ([GM]). Let J0 be an almost complex structure such that the stabilizer G of J0 in H has positive dimension and the infinitesimal representation of G on Qk (J0 ) is non-trivial. Then there is no natural projectively flat connection on Vk . Remark H.13. The theorem is a more or less an immediate consequence, already on the infinitesimal level, of the following well–known fact. Denote by A the Lie algebra Cc∞ (W ) of compactly supported functions on a (not necessarily compact) symplectic manifold W . Then the commutant {A, A} is the only ideal of finite codimension in A. This result, closely related to the Gr¨ oenewald–van Hove theorem, holds for many other Poisson algebras; see, e.g., [GM, GGG] for further

312

H. CHARACTERISTIC NUMBERS

references. Theorem H.12 follows from the observation that if the connection existed, it would result in a non-trivial finite–dimensional projective representation of A. The kernel of this representation would be an ideal of finite codimension, which is different from the commutant. Remark H.14. Theorem H.12 holds already for the restriction of V to a neighborhood of J0 in the H-orbit of J0 . Note that if J0 is K¨ ahler, the entire orbit is comprised of K¨ ahler complex structures. It would be interesting to see if the theorem remains correct for any compact symplectic manifold, i.e., without the additional assumption on the stabilizer G. Example H.15. The simplest manifold to which Theorem H.12 applies is W = S 2 with ω being the standard symplectic form and J0 being the standard complex structure. In this case one can take as U the entire H-orbit of J0 . The stabilizer G is SU(2) and its representation on Qk (J0 ) is just the standard representation of SU(2) on the space of homogeneous polynomials of degree k on C2 . More generally, the theorem applies to coadjoint orbits of compact Lie groups. Remark H.16. In conclusion note that usually the bundle V = Vk over U = Uk admits many natural connections, which are, of course, non-flat. The following connection ∇ seems to be of particular interest. To define ∇ observe first that for any J there exists a natural inclusion Qk (J) ,→ C ∞ (W ; L), where L is the quantization line bundle; see [GU, BU]. (For example, when J is K¨ ahler, by definition, Qk (J) ⊂ C ∞ (W ; L).) Thus V is a subbundle of the trivial bundle U × C ∞ (W ; L). The connection ∇ comes from the flat connection on the trivial bundle. Namely, let s be a section of V and let J(t) be a path in U. Denote by P : C ∞ (W ; L) → Qk (J(0)) the orthogonal projection. Then we set   ds(J(t)) (∇J˙(0) s)(J(0)) = P . dt t=0

2

It would be extremely interesting, already for W = S , to find an explicit formula for the curvature of this connection and to see how the curvature behaves as k → ∞. As pointed out by Alejandro Uribe, the problem should be accessible using the techniques developed in [BdMG]. (See also the results of [Da] related to the case where W = S 2 .) 5.3. Quantization and cobordism. In this section we outline the construction, due to Kopr´ as and Uribe [KU], of a homeomorphism between quantization spaces, associated with a cobordism. Let (M, σ) be a cobordism between symplectic manifolds (W0 , ω0 ) and (W1 , ω1 ). Assume, for the sake of simplicity, that the orientation of W0 , as a part of ∂M , is symplectic, and the orientation of W1 is the opposite of symplectic. Let us equip M with a Riemannian metric which is compatible with σ in the natural sense and which is cylindrical near ∂M ; see [KU]. Such a metric gives rise to a Spinc -Dirac operator DM on M which depends on the integer parameter k > 0 as σ gets replaced by kσ. Set Qk (M ) = ker DM . The restriction of Qk (M ) to ∂M gives rise (in a non-obvious way) to a linear subspace in Qk (∂M ) = Qk (W0 ) ⊕ Qk (W1 ). This subspace is the graph of the operator UM : Qk (W0 ) → Qk (W1 ) which corresponds to the cobordism M . Let us be more specific. Denote by Λ the linear space comprised of the restrictions to ∂M of solutions of the equation DM ψ = 0. Let H≥ (respectively, H> ) be the space spanned by the eigenvectors of the Spinc -Dirac operator

5. GEOMETRIC QUANTIZATION

313

D∂M on ∂M with non-negative (respectively, positive) eigenvalues. The quotient H0 = H≥ /H> is then naturally isomorphic to the quantization of ∂M . In other words, H0 = Qk (W0 ) ⊕ Qk (W1 ). Consider the image Λred of Λ ∩ H≥ under the projection H≥ → H0 . The operator H0 → H0 given by the orthogonal reflection of H0 in the space Λred is unitary and sends Qk (W0 ) to Qk (W1 ); see [KU]. By definition, the restriction UM : Qk (W0 ) → Qk (W1 ) of this operator corresponds to the cobordism M . Moreover, the semiclassical properties of UM turn out to be related to the Lagrangian relation associated with M . This Lagrangian relation is the graph of an (almost-everywhere defined) symplectic mapping, and in this case UM can be viewed as a quantization of this symplectic mapping. (An interesting example, described by Kopr´ as and Uribe in their paper, is a symplectic cobordism between two 2-dimensional tori for which the symplectic map is the classical “Baker’s transformation.”) The analytical aspects, entirely omitted here, of the construction we have just outlined are quite non-trivial. The reader interested in details and examples should consult [KU]. Remark H.17. The results of this section extend to the equivariant setting in a straightforward way: when the manifolds are equipped with Hamiltonian actions of a Lie group G, quantization spaces become representations of G and the operators corresponding to cobordisms commute with the G-action.

APPENDIX I

The Kawasaki Riemann–Roch formula The Kawasaki Riemann–Roch theorem is a version of the Riemann–Roch theorem which applies to orbifolds. The goal of this appendix is to explain the formulation of this theorem. 1. Todd classes Throughout this appendix we will make frequent use of the “splitting principle”. For a precise formulation and justification of this principle we refer to [BT1, Section 21] or Section 5 of Appendix C. We refer to Appendix C for an introduction to equivariant cohomology and equivariant characteristic classes. A somewhat free translation of the splitting principle is as follows. Splitting Principle: If M is a compact manifold and E → M a complex vector bundle one can “in making computations” assume that E splits into a direct sum of complex line bundles E = L1 ⊕ . . . ⊕ Lr . We will also need the equivariant version of the splitting principle: Let G be an ndimensional torus, M a G-manifold M , and E → M a G-equivariant vector bundle. (This means that E is equipped with a lift of the G-action on M .) Then one can assume “in making computations” that the above splitting is G-equivariant. This principle greatly facilitates the computation of the basic topological invariants of E. For instance, the total Chern class c(E) = 1 + c1 (E) + . . . + cr (E) of E (where r = dimC M ) can be found by the product formula: (I.1)

c(E) =

r Y

(1 + c1 (Lk )) .

k=1

In other words, cm (E) is the elementary symmetric polynomial of degree m in c1 (L1 ), . . . , c1 (Lr ). Note also that 1 + c1Q (Lk ) is the total Chern class of Lk and hence this formula can be read as c(E) = c(Lk ). A slightly more complicated invariant is the Todd class of E which is defined as follows. Let x τ (x) = 1 − e−x be the “classical” Todd function. Recall that its Taylor series expansion about the origin is ∞ x X Bk 2k τ (x) = 1 + + (−1)k−1 x , 2 (2k)! k=1

where Bk is the kth Bernoulli number. Since c1 (Li )k = 0 for k > 21 dim M , the substitution x = c1 (Li ) converts this series into a finite sum, and we define the 315

316

I. THE KAWASAKI RIEMANN–ROCH FORMULA

Todd class of E to be (I.2)

Td (E) =

r Y

τ (c1 (Li )) .

i=1

Notice also that the right-hand side can be written as a polynomial in the elementary symmetric functions of the first Chern classes c1 (Li ) and hence, by the product formula, as a polynomial in the Chern classes ck (E). One useful consequence of the splitting principle is the following multiplicative property of the Todd class. Theorem I.1. Let E 0 and E 00 be vector bundles over M . Then (I.3)

Td (E 0 ⊕ E 00 ) = Td (E 0 ) Td (E 00 ).

Proof. Let E = E 0 ⊕ E 00 . From splittings E 0 = ⊕L0i and E 00 = ⊕L00i we obtain a splitting of E into the sum of the line bundles L0i and L00i . Hence Y  Y  Td (E) = τ (c(L0i )) τ (c(L00i )) = Td (E 0 ) Td (E 00 ).

In particular, Theorem I.1 implies that the Todd class is stable: Td (E ⊕ Ck ) = Td (E).

Now let M be a compact pre-quantizable manifold with a pre-quantization (L, h, i , ∇) and a stable complex structure J. (See Section 2 of Chapter 6.) In Section 7 of Chapter 6 we defined the stable complex quantization Q(M ) of M as the index of an associated Dirac operator D. That is, Q(M ) is a finite– dimensional virtual vector space equal to ker D − coker D. However, since the Grothendieck group of virtual vector spaces is canonically isomorphic to Z via the isomorphism dim, we can think of Q(M ) as the integer dim Q(M ), which is, by definition, equal to the index of D. The Atiyah–Singer index theorem applied to this operator implies that Z (I.4) Q(M ) = ec1 (L) Td (T M ). Remark I.2. Probably the simplest of the many proofs of this is the Ktheoretic proof in [ASe]. See also [BGV, Chapter 4] and [Du]. 2. The Equivariant Riemann–Roch Theorem In this section we discuss the equivariant version of the Atiyah–Singer formula (I.4). Let us begin by recalling how the equivariant Chern class of a line bundle is defined. (See also Section 7 of Appendix C for a different construction.) 2.1. Equivariant Todd classes. Let G be an n-torus, M a G-manifold, and π : P → M a principal S 1 -bundle on which G acts by bundle automorphisms. Pick a G-invariant connection form Θ ∈ Ω1 (P )G on P (which we assume to be realvalued). Then the first equivariant Chern class of P , with real coefficients, is the 1 [ω G ], where ω G is the equivariant curvature, which equivariant cohomology class 2π 2 is defined by dG Θ = −π ∗ ω G . We denote this class by cG 1 (L) ∈ HG (M ), where L = P ×S 1 C is the complex line bundle associated with P . (See Section 2 of

2. THE EQUIVARIANT RIEMANN–ROCH THEOREM

317

Chapter 6 and Appendix A.) It is easy to see that although ω G depends on the choice of Θ, the class [ω G ] is well defined. If G acts trivially on M , this class is particularly easy to describe. (See also Example C.41.) In this case the group G acts on Lp for all p ∈ M , and (if M is connected) this action is independent of p. Denote by α the weight of this action. ∗ Recall that, since the G-action on M is trivial, HG (M ) can be thought of as the ∗ space of polynomials on g with values in H (M ). (See Example C.6.) In particular, 2 HG (M ) = g∗ ⊕ H 2 (M ). Lemma I.3. For the trivial G-action, cG 1 (L) = c1 (L) − ordinary first Chern class of L.

1 2π α,

where c1 (L) is the

Proof. The action of g = exp(tξ) on P is just multiplication by e Hence ∂ ξP = α(ξ) , ∂θ

√ −1tα(ξ)

.

where ∂/∂θ is the generator of the action of S 1 on P . Thus dG Θ(ξP ) = ι(ξP )Θ + dΘ = α(ξ) − π ∗ ω, where Θ is a connection form and ω is the curvature for Θ. Because the lemma follows.

1 2π [ω]

= c1 (L),

Next, let E be a complex G-equivariant vector bundle over a G-manifold M . By the splitting principle (see Section 1 of this appendix and Section 5 of Appendix C), we can assume that there is an equivariant splitting E = L1 ⊕ . . . ⊕ Lr . Thus we define the equivariant Chern classes cG k (E) of E, as before, to be the symmetric polynomials in cG (L ), i = 1, . . . , r. Equivalently, it may be defined by i 1 (I.1) with all Chern classes being now equivariant. To define the equivariant Todd class of Li , we substitute cG 1 (Li ) into the infinite series for τ (x) and, similarly to (I.2), use the splitting principle (and the product formula) to define the equivariant Todd class of E. Thus, (I.5)

G

Td (E) =

r Y

τ (cG 1 (Li )).

i=1

This definition deserves some discussion. As we have pointed out above, the usual Todd class of E is a polynomial in the Chern classes. This is no longer the case for the equivariant Todd class. Since the high powers of cG 1 (Li ) are no longer zero in G general, Td (E) is an infinite series, not just a polynomial, in cG k (E) or in c1 (Li ). G Thus, Td (E) should be considered as an element of the equivariant cohomology of M over the ring of infinite power series on g; see Remark C.14. Furthermore, taking the representatives for c1 (Li ) described above, we obtain a differential form representing Td G (E) which is a power series on g convergent for small values of ξ ∈ g. Hence, Td G (E) can also be thought of as an element of the equivariant cohomology of M over the ring of (the germs of) analytic functions on g. This is the point of view which we will adopt in what follows.

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2.2. The equivariant index theorem. Suppose now that M is a G-manifold equipped with pre-quantization data (L, h, i , ∇) and with a G-equivariant stable complex structure. From this data we obtain a virtual representation Q(M ) of G, as described in Section 7 of Chapter 6 and Section 3.5 of Appendix D. Denote by χ the character of this representation. Recall that χ is a smooth function (in fact a trigonometric polynomial) on G. The composition χ ◦ exp is then a smooth function on g. Theorem I.4 (The equivariant index theorem). On a neighborhood of 0 ∈ g, Z G (I.6) ec1 (L) Td G (T M ). χ ◦ exp = M

This version of the equivariant index theorem is due to Bismut (for the standard Dirac operator); see, e.g., [Bi] and also [BGV, Chapter 8] and references therein. An adaptation of his proof to the Dolbeault-Dirac operator case can be found in [Du]. Remark I.5. Let us compare formula (I.6) with the topological Riemann– Roch formula (6.42) which has the same right-hand side as (I.6). The topological Riemann–Roch formula (6.42) establishes an equality of two topological objects, namely, that the right-hand side of (I.6) is equal to ch G (p! ([L])). This is equivalent to that the Taylor expansion of the character of p! ([L]) is given by the integral on the right-hand side of either of these formulas. On the other hand, the index theorem states that the virtual representation obtained as the equivariant index of the Dirac operator has the character given by this integral. Matching these formulas, we see that Q(M ) = p! ([L]), which was the main motivation for the topological definition of the quantization as the push–forward in Section 8 of Chapter 6. The proof of the topological Riemann–Roch formula (6.42) is relatively straightforward and purely algebraic topological, while the proof of (I.6) requires a considerable effort and relies on a heavy use of analysis. Remark I.6. More explicitly, Theorem I.4 states that Z √ √ G χ(exp ξ) = ec1 (L)( −1ξ) Td G (T M )( −1ξ). (I.7) M

Note that the right-hand side of (I.7) is a germ of an analytic √ function on g, evaluated on ξ. (When g is identified with Rn , one has to plug −1ξ in the right-hand side of (I.7) rather than ξ.) This identity should be treated with care. The integrand on the right-hand side of (I.7) is not well defined; in general, a non-zero element ξ ∈ g cannot be plugged in a cohomology class. However, the integral in (I.7), when evaluated at ξ is well defined as pointed out in Remark C.56. For (I.7) to literally make sense, one should replace the cohomology class of the integrand by an equivariant differential form representing it. (See Remark C.56.)

Applying the Atiyah–Bott–Berline–Vergne localization theorem (see Theorem C.53) to the integrand of (I.6) (or (I.7)), one gets a localized version of the equivariant localization theorem, due to Atiyah–Segal–Singer, [ASe, ASi], as follows. Let F be a connected component of M G and let j : F → M be the inclusion map. Denote by T M and T F the stable tangent bundles to M and F , respectively. Note that since M carries a G-equivariant stable complex structure, these are complex vector bundles. By the splitting principle we assume that j ∗ T M splits equivariantly

2. THE EQUIVARIANT RIEMANN–ROCH THEOREM

319

into a sum of line bundles j ∗ T M = L1 ⊕ . . . ⊕ Ln .

(I.8)

We can also assume that the first r summands span the normal bundle of F and the remaining n − r the tangent bundle of F . Let αi be the weight of the isotropy representation of G on Li . Furthermore, the weight of the isotropy representation of G on the fibers of L over F is the constant value ΦF = Φ(F ). Theorem I.7 (The localized equivariant index theorem, I). On an open and dense subset of g, where the right-hand side is defined, Z r  −1 Y X ∗ ΦF /2π (I.9) 1 − e−αi /2π ec1 (Li ) . χ ◦ exp = ec1 (j L) Td (T F ) e F

F

i=1

Proof. From the Atiyah–Bott–Berline–Vergne localization theorem and Theorem I.4 we conclude that ∗ G ∗ X Z e cG 1 (j L) Td (j T M ) Qr χ ◦ exp = (I.10) . G (L ) c i F i=1 1 F By the product formula for the Todd class (Theorem I.1), Td G (j ∗ T M ) = Td (T F ) By Lemma I.3,

r Y

cG 1 (Li ) . G 1 − e−c1 (Li ) i=1

1 αi (ξ) 2π for 1 ≤ i ≤ r and j ∗ cG 1 (Li ) = c1 (Li ) for r < i ≤ d. Furthermore, j ∗ cG 1 (Li ) = c1 (Li ) −

1 ΦF . 2π After some cancelations (I.10) becomes (I.9). By analytic continuation, Theorem I.7 follows from Theorem I.4. ∗ cG 1 (j L) = c1 (L) −

Formula (I.9) holds only over an open and dense subset of g. One can, however, extend it to all of g with the sum now depending on ξ. (This is the standard form of the localized equivariant index theorem.) Namely, recall that M ξ is by definition the submanifold {ξM = 0}. For connected components F of M ξ we still assume that the splitting (I.8) exists. Let us as usual identify g with Rn . Theorem I.8 (The localized equivariant index theorem, II). For every ξ ∈ g, (I.11) χ(exp ξ) =

X F

e

√ −1ΦF (ξ)/2π

Z

ec1 (j F

∗

L)

Td (T F )

r  −1 Y √ 1 − e −1αi (ξ)/2π e−c1 (Li ) . i=1

where F runs through the connected components of M ξ .

For a generic ξ, we have M ξ = M G and formula (I.11) turns into (I.9) evaluated at ξ. The proof of this theorem follows the same line as the proof of Theorem I.7, but Theorem C.57 is used in place of Theorem C.53.

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Remark I.9. Theorems I.4 and I.8 are stated here in the order opposite of how these results are proved. Theorem I.8 is usually proved first, and Theorem I.4 is derived from it. In the next section we will discuss a version of the equivariant index theorem for discrete groups. In order to see how this is related to Theorem I.8, we will rewrite the first factor in (I.11) in √a slightly different form. For g = exp ξ, let √ γL (g) = e −1ΦF (ξ)/2π and γk (g) = e −1αk (ξ)/2π . These functions are the characters of the isotropy representation of G on L |F and on the line bundles Lk , respectively. Theorem I.8 immediately implies Corollary I.10. Z r  −1 X Y ∗ χ(g) = (I.12) γL (g) ec1 (j L) Td (T F ) 1 − γk (g)e−c1 (Lk ) , F

F

k=1

where F runs through the connected components of M g .

Remark I.11. This follows from (I.11) when M g = M ξ , but is in fact true for all g ∈ G. Remark I.12. It is clear that the products in (I.9), (I.11), and (I.12) can be expressed purely in terms of the equivariant (but not ordinary) Chern classes of the normal bundles VF to the connected components F . In this form, Theorems I.7, I.8, and Corollary I.10 do not rely on the assumption that the bundles VF split. The role of the splitting in these results is, however, more serious than a mere technical simplification of the proofs. The advantage of the statements given above is that in (I.9), (I.11), and (I.12) the topology of VF is separated from the G-action on VF . This feature is inevitably lost when the splitting is not assumed. 3. The Kawasaki Riemann–Roch formula I: finite abelian quotients In [ASi] Atiyah and Singer discuss an equivariant index theorem for finite groups. If Γ is a finite abelian group, this theorem reads Theorem I.13. Let Γ be an abelian group of symmetries of the Spinc -Dirac operator, and let χ : Γ → C be the character of the virtual representation Q(M ). Then, for g ∈ Γ, the value χ(g) is given by (I.12). In this section we will derive from this result the Kawasaki Riemann–Roch formula for the orbifold M/Γ. This formula will be, in some sense, a “trivial” case of the Kawasaki Riemann-Roch theorem since most orbifolds do not have global presentations of the form M/Γ with Γ finite. However, the Kawasaki Riemann– Roch theorem is an example of a theorem which is “almost as hard to state as to prove”, and all the complications involved in the statement of the theorem are already in evidence for the orbifold M/Γ. In fact, the simplest formulation of the Kawasaki Riemann–Roch theorem may be interpreted as the assertion that “the index theorem for an arbitrary orbifold is identical with the index theorem for orbifolds of the form M/Γ.” To compute the stable complex quantization of M/Γ with no group acting on M/Γ, we must first define this quantization and this we will do by simply defining it to be the dimension of the virtual vector space Q(M )Γ . In other words, we will define it by assuming that quantization and reduction commute, i.e., Q(M/Γ) = Q(M )Γ .

3. THE KAWASAKI RIEMANN–ROCH FORMULA I

321

By Frobenius’ theorem (see, e.g., [Ki2]), dim Q(M )Γ =

(I.13)

1 X χ(g). |Γ| g∈Γ

In principle, we already know how to compute all the summands in this formula by Corollary I.10. In what follows we will do this computation carefully, so as to make the “bookkeeping” involved as simple as possible. To this end, consider the orbit type stratification of M (see Appendix B) and denote by S the set of its orbit type strata. For each each S ∈ S the stabilizer group Γp with p ∈ S, is independent of p since Γ is abelian. We will denote this group by ΓS and call it the isotropy group of the stratum S. In addition, we will denote by Γ# S the set of elements g ∈ ΓS with the property that for some ΓS -invariant neighborhood U of the closure F = S of S, we have U g = S. In other words, Γ# S is the set of g ∈ Γ for which the closure F is a connected component of M g . (Note # that Γ# S is not, in general, a subgroup. For example, e ∈ ΓS if and only if S is the open and dense stratum. For this stratum we have Γ# S = {e} and ΓS = Γ.) Clearly, Γ# ⊂ Γ . S S We have a natural bijection g {(S, g) | s ∈ S, g ∈ Γ# S } ←→ {(g, F ) | g ∈ Γ, F a component of M }

obtained by (S, g) 7→ (g, F = S). We can now state a preliminary version of the Kawasaki Riemann-Roch theorem for M/Γ. For S ∈ S let j : F = S → M be the inclusion map and VF = j ∗ T M/T F the normal bundle of F . Note that here T M and T F are stable tangent bundles, but VF is the actual normal bundle to F in M . Furthermore, VF is an equivariant complex bundle over F . By the splitting principle, we assume that VF splits equivariantly into a sum of line bundles: VF = L 1 ⊕ . . . ⊕ L r .

Let γi be the character of the representation of ΓS on Li and γL the character of the representation of ΓS on L |F . Also let Z r  −1 X Y ∗ Ind(S) = (I.14) ec1 (j L) Td (T F ) 1 − γi (g)e−c1 (Li ) . γL (g) F

g∈Γ# S

i=1

Observe that Γ acts on S and that Ind(S) = Ind(gS). Finally, for S ∈ S denote by ΓS the subgroup of all g ∈ Γ which fix S, i.e., gS = S. Since Γ is abelian, ΓS = ΓgS and ΓS = ΓgS for all S ∈ S and g ∈ Γ. As a result, Ind(Y ) and the subgroups ΓY and ΓY are well defined for an equivalence class Y ∈ S/Γ. Proposition I.14. (I.15)

dim Q(M/Γ) =

X

Y ∈S/Γ

1 Ind(Y ). |ΓY |

Proof. First note that (I.16)

dim Q(M/Γ) =

1 X Ind(S). |Γ| S∈S

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I. THE KAWASAKI RIEMANN–ROCH FORMULA

This can be easily proved by expressing dim Q(M/Γ) as a double sum using (I.13) and (I.12) and comparing the result with the double sum given by (I.16) and (I.14). We leave the details to the reader as an exercise. Since Ind(S) is independent of S representing a given class Y , each Ind(Y ) occurs on the right-hand side of (I.16) exactly [Γ : ΓY ] times. Thus the right-hand side of (I.16) is equal to the right-hand side of (I.15). Our final version of the Kawasaki Riemann-Roch theorem for M/Γ consists simply of interpreting (I.15) in the language of orbifolds. The orbit type stratification of M descends to a stratification of M/Γ indexed by the poset S/Γ. (This stratification is called the orbifold stratification of M/Γ). Recall that the strata of this stratification are   G X= gS  /Γ = S/ΓS = S/(ΓS /ΓS ). g∈Γ

In particular, all S representing the same class Y ∈ S/Γ project to the same X. (Note that ΓS does not act freely on S but ΓS /ΓS does and hence X is a smooth manifold.) The closure X of X in M/Γ is the orbifold F/ΓS . For each of the orbifolds X, consider the integral on the right-hand side of (I.12). The integrand is represented by a ΓS -invariant differential form, and so it can be regarded as an (orbifold) form on the quotient space X = F/ΓS . Dividing the integral by |ΓS |, we get the integral in the orbifold sense of this form over the (non-effective) orbifold X. Denote the sum of the resulting orbifold integrals by Ind(X), so that Ind(X) = Ind(Y )/|ΓY |. We will refer to Ind(X) as the “local Kawasaki index” of the stratum X. From Proposition I.14, we immediately obtain Corollary I.15. dim Q(M/Γ) =

(I.17)

X

Ind(X),

where the sum extends over the strata of the orbifold stratification of M/Γ. Remark I.16. Frobenius’ theorem can also be used to obtain a formula for the ordinary Euler characteristic of M/Γ: 1 X (I.18) χ(M/Γ) = χ(M g ), |Γ| g∈Γ

where Γ is a finite, but not necessarily abelian, group acting on M . Let us outline the proof of (I.18); see also [Sh]. Observe first that H ∗ (M/Γ) = H ∗ (M )Γ by Corollary B.36. (Here the cohomology is taken with real coefficients and since Γ is discrete basic forms are just Γ-invariant forms.) Hence, X χ(M/Γ) = (−1)k dim H k (M )Γ .

By applying Frobenius’ theorem as in (I.13), we see that 1 X χ(M/Γ) = (I.19) trace(g ∗ ), |Γ| g∈Γ

∗

∗ where trace(g P ) stands for the trace of the linear operator g on the virtual representation (−1)k dim H k (M ). These traces can be found using the Lefschetz formula as follows. For a diffeomorphism f : M → M with non-degenerate isolated P fixed points the Lefschetz formula asserts that trace(f ∗ ) = x∈M G σx (f ). Here

4. THE KAWASAKI RIEMANN-ROCH FORMULA II

323

σx (f ) is the parity of the number of real eigenvalues of df : Tx M → Tx M which are greater than one. By taking a small, tangent to M G , perturbation of g : M → M , P ∗ we conclude that trace(g ) = F σF (g)χ(F ), where F runs through connected components of M G and σF (g) is the parity of the number of real eigenvalues of dg restricted to a fiber of VF . Since Γ is compact, dg has no real eigenvalues greater than one and σF (g) = 1. As a result, X trace(g ∗ ) = χ(F ) = χ(M g ). F

Combining this formula with (I.19), we obtain (I.18).

4. The Kawasaki Riemann-Roch formula II: torus quotients 4.1. The Kawasaki Riemann-Roch formula. In this section we will describe how to compute the dimension of the quantization space Q(Z/G) for orbifolds which have presentations of the form Z/G, where G is a compact connected abelian Lie group acting on a compact manifold Z in a locally free fashion. We will see that as far as “bookkeeping” is concerned, this computation is virtually identical with the computation of dim Q(M/Γ) carried out in the previous section. Consider, as in the previous section, the orbit type stratification of Z. (See Appendix B for the definition.) In what follows we keep the notation from Section 3. Thus, S is the poset of the strata of the orbit type stratification; GS = Gp is the stabilizer of p ∈ S ∈ S (as before, it is independent of p ∈ S), and G# S is the set of all g ∈ GS for which the closure F = S is a connected component of Z g . We will refer to GS as the isotropy subgroup of S. Note that since Z and G are compact, S is finite as are all subgroups GS (see Appendix B). Furthermore, since G is connected, the G-action on Z preserves S. Hence, in the notation of the previous section, GS = G for any S. Moreover, it is clear that G/GS acts freely on S and hence the quotient space S/G is a submanifold of Z/G. These submanifolds give rise to the orbifold stratification of Z/G. For each stratum S/G, its closure F/G in Z/G is a sub-orbifold of Z/G. Let Z/G be equipped with a closed two-form ωred , a pre-quantization line bundle Lred and a stable complex structure. Via the projection π : Z → Z/G we can pull-back these to Z. As a result, we obtain a closed two-form ω = π ∗ ωred , a complex line bundle L = π ∗ Lred , and a stable complex structure on Z. Let us define, for each of the sub-orbifolds F/G, a local Kawasaki index as follows. Denote by j : F → Z the inclusion map. By the splitting principle, we assume that the restriction of the stable tangent bundle T Z to F splits equivariantly into a sum of complex line bundles j ∗ T Z = L1 ⊕ . . . ⊕ Lm . Without loss of generality, we can also assume that the first r summands on the right-hand side span the normal bundle of F . Let us denote by γi , for i = 1, . . . , m, the character of the representation of ΓF on Li . Note that γi ≡ 1 for i = r+1, . . . , m and (I.20)

γi (g) 6= 1 for all i = 1, . . . , r

if and only if g ∈ G# S . We also denote by γL the character of the representation of GS on the line bundle L.

324

I. THE KAWASAKI RIEMANN–ROCH FORMULA

Since G acts on Z (and hence on F ) in a locally free fashion, the map ∗ π ∗ : H ∗ (F/G) → HG (F )

is an isomorphism. (See Appendix C.) Therefore, for each of the equivariant Chern 2 2 ∗ classes cG 1 (Li ) ∈ HG (F ) there exists an element ui ∈ H (F/G) such that π ui = 2 cG 1 (Li ). In addition, there exists an orbifold Chern class c1 (Lred ) ∈ H (F/G) associated with the orbifold line bundle Lred for which we have π ∗ c1 (Lred ) = cG 1 (L) since L = π ∗ Lred . Now we are in a position to define the local Kawasaki index of the orbifold F/G to be the integral Z (I.21) τFnor τFtan , Ind(F/G) = F

where

(I.22)

τFnor =

X

g∈G# F

γL (g)

r Y

i=1

and τFtan = ec1 (Lred )

1 − γi (g)e−ui


−1

m Y

ui . 1 − e−ui i=r+1

Remark I.17. The inverses (1 − γi (g)e−ui )−1 in (I.22) are indeed defined. To show this, we first note that γi (g)e−ui = γi (g) + . . . with the dots indicate cohomology classes of degree greater than zero. By (I.20), γi (g) 6= 1. Hence the expansion of (1 − γi (g)e−ui )−1 does not have 1/0 as its leading term. Also notice that if we deleted the normal contribution τFnor from the integrand in (I.21) we would be just computing the “classical” Riemann–Roch number of F/G. In particular, this is exactly what Ind(F/G) is for the largest stratum, i.e., for F = Z/G. The definitions of quantization given in Chapter 6 make perfectly good sense for orbifolds. (See, in particular, Section 7 of Chapter 6.) In fact, for orbifolds of the form Z/G the Dolbeault and Spinc -Dirac operators can be defined on Z by means of the pre-quantization structure and the complex structure on Z. This operator will not be elliptic, but it will be transversally elliptic in the sense of Atiyah (see, e.g., [At1, Ve3, BV3, BV4]). Thus the dimension of Q(Z/G) can, in this case, be computed by using, instead of the Kawasaki Riemann–Roch formula, a somewhat simpler index theorem for transversally elliptic operators proved by Atiyah in [At1]. We will not attempt to give here a precise description of Atiyah’s result since we are not concerned with the details of the proof of the Kawasaki theorem, but only with the details of the statement of this theorem. For orbifolds with presentations of the form Z/G the Kawasaki Riemann–Roch theorem asserts: Theorem I.18. Let Q(Z/G) be the Spinc quantization of Z/G. Then X dim Q(Z/G) = (I.23) Ind(F/G), S∈S

where F = S and Ind(F/G) is defined by (I.21).

4. THE KAWASAKI RIEMANN-ROCH FORMULA II

325

We will call the terms on the right-hand side of (I.23) the Kawasaki invariants of Z/G. 4.2. Cobordism invariance of geometric quantization. In this section we show that the geometric quantization of a reduced space is an invariant of cobordism. Consider a Hamiltonian G-manifold (M, ω, Φ) equipped with a G-equivariant stable complex structure J, for G a torus. Here and in what follows we assume that Φ is proper. Let c ∈ Z∗G be a regular value of Φ. Then the quotient space Mred = Φ−1 (c)/G is a stable complex oriented orbifold (See Section 2.3 of Chapter 5 for a discussion of the reduction of stable complex structures). Furthermore, the form ω descends to this orbifold. Hence, we have sufficient data to define the stable complex quantization Q(Mred ). Next, let (M0 , ω0 , Φ0 , J0 ) and (M1 , ω1 , Φ1 , J1 ) be two such Hamiltonian Gmanifolds which are cobordant via a G-equivariant stable complex G-manifold with proper moment map Φ in the sense of Chapter 2. Let c ∈ Z∗G be a regular value of both Φ0 and Φ1 . As before, the geometric quantizations Q((M0 )red ) and Q((M1 )red ) are defined. Theorem I.19. Let c be a regular value of Φ. Then Q((M0 )red ) = Q((M1 )red ). Proof. Since c is a regular value of Φ, the cobordism between (M0 , ω0 , Φ0 , J0 ) and (M1 , ω1 , Φ1 , J1 ) gives rise to an orbifold cobordism of oriented orbifolds (M0 )red and (M1 )red (see Chapter 5). The stable complex structure descends to this cobordism as does the cohomology class of the equivariant two-form. Similarly, we have orbifold cobordisms between closures of strata of (M0 )red and (M1 )red . It is an immediate consequence of Stokes’ formula and (I.21) that Ind(F0 /G) = Ind(F1 /G), where F0 /G and F1 /G are cobordant closures of strata in (M0 )red and (M1 )red , respectively. The theorem follows now from (I.23). Alternatively, the cobordism invariance of the index can be proved directly as in Appendix J. In a similar vein one can use the Kawasaki Riemann–Roch theorem to prove the quantum shift formula (Proposition 6.49). The argument requires a careful (and non-trivial) calculation which equates the Kawasaki invariant for a stratum computed for J0 and L with that computed for J1 and L ⊗ Lδ .

APPENDIX J

Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman

1. The SpinC -Dirac operator and the SpinC -quantization In this section we reformulate the “SpinC -quantization commutes with cobordism” principal in a more analytic language. In Subsection 1.1, we briefly recall the notion of SpinC -Dirac operator. We refer the reader to [BGV, Du] for details. We also express the SpinC -quantization of an orbifold M = X/G in terms of the index of the lift of the SpinC -Dirac operator to X. In Subsection 1.2, we explain the relationship between such lifts associated with cobordant Spin C -structures. This leads us to a notion of a cobordism between transversally elliptic operators. To show that SpinC -quantization of orbifolds commutes with cobordism, it is then enough to prove that the index of transversally elliptic operators commutes with cobordisms, which will be shown in the subsequent sections. 1.1. A Dirac operator associated with a SpinC -structure. Suppose M is a compact oriented m-dimensional orbifold. Let (P, p) be a SpinC -structure on M . Recall from Section D.2 that here P is a principal Spin C (n)-bundle over M and p : P → GL+ (T M ) is a SpinC (n)-equivariant map, which gives rise to an SO(n)-structure on T M and, thus, to a Riemannian metric g M on M . The group SpinC (n) has a canonical unitary representation S, called the space of spinors. The Hermitian orbibundle S = P ×SpinC (n) S is called the spinor bundle over M . Example. If the SpinC -structure on M is given by an almost complex structure and a complex line bundle L (cf. Section D.5), then the spinor bundle is isomorphic to Λ• (T 0,1 M )∗ ⊗ L, where Λ• (T 0,1 M )∗ is the bundle of anti-holomorphic forms on M. There is a canonical action c : T M → End(S) of the tangent bundle T M on S by skew-adjoint endomorphisms, such that c(v)2 = −|v|2 where |v| denotes the norm of the vector v ∈ T M with respect to the Riemannian metric g M . Let ∇S be a Hermitian connection on S. The SpinC -Dirac operator on S is the first order differential operator (J.1)

D :=

n X i=1

c(ei )∇Sei : C ∞ (M, S) → C ∞ (M, S), 327

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J. COBORDISM INVARIANCE OF THE INDEX

where e1 , . . . , em is an orthonormal frame of T M (the operator D is independent of the choice of this frame). If the connection ∇S was properly chosen, which we will henceforth assume, then the operator D is self-adjoint; cf. [BGV, Proposition 3.44]. Suppose now that m = dim M is even. Then the spinor bundle possesses a natural grading S = S + ⊕ S − such that D : C ∞ (M, S ± ) → C ∞ (M, S ∓ ).

We denote by D± the restriction of D to the space C ∞ (M, S ± ). Note that (D+ )∗ = D− . By definition, the SpinC -quantization Q(M ) of M is equal to the index the operator D+ : Q(M ) := dim Ker D+ − dim Coker D+ = dim Ker D+ − dim Ker D− . Suppose now that M = X/G is a presentation of the orbifold M . Here X is a smooth compact manifold with a locally free action of a compact group G. Let g X be the lift of the Riemannian metric on M to X. Then the action of G on X preserves g X . ˜ ± the lift of D± to X. Then D ˜ ± is a G-invariant transversally Denote by D elliptic operator on X (see Subsection 3.1 for a definition of a transversally ellip˜ ± and Ker D± is naturally tic operator). The group G acts on the kernel of D
± G ˜ ± ; cf. [Kaw2, §1]. Hence, ˜ of Ker D isomorphic to the G-invariant part Ker D

˜ + G − dim Ker D ˜ − G. Q(M ) = dim Ker D

1.2. A cobordism of Dirac operators. Let M0 = X0 /G and M1 = X1 /G be presented orbifolds of the same even dimension m = 2k, endowed with Spin C structures. Assume that there is given a SpinC -cobordism between M0 and M1 . In other words, we assume the following data: A compact oriented manifold W with boundary; a locally free action of a compact Lie group G on W ; the representation of the boundary of W as a disjoint union of −X0 and X1 (here, −X0 is the manifold X0 with the opposite orientation); the SpinC -structure on W/G, which, by restriction, induces SpinC -structures on Xi /G, i = 0, 1; the isomorphisms between the orbifolds Mi and Xi /G, i = 0, 1, which carry the SpinC -structures on Mi to the SpinC -structures on Xi /G. Recall from the previous subsection, that there is a natural Riemannian metric on W . Using this metric we can identify the union of the cylinders X0 × [0, ε) and X1 × (−ε, 0] with a neighborhood U ⊂ W of the boundary of W . We denote by t : U → R the projection onto the second factor. By a slight abuse of notation, we denote by the same latter t the induced map U/G → R. Let dt ∈ T (U/G) ⊂ T (W/G) be the corresponding vector (we use the Riemannian metric to identify the tangent and the cotangent bundles to W/G). Set γ := c(dt). Let S, Si denote the spinor bundles on W/G and Mi = Xi /G respectively. Recall from the previous section that there is a natural grading Si = Si+ ⊕ Si− . There is a natural isomorphism between √ the restriction of S to Mi and Si . Under this isomorphism we have γ|S ± = ± −1. i ˜ D ˜ i denote the lifts Fix a connection on S. It induces connections on Si . Let D, of the corresponding Dirac operators to W and Xi respectively. From (J.1), we see

2. THE SUMMARY OF THE RESULTS

329

˜ to the neighborhood of Xi , i = 0, 1 has the form that the restriction of D (J.2)

˜ i, ˜ = γ∂ + D D ∂t

∂ where ∂t denotes the covariant derivatives along t. Thus to show that the SpinC -quantizations commute with cobordism, it is ˜ i on enough to prove that the indexes of the transversally elliptic operators D Xi , i = 0, 1 coincide, whenever there exists a G-equivariant cobordism W be˜ on W , whose tween X0 and X1 and a G-invariant transversally elliptic operator D restriction to a neighborhood of the boundary satisfies (J.2). This statement will be made more precise and proven below.

2. The summary of the results 2.1. Let X be a compact n-dimensional Riemannian manifold on which a compact Lie group G acts by isometries. Let E + , E − be G-equivariant Hermitian vector bundles over X. Let A+ : C ∞ (X, E + ) → C ∞ (X, E − ) be a G-invariant transversally elliptic differential operator of order 1 (cf. [At1] or Section 3 of this appendix). Let A− : C ∞ (X, E − ) → C ∞ (X, E + ) be the formal adjoint of A+ and consider the operator   0 A− : C ∞ (X, E + ⊕ E − ) → C ∞ (X, E + ⊕ E − ). A := A+ 0 2.2. The distributional index. The kernel Ker(A± ) ⊂ L2 (X, E ± ) is a closed G-invariant subspace, and, hence, can be considered as unitary representations of G. ˆ the set of all equivalence classes of irreducible represenLet us denote by G ˆ we denote by Kerρ (A± ) := HomG (ρ, Ker(A± )) the ρtations of G. For ρ ∈ G, ± component of Ker(A ). ˆ the dimension of the spaces Atiyah, [At1], showed that, for each ρ ∈ G, ± Kerρ (A ) is finite. Moreover, the formal sum (J.3)

char Ker(A± ) :=

X

ˆ ρ∈G

dim Kerρ (A± ) · char ρ

converges to a distribution on G. The distributional index indG (A) is defined by (J.4)

indG A := char Ker(A+ ) − char Ker(A− ) ∈ D0 (G)inv ,

where D0 (G)inv denotes the space of distributions on G invariant under the inner + + automorphisms of G. Set indG ρ A := dim Kerρ (A ) − dim Kerρ (A ). Then (J.5)

indG A =

X

ˆ ρ∈G

indG ρ (A) · char ρ.

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2.3. The case when X is a boundary. Suppose now that X is a boundary of a compact G-manifold W and that F is a G-equivariant vector bundle over W , whose restriction to X is G-equivariantly isomorphic to E. Note, that we do not assume that the bundle F is graded. We choose an equivariant identification of a neighborhood U of the boundary of W with the product X × (−ε, 0] and we denote by t : X × (−ε, 0] → (−ε, 0] the projection. We fix a G-equivariant connection on F , so that the operator ∂/∂t acts on the sections of the restriction of F to U . The main result of this appendix is the following Theorem 1. Assume that there exists a self-adjoint G-invariant transversally elliptic symmetric differential operator B : C ∞ (W, F ) → C ∞ (W, F ), whose restriction to U has the form ∂ (J.6) + A, B = γ ∂t √ where γ is a skew-adjoint bundle map, such that γ|E ± = ± −1 . Then the index indG A = 0. Remark 1. By (J.5), the theorem is equivalent to the statement that indG ρ A = ˆ 0 for all ρ ∈ G. 2.4. The cobordism invariance. Theorem 1 implies the cobordism invariance of the index in the following sense. Assume that Xi , i = 0, 1 are compact Riemannian G-manifolds and that Ei± + ∞ are G-equivariant Hermitian vector bundles over Xi . Let A+ i : C (Xi , Ei ) → − C ∞ (Xi , Ei− ) be transversally elliptic differential operators. Set Ai = A+ i ⊕ Ai . Suppose W is a compact G-manifold, whose boundary is the disjoint union of X0 and X1 . Then we can and will identify a neighborhood U of the boundary of W with the disjoint union of the cylinders X0 × [0, ε) and X1 × (−ε, 0]. We denote by t : U → R the projection onto the second factor. Assume that there exist a G-equivariant Hermitian vector bundle F over W , whose restriction to the boundary is isomorphic to the bundle induced by Ei = Ei+ ⊕ Ei− , and an operator B : C ∞ (W, F ) → C ∞ (W, F ), which near the boundary takes the form (J.6). In this situation we say that the operators A0 and A1 are cobordant. Let E0op = E0op+ ⊕ E0op− be the bundle E0 with the opposite grading, i.e., op op ∞ ∞ E0op± = E0∓ . Then A0 defines the operator Aop 0 : C (X0 , E0 ) → C (Xi , E0 ). G op G Clearly, ind A0 = − ind A0 . Set X = X0 t X1 and let E = E + ⊕ E − be the graded bundle over X induced by E0op and E1 . Let A : C ∞ (X, E) → C ∞ (X, E) be the operator induced by Aop 0 and A1 . Then the operators A and B satisfy the condition of Theorem 1. Hence, indG A1 − indG A0 = indG A = 0. Thus we proved the following Corollary 1. The distributional indexes of cobordant transversally elliptic operators coincide. Combining this corollary with the discussion in Subsection 1.2 we also obtain the following

3. TRANSVERSALLY ELLIPTIC OPERATORS AND THEIR INDEXES

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Corollary 2. SpinC quantization of orbifolds commutes with SpinC cobordism. 2.5. The plan of the proof of Theorem 1. We apply the method of [Brav3] with necessary modifications. Choose a G-invariant Riemannian metric on W , which induces the product ˜ denote the complete non-compact Riemannian metric on U = X × (−ε, 0]. Let W manifold obtained from W by attaching the semi-infinite cylinder X × [0, ∞) to the ˜ in the obvious way. boundary. We extend the bundle F and the operator B to W 1 Consider two linear operators cL and cR on the exterior algebra Λ• C = Λ0 C ⊕ 1 Λ C, defined by the formula (J.7)

cL ω = 1 ∧ ω − ι1 ω;

cR ω = 1 ∧ ω + ι1 ω,

ω ∈ Λ• C.

(Here we consider 1 as a vector in C and denote by ι1 the interior multiplication by this vector.) These operators anti-commute with each other, cL cR + cR cL = 0. They also satisfy c2L = −1, c2R = 1. Set F˜ = F ⊗ Λ• C and consider the operator √ ˜ , F˜ ) → C ∞ (W ˜ , F˜ ). ˜ := −1 B ⊗ cL : C ∞ (W (J.8) B ˜ is symmetric, since c∗ = −cL . Note, that the operator B L ˜ → R be a G-invariant map, whose restriction to X × (1, ∞) is Let p : W the projection on the second factor, and such that p(W ) = 0 (see Subsection 4.1 for a convenient choice of this function). For any a ∈ R, consider the operator ˜ − (p(x) − a) ⊗ cR . Then (cf. Lemma 1) Ba := B (J.9)

B2a = B 2 ⊗ 1 − R + |p(x) − a|2 ,

˜ , F˜ ) → C ∞ (W ˜ , F˜ ) is a bounded operator. where R : C ∞ (W ± Let Ba denote the restriction of Ba to the spaces F ⊗ Λ0 C and F ⊗ Λ1 C ˆ the index indG Ba := respectively. In Subsection 4.2 we show that, for each ρ ∈ G, ρ + − Kerρ (Ba ) − Kerρ (Ba ) is well defined and is independent of a. It follows from (J.9) that, if a  0, then the operator B2a is strictly positive. In particular, its kernel is empty and indG ρ Ba = 0. Also, if a  0, then all the sections in Ker B2a are concentrated on the cylinder X × (0, ∞), not far from X × {a} (this part of the proof essentially repeats the arguments of Witten in [Wi1]). Hence, the calculation of Ker B2a is reduced to a problem on the cylinder X × (0, ∞). It is not G ˆ difficult now to show that indG ρ Ba = indρ A for a  0, ρ ∈ G. G Theorem 1 follows now from the fact that indρ Ba is independent of a. 3. Transversally elliptic operators and their indexes In this section we recall the definitions and some properties of transversally elliptic operators on a compact G-manifold; cf. [At1]. Since in the proof of Theorem 1 we apply some of the constructions which Atiyah used to prove that the index of a transversally elliptic operator is well defined, we will briefly recall this proof in Subsection 3.2. 1 These operators generate two actions of the Clifford algebra of C on Λ • C, called, respectively, left and right actions. This is the motivation for the subscripts “L” and “R” in our notation.

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Throughout the section X is a Riemannian G-manifold, E, F are G-equivariant Hermitian vector bundles over X and D : C ∞ (X, E) → C ∞ (X, F ) is a G-invariant pseudo-differential operator of order 1. 3.1. Transversally elliptic operators. Recall that the leading symbol σ(D) of D is a function on the cotangent bundle T ∗ X taking values in Hom(E, F ). Let TG∗ X ⊂ T ∗ X denote the subbundle of covectors which vanish on vectors tangent to the orbits of G. We will identify X with the zero section of TG∗ X. Definition 1. The operator D is called transversally elliptic if σ(D) is invertible when restricted to TG∗ X\X. Fix a bi-invariant Riemannian metric on G, and let Y1 , . . . , Yk be an orthonormal basis for the Lie algebra g = Lie G. Denote by Y˜1 , . . . , Y˜k the corresponding first order differential operators defined by the action of G on E + , and form the operator ∆G := 1 −

k X i=1

Y˜i2 : C ∞ (X, E) → C ∞ (X, E).

Consider the second order pseudo-differential operator ¯ := (D, ∆1/2 ) : C ∞ (X, E) → C ∞ (X, F ) ⊕ C ∞ (X, E). D G One immediately sees that the operator D is transversally elliptic if and only if the ¯ is injective on T ∗ X\X. Equivalently, the operator principal symbol of D ¯ ∗D ¯ = D∗ D + ∆G : C ∞ (X, E) → C ∞ (X, E) D

¯ ∗ denote the formal adjoints of the operators D and D, ¯ reis elliptic (here D∗ , D spectively). ˆ is an irreducible 3.2. The distributional index. Suppose now that ρ ∈ G representation of G and let L2ρ (X, E) := HomG (ρ, L2 (X, E)) ⊗ ρ be the ρ-component of the space L2 (X, E ± ) of square-integrable sections of E. Clearly, the restriction of ∆G to L2ρ (X, E) is bounded by a constant C(ρ). It follows that the space Kerρ (D) := Ker D∩L2ρ (X, E) is a subset of the space  ¯ ∗ D) ¯ C(ρ) := u ∈ L2ρ (X, E) : hD ¯ ∗ Du, ¯ ui ≤ C(ρ) . (D

By the standard theory of elliptic operators (cf., for example, [Sh1] or [At1, ¯ C(ρ) is finite dimensional. Hence, so is Kerρ (D) ¯ ∗ D) Lemma 2.3]), the space (D and we have the inequality ¯ C(ρ) . ¯ ∗ D) dim Kerρ (D) ≤ dim(D ¯D ¯ ∗ )C(ρ) . With just a little more work (cf. [At1, Similarly, dim Kerρ (D∗ ) ≤ dim(D p. 13]), one shows that the sum (J.3) converges to a distribution on G. Thus the sum (J.4) also converges to a distribution on G, called the distributional index of A.

4. INDEX OF THE OPERATOR BA

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4. Index of the operator Ba 4.1. The calculation of B2a . We will use the notation of Subsection 2.5. In particular, U ' X × (−ε, 0] is a neighborhood of ∂W , t : U → (−ε, 0] is ˜ is the manifold obtained from W by attaching a cylinder, F˜ = the projection, W ˜ is the operator defined in (J.8). Recall that in Subsection 2.3 F ⊗ Λ• C and B we have chosen a connection on F . This connection defines a trivialization of the restriction of F to U along the fibers of t. Hence the Hermitian metric on E induces a metric on F |U . We extend this metric to a G-invariant Hermitian metric on F . This metric induces a Hermitian metric on F˜ in the obvious way. Let s : R → [0, ∞) be a smooth function such that s(t) = t for |t| ≥ 1, and ˜ → R such that p(y, t) = s(t) for s(t) = 0 for |t| ≤ 1/2. Consider the map p : W (y, t) ∈ X × (0, ∞) and p(x) = 0 for x ∈ W . Recall that the operator cR is defined in (J.7) and define the operator ˜ − (p(x) − a) ⊗ cR . Ba := B

(J.10)

Lemma 1. Let Πi : F˜ → F ⊗ Λi C, (i = 0, 1) be the projections. Then (J.11)

B2a = B 2 ⊗ 1 − R + |p(x) − a|2 ,

where R : F˜ → F˜√is a uniformly bounded bundle map, whose restriction to X × (1, ∞) is equal to −1 γ(Π1 − Π0 ), and whose restriction to W vanishes. Proof. Note, first, that p(x) − a ≡ −a on W . Thus, since cR anti-commutes ˜ we have B2a |W = B ˜ 2 |W + a2 = B 2 ⊗ 1|W + a2 . Hence, (J.11) holds, when with B, restricted to W . We now consider the restriction of B2a to the cylinder X × (0, ∞). Since the operators cL and cR anti-commute, we obtain √ B2a |X×(0,∞) = B 2 ⊗ 1 + −1 s0 γ ⊗ cL cR + |s(t) − a|2 .
√ Since cL cR = Π1 −Π0 , it follows, that (J.11) holds with R = s0 −1 γ Π1 −Π0 . 4.2. An estimate on the kernel of Ba . As in Subsection 3.1, we choose an orthonormal basis Y1 , . . . , Yk for the Lie algebra g = Lie G, and we denote by Y˜1 , . . . , Y˜k the corresponding first order differential operators defined by the action P ˜ , F˜ ). Set ∆G := 1 − k Y˜ 2 and consider the second order pseudoof G on C ∞ (W i=1 i differential operator ¯ a := (Ba , ∆1/2 ) : C ∞ (W ˜ , F˜ ) → C ∞ (W ˜ , F˜ ) ⊕ C ∞ (W ˜ , F˜ ). B G

Using Lemma 1, we have (J.12)

¯ ∗a B ¯a = B



B 2 ⊗ 1 + ∆G



+



 |p(x) − a|2 − R .

¯ ∗a B ¯ a as an operator acting on the space of square-integrable sections We consider B ˜ of F . ¯ ∗a B ¯ a is self-adjoint and has discrete spectrum. Lemma 2. The operator B Proof. Since the operator B is transversally elliptic, the operator (J.12) is elliptic. Hence (cf., for example, [Sh3, Lemma 6.3]), the Lemma is equivalent to

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˜ , such the following statement: For any ε > 0 there exists a compact set K ⊂ W ˜ that if u is a smooth compactly supported section of F , then Z Z ¯ ∗B ¯ (J.13) hB |u|2 dµ < ε a a u, ui dµ. ˜ W

˜ \K W

˜ , and h·, ·i denotes the Hermitian Here, dµ is the Riemannian volume element on W ˜ scalar product on the fibers of F . Set V (x) = |p(x) − a|2 − R. To prove (J.13) note that, since R is bounded, ˜ , such that V > 1/ε on W ˜ \K, i.e., there exists a compact set K ⊂ W Z Z ˜ , F˜ ). |u|2 dµ, for all u ∈ L2 (W hV u, uidµ > ˜ \K W

˜ \K W

Note, also, that the first summand in (J.12) is a non-negative operator. Hence, we have Z Z Z Z ¯ ∗B ¯ |u|2 dµ < ε hV u, ui dµ ≤ ε hV u, ui dµ ≤ ε hB a a u, ui dµ. ˜ \K W

˜ \K W

˜ W

˜ W

We apply the method of Subsection 3.2, to study the kernel of Ba . ˆ denote by L2 (W ˜ , F˜ ) the ρ-component For an irreducible representation ρ ∈ G, ρ ˜ , F˜ ). of the space of square-integrable sections, and by Kerρ (Ba ) = Ker Ba ∩ L2ρ (W ˜ , F˜ ) is Lemma 3. The spectrum of the restriction of the operator Ba to L2ρ (W discrete. In particular, dim Kerρ (Ba ) < ∞. ˆ Proof. As in Subsection 3.2, the equation (J.12) implies that, for each ρ ∈ G, there is a constant C(ρ) such that, for all a, λ ∈ R we have   ¯ ∗B ¯ a u, ui ≤ C(ρ) + λ . u ∈ L2 (X, E) : hB2 u, ui ≤ λ ⊂ u ∈ L2 (X, E) : hB ρ

a

ρ

a

By Lemma 2, the dimension of the right-hand side of this formula is finite. Hence, so is the dimension of the left-hand side. Set F˜ + := F ⊗ Λ0 C, F˜ − := F ⊗ Λ1 C, B± ˜ ,F˜ ± ) and define a := Ba |L2 (W (J.14)

+ − indG ρ Ba = dim Kerρ (Ba ) − dim Kerρ (Ba ).

Remark 2. One can obtain estimates on the growth of the numbers ind G ρ Ba with ρ and prove that the distributional index indG Ba is defined. The direct proof of this fact will be more involved than the proof in [At1], since we have to work ˜ . We will show, however, that indG Ba = 0 for all on a non-compact manifold W ρ ˆ a ∈ R. ρ ∈ G, ˆ the index indG Lemma 4. For each ρ ∈ G, ρ Ba is independent of a.

Proof. From (J.10), we see that Bb − Ba = (b − a)⊗cR is a bounded operator; depending continuously on b − a ∈ R. Since indG ρ Ba coincides with the usual index ˜ , F˜ ), the lemma follows from the stability of the restriction of B+ to the space L2ρ (W of the index of a Fredholm operator; cf., for example, [Sh1, §I.8]. ˆ Lemma 5. indG ρ (Ba ) = 0 for all ρ ∈ G, a ∈ R.

5. THE MODEL OPERATOR

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Proof. By Lemma 4, it is enough to prove the proposition for one particular value of a. But it follows from Lemma 1 that, if a is a negative number such that 2 2 a2 > supx∈W ˜ kR(x)k, then Ba > 0, so that Ker Ba = 0. G To prove Theorem 1 it is enough now to show that ind G ρ Ba = indρ A. This is done in two steps: first, in Section 5, we construct a “model” operator Bmod on the cylinder X × (−∞, ∞), whose index is equal to indG ρ A. Then, in Section 6, we G mod G B . B = ind show that indρ a ρ

5. The model operator The bundles E ± lift to Hermitian vector bundles over the cylinder X × R, which we will denote by the same letters. Consider the Hermitian vector bundle F˜ := (E + ⊕ E − ) ⊗ Λ• C and the operator Bmod : C ∞ (X × R, F˜ ) → C ∞ (X × R, F˜ ) defined by Bmod :=

√ √ ∂ + t ⊗ cR , −1 A ⊗ cL + −1 γ ⊗ cL ∂t

where t is the coordinate along the axis of the cylinder. We refer to Bmod as the model operator ; cf. [Sh2]. It follows from Lemma 3 that the spectrum of the restriction of Bmod to the space L2ρ (X × R, F˜ ) is discrete (To see this, one can set W = X × [0, 1], and view X × R as a manifold obtained from W by attaching a cylinder.) We define ind Bmod by (J.14). Lemma 6. The kernel of the model operator Bmod is G-equivariantly isomormod phic (as a graded space) to Ker(A). In particular, indG = indG ρ B ρ A for all ˆ ρ ∈ G. Proof. Repeating the arguments of Lemma 2 we see that the model operator 2 B is self-adjoint. Hence, its kernel coincides with Ker Bmod . Also, from Subsection 3.2, we know that the kernel of the transversally elliptic operator A is a direct sum of the kernels of bounded operators A|L2ρ (X,E) . It follows that Ker A = Ker A2 . mod

2

Therefore, to prove the lemma it is enough to show that Ker Bmod is equivariantly isomorphic to Ker A2 . The same calculations as in the proof of Lemma 1 show that   ∂2 (J.15) (Bmod )2 |L2 (X×R,E ± ⊗Λ• C) = A2 ⊗ 1 + 1 ⊗ − 2 ± (Π1 − Π0 ) + t2 . ∂t

Both summands on the right-hand side of (J.15) are non-negative. Hence, the kernel of (Bmod )2 is given by the tensor product of the kernels of these operators.
∂2 2 is one-dimensional and is spanned by The space Ker − ∂t 2 + Π1 − Π0 + t
2 ∂2 2 is onethe function ρ+ (t) := e−t /2 ∈ Λ0 R. Similarly, Ker − ∂t 2 + Π0 − Π1 + t − −t2 /2 dimensional and is spanned by the one-form ρ (t) := e ds, where we denote by ds the generator of Λ1 C. It follows that n o Ker(Bmod )2 ∩ L2 (X × R, E ± ⊗ Λ• C) ' σ ⊗ ρ± (t) : σ ∈ Ker A2 ∩ L2 (X, E ± ) .

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5.1. The shifted model operator. Let Ta : X × R → X × R, Ta (x, t) = (x, t + a) be the translation, and consider the pull-back map Ta∗ : L2 (X × R, F˜ ) → L2 (X × R, F˜ ). Set
∗ Bmod := T−a ◦ Bmod ◦ Ta∗ = B ⊗ 1 − 1 ⊗ cR t − a . a

mod mod Then indG = indG , for any a, ρ ∈ R. ρ Ba ρ B

6. Proof of Theorem 1 ˆ All the operators studied in this section are In this section we fix ρ ∈ G. restricted to the ρ-component of the space of square-integrable sections. For simplicity, we omit the subscript “ρ” from the notion for these operators. In particular, 2 ˜ ˜± mod mod B± a denote the restriction of Ba to the spaces Lρ (W , F ). Similarly, let B± , B±,a denote the restriction of the operators Bmod , Bmod to the spaces L2ρ (X × R, F˜ ± ). a Note that, with this notation, we have + − indG ρ Ba = dim Ker Ba − dim Ker Ba ;

mod mod indG = dim Ker Bmod ρ Ba +,a − dim Ker B−,b .

If A is a self-adjoint operator with discrete spectrum and λ ∈ R, we denote by N (λ, A) the number of the eigenvalues of A not exceeding λ (counting multiplicities). 2 Proposition 1. Let λ± denote the smallest non-zero eigenvalue of (Bmod ± ) . Then, for any 0 < ε < min{λ+ , λ− }, there exists A = A(ε, V ) > 0, such that
2 2 N λ± − ε, (B± = dim Ker(Bmod (J.16) for all a > A. a) ± ) ,

Before proving the proposition let us explain how it implies Theorem 1.

± ˜ , F˜ ± ) denote the vector space 6.1. Proof of Theorem 1. Let Vε,a ⊂ L2ρ (W 2 spanned by the eigenvectors of the operator (B± a ) with eigenvalues smaller or equal ± ± ∓ to λ± − ε. The operator Ba sends Vε,a into Vε,a . It follows that − + − dim Ker B+ a − dim Ker Ba = dim Vε,a − dim Vε,a .

By Proposition 1, the right-hand side of this equality equals dim Ker Bmod + G G mod − dim Ker Bmod . Theorem 1 follows now from Lem− . Thus indρ Ba = indρ B mas 5 and 6.  The rest of this section is occupied with the proof of Proposition 1. 2 6.2. Estimate from above on N (λ± − ε, (B± a ) ). We will first show that

(J.17)

2

mod N (λ± − ε, (B±) a )) ≤ dim Ker B± .

To this end we will estimate the operator B2a from below. We will use the technique of [Sh2, BF], adding some necessary modifications.

6. PROOF OF THEOREM 1

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6.3. The IMS localization. Let j, ¯j : R → [0, 1] be smooth functions such that j 2 + ¯j 2 ≡ 0 and j(t) = 1 for t ≥ 3, while j(t) = 0 for t ≤ 2. Set ja (t) = j(a−1/2 t), ¯ja (t) = ¯j(a−1/2 t). These functions induce smooth functions on the cylinder X × [0, 1], which we denote by the same letters. By a slight abuse of ˜ given notation we will denote by the same letters also the smooth functions on W −1/2 −1/2 ¯ ¯ by the formulas ja (x) = j(a p(x)), ja (x) = j(a p(x)). The following version of the IMS localization formula is due to Shubin [Sh2, Lemma 3.1] (The abbreviation IMS stands for the initials of R. Ismagilov, J. Morgan, I. Sigal and B. Simon). Lemma 7. The following operator identity holds: 1 1 (J.18) B2a = ¯ja B2a ¯ja + ja B2a ja + [¯ja , [¯ja , B2a ]] + [ja , [ja , B2a ]]. 2 2 2 2 ¯ Proof. Using the equality ja + ja = 1, we can write B2a = ja2 B2a + ¯ja2 B2a = ja B2a ja + ¯ja B2a ¯ja + ja [ja , B2a ] + ¯ja [¯ja , B2a ]. Similarly, B2a = B2a ja2 +B2a¯ja2 = ja B2a ja + ¯ja B2a ¯ja −[ja , B2a ]ja −[¯ja , B2a ]¯ja . Summing these identities and dividing by 2, we come to (J.18). We will now estimate each of the summands in the right-hand side of (J.18). Lemma 8. There exists A > 0, such that ¯ja B2a ¯ja ≥

a2 ¯2 8 ja ,

for all a > A.

Proof. Note that p(x) ≤ 3a1/2 for any x in the support of ¯ja . Hence, if 2 a > 36, we have ¯ja2 |p(x) − a|2 ≥ a4 ¯ja2 .  1/2 Set A = max 36, 4 supx∈W and let a > A. Using Lemma 1, we ˜ |R| obtain 2 ¯ja B2a¯ja ≥ ¯ja2 |p(x) − a|2 − ¯ja R¯ja ≥ a ¯ja2 . 8 6.4. Let Pa : L2ρ (X × R, F˜ ) → Ker Bmod be the orthogonal projection. Let a ± 2 Pa denote the restriction of Pa to the space Lρ (X × R, F˜ ± ). Then Pa± is a finite rank operator and its rank equals dim Ker Bmod ±,a . Clearly, ± Bmod ±,a + λ± Pa ≥ λ± .

(J.19)

˜ , we can and will By identifying the support of ja in X × R with a subset of W 2 mod ˜ consider ja Pa ja and ja Bmod j as operators on W . Then j ja . a a Ba j a = j a Ba a Hence, (J.19) implies the following: Lemma 9.

2 ± mod ± j a B± a ja + λ± ja Pa ja ≥ λ± ja , rkja Pa ja ≤ dim Ker B± .

˜ , F˜ ) → L2ρ (W ˜ , F˜ ), we denote by kAk its norm. For an operator A : L2ρ (W n o Lemma 10. Let C = 2 max max{|j 0 (t)|2 , |¯j 0 (t)|2 } : t ∈ R . Then

(J.20)

k[ja , [ja , B2a ]]k ≤ Ca−1 ,

k[¯ja , [¯ja , B2a ]]k ≤ Ca−1 ,

Proof. From Lemma 1 we obtain |[ja , [ja , B2a ]]| = 2|ja0 (t)|2 = 2a−1/2 |j 0 (a−1/2 t)|, |[¯ja , [¯ja , B2a ]]| = 2a−1/2 |j 0 (a−1/2 t)|.

for all a > 0.

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From Lemmas 7, 9 and 10 we obtain the following: Corollary 3. For any ε > 0, there exists A = A(ε, V ) > 0, such that, for all a > A, we have (J.21)

± B± a + λ± ja Pa ja ≥ λ± − ε,

rkja Pa± ja ≤ dim Ker Bmod ± .

The estimate (J.17) follows from Corollary 3 and the following general lemma [RS, p. 270]: Lemma 11. Assume that A, B are self-adjoint operators in a Hilbert space H such that rkB ≤ k and that there exists µ > 0 such that h(A + B)u, ui ≥ µhu, ui for any u ∈ Dom(A). Then N (µ − ε, A) ≤ k for any ε > 0.

2 6.5. Estimate from below on N (λ± − ε, (B± a ) ). To prove Proposition 1 it remains now to show that

(J.22)

2 mod N (λ± − ε, (B± ≡ dim Ker Bmod a ) ) ≥ dim Ker B± ±,a .

± ˜ , F˜ ) denote the vector space spanned by the eigenvectors of the Let Vε,a ⊂ L2ρ (W 2 ± 2 ˜ ˜± operator (B± a ) with eigenvalues smaller or equal to λ± −ε. Let Πε,a : Lρ (W , F ) → ± ± 2 Vε,a be the orthogonal projection. Then rkΠε,a = N (λ± − ε, (B± a ) ). As in ± 2 Subsection 6.4, we can and will consider ja Πε,a ja as an operator on Lρ (X × R, F˜ ± ). The proof of the following lemma does not differ from the proof of Corollary 3.

Lemma 12. For any ε > 0, there exists A = A(ε, V ) > 0, such that, for any a > A, we have (J.23)

± Bmod ±,a + λ± ja Πa ja ≥ λ± − ε,

± 2 rkja Π± a ja ≤ dim N (λ± − ε, (Ba ) ).

The estimate (J.22) follows now from Lemmas 12 and 11. The proof of Proposition 1 is complete. 

Department of Mathematics, Northeastern University, Boston, MA 02115, USA E-mail address: [email protected]

Bibliography [Ad] [Al]

[ACG]

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Index convexity theorem, 21 curvature, 169 equivariant, 172 curvature class, 171 equivariant, 172

abstract moment map, 31 exact, 32 Hamiltonian, 32 non-degenerate, 289 polarized, 34 proper, 33 action, 173 effective, 174 faithful, 174 free, 174 locally free, 174 proper, 173 additivity techniques, 203 alcove, 21, 77 assignment, 259 η-polarized, 261 associated with a moment map, 31 minimal stratum, 263 moment, 279 assignment cohomology, 279 relative, 283 Atiyah–Bott Lefschetz fixed point formula, 116

Delzant construction complex, 76 symplectic, 75 polytope, 75 space, 23, 70, 76 determinant line bundle, 113 Dirac axioms, 89 distinguishing cohomology class, 247 distribution, 103 complex, 103 Liouville, 27 real, 103 distributional character, 132 Dolbeault complex, 105 Duistermaat–Heckman distribution, 27 formula, 60, 222 function, 70 integral, 24 measure, 17 polynomial, 72 theorem, 69

Bargmann space, 120 basic form, 64, 184 Borel construction, 197 Cartan model, 198 Chern class, 171, 315 equivariant, 212, 316 Chevalley’s theorem, 200 cobordism, 34 complex oriented, 235 equivariant, 35 Hamiltonian compact, 25 complex, 30 polarized, 28 proper, 27 non-degenerate, 308 proper, 35 compatible almost complex structure and symplectic structures, 111 complex and symplectic structures, 103 convexity, 294

Ehresmann’s lemma, 69 equivariant characteristic classes, 212 characteristic numbers, 218, 304 mixed, 304 cohomology, 197 complex line bundles, 214 de Rham Theorem, 198 differential forms, 198 homology, 226 Mayer–Vietoris exact sequence, 203 Poincar´e duality, 223 vector bundles, 212 Fock space, 120 formality, 206, 290 349

350

generating vector field, 177 group cohomology, 193 Guillemin–Lerman–Sternberg formula, 59 Hamiltonian G-manifold, 16 stable complex, 29 index theorem, 316 equivariant, 115, 318 Kawasaki invariants, 325 Kawasaki Riemann–Roch formula, 324 Kirwan’s epimorphism, 292 L2 -cohomology, 106 lattice group, 168 weight, 168 linearization theorem for abstract moment maps, 51 Hamiltonian, 47 quantum, 134 Liouville measure, 16, 17 lit set, 153 local linearization theorem, 181 localization Jeffrey–Kirwan, 82 localization theorem Atiyah–Bott–Berline–Vergne, 218, 219 Borel’s, 204 moment cone, 22 map, 15 polarized, 27 polytope, 21 Mostow–Palais embedding theorem, 189 orbit type, 187 principal, 188 stratification, 188 infinitesimal, 188 Pinc structure, 244 Poisson bracket, 19 polarization, 103 polarized function to g∗ , 34 polarizing vector, 57 polytope simple, 75 pre-quantization, 91 data, 91 equivariant, 94 line bundle, 90 equivariant, 93 principal bundle, 178 push-forward, 219 equivariant, 221

INDEX

[Q, R] = 0, 139 quantization, 104 Dolbeault, 111 Spinc , 113 quantization commutes with reduction regular values, 140 singular values, 141 reduced space, 63 reduction for abstract moment maps, 65 symplectic, 64 Riemann–Roch formula, 112 number, 83 Riemann–Roch theorem equivariant, 316 topological, 118 rigidity, 191 ring structure on set of polarized cobordism classes, 37 shift formula, 114, 253 slice theorem, 180 Spinc Dirac operator, 113 Spinc structure, 113, 238 bundle equivalence, 240 destabilization, 252 homotopy, 241 homotopy classification, 246 metric, 242 splitting principle, 208, 315 stability theorem, 22 stabilizer, 174 infinitesimal, 178 stable complex structure, 229 bundle equivalence, 230 equivariant, 229 homotopy, 230 symplectic cutting, 84 tame G-manifold, 37 Todd class, 112, 315 equivariant, 317 uniqueness lemmas for abstract moment maps, 37, 51 for Hamiltonian actions on vector bundles, 49 weights, 22, 52 isotropy, 22 polarized, 53, 57, 126 real, 52