HAMILTONIAN ACTIONS OF LIE
GROUPS
A thesis presented
by
Yael Karshon
to
The Department of Mathematics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Mathematics.
Harvard University
Cambridge, Massachusetts
April, 1993
1
χ↑′D
c 1993 by Yael Karshon
All Rights Reserved
Contents
Introduction
1
Chapter 1. Preliminaries
1. Group actions
2. The torus
3. (Pre-)symplectic forms
4. Poisson structures
5. Moment maps
6. Examples
7. Geometry of the moment map
8. Pre-symplectic Duistermaat-Heckman
9. The Guillemin-Sternberg-Marle local normal form
5
5
7
9
10
10
12
14
15
17
Chapter 2. Completely integrable actions on presymplectic spaces
1. Introduction
2. A first example
3. Local structure of M/T
4. Local description of the D-H measure
5. Degree of the moment map
6. Examples of completely integrable spaces
7. The Guillemin-Lerman-Sternberg formula
23
23
24
25
27
29
31
34
Chapter 3. Index theory for completely integrable spinc manifolds 35
1. Introduction
35
2. Preliminary notation and examples
37
3. Next Examples: CP1 and CPn
39
4. The index
43
5. Statement of the Theorem
44
6. Outline of the proof
45
7. The elliptic operators
46
8. Lifting actions
50
9. The Lefschetz formula of Atiyah and Bott
52
10. Expanding the terms
56
11. Examples
60
5
Chapter 4. Periodic Hamiltonian flows on four dimensional manifolds
63
1. Introduction
63
2. Constructing a graph
64
3. Gradient spheres
70
4. Combinatorics
76
5. Uniqueness – I
81
6. Uniqueness – II
84
7. Classification
86
Chapter 5. Poisson Duality
1. Introduction
2. Proof of Theorem 5.1 – Part I
3. Proof of Theorem 1 - Part II.
87
87
89
94
Bibliography
97
Acknowledgements
First I would like to thank my advisor Shlomo Sternberg for his
advice and support all along the way. I would like to thank Victor
Guillemin for many stimulating discussions. I would like to thank
Raoul Bott for his inspiring teaching.
Chapters II and III of this thesis consist of joint work with Michael
Grossberg and with Susan Tolman. I deeply thank both of them for
very intensive, pleasant and fruitful collaborations. I thank Reyer Sjamaar and Jiang-Hua Lu for some very helpful conversations. I thank
Ilya Zakharevich for his helpful comments on this manuscript. I learned
quite a lot from numerous discussions with my fellow graduate students.
It would be impossible to list them all, but thanks!
I would like to thank my parents, Roni and Uri Karshon, for being such great parents. I would like to thank my grandmother Riva
Sapporta, my sister Nurit and my brother Yizhar for being such great
grandma, sister and brother respectively.
I would like to thank Dror Bar-Natan for being the most supportive,
warm, sexy, funny, wise, generous and fertile friend and husband in the
whole world.
I would like to thank the Harvard University Math department for
the opportunity I had to spend several years in this pleasant environment. I am currently supported by an Alfred P. Sloan Dissertation
Fellowship. I would like to thank the Math department of the Weizmann Institute of Science for their support in the summer of 1992.
Introduction
In this thesis we discuss Hamiltonian actions of compact Lie groups
on compact symplectic and presymplectic manifolds. A presymplectic
manifold is a manifold M equipped with a closed 2-form ω. It is symplectic if the 2-form is nondegenerate. An action of a Lie group G
on (M, ω) is Hamiltonian if there is a moment map Φ : M −→ g∗
whose components generate the corresponding flows on M; i.e., if ξM
is the vector field on M corresponding to ξ ∈ g then ι(ξM )ω = dhΦ, ξi.
Moreover, Φ respects the Lie structure in that it is G-equivariant with
respect to the G-action on M and the coadjoint action on g∗ .
We now give an overview of the main results in this thesis. Additionally, each one of the chapters 2 – 5 contains an introduction to that
particular chapter.
Chapter 1 contains basic definitions, notation and theorems which
will be used in the subsequent chapters. These include actions of compact Lie groups, presymplectic forms, moment maps, the DuistermaatHeckman measure, and the local normal form of Guillemin, Sternberg
and Marle for a Hamiltonian action.
Both chapters 2 and 3 were obtained in joint work with S. Tolman
and with M. Grossberg and will appear in [KT] and [GK2].
In Chapter 2 we discuss completely integrable torus actions on
presymplectic manifolds; i.e., G = T is a torus and dim T = 12 dim M.
The Duistermaat-Heckman (D-H) measure on t∗ is the push-forward
of Liouville measure ω n /n! via Φ. We give a topological description of
this push-forward measure as follows. The moment map descends to a
map Φ : M/T −→ t∗ . Topologically, the connected components of the
boundary ∂(M/T ) are orbifolds Nj . If α ∈ t∗ is not in the image of Nj
then the winding number of Φ : Nj −→ t∗ around α is a well defined
integer; dj (α). The D-H measure has a density function ρ : t∗ −→ Z
defined outside a finite union of hyperplanes. We prove that ρ(α) is the
sum of the winding numbers dj (α). The basic examples are toric manifolds, i.e., smooth toric varieties. If (M, T ) is a compact toric manifold
then M/T is homeomorphic to a closed ball. We describe a construction called equivariant connected sum which gives new examples which
1
2
INTRODUCTION
are not toric manifolds. In this construction we start with two spaces
(M1 , T ) and (M2 , T ) and we construct a third space (M1 ♥M2 , T ) for
which the quotient M1 ♥M2 /T is the usual connected sum of M1 /T and
M2 /T .
In Chapter 3 we explain the relation of the moment map to certain
quantities which are constructed from other structures on the manifold.
Explicitly, we discuss three possible structures, of increasing generality
– (i) a complex structure and a holomorphic line bundle; (ii) an almost
complex structure and a smooth line bundle; or (iii) a spinc structure.
We assume that we have a torus action on the manifold and on the
extra structure, with dim T = 12 dim M. To each of the structures (i) –
(iii) corresponds a natural elliptic differential operator. Its equivariant
index is the character of a virtual T -representation. It can be described
in terms of a multiplicity function mult : ℓ∗ −→ Z where ℓ∗ is the
integral weight lattice in t∗ . We show that, for appropriate choices of
a presymplectic form and a moment map, mult (α) = ρ(α) where ever
both are defined. In case (iii) the choices will be such that ρ is defined
for all α ∈ ℓ∗ .
The results of Chapter 3 combined with those of Chapter 2 give the
following topological description for the equivariant index:
Theorem 0.1. Let M be a compact spinc manifold. Suppose that
the torus T acts on M, that dim T = 21 dim M and that the action
lifts to the spinc structure. Consider the Dirac spinc operator, ∂\c . Its
equivariant index is a virtual T -representation, determined by a multiplicity function mult : ℓ∗ −→ Z, where ℓ∗ is the integral weight lattice
in t∗ . This multiplicity function can be given the following topological
description. We have a continuous map Φ : M/T −→ t∗ . The quotient
M/T is an orbifold with boundary, and the image of the boundary does
not meet the weight lattice ℓ∗ . For all α ∈ ℓ∗ , mult (α) is the winding
number of Φ : ∂(M/T ) −→ t∗ around α.
In Chapter 4 we give a classification of all periodic Hamiltonian
flows on compact four dimensional manifolds, with isolated fixed points.
In other words, we classify the 4 dimensional Hamiltonian spaces (M, S 1 , ω, Φ)
with isolated fixed points. Here we always assume that ω is symplectic.
Delzant classified all Hamiltonian torus actions which are completely
integrable, i.e., with dim T = 12 dim M. The dimensions dim T = 1
and dim M = 4 are the lowest for which the action is not completely
integrable. We follow the lines of earlier work of M. Audin and of
K. Ahara and A. Hattori, who described all the underlying S 1 -spaces
which admit a symplectic form and a moment map, in dimension 4.
We show that in the case of isolated fixed points, every such space is
INTRODUCTION
3
equivariantly symplectomorphic to a Delzant space, i.e., to a symplectic toric variety. In particular, every such Hamiltonian space is Kähler,
and the S 1 action extends to an action of T 2 = S 1 × S 1 which preserves the Kähler structure. Conversely, from each Delzant space we
get a periodic Hamiltonian flow by restricting the action to a subcircle
S 1 ⊆ T 2 . In dimension 2n = 4, Delzant spaces are classified by a certain family of polygons in R2 . The periodic Hamiltonian flows are then
classified by certain equivalence classes of polygons. In a subsequent
paper [K] we plan to give the full classification which includes the cases
of non-isolated fixed points.
In Chapter 5 we meet Hamiltonian spaces (M, G, ω, Φ) with ω symplectic and G a compact Lie group, not necessarily abelian. The Poisson algebra C ∞ (M) contains two interesting subalgebras. The first is
the algebra of G-invariant functions. The second is the algebra of collective Hamiltonians, i.e., pull-backs of smooth functions on g∗ via Φ.
The centralizer in C ∞ (M) of the collective Hamiltonians is exactly the
G-invariants. The centralizer of the G-invariants can be larger than
the collective Hamiltonians. One now considers the functions which
are smooth on M and constant on the level sets of Φ. This third subalgebra contains the collective Hamiltonians but might be larger. In
this chapter we show that this algebra and the algebra of G-invariants
are mutual centralizers in the Poisson algebra C ∞ (M).
4
INTRODUCTION
CHAPTER 1
Preliminaries
1. Group actions
We work in the smooth (C ∞ ) category, in particular all our groups
are Lie groups and all our manifolds are smooth. An action of a group
G on a manifold M is a homomorphism from G to the group Diffeo(M)
of diffeomorphisms of M such that the associated map G × M −→
M is smooth. We write the action of g ∈ G as g : m 7→ g · m.
The manifold M together with the action of G are denoted by the
pair (M, G) and called a G-space. An isomorphism of G-spaces is an
equivariant diffeomorphism. The word locally in this context refers to
small equivariant neighborhoods of an orbit. The action is effective if
the homomorphism G −→ Diffeo(M) is one to one.
If G is compact then, locally, (M, G) is completely described by
Koszul’s slice theorem [Ko]. To state it, we first need some notation.
Let p ∈ M and let O = G · p be its orbit. Let H ⊆ G be the stabilizer
of p. H acts linearly on the tangent space Tp M and preserves Tp O,
thus it acts on the quotient W = Tp M/Tp O. We can then form the
associated bundle G ×H W over G/H. This is a homogeneous vector
bundle, on which G acts by left multiplication on the first factor.
Theorem 1.1. There is a G-equivariant diffeomorphism between a
neighborhood of the zero section in G ×H W and a neighborhood of O
in M, which maps the zero section onto O in the obvious manner.
Proof. Choose a norm on W and denote by B(δ) the ball in W
of radius δ. With the help of local coordinates we can define a map
ϕ : B(δ) −→ M which sends 0 to p and whose differential at 0 is a linear
isomorphism of W with a complementary subspace to Tp O in Tp M. We
define a map ϕ̃ from G ×H B(δ) to M by sending ϕ̃ : [g, w] 7→ g · ϕ(w).
The differential of ϕ̃ at [e, 0] is a linear isomorphism. If follows that
every neighborhood of the zero section in G×H W is mapped onto some
neighborhood of O in M. We now show that ϕ̃ is one to one on a small
enough neighborhood of the zero section.
Since dϕ̃|[e,0] is an isomorphism, ϕ̃ is one to one on a neighborhood of
[e, 0]. Thus there is a neighborhood U1 of e in G and there is 0 < ǫ1 < δ
5
6
1. PRELIMINARIES
such that for g ∈ U1 and |wi | < ǫ1 , if g · ϕ(w1 ) = ϕ(w2 ) in M then
[g, w1] = [e, w2 ] in G ×H B(δ). Let U2 be an open neighborhood of e
whose closure is contained in U1 . Using the compactness of G, we can
find 0 < ǫ2 < δ with the following property. Denote Ũ2 = {g·ϕ(w) | g ∈
U2 , |w| < ǫ2 }. Then if g 6∈ U1 and |w| < ǫ2 then g · ϕ(w) 6∈ Ũ2 . Let
ǫ = min(ǫ1 , ǫ2 ), then ϕ̃ is one to one on G ×H B(ǫ).
Remark 1.1. If G is connected and M is oriented then the action of H on W preserves orientation. Thus if we have a G-invariant
Riemann metric on M then H acts on W by a subgroup of SO(W ).
Proof. If G is connected then the action of each element of G
preserves the orientation on M. At p we have Tp M ∼
= Tp O ⊕ W . Each
element of H preserves the orientation on Tp M and acts trivially on
Tp O, and therefore preserves orientation on W .
Corollary 1.2. Let FG be the set of points in M which are fixed
by G. Then FG is a closed submanifold of M.
Proof. FG is clearly a closed subset. If p ∈ FG then by Koszul’s
theorem a neighborhood of p in M is isomorphic to Tp M with G acting linearly. The set of fixed points is a linear subspace of Tp M, in
particular it is a submanifold in a neighborhood of the origin.
Corollary 1.3 (Effective ⇔ locally effective). Suppose that G is
a compact Lie group which acts effectively on a connected manifold M.
Then G acts effectively on any invariant open subset of M.
Proof. Let U ⊆ M be an invariant open subset and let K =
ker(G −→ Diffeo(U)). Then K is a closed subgroup of G. Let FK be
the set of points of M fixed by K. By Corollary 1.2 applied to the
group K, FK is a closed submanifold of M. Since it contains the open
set U, we must have FK = M. By effectiveness, K is trivial.
Definition 1.4. The regular points of M are those points whose
G-orbit has the maximal dimension among the G-orbits in M.
Proposition 1.5. The set of regular points is open and dense in
M.
Proof. Suppose that the orbit of p has dimension d. Let N =
G ×H W be its normal bundle, in the notation of Theorem 1.1. Let
H0 be the identity component in H and let W ′ ⊆ W be the subspace
fixed by H0 . Then the set of points in N whose orbit has dimension d
is G ×H W ′ . This is a submanifold of N .
If W ′ 6= W then H acts on W/W ′ nontrivially. By Remark 1.1, this
action is by orthogonal transformations. It follows that dim W/W ′ ≥ 2
2. THE TORUS
7
because the group SO(1) is trivial. In this case there are points near p
with orbits of dimension > d.
Denote by Md the set of points in M whose orbit has dimension d.
From the above
S and Theorem 1.1, it follows that Md is a locally finite
union Ud ∪ j Md,j where Md,j are submanifolds of codimension ≥ 2
in M and where Ud is open in M. If we remove from M all the Md,j ’s
then we are left with an open dense subset M0 of M which is connected.
Therefore there can be exactly one Ud , say, with d = d0 , which fills up
all of M0 . From what we showed for N it follows that near every Md,j
there are points whose orbits have dimension > d. Therefore d0 must
be the maximal possible dimension, so that M0 is the set of regular
points.
The purpose of the rest of this section is to set our sign conventions.
If γ(t) is a smooth curve in M with γ(0) = p then the tangent
vector γ̇|t=0 ∈ Tp M defines a linear functional on C ∞ (M) by γ̇f =
d
| f (γ(t)). As we let p vary, we get that every vector field on M
dt t=0
acts on C ∞ (M) by derivations. If X, Y are vector fields on M then
their Lie bracket is defined to be the vector field [X, Y ] which acts on
C ∞ (M) by [X, Y ]f = X(Y f ) − Y (Xf ).
The Lie algebra of G is g = Te G. We define the Lie bracket on g in
the following way; if we identify g with the left invariant vector fields
on G then the Lie bracket of elements of g is equal to the Lie bracket
of the corresponding vector fields.
Given an action of G on M, every ξ ∈ g defines a vector field ξM
by ξM |p = dtd (exp(tξ) · p). The map ξ 7→ −ξM is a homomorphism of
Lie algebras;
g −→ Vect(M)
(1.6)
where Vect(M) are the vector fields on M. We also denote ξM |p = ξ · p;
we think of it as the infinitesimal action of g at p.
2. The torus
In this section we assume that our group is a torus; G = T . We
will apply Koszul’s slice theorem, Theorem 1.1, to some examples which
will appear in subsequent chapters. We first set some notation.
The circle group is S 1 = {λ ∈ C | |λ| = 1}. We identify its
Lie algebra with R via the exponential map exp : β 7→ e2πiβ . The
standard n-dimensional torus is T n = S 1 × . . . × S 1 (n factors). An
n-dimensional torus T is a group which is isomorphic to T n . Denote
its Lie algebra by t. We have the exponential map exp : t −→ T . Its
kernel is a lattice ℓ ⊂ t. The integral weight lattice is the dual lattice;
8
1. PRELIMINARIES
ℓ∗ = {α ∈ t∗ | hα, βi ∈ Z ∀β ∈ ℓ}. For the standard torus we have
t = t∗ = Rn and ℓ = ℓ∗ = Zn .
Every α ∈ ℓ∗ defines a homomorphism ρ : T −→ S 1 by exp(β) 7→
e2πihα,βi for β ∈ t. Equivalently, we write ρ(λ) = λα for λ ∈ T . For
the standard torus this makes sense as multi-index notation, so that
α(1)
α(n)
α ∈ Zn and λα = λ1 · · · λn . The homomorphism ρ gives a one
dimensional representation of T by λ : z 7→ ρ(λ)z. We denote this
representation by Cα .
Suppose that we are given a linear representation of T on a vector
space W . If dim W is even then there exist weights α1 , . . . , αr ∈ ℓ∗
such that W ∼
= Cα1 ⊕ . . . ⊕ Cαr as real representations of T . If dim W
is odd then we can write W ∼
= Cα1 ⊕ . . . ⊕ Cαr ⊕ R and T acts trivially
on the last factor R. The weights αi are determined only up to sign,
because as real representations, Cα and C−α are isomorphic.
Suppose now that dim T = n and dim W = 2n, then the linear
T -action is effective if and only if det(α1 . . . αn ) = ±1. In particular it
follows that if T acts effectively then no vector is fixed by the action
besides the origin. If we are given an orientation of W then we require
the isomorphism W ∼
= Cn to respect this orientation. In this case the
weights are determined up to a simultaneous change of the signs of an
even number of weights.
Example 2.1. Let S 1 act on a four dimensional manifold M. Then
there are two possibilities.
(i) Suppose p is a fixed point. Then M is isomorphic near p to C2
with the action λ · (z, w) = (λm z, λn w) for some integers m, n.
The action is effective if and only if m and n are relatively prime.
(ii) Suppose that the stabilizer of p is the cyclic group ZK = {λ ∈
C | λK = 1}. Then near the orbit of p, M is isomorphic to
S 1 ×ZK (R × C) where ZK acts on R × C by λ · (x, z) = (x, λl z)
for some 0 ≤ l < K.
Example 2.2. Suppose that dim M = 2n and dim T = n. Then
∼
H = T l × Γ where Γ is a finite group. Near the orbit of p we have
M = T n−l ×Γ (Rn−l ⊕ Cl )
(2.3)
where Γ ⊂ T n−l acts on Rn−l as a finite subgroup of SO(n − l). We
have T ∼
= T l × T n−l where T n−l acts on (2.3) by left multiplication and
where T l acts on Cl diagonally.
Example 2.4. A particular case of Example 2.2 is when dim M =
2n, dim T = n and p is a fixed point. Then near p, the action of T
on M looks like its linear isotropy action on V = Tp M. From the
3. (PRE-)SYMPLECTIC FORMS
9
above discussion it follows that M ∼
= Cn near p and that T acts by
λ · (z1 , . . . , zn ) = (λα1 z1 , . . . , λαn zn ). The exponents α1 , . . . , αn ∈ ℓ∗
are called the isotropy weights at p. If T acts effectively on M with
dim T = 21 dim M then all the fixed points are isolated. If M is oriented
then the αi ’s are determined up to a simultaneous change of the sign
of an even number of them.
Example 2.5. Another special case of Example 2.2 is when the
stabilizer of T is trivial. Then a neighborhood of the orbit of p is
isomorphic to T × D when D is an n dimensional disc and T acts by
multiplication on the first factor.
3. (Pre-)symplectic forms
A symplectic form ω on a manifold M is a closed, non-degenerate
differential 2-form. Closed means that its exterior differential vanishes;
dω = 0. Non-degenerate means that at each p ∈ M, the bilinear
form ωp is nondegenerate, i.e., that for u ∈ Tp M, if ι(u)ωp = 0 then
u = 0. This condition implies that the dimension of M is even, say 2n.
Nondegeneracy is equivalent to ω n never vanishing. We can choose on
M an orientation consistent with ω n . Then, by integration, ω n defines
a positive Rmeasure mω on M, called Liouville measure. It is defined by
mω (A) = A ω n /n! for every open subset A ⊆ M.
By Darboux’s theorem, in every symplectic manifold (M, ω) we can
find local coordinates x1 , y1 , . . . , xn , yn in which ω can be written as
ω = dx1 ∧ dy1 + . . . + dxn ∧ dyn . In these coordinates, Liouville measure
is mω = |dx1 ∧ dy1 ∧ . . . ∧ dxn ∧ dyn |.
A pre-symplectic form ω on M is just a fancy name for a closed 2form. We use this name because to these forms we shall apply certain
machinery (e.g., the moment map) which is traditionally applied to
symplectic forms. For a pre-symplectic form, there is no local formula
such as Darboux’s which describes neighborhoods of arbitrary points.
Now let (M, G) be a G-space. A differential form on (M, G) is
defined to be a differential form on M which is G-invariant. A (pre)symplectic form on (M, G) is a G-invariant (pre-)symplectic form on
M.
Note that if G is compact and connected then, by averaging, every
De Rham class in H 2 (M, R) can be realized by a pre-symplectic form
on (M, G). Also note that if P −→ M is a principal S 1 -bundle and
we have a lifting of the G action to P , then the curvature of any Ginvariant connection is a pre-symplectic form on (M, G).
10
1. PRELIMINARIES
4. Poisson structures
Let (M, ω) be a symplectic manifold. To every smooth function f :
M −→ R we associate a vector field Xf on M, defined by the condition
df = −ι(Xf )ω; for this we must use the nondegeneracy of ω. The
Poisson bracket of two smooth functions f, g on M is a third function,
defined by {f, g} = LXg f . This defines on C ∞ (M) the structure of a
Lie algebra. The ordinary multiplication and the Poisson bracket fit
together to form the structure of a Poisson algebra on C ∞ (M); this
means that {f, gh} = {f, g}h + g{f, h} for all f, g, h ∈ C ∞ (M).
More generally, a Poisson structure on a manifold M is a Poisson
structure on the algebra C ∞ (M). A Poisson structure is given by a bivector field on M which satisfies a certain closure condition; see [W2].
This is a weakening of the notion of symplectic structure which is different than the pre-symplectic structures (i.e., closed 2-forms) which were
discussed in the previous section. In this thesis we will only discuss nondegenerate Poisson structures, i.e., those which come from symplectic
structures. These appear in Chapter 5.
5. Moment maps
Let (M, G, ω) be a pre-symplectic G-space. Recall, for every ξ ∈ g
we have a vector field ξM . The contraction α = ι(ξM )ω is a 1-form,
given by α(u) = ω(ξM , u) for every tangent vector u. If ω is Ginvariant then this 1-form is closed, because being G-invariant implies that the Lie derivative LξM ω vanishes for all ξ ∈ g, and because
LξM ω = dι(ξM )ω + ι(ξM )dω = dι(ξM )ω + 0 by Weil’s formula and by
the closedness of ω. We then ask whether −ι(ξM )ω is exact, i.e., is
equal to df for some function f : M −→ R. If the answer is yes for all
ξ then f can be chosen to depend linearly on ξ, so that the different
f ’s fit together to a function Φ : M −→ g∗ .
In the symplectic case, we can view this situation from a different
direction. The Poisson bracket which was defined in §4 makes C ∞ (M)
into a Lie algebra. The map f 7→ Xf (see §4) is a homomorphism of
Lie algebras;
C ∞ (M) −→ Vect(M).
(5.1)
Denote by Φξ (p) = hΦ(p), ξi the ξ’th component of Φ. The map ξ 7→ Φξ
defines a lifting of (1.6) via (5.1) to a linear map g −→ C ∞ (M). If this
lifting is a homomorphism of Lie algebras then we call Φ a moment map.
If G is connected, then this condition is equivalent to the equivariance
of Φ : M −→ g∗ with respect to the action of G on M and to the
coadjoint action of G on g∗ .
5. MOMENT MAPS
11
If ω is only pre-symplectic then the map C ∞ (M) −→ Vect(M) is
not defined and we cannot use ω to define a Poisson bracket, but the
requirement on Φ to be equivariant still makes sense. Thus we make
the following definition.
Definition 5.2. Let (M, G, ω) be a pre-symplectic G-space. A
moment map for (M, G, ω) is a map Φ : M −→ g∗ such that
dΦξ = −ι(ξM )ω
for all ξ ∈ g
(5.3)
and such that Φ is equivariant with respect to the action of G on M
and to the coadjoint action of G on g∗ .
A Hamiltonian G-space is a symplectic G-space with a moment
map. We denote it by (M, G, ω, Φ). An isomorphism of Hamiltonian
spaces is an equivariant symplectomorphism which preserves Φ. We
will only use the word Hamiltonian when ω is nondegenerate.
Remark 5.4. If G = T is a torus then (5.3) already implies that
Φ is T -invariant on M.
Proof. Let ξ, η ∈ t, then
dLηM Φξ = −LηM ι(ξM )ω = −ι([ηM , ξM ])ω − ιξM LηM ω = 0,
therefore LηM Φξ is constant. The restriction of Φξ to an orbit has a
critical point by compactness, and there we have LηM Φξ = 0.
Corollary 5.5. If G = T is a torus then
(i) The only obstructions for having a moment map lie in the first
De-Rham cohomology group of M. In particular, if M is simply
connected then a moment map exists.
(ii) If a moment map exists then it is determined up to a translation
in t∗ , and any translation is allowed.
(iii) The pullback of ω to any orbit is zero.
Proof. From the definition of the moment map, ω(ξ · p, η ·
p) = −LηM Φξ which we showed is zero.
Here is one particularly important situation in which moment maps
arise. Take a manifold M with an action of a torus T . Take a line
bundle π : L −→ M and a lifting of the action to L. Let Θ be an
invariant connection and ω its curvature. Then Θ is a 1-form on L
(defined outside the zero section) and ω is a 2-form on M. They are
related by π ∗ ω = dΘ. Our convention is that on a fiber of L with a
1
complex coordinate z we have Θ = 2πi
dz/z. Although ω might not be
12
1. PRELIMINARIES
symplectic, it is always closed and invariant. Define Φ : M −→ t∗ by
the condition
hΦ ◦ π, ξi = hΘ, ξL i for all ξ ∈ t
(5.6)
where ξL is the vector field on L determined by ξ. Then Φ is a moment
map for (M, T, ω). If p ∈ M is a fixed point then Φ(p) is equal to the
weight with which T acts on the fiber over L, this follows from (5.6).
6. Examples
In this section we describe three important examples of symplectic
manifolds with Hamiltonian group actions. In §9 we will see that every
Hamiltonian action of a compact connected Lie group is locally given
by a combination of these three models.
We first set some notation. G is a connected Lie group, g = Te G
and g∗ = Te G. For g ∈ G and ξ ∈ g we denote by gξ and ξg the left
and right translation of ξ by g respectively; these are elements of Tg G.
Similarly, for α ∈ g∗ we define gα and αg in Tg∗ G to be the values at g
of the left- and right-invariant 1-forms whose values at e is α. If h·, ·ig
denotes the pairing between Tg∗ G and Tg G then we have hgα, gξig =
hαg, ξgig = hα, ξi. We rather not fix particular identifications of Tg G
and Tg∗ G with g and g∗ , in order to avoid confusing left and right in
the subsequent computations.
The adjoint and coadjoint actions of G on g and g∗ respectively are
defined by g : ξ 7→ gξg −1 for ξ ∈ g and g : β 7→ gβg −1 for β ∈ g∗ .
These actions of G give rise to infinitesimal actions of g. Fix ξ ∈ g
and β ∈ g∗ ; then the infinitesimal coadjoint action of ξ at β produces
a vector in Tβ g∗ which we denote [ξ, β]. It is defined to be the tangent
at t = 0 to the curve etξ βe−tξ . We identify Tβ g∗ with g∗ , so [ξ, β] is an
element of g∗ . Similarly, we define the infinitesimal adjoint action of
∂
ξ at ζ ∈ g to be ∂t
|t=0 etξ ζe−tξ . This is equal to the usual Lie bracket
[ξ, ζ]. Note that h[ξ, β], ζi = hβ, [ζ, ξ]i.
6.1. The cotangent bundle of G. The left action of G on G
naturally lifts to an action of G on T ∗ G. On every cotangent bundle
there is a natural symplectic form. The corresponding moment map on
T ∗ G is ΦL (gα) = gαg −1 for α ∈ g∗ , gα ∈ Tg∗ G. We are also interested
in the right action of a subgroup H ⊆ G. The corresponding moment
map is the restriction ΦR (gα) = −restrhα. This was proven in [KKS,
§2]. We now present the details.
∼
=
The map (g, α) 7→ gα defines an isomorphism G × g∗ −→ T ∗ G.
We use this isomorphism to identify the tangent space Tgα (T ∗ G) with
6. EXAMPLES
13
Tg G × Tg∗ G. For example, the vector (gξ, gβ) represents the tangent at
t = 0 to the curve (getξ )(α + tβ).
The canonical 1-form θ on T ∗ G is θgα (gξ, gβ) = hgα, gξig . The
symplectic form on cotangent space is ω = dθ. Take two tangent fields
ui = (gξi , gβi ), i = 1, 2; then (dθ)gα (u1 , u2) = u1 θ(u2 ) − u2 θ(u1 ) −
θ([u1 , u2 ]). Now, θgα (u2 ) = hα, ξ2i, and its derivative in the direction
of u1 is simply hβ1 , ξ2 i. Similarly, u2 θ(u1 ) = hβ2 , ξ1 i. Finally, we have
[u1 , u2 ] = (g[ξ1, ξ2 ], gβ ′) for some β ′ . So then
ωgα (u1 , u2 ) = hβ1 , ξ2 i − hβ2 , ξ1i − hα, [ξ1, ξ2 ]i.
In particular, at the zero element of Te∗ G, the tangent space to T ∗ G
is g ⊕ g∗ and the 2-form is ω((ξ1, β1 ), (ξ2 , β2 )) = hβ1 , ξ2 i − hβ2 , ξ1i.
G acts on T ∗ G on the left by a : gα 7→ agα. This action is generated by the vector fields ξL |gα = (ξg, 0) for ξ ∈ g. We now verify that
Φ(gα) = gαg −1 is a moment map. Fix ξ ∈ g. First, for u = (ζg, 0)
we have uΦξ = h[ζ, gαg −1], ξi = hgαg −1, [ξ, ζ]i = hgα, [ξg, ζg]ig =
−ωgα (ξL , u). Secondly, for u = (0, gβ) we have uΦξ = hgβg −1, ξi =
hgβ, ξgig = −ωgα (ξL , u).
Any subgroup H ⊆ G acts on the right by a : gα 7→ gαa−1 =
−1
(ga )(aαa−1 ). The generating vector fields are ξR |gα = (g(−ξ), g[ξ, α])
for ξ ∈ h. We now verify that Φ(gα) = −restrhα is a moment map.
Fix ξ ∈ h. First, for u = (gζ, 0) we have uΦξ = 0 and −ωgα (ξR , u) =
−h[ξ, α], ζi + 0 + hα, [−ξ, ζ]i = 0. Secondly, for u = (0, gβ) we have
uΦξ = −hβ, ξi = −ωgα (ξR , u).
6.2. A coadjoint orbit of G. Our second model is the coadjoint
orbit Oα of some fixed α ∈ g∗ ; on it we have the Kirillov-KostantSouriau symplectic form. Every subgroup H ⊆ G acts on Oα on the left
by the coadjoint action, and a corresponding moment map is Φ(β) =
restrhβ. We now give more details
The coadjoint orbit is Oα = {gαg −1 | a ∈ G}, and its tangent space
at β = gαg −1 is Tβ (Oα ) = {[ξ, β] | ξ ∈ g}. Note that [ξ, β] = 0 if
and only if hβ, [ζ, ξ]i = 0 for all ζ ∈ g. The Kirillov-Kostant-Souriau
symplectic form is ωβ ([ξ1 , β], [ξ2, β]) = −hβ, [ξ1 , ξ2]i. Note that it is
well defined on Tβ Oα .
The action of H is given by a : β 7→ aβa−1 . It is generated by
the vector fields ξOα |β = [ξ, β] for ξ ∈ h. We now verify that Φ(β) =
restrh(β) is a moment map. Fix ξ ∈ h. Then for u = [ζ, β] we have
uΦξ |β = h[ζ, β], ξi = hβ, [ξ, ζ]i = −ωβ (ξOα , u).
6.3. A linear symplectic action. Our third and last model is
a symplectic vector space (V, ωV ). We assume that H acts linearly
14
1. PRELIMINARIES
and preserves ωV . Denote by SP (V ) the group of symplectic automorphisms of V . Its Lie algebra sp(V ) consists of the endomorphisms A
of V which satisfy ωV (Au, v) + ωV (u, Av) = 0 for all u, v ∈ V . The
action H −→ SP (V ) gives rise to a map h −→ sp(V ). For ξ ∈ h
we denote by Aξ the corresponding endomorphism of V . If we identify
Tv V ∼
= V then ξV |v = Aξ v is the vector field which generates the action
of exp(tξ). The moment map is then
1
ΦξV (v) = − ωV (Aξ v, v).
2
We now verify this. Take u ∈ V , viewed as an element of Tv V . Then
uΦξV = − 21 ωV (Aξ u, v) − 12 ωV (Aξ v, u) = −ωV (ξV |v , u).
7. Geometry of the moment map
Let G = T be a torus and let (M, T, ω) be a Hamiltonian T -space
with M compact (and ω symplectic). Then there are some beautiful
theorems describing the geometry of the moment map Φ.
The critical points for Φ are the points on M with a stabilizer of
a positive dimension. The points which are stabilized by any given
subgroup H ⊆ T are sent by Φ into a finite union of planes in t∗ , each
of which is parallel to the annihilator of Lie(H) in t∗ . This follows
from the definition of a moment map, see [GS2]. Only a finite number
of subgroups H occur as stabilizers, this follows from Koszul’s normal
form.
If T = S 1 acts with isolated fixed points, then Φ is a Morse function
on M whose indices are even. This idea already appears in [F]. Therefore Φ is a perfect Morse function, H ∗ (M, Z) has no torsion, and the
odd cohomology groups of M vanish. Other consequences are that the
level sets of Φ are connected, and that Φ has no local extrema besides
its global minimum and its global maximum; see [GS2].
The convexity theorem of Guillemin and Sternberg and of Atiyah
states that the image of Φ is a convex polytope in t∗ . Its vertices are
images of fixed points for the T -action. Moreover, the set of the regular
values for Φ is open and each of its connected components is itself a
polytope.
From the polytope image (Φ) one can read certain information about
(M, T, ω). There is a particularly strong result when dim T = 1/2 dim M.
The space is then called completely integrable. For completely integrable Hamiltonian actions, Delzant [De] has shown that the polytope
image (Φ) actually determines the space (M, T, ω, Φ) up to isomorphism. He gave a characterization of those polytopes which arise in
8. PRE-SYMPLECTIC DUISTERMAAT-HECKMAN
15
this way. He has shown that all such Hamiltonian spaces are symplectic
toric varieties, in particular they all admit invariant Kähler structures.
In every effective Hamiltonian action we have dim T ≤ 12 dim M =
say, n. In the case of a strict inequality it is useful to consider a more
refined object than image (Φ), namely, the Duistermaat-Heckman (DH) measure on t∗ . This is defined to be the push-forward of Liouville
measure mω by the moment map. It is thus a positive measure which
is supported on image (Φ) and which is defined by
Z
mDH (A) =
ω n /n!
(7.1)
Φ−1 A
∗
for every open subset A ⊆ t . We denote mDH = Φ∗ mω or, by abuse
of notation, mDH = Φ∗ ω n /n!.
If the action is effective then mDH is absolutely continuous with
respect to Lebesgue measure |dα|, so we can write mDH = ρ(α)|dα|
for a density function ρ : t∗ −→ R. The theorem of Duistermaat and
Heckman states that ρ is piecewise polynomial; it is a polynomial on
each component of regular values of Φ. The degree of the polynomial
is bounded by (n − dim T ).
As a corollary of the polynomial nature of mDH , Duistermaat and
Heckman proved an exact stationary phase formula for the Fourier
transform of mDH . In the case of isolated fixed points, this formula is
Z
X eihΦ(p),ξi
Qn
eihΦ,ξi ω n /n! = (−2πi)n
(7.2)
α
(ξ)
jp
M
j=1
p
where we sum over the fixed points in M.
8. Pre-symplectic Duistermaat-Heckman
The moment map was originally defined for symplectic spaces. The
point of view was to start with a symplectic manifold (M, ω) and then
to consider a group action which preserves ω. In particular, the nondegeneracy of ω was crucial in the proof of (7.2) in [DH].
Later, the Duistermaat-Heckman theorem was put in a more general
setting of equivariant cohomology by Atiyah, Bott, Berline and Vergne;
see [AB3]. In this situation we view the group-action (M, G) as being
the fundamental object and later we introduce an invariant 2-form ω.
Within this new framework ω can be degenerate, and (7.2) remains
true.
Suppose now that (M, T, ω) is pre-symplectic and admits a moment
map Φ. Given an orientation of M, we again define the D-H measure
by (7.1). This is now a signed measure on t∗ . If ω is not symplectic
then, in contrast to the results in Section 7, we no longer can say
16
1. PRELIMINARIES
anything about image (Φ). Nevertheless; the D-H measure is still given
by polynomials with the same bound as before on their degrees; this
still follows from the proof in [DH]. Moreover, (7.2) is still a formula
for the Fourier transform of mDH .
Theorem 1.2 (Guillemin and Sternberg). Φ∗ ω n only depends on
the cohomology class of ω in H 2 (M, R).
Proof. This was proven by Guillemin and Sternberg in [GS7].
Since their paper is not published, we sketch their proof here. Suppose
that ω1 and ω2 are cohomologeous and let Φ1 , Φ2 be corresponding
moment maps. It is sufficient to show that the Fourier transforms of
the push-forward measures coincide, i.e., that
Z
Z
ihΦ1 ,ξi n
e
ω1 =
eihΦ2 ,ξi ω2n for all ξ ∈ t.
(8.1)
M
M
It is sufficient to prove (8.1) for ξ which generates a circle S 1 ⊆ T ,
because the set of such ξ’s is dense. Define ϕj = hΦj , ξi, j = 1, 2. We
have ω2 = ω1 + dβ and by averaging over T we can assume that β is
an invariant 1-form. Then ϕ1 − ι(ξM )β is a moment map for ω2 , so it
differs from ϕ2 by a constant, and we assume that it is equal to ϕ2 .
Now consider the graded ring A∗S 1 ⊗ C[u] where A∗S 1 denotes the S 1 invariant differential forms on M and C[u] is the ring of polynomials
in the variable u, with degree(u) = 2. Define a differential d˜ by
˜ ⊗ p(u)) = dα ⊗ p(u) + ι(ξM )α ⊗ up(u). One can explicitly check
d(α
the following facts.
• d˜2 = 0.
˜
˜
Denote HS∗ 1 (M) = ker(d)/image
(d).
∗
• The ring structure on AS 1 ⊗ C[u] descends to HS∗ 1 (M).
• Integration over M Rdefines a map A∗S 1 ⊗ C[u] −→ C[u] which
descends to a map M : HS∗ 1 (M) −→ C[u]. We note that if
rank(α)
6= n then we define
R
α
= 0.
M
˜
• ω̃j := ωj ⊗ 1 + ϕj ⊗ u are d-closed
for j = 1, 2; and ω̃2 = ω̃1 +
˜
d(β ⊗ 1)
R
R
These facts imply Rthat M ω̃1nR = M ω̃2n for all n. Define eω̃ = 1 + ω̃ +
ω̃ 2 /2! + . . . , then M eω̃1 = M eω̃2 . Restricting to the components of
eω̃j whose form-parts have degree n, we get
Z
Z
uϕ1 n
e ω1 /n! =
euϕ2 ω2n /n!
M
M
as an equality between power series in u. Setting u = i gives (8.1).
9. THE GUILLEMIN-STERNBERG-MARLE LOCAL NORMAL FORM
17
9. The Guillemin-Sternberg-Marle local normal form
Let (M, G, ω, Φ) be a Hamiltonian G-space. If G is compact, then
Koszul’s theorem and Darboux’s theorem can be combined to give a
complete local description of (M, G, ω, Φ). Namely, let O = G · p be a
G-orbit in M. In this section we will completely describe a G-invariant
neighborhood U of O: The symplectic structure on U, the group action on it, and the moment map. This normal form was discovered by
Guillemin and Sternberg [GS6, §41], [GS5] and by Marle [M]. Our
current exposition is inspired by R. Sjamaar’s description of a neighborhood of a coisotropic orbit in [Sj].
A neighborhood of O in M will be isomorphic to a neighborhood
of the zero section in a vector bundle over O. This vector bundle is
the sum of two bundles; the co-null bundle and a symplectic vector
bundle. The null-bundle is a subbundle of the tangent bundle whose
fiber over q ∈ O is the null-space of the restriction of ω to Tq O. The
co-null bundle is the dual bundle.
We will first construct the local model for a Hamiltonian G-space
and will then show that every Hamiltonian G-space is locally isomorphic to such a model. The construction uses the following ingredients.
1. A vector α ∈ g∗ .
2. A subgroup H ⊆ Gα where Gα is the stabilizer of α under the
coadjoint action.
3. A symplectic vector space (V, ωV ) on which H acts by linear
symplectic transformations.
4. A complementary subspace m to h in gα which is invariant under
the adjoint action of H. Here h and gα denote the Lie algebra of
H and Gα .
5. A complementary subspace m′ to gα in g which is invariant under
the adjoint action of Gα .
Denote N = gα /h. The adjoint action of H on g induces an action
of H on the vector space N, and dually, an action of H on N ∗ . The
choices of m and m′ determine embeddings of h∗ in g∗α and of g∗α in g∗ ,
so that we think of h∗ and N ∗ as subspaces of g∗ .
As a preliminary model, consider
G × (h∗ ⊕ N ∗ ⊕ V ).
We have an action of G on this space by left multiplication on the
first factor. We also have an action of H given by h : (g, µ, ν, v) 7→
(gh−1 , hµh−1 , hνh−1 , h · v). These two actions commute.
Recall, in section 1.6 we described three examples of symplectic
spaces with actions of the groups G and H and with moment maps.
18
1. PRELIMINARIES
These were the coadjoint orbit Oα of α, the cotangent bundle T ∗ G,
and a symplectic vector space V .
We now describe maps from G × (h∗ ⊕ N ∗ ⊕ V ) to those three spaces,
which are equivariant with respect to the actions of G and of H. A
map to Oα is given by f1 (g, µ, ν, v) = gαg −1. It is equivariant when G
acts on Oα by the coadjoint action and H acts on Oα trivially. A map
to T ∗ G is given by f2 (g, µ, ν, v) = g(µ + ν). It is equivariant when G
and H act on T ∗ G on the left and right respectively. A map to V is
given by f3 (g, µ, ν, v) = v. It is equivariant with respect to the linear
action of H on V and with G acting on V trivially.
Let ωα , ωG and ωV be the symplectic forms on the spaces Oα , T ∗ G
and V respectively. Consider the 2-form ω̃ = f1∗ ωα + f2∗ ωG + f3∗ ωV on
G × (h∗ ⊕ N ∗ ⊕ V ). We compute the corresponding moment maps for
the actions of G and H by taking the moment maps on the spaces Oα ,
T ∗ G and V as described in section 1.6; pulling back via the fi ’s and
summing. This gives:
Φ̃G (g, µ, ν, v) = g(α + µ + ν)g −1
(9.1)
Φ̃H (g, µ, ν, v) = −µ + ΦV (v).
We can embed G × (N ∗ ⊕ V ) in G × (h∗ ⊕ N ∗ ⊕ V ) as the zero level
set of the moment map Φ̃H , by
(g, ν, v) 7→ (g, ΦV (v), ν, v).
(9.2)
Remark 9.3. The tangent space to G × (N ∗ ⊕ V ) at (e, 0, 0) is
g ⊕ N ∗ ⊕ V . It is mapped by (9.2) isomorphically to the subspace g ⊕
{0} ⊕ N ∗ ⊕ V of the tangent space to G × (h∗ ⊕ N ∗ ⊕ V ) at (e, 0, 0, 0);
the h∗ component is zero because ΦV is quadratic.
The pullback of ω̃ to G×(N ∗ ⊕V ) via (9.2) descends to a closed form
ωN on the quotient mod H; this follows from Marsden and Weinstein’s
procedure of symplectic reduction. This quotient is the homogeneous
vector bundle:
N = G ×H (N ∗ ⊕ V ).
(9.4)
The composition of (9.2) and (9.1) gives a formula for a moment map
on N ;
ΦG (g, ν, v) = Ad∗ (g)(α + ν + ΦV (v)).
(9.5)
Note that ΦG sends the point [e, 0, 0] to α.
We compute the 2-form ωN on the tangent at [e, 0, 0] using Remark
9.3 and by keeping track of the 2-forms which were involved in the
definition of ω̃. We get:
9. THE GUILLEMIN-STERNBERG-MARLE LOCAL NORMAL FORM
19
Remark 9.6. The tangent space to N at [e, 0, 0] can be written as
g/gα ⊕ (N ⊕ N ∗ ) ⊕ V.
(9.7)
The three summands in (9.7) are ωN -orthogonal. Note that the first
summand is isomorphic to the tangent space to Oα at α. On this
space, ωN is equal to the Kirillov-Kostant-Souriau form. In the second
summand, N and N ∗ are isotropic, and ωN |N ∗ ×N is the pairing between
N ∗ and N. On the third summand we have ωN = ωV .
In particular it follows that ωN is nondegenerate at [e, 0, 0]. Therefore ωN is nondegenerate on a whole neighborhood of the orbit of
[e, 0, 0], i.e., on the zero section of the bundle G ×H (N ∗ ⊕ V ). Finally, our model is a symplectic neighborhood of this zero section, with
the 2-form ωN and the moment map (9.5).
We now proceed in showing that any Hamiltonian G-space is locally
isomorphic to one of the models which were constructed above. Let
(M, G, ω, Φ) be a Hamiltonian G-space. Fix p ∈ M and denote by
O = G · p the orbit of p in M. The ingredients for the local model are
determined as follows.
1. α = Φ(p).
2. H = Stab(p) is the stabilizer of p in G.
3. V is the symplectic normal slice to O at p. It is defined in the
following way. Denote by A = Tp O the tangent to the orbit
at p and by Aω its ω-orthogonal complement in Tp M. Then
V = Aω /(Aω ∩ A). The linear isotropy action of H on Tp M
induces a linear symplectic action on V .
The subspaces m and m′ can be chosen arbitrarily. For example, we can
take m = h⊥ ∩ gα and m′ = g⊥
α with respect to any Ad-invariant inner
product on g. As before, we define N = gα /h and we get embeddings
of N ∗ and of h∗ in g∗ .
Theorem 1.3 (V. Guillemin, S. Sternberg and C. M. Marle). There
is a G-invariant neighborhood of O in M which is isomorphic, as a
Hamiltonian G-space, to a neighborhood of the zero section in G ×H
(N ∗ ⊕ V ). The moment map is given by the formula:
Φ([g, ν, v]) = Ad∗ (g)(α + ν + ΦV (v))
(9.8)
Proof of Theorem 1.3. First note that by G-equivariance, Φ
sends the orbit O to the coadjoint orbit Oα . Denote by i : O ֒→ M
the inclusion map. Denote by ωα the symplectic form on Oα .
Lemma 9.9. The pullback of the symplectic form on M to the orbit
O is equal to the pullback of the symplectic form on Oα via the moment
map; i∗ ω = Φ∗ ωα .
20
1. PRELIMINARIES
Proof. This was proved in [KKS, §1], we now present the details.
We shall use the notation of §6.2. By (5.3), for any ξ, η ∈ g; ω(ξ ·
p, η · p) = −(η · p)Φξ = −h[η, α], ξi = −hα, [ξ, η]i = ωα ([ξ, α], [η, α]) =
(Φ∗ ωα )(ξ·p, η·p). The second and fifth equalities use the G-equivariance
of Φ.
Corollary 9.10. Consider the restriction of ωp to the tangent
Tp O. Its null-space is then the tangent to the Gα -orbit through p. If we
identify O ∼
= G/H then the tangent to the Gα orbit becomes N = gα /h.
Denote A = Tp O; then we have A ∩ Aω = N.
Lemma 9.11. Let H be a compact group which acts linearly on a
symplectic vector space (T, ω) and preserves ω. Let A ⊆ T be an Hinvariant linear subspace. Let N = A ∩ Aω be the null space of ω|A
(and of ω|Aω ). Let V = Aω /N and let B = A/N. Then there is an
H-equivariant symplectic isomorphism
(9.12)
T ∼
= B ⊕ (N ⊕ N ∗ ) ⊕ V
where the three summands are ω-orthogonal, where N and N ∗ are
isotropic and ω|N ∗ ×N is the pairing between N ∗ and N, and where each
one of the spaces B, N, N ∗ and V is H-invariant.
Proof. We can embed B ֒→ A, say, as B = A ∩ N ⊥ with respect
to any H-invariant inner product. Similarly we can embed V ֒→ Aω .
Finally, (B ⊕ V )ω is a symplectic space containing N as a lagrangian
subspace. We can embed N ∗ in (B ⊕ V )ω as any complementary Lagrangian subspace, such that the symplectic form on N ∗ × N will give
the usual pairing.
Applying Corollary 9.10 and Lemma 9.11 to T = Tp M and A =
Tp O, we get:
(9.13)
Tp M ∼
= Tp O ⊕ N ∗ ⊕ V
∗
= g/gα ⊕ (N ⊕ N ) ⊕ V
(9.14)
where V is the symplectic normal slice to O at p. Note that (9.14) is
an isomorphism as symplectic vector spaces with H-actions.
Equation (9.13) implies that Tp M/Tp O ∼
= N ∗ ⊕ V . From the slice
theorem 1.1 there is an invariant neighborhood U of the zero section
in G ×H (N ∗ ⊕ V ) and a G-equivariant diffeomorphism ρ from U to a
neighborhood of O in M. From (9.14) together with Remark 9.6 we
can arrange that the differential dρ[e,0,0] be an H-equivariant symplectic
isomorphism from T[e,0,0] U to Tp M.
By G-equivariance it then follows that ρ∗ ω is equal to ωN on T N |{the zero section} .
By a theorem of A. Weinstein [W1] it follows that we can compose ρ
9. THE GUILLEMIN-STERNBERG-MARLE LOCAL NORMAL FORM
21
with an equivariant diffeomorphism of a smaller neighborhood of the
zero section, U ′ ⊆ U, and get an equivariant symplectic diffeomorphism
ρ from (U ′ , ωN ) to a neighborhood of the orbit O in M. (Actually, this
theorem without the G-action is due to A. Weinstein. Guillemin and
Sternberg have observed that the proof also applies under the action
of a compact group). Moreover, since ΦG ([e, 0, 0]) = Φ(p) = α, this
diffeomorphism also interwines the moment maps.
We have completed the proof of Theorem 1.3. We now give examples of some special cases which we will use in the subsequent chapters,
and in which the group is a torus; G = T .
Example 9.15. In the two cases of Example 2.1 we have
(i) The symplectic form is ω = 2i1 (dz ∧ dz + dw ∧ dw); the moment
map is Φ(z, w) = Φ(p) − m2 |z|2 − n2 |w|2.
(ii) The symplectic form is ω = dx ∧ dθ + 2i1 dz ∧ dz where a = e2πiθ ;
the moment map is Φ[a, x, z] = Φ(p) + x.
Example 9.16. Suppose that dim T = n, dim M = 2n and p has a
trivial stabilizer, as in Example 2.5. If we write T = Rn /Zn , then M =
Rn /Zn × Rn near a neighborhood of p. Let x1 , . . . , xn be coordinates
modulo Z on Rn /Zn and let y1 , . . . , yn be coordinates on the second
Rn . Then the symplectic form is ω = (dy1 ∧ dx1 + . . . + dyn ∧ dxn )
and the moment map is (x1 , . . . , xn , y1, . . . , yn ) 7→ (y1 , . . . , yn ). In a
coordinate free language, M is isomorphic to the cotangent bundle of
the orbit of p, which is simply T × t∗ , and Φ is the projection to the t∗
component.
22
1. PRELIMINARIES
CHAPTER 2
Completely integrable actions on presymplectic
spaces
1. Introduction
Let (M, T ) be a completely integrable compact T -space. Recall,
this means that we have an effective action of a torus T on a compact manifold M where dim T = n and dim M = 2n. Let ω be a
presymplectic form on (M, T ) and let Φ : M −→ t∗ be a corresponding
moment map as defined in sections 1.3 and 1.5. Then we can define
the D-H measure on t∗ as in §1.7, recall that this is the push-forward
of Liouville measure;
mDH = Φ∗ mω = ρ(α) · |dα|
where mω is defined by integrating the volume form ω n /n!. By the
Duistermaat-Heckman theorem, the density function ρ(α) is piecewise
constant on t∗ . In this chapter we give a topological description of the
function ρ, in terms of winding numbers of a certain map into t∗ .
This result was motivated by work of M. Grossberg, [G]. It was
proved for toric varieties in my joint paper with S. Tolman, [KT]. In
its current generality it will appear in my paper with M. Grossberg,
[GK2].
By Remark 1.5.4 the moment map is T -invariant, it therefore descends to a function Φ : M/T −→ t∗ . In section 3 we will describe
M/T as an n-dimensional orbifold with boundary components Ni which
themselves are orbifolds. Their images under Φ lie in a finite union of
hyperplanes in t∗ . If α ∈ t∗ is not contained in any of these hyperplanes
then for every i we have a map
Φ|Ni : Ni −→ t∗ r {α}.
The winding number of this map is an integer di (α) defined in §5.
Theorem 2.1. The density function for the D-H measure is
X
ρ(α) =
di (α).
i
23
24 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
Theorem 2.1 is proved in section 5, following a sequence of lemmas
in sections 4 and 5. In section 7 we describe a formula, which is due
to Guillemin, Lerman and Sternberg, for the density function ρ(α) in
terms of the isotropy actions of T at the fixed points. We will use
this formula later, in chapter 3, to relate the D-H measure to certain
quantities which come from other structures on M.
2. A first example
Take M = C with the S 1 action λ : z 7→ λz. Although C is not
compact, this example does illustrate the main idea in Theorem 2.1.
We will discuss several presymplectic forms ω on (M, T ). Remember,
the moment map is now a map Φ : C −→ R and is determined by
∂
)ω (and, say, Φ(0) = 0), where z = re2πiθ .
dΦ = −ι( ∂θ
1
Example 2.1. On M = C we have the symplectic form ω = 2π
dy ∧
dx = rdθ ∧ dr. The moment map is Φ(re2πiθ ) = − 12 r 2 . The quotient
M/T is a ray, parametrized by r;
C
It is sent by Φ to the negative ray in t∗ = R;
We can write ω = d(−r 2 /2) ∧ dθ = Φ∗ dx ∧ dθ where x is the
∂
∂
coordinate on R. We choose the orientation on C such that { ∂y
, ∂x
}
is an oriented frame. This convention is explained in Remark 3.3.6.
Then Liouville measure is positive and is given by mω = |ω|. Pushingforward via Φ : M −→ t∗ amounts to integrating over the θ variable,
which gives Φ∗ mω = |dx| on the set {x ≤ 0} and zero outside.
2
Example 2.2. Let ω = 7rdθ ∧ dr, then Φ(reiθ ) = −7 r2 and, as in
Example 2.1, ρ(x) is equal to one on {x ≤ 0} and zero outside. Notice
that Liouville measure is seven times denser than in Example 2.1. This
is not felt in the push-forward measure because the moment map now
spreads the same regions in C over larger intervals in R.
3. LOCAL STRUCTURE OF M/T
25
Example 2.3. Let ω = f (r 2 )rdθ ∧ dθ where f is a smooth function
which takes positive values, then again ρ(x) = 1{x≤0} , by a similar
argument.
Example 2.4. Let ω = (r 2 − 1)rdθ ∧ dr. Then inside the unit disc
∂
mω is a negative measure. Since ι( ∂θ
)ω = −(r 2 − 1)rdr = dΦ, we can
2
take Φ(reiθ ) = r4 (2 − r 2 ). If we identify M/T = R+ via r 2 then the
map Φ : R+ −→ R “folds” at r 2 = 1;
r2
Φ
−3
−2
−1
0
1
4
1
2
3
t∗ =R
The contributions of the overlapping pieces cancel, so again ρ(x) =
1{x≤0} .
The above examples reflect the general case. Locally, regions in
M on which ω is nondegenerate contribute ±Lebesgue measure on t∗
to the D-H measure. These contributions combine to give the degree
function on t∗ , defined in Lemma 5.2. It turns out that we can ignore
the regions in M where ω is degenerate, see Lemma 4.7. In the case of
a compact manifold, the degree function can be expressed in terms of
winding numbers of the map Φ restricted to the boundary components
of M/T , as stated above.
3. Local structure of M/T
Let (M, T ) be a completely integrable space. The smooth structure
of M/T is defined by declaring a function smooth if its pullback to M is
smooth. A diffeomorphism is, by definition, a homeomorphism which
induces a bijection on the sets of smooth functions. For example, by [],
any S 1 invariant smooth function on C is of the form f (|z|2 ) where f
is smooth on R. Therefore the map z 7→ |z|2 defines a diffeomorphism
of C/S 1 with R+ , where the smooth functions on R+ are defined as the
restrictions of smooth functions on R.
We begin by computing the quotients M/T in a two simple examples.
Example 3.1. M = S 2 , T = S 1 acts by rotations. For (x1 , x2 , x3 ) ∈
S 2 we can write x1 + ix2 = ζ and x3 = h with |ζ|2 + h2 = 1. Then the
26 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
circle action is λ · (ζ, h) = (λζ, h) for |λ| = 1.
The quotient M/T is a closed interval, parametrized by the height
function h.
Example 3.2. M = CP 2 and T = (S 1 )2 with the action (λ1 , λ2 ) ·
[z0 , z1 , z2 ] = [z0 , λ1 z1 , λ2 z2 ]. If we take CP 2 to be the unit sphere in C3
modulo the diagonal S 1 -action then dividing by our T -action amounts
to forgetting the phases of z0 , z1 , z2 and only remembering their norms.
Thus the map (z0 , z1 , z2 ) 7→ (|z0P
|2 , |z1 |2 , |z2 |2 ) identifies CP 2/T with
the triangle {(s0 , s1 , s2 ) | si ≥ 0,
si = 1} in R3 .
In the above examples, M/T is topologically a manifold with boundary. It is homeomorphic to the n dimensional ball where n = dim T =
1
dim M. The boundary is not smooth; it has corners.
2
Remark 3.3. Example 3.2 illustrates how the structure of M/T
reflects the T -action on M. First, the three vertices of the triangle M/T
correspond to the three fixed points in M; [1, 0, 0], [0, 1, 0] and [0, 0, 1].
Next, the three edges correspond to sets of points with one dimensional
stabilizers; {[z0 , z1 , 0]}, {[z0 , 0, z2]} and {[z0 , z1 , 0]}. Finally, the interior
of the triangle corresponds to the set of points [z0 , z1 , z2 ] with zi 6= 0
for all i; on which the action is free. We now proceed with the general
case.
Lemma 3.4. Let (M, T ) be an integrable T -space, then the quotient
M/T is locally diffeomorphic to (R+ )l × (Rn−l /Γ) where Γ is a finite
subgroup of SO(n − l).
Proof. By Example 1.2.2 we can write locally M/T = Rn−l /Γ ×
C /T l where Γ is a finite abelian subgroup of SO(n − l).
l
A topological orbifold is a topological space X with an open covering
U, and for each U ∈ U we are given a homeomorphism of U with a
neighborhood of the origin in Rm /Γ; where Γ is a finite group acting
linearly on Rm . These must be compatible on the intersections. The
orbifold structure is defined to be an equivalence class of such coverings.
4. LOCAL DESCRIPTION OF THE D-H MEASURE
27
Similarly, an orbifold with boundary is defined as a space for which
we have local homeomorphisms with (R+ × Rm−1 )/Γ; where Γ acts
on Rm and preserves the half-space R+ × Rm−1 . Again we require
compatibility on the intersections.
Given a topological orbifold X with boundary, its boundary is defined as the points in X which correspond to the points of ({0} ×
Rm−1 )/Γ in the given local homeomorphisms. This boundary is itself
an orbifold.
For an integrable T -space, the quotient M/T is a topological orbifold; this follows from Lemma 3.4 because (R+ )l is homeomorphic (but
not diffeomorphic) to R+ × Rl−1 . Thus the boundary of M/T are those
points which correspond to ∂((R+ )l ) × (Rn−l /Γ). From the proof of
Lemma 3.4 and from Example 1.2.2 it follows that these points are exactly those which come from points in M whose stabilizer has a positive
dimension. We get:
Remark 3.5. The boundary of M/T is an orbifold. It is equal to
Ms /T where Ms consists of the points in M with a stabilizer of positive
dimension.
4. Local description of the D-H measure
Throughout this section we assume that (M, T, ω) is a
• oriented, compact,
• completely integrable,
• presymplectic
T -space with a moment map Φ : M −→ t∗ . Our goal is to arrive at the
local description of the D-H measure given in Proposition 4.9.
Lemma 4.1. Let p ∈ M be such that
• The stabilizer of p is trivial,
• ω is nondegenerate at p and ωpn is compatible with the orientation
of M.
Then there exists an invariant neighborhood U of the orbit T · p, such
that
Φ∗ (ω n /n!)|U = Lebesgue measure on Φ(U).
Proof. This is a special case of the Duistermaat-Heckman theorem. By Example 1.9.16 we can write M = (R/Z)n ×(R)n and ω n /n! =
dx1 ∧dy1 ∧. . .∧dxn ∧dyn . The moment map is (x1 , . . . , xn , y1 , . . . , yn ) 7→
(y1 , . . . , yn ). Pushing forward thus amounts to integrating over the xi
variables. The push-forward measure is then |dy1 ∧ . . . ∧ dyn | which is
what we wanted.
28 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
Convention 4.2. Recall that M is oriented. Choose an orientation on T . This induces an orientation on t, on t∗ , and on any
free T -orbit in M. Orient M/T in such a way that the orientation of
M/T , followed by that of a free orbit, gives the orientation on M times
n(n−1)
(−1) 2 ; this strange convention will save us trouble later on.
Definition 4.3. Suppose that p is a regular point of Φ with a
trivial stabilizer. Then [p] is a smooth point of M/T and the differential
dΦp is an isomorphism from T[p] M/T to t∗ . Then sign(dΦp ) is defined to
be 1 or −1 according to whether Φ : M/T −→ t∗ preserves or reverses
orientation at p, where the orientations follow Convention 4.2. This is
independent on the orientation of T .
Definition 4.4. Suppose that ω is non-degenerate at p. Then
sign(ωpn ) is defined as ǫ = ±1 such that ǫωpn is compatible with the
orientation of M.
Lemma 4.5. Suppose that
• The stabilizer of p is trivial,
• ωp is nondegenerate.
Then
sign(ωpn ) = sign(dΦp ).
Proof. Both sides of the equality switch sign when the orientation of M is reversed. It is thus sufficient to prove the equality when
sign(ωpn ) = 1. We can apply the local model 1.9.16. We can choose
the orientation on T such that ∂x∂ 1 , . . . , ∂x∂n is an oriented basis on T
or on the orbit. Since ∂x∂ 1 . . . , ∂x∂n and ∂y∂ 1 , . . . , ∂y∂n are dual bases in
t and in t∗ , we get that ∂y∂ 1 , . . . , ∂y∂n is an oriented basis of t∗ . Since
ωpn /n! = dy1 ∧dx1 ∧. . .∧dyn ∧dxn , we have that ∂y∂ 1 , ∂x∂ 1 , . . . , ∂y∂n , ∂x∂n is
an oriented basis on M. Therefore, ∂y∂ 1 , . . . , ∂y∂n , ∂x∂ 1 , . . . , ∂x∂n gives the
n(n−1)
orientation on M times (−1) 2 . By Convention 4.2, ∂y∂ 1 , . . . , ∂y∂n
give the positive orientation on M/T . It follows that the function
Φ : M/T −→ t∗ sends an oriented basis to an oriented basis, i.e.,
sign(dΦp ) = 1.
Corollary 4.6. Let p ∈ M be such that
• The stabilizer of p is trivial,
• ωp is nondegenerate.
Then on an invariant neighborhood U of the orbit T ·p we have Φ∗ (ω n /n!)|U =
sign(dΦp ) · (Lebesgue measure on Φ(U)).
5. DEGREE OF THE MOMENT MAP
29
Proof. From Lemma 4.1 it follows that Φ∗ (ω n /n!)|U = sign(ωpn ) · (
Lebesgue measure on Φ(U)). The rest follows from Lemma 4.5.
Lemma 4.7. Suppose that p ∈ M is a regular point for Φ. Then
• The stabilizer of p is discrete,
• ωp is nondegenerate.
Proof. Let ξ1 , . . . , ξn be a basis of t and let e1 , . . . , en be a dual
basis in t∗ . Being a regular point of Φ means that dΦp is onto t∗ , so
there exist vectors v1 , . . . , vn ∈ Tp M such that dΦp (vi ) = ei . Applying
the definition of the moment map, we get that ω(ξi · p, vj ) = δij . It
follows that the vectors ξi · p are linearly independent, and therefore
that the stabilizer of p is discrete. In terms of the basis ξ1 · p, . . . , ξn ·
p,
n of Tp M, ωp is given by a 2n × 2n matrix of the form
v1 , . . . , v
0 I
. The block of zeroes follows from the fact that a T -orbit
−I A
is always isotropic, see Corollary 1.5.5. In particular it follows that ωp
is nondegenerate.
Lemma 4.8. Assume that p ∈ M satisfies
• The stabilizer of p is discrete,
• ωp is nondegenerate.
Then the stabilizer of p is trivial.
Proof. Let Γ ⊆ T be the finite stabilizer of p. By the normal form
in Example 1.2.2 we can write near p that M = T ×Γ t∗ with Γ acting
on t∗ linearly and effectively. We have a finite covering M̃ −→ M
where M̃ = T × t∗ . We pull back to M̃ the symplectic form and the
moment map. By applying to M̃ the normal form in Example 1.9.16,
we have that the moment map is, up to a translation, projection to the
t∗ component. For this projection to descend to M we must have that
Γ act trivially on t∗ , and since this action is effective, we must have
Γ = 0.
Proposition 4.9. Let p ∈ M be a regular point for Φ. Then there
is a T -invariant neighborhood U of p in M such that the push-forward
of Liouville measure from U to t∗ via Φ is equal to sign(dΦp ) times
Lebesgue measure on Φ(U).
Proof. This follows immediately from Lemma 4.7, Lemma 4.8 and
Corollary 4.6.
5. Degree of the moment map
Let (M, T ) be a completely integrable compact T -space, let ω be
a presymplectic form on (M, T ) and let Φ be a corresponding moment
30 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
map. By Remark 1.5.4, Φ descends to a function Φ : M/T −→ t∗ .
Let N1 , N2 , . . . be the boundary components of M/T . By Remark 3.5,
each Ni is an n − 1 dimensional orbifold. Suppose that α ∈ t∗ is not in
the image of any Ni . Denote α = {α} and consider
Φ|Ni : Ni −→ t∗ r α.
On the level of homology this gives a map between the homology
groups;
H̃n−1 (Ni ) −→ H̃n−1 (t∗ r α).
(5.1)
We are taking here the reduced homology groups, this makes a difference when n = 1. Then both sides of (5.1) are isomorphic to Z. We
choose isomorphisms in the following way.
Remember that an orientation of T determined orientations of M/T
and of t∗ , as in Convention 4.2. We choose an orientation on Ni by
demanding that an outward normal to M/T followed by the orientation
of ∂(M/T ) gives the orientation of M/T . This orientation picks for us
the generator of H̃n−1 (Ni ). A similar relation picks a generator of
H̃n−1 (t∗ r α).
Thus the map (5.1) is given by multiplication by an integer. This
integers is defined to be the winding number of Φ|Ni around α and it
is denoted by di (α). It does not depend on the orientation of T .
Lemma 5.2. Let α ∈ t∗ be a regular value for Φ. From Lemma 4.7
and Remark 3.5 it follows that α is not in the image of ∂(M/T ), so
that di (α) is defined for all i. Then
X
X
di (α) =
sign(dΦ[p]).
(5.3)
i
−1
[p]∈Φ
(α)
Proof. The map Φ : M/T −→ t∗ rα induces a map H(Φ) between
the relative homology groups Hn (M/T, ∂(M/T )) and Hn (t∗ , t∗ r α).
These groups are isomorphic to Z, the isomorphisms being determined
by the orientations of M/T and of t∗ . Therefore the map H(Φ) is given
by multiplication by an integer, say, d.
From the long exact sequences in homology for the pairs (M/T, ∂(M/T ))
and (t∗ , t∗ r α), it follows that the left term in (5.3) is equal to this
same integer d.
Let α be a regular value of Φ and let p ∈ Φ−1 (α). By Lemmas 4.7
and 4.8, T acts freely near p. Thus M/T is a smooth n dimensional
manifold near [p] and dΦ[p] : T[p] (M/T ) −→ Tα (t∗ ) is an isomorphism.
−1
Therefore there is a neighborhood U of α in t∗ such that Φ (U) is a
disjoint union of open sets which are mapped diffeomorphically to U
6. EXAMPLES OF COMPLETELY INTEGRABLE SPACES
31
by Φ. By excision, the image of 1 under the map H(Φ) is equal to the
right hand side of (5.3).
Can now prove the theorem which was stated in the introduction
to this chapter.
Proof of Theorem 2.1. Let α ∈ t∗ be a regular value of Φ.
From Proposition 4.9 and
P Lemma 5.2 it follows that α has a neighborhood on which mDH = i di (α)|dα|. The theorem follows, because the
singular values of Φ have zero Lebesgue measure.
6. Examples of completely integrable spaces
In this section we discuss some examples of completely integrable
T -spaces (M, T ).
Example 6.1. Take M = S 2 on which T = S 1 acts by rotations.
Following the notation of Example
3.1, we can introduce cylindrical
√
coordinates (ϕ, h) where ζ = 1 − h2 e2πiϕ . The circle action is gener∂
. The standard area form on S 2 with total
ated by the vector field ∂ϕ
area 4π is equal to 2πdh ∧ dϕ. An invariant 2-form can be written as
ω = f (h)dh ∧ dϕ. A corresponding moment map is Φ(ϕ, h) = F (h)
where ∂F/∂h = f (h). The function F is increasing or decreasing according to the sign of f . The measure mω is positive or negative again
according to the sign of f . The push-forward measure is supported
on the interval [F (−1), F (1)] and has a density = ±1 according to
whether F (−1) < F (1) or F (−1) > F (1). As for the degree function,
the boundary of M/T is a zero-sphere, i.e., two points. The winding
number of Φ : ∂(M/T ) −→ R around α ∈ R is equal to ±1 for α
between the images of these points and zero outside.
Our M is isomorphic to CP1 with the action λ · [z, w] = [z, λw].
Example 6.2. M = CP2 and T = (S 1 )2 with the action (λ1 , λ2 ) ·
[z0 , z1 , z2 ] = [z0 , λ1 z1 , λ2 z2 ], as in Example 3.2. The Fubini-Study metric on CP2 defines an invariant symplectic form. Symplectically, we
can describe this 2-form in the following way. We realize CP2 as the
unit sphere S 5 in C3 modulo the diagonal action of S 1 . Denote by Ω
the standard symplectic form on C3 . Then the restriction of ω to S 5
is equal to the pull-back of ω via S 5 −→ CP2 . (This is an example of
symplectic reduction of (C3 , Ω) with respect to the diagonal S 1 action).
The moment map on (CP2 , T, ω) is [z0 , z1 , z2 ] 7→ (− 12 |z1 |2 , − 12 |z2 |2 ).
Example 6.3 (Delzant spaces). A Delzant space is a Hamiltonian
T -space (M, T, ω, Φ) with dim T = 12 dim M and ω nondegenerate.
Delzant [De] classified these spaces. A Delzant space is determined
32 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
by a convex polytope in t∗ , namely image (Φ), and there are explicit
conditions to determine which polytopes are allowed. The level sets
of the moment map are precisely the T -orbits in M. Therefore, Φ descends to a map Φ : M/T −→ t∗ which is a diffeomorphism with its
image; where the smooth structures are defined in the sense of section
3. Topologically, M/T ∼
= image (Φ) is a closed n-ball. Its boundary N
is an (n − 1)-sphere.
If we keep (M, T ) but vary ω then the faces of the corresponding polytope get parallel translated, but their directions remain the
same, and so does the combinatorial information of which face intersects which – as long as ω remains symplectic.
There are completely integrable spaces (M, T ) for which M is simply connected but M/T is not a topological ball. These will be described later in this section. By the above discussion, these manifolds
do not admit invariant symplectic structures. (By simple-connectedness,
there will always be a moment map, so we could then apply Delzant’s
theorem). Theorem 2.1 applies to presymplectic forms on such manifolds.
Example 6.4 (Toric varieties). Toric varieties are a special collection of algebraic varieties. They are explicitly constructed from a given
combinatorial object called a fan. A toric variety of complex dimension
n comes equipped with a holomorphic action of an n dimensional torus.
We are interested in those varieties which are compact and smooth.
In fact, every n-dimensional smooth complex algebraic variety with a
holomorphic action of an n-torus is a toric variety.
Every integral convex polytope determines a fan. Integral means
that the sides have rational slopes. The fan encodes the directions of
the faces and which faces intersect. It does not remember the actual
location of the faces.
Every symplectic smooth toric variety is a Delzant space. All
Delzant spaces arise in this way. Not all toric varieties admit invariant symplectic structures, but they still share many properties with
Delzant spaces. Here are some properties of toric varieties which are
relevant for us.
(i) M is simply connected.
(ii) M/T is homeomorphic to a closed ball.
(iii) The stabilizer of any p ∈ M is a connected subgroup of T .
Thus for toric varieties, the D-H measure is given by the winding
numbers of a map S n−1 −→ t∗ around the points in t∗ .
6. EXAMPLES OF COMPLETELY INTEGRABLE SPACES
33
Example 6.5. This example was discovered by S. Tolman and it
has lead us to Theorem 2.1. Let T = (S 1 )2 act on CP2 by (λ1 , λ2 ) ·
[z0 , z1 , z2 ] = [z0 , λ1 z1 , λ2 z2 ]. Let M be the blow-up of CP2 at the three
fixed points, then the action of T extends to M. On (M, T ) there is a
presymplectic form for which the D-H measure is given by the following
picture.
This is explained in [KT].
Here are some examples of integrable T -spaces (M, T ) where M/T
is not a ball.
Example 6.6. M = S 1 × S 1 is a torus, S 1 acts on the left factor
and ignores the right factor. Then M/T is a circle, not an interval.
There is a symplectic structure (the area form) but there is no moment
map; an obstruction lies in H 1 (M, R).
Example 6.7. Equivariant connected sums One can construct
new examples from old ones by a procedure which we call equivariant
connected sum. From two integrable T -spaces, M1 and M2 , we get a
third space, M1 ♥M2 , for which the quotient M1 ♥M2 /T is the usual
connected sum of the two quotients M1 /T and M2 /T . In particular, if
M1 and M2 are toric varieties then M1 ♥M2 /T is the connected sum of
two closed balls, thus M1 ♥M2 is a completely integrable space which
is not a toric variety. If Mi = CP2 for i = 1, 2 then Mi /T are triangles,
see Example 6.2. M1 ♥M2 /T is then “two kissing triangles”, and is
homeomorphic to I × S 1 .
We now describe the construction. Let (M1 , T ) and (M2 , T ) be two
completely integrable spaces of dimension 2n. A neighborhood Ui of
a free orbit in Mi is isomorphic to T × Di where Di is an n-disc and
where T acts by left multiplication on the first factor; see Example
1.2.5. Then Mi r Ui are manifolds with a boundary T × S n−1 . Take a
map ψ : S n−1 −→ S n−1 which reverses orientation. We glue the M1 rU1
to M2 r U2 along their boundaries via the map id × ψ : T × S n−1 −→
T × S n−1 . This procedure can also be carried out smoothly. Note that
the T -action is well defined on the resulting space.
34 2. COMPLETELY INTEGRABLE ACTIONS ON PRESYMPLECTIC SPACES
Example 6.8. Let M = CP2 ♥CP2 as in the previous example.
Take the six dimensional manifold M̃ = M × CP1 with the action of
T̃ = T × S 1 , then M̃ /T̃ is topologically a solid torus because M̃ /T̃ =
M/T × CP1 /S 1 = (I × S 1 ) × I.
In a similar way we can obtain examples with ∂(M̃ /T̃ ) being a
surface of any genus. We start by connecting a sequence of g + 1 copies
of CP2 ; M = CP2 ♥ . . . ♥CP2 . The quotient M/T is homeomorphic
to a disc with g holes. As before we construct M̃ = M × CP1 , then
M̃ /T̃ is the product of the disc with n holes with an interval. This is
a handlebody with g handles.
7. The Guillemin-Lerman-Sternberg formula
Duistermaat and Heckman [DH] wrote a formula which expresses
the Fourier transform of the push-forward measure Φ∗ ω n in terms of the
fixed point data of the torus action. Guillemin, Lerman and Sternberg
[GLS1, GLS2] have transformed this into a formula for the pushforward measure itself. We will now describe their formula in the special case of a completely integrable action.
Let (M, T, ω) be a presymplectic completely integrable oriented
space. Let p be a fixed point and consider the isotropy action of T
on Tp M; this is λ · (z1 , . . . , zn ) = (λα1 z1 , . . . , λαn zn ) in the notation of
Example 2.4, where αj = αj,p ∈ ℓ∗ are the isotropy weights at p.
Choose v ∈ t such that hαj,p, vi =
6 0 for all 1 ≤ Q
j ≤ n and for
every fixed point p. Define ǫj,p = signhmj,p, vi and ǫp = j ǫj,p . Denote
′
′
αj,p
= ǫj,p αj,p so that hαj,p
, vi > 0. Consider the following function on
∗
t;
(
P
ǫp if α = Φ(p) − nj=1 rj αj,p for some rj ≥ 0
′
νp (α) =
0 otherwise.
Its support is a polyhedral cone emanating from Φ(p) and pointing in
the direction of −v.
Theorem 2.2 (Guillemin-Lerman-Sternberg). The push-forward mean
sure Φ∗ ωP
/n! is given by Lebesgue measure on t∗ times the density
function p νp′ (α).
CHAPTER 3
Index theory for completely integrable spinc
manifolds
or:
Lots of fancy words but it’s not really that
bad.
1. Introduction
In this chapter I describe joint work with M. Grossberg [GK2]
which followed earlier joint work with S. Tolman [KT]. Let (M, T ) be
a completely integrable space. Recall, this means that a torus T acts
effectively on a compact manifold M and that dim T = 12 dim M = say,
n. In Chapter 2 we have shown how, given a closed invariant 2-form
on M and a moment map, we get a measure on t∗ ; the D-H measure.
Its density function ρ(α) depended only on the cohomology class of
the 2-form. In Theorem 2.1 we gave a topological description for the
function ρ(α) in terms of certain winding numbers of a map into t∗ .
In this chapter we will discuss several kinds of extra structures
that (M, T ) might admit. The first is a complex structure on M and
a holomorphic line bundle over M on which T acts. The others are
generalizations of this structure. One generalization is to consider an
almost complex structure on M and a smooth line bundle over M. A
further generalization is to consider a spinc structure on M on which
the torus acts.
In t∗ we have the integral weight lattice, ℓ∗ . Each one of the above
structures gives rise to a map mult : ℓ∗ −→ Z. This map is essentially
the equivariant index of a certain elliptic differential operator which is
constructed from the extra structure on M. These are the twisted Dolbeault operator, the rolled-up twisted Dolbeault operator, or the Dirac
spinc operator. Moreover, in each case we choose a certain presymplectic form on M and a moment map; these give rise to a function
ρ(α) on t∗ , as described above. Our main theorem, Theorem 3.1, is
that mult (α) = ρ(α) for α ∈ ℓ∗ . Combining this with Theorem 2.1,
we get a topological description for the index of our elliptic differential
operator, in terms of certain winding numbers of a map into t∗ . This
was stated in Theorem 0.1 in the introductory chapter of this thesis.
35
36 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
The 2-form ω is constructed in the following way. In the presence
of a complex or almost complex structure and a line bundle L, we take
ω to be the curvature of L (with respect to any invariant connection)
and the moment map is determined by the lifting of the action to L.
In the case of a spinc structure – such a structure determines a line
bundle L̃. We then take ω to be half of the curvature of L̃.
The relationship between ρ(α) and mult (α) can be thought of as a
generalization of the procedure of geometric quantization. If we have
a holomorphic line bundle L over M whose curvature ω is Kähler then
the function mult (α) describes the space of holomorphic sections of L
as a representation of T . This is then related to a symplectic invariant
of ω – the function ρ(α). In general, if we are given (M, T, ω), then
we can think of the extra structure on (M, T ) as a “polarization”. We
require a certain compatibility between the extra structure and the
cohomology class of ω; we do not require any compatibility with the
2-form ω itself.
I now present some background that lead to this work. In the
paper [KT], Tolman and I discussed a special kind of manifolds – toric
varieties. In particular we computed the twisted Dolbeault index of a
toric variety with a holomorphic line bundle. We have shown that the
functions mult (α) and ρ(α) are both given by certain winding numbers
which come from a map of M/T into t∗ . In particular it follows that
mult (α) = ρ(α) when both are defined. They form the shape of a
twisted polytope in t∗ .
The motivation to look for winding numbers came from the thesis
of M. Grossberg [G]. He computed the twisted Dolbeault index in the
special case of Bott towers and obtained shapes which he called twisted
cubes, also see [GK1]. The twisted polytopes described in [KT] are
generalizations of these.
The Dolbeaux cohomology for a toric variety was already computed
by Danilov [Da]. In the Kähler case, mult (α) = 1 for α in a certain
polytope and mult (α) = 0 otherwise. Atiyah observed that this polytope coincides with image (Φ). Moreover, ρ(α) = 1 on image (Φ), e.g.,
by [DH]. Our computation in [KT] gives a topological description for
mult (α) in the non-Kähler case.
The computation of the index in [KT] was done directly and explicitly and is due to S. Tolman. These results were later generalized by
M. Grossberg and myself to other manifolds with torus actions which
are not toric varieties. This chapter contains this more general result.
The twisted Dolbeault index is then replaced by the Dirac spinc index.
In this context, we compute the index using a localization formula of
Atiyah and Bott. On the other hand, the function ρ(α) is given by a
2. PRELIMINARY NOTATION AND EXAMPLES
37
localization formula of Guillemin, Lerman and Sternberg. These two
formulas coincide term by term; it follows that mult = ρ.
The same proof would also gives a relationship between the functions mult (α) and ρ(α) in the non-integrable case; i.e., when dim T <
1
dim M. We then get that ρ(α) is, in a certain sense, an approxima2
tion to mult (α). This generalizes results of Heckman [He] for the case
of flag manifolds, and of Guillemin and Sternberg [GS3] for general
Kähler manifolds. We will explain this in greater detail in [GK2].
A large part of this chapter is based on notes that M. Grossberg
and I have written jointly for talks that we gave in Tel-Aviv University
in August 1992.
2. Preliminary notation and examples
We begin with a simple example. Let M = C and let S 1 act
by λ : z 7→ λz. Although this is not compact, it does capture the
phenomena which we would like to study.
Recall, in Example 2.2.1 we took the symplectic form ω = rdθ ∧ dr
on C. The image of the moment map was the negative ray in R;
The D-H measure was Lebesgue measure on the set {x ≤ 0} and zero
elsewhere.
Now consider the vector space V = {holomorphic functions on C}.
Any f ∈ V can be written as
f (z) = a0 + a1 z + a2 z 2 + · · ·
We have a representation of S 1 on V by (λf )(z) = f (λ−1 z). In particular, if f (z) = z −k then (λf )(z) = λ−k f (z), so the monomial z k is
a weight vector with weight −k. Thus the weights which occur in the
representation are the non-positive integers, and each of these occurs
with multiplicity 1. We then draw the multiplicity diagram for the
representation:
t∗ =R
−3
−2
−1
0
1
2
Note that we have a differential operator ∂ on the space of holomorphic
functions on C, and that our representation space V is its kernel.
Now take M = C × C with the diagonal action of S 1 × S 1 ; then
the image of the moment map is the negative quadrant in R2 . The
38 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
function ρ is equal to 1 on this quadrant and 0 elsewhere;
Take V = {holomorphic
on C2 }. Then for every f ∈ V
P functions
i j
we can write f (z, w) = i,j aij z w . We get a representation of S 1 ×S 1
with a multiplicity diagram:
t∗ = R2
Now, let T be a torus. We follow the notation of section 1.2. In
particular, t is the Lie algebra of T and ℓ∗ is the integral weight lattice
in t∗ . In Chapter 1, the space t∗ appeared in two different contexts:
(i) Symplectic geometry; the moment map
(ii) Representation theory of the torus
In the above examples, symplectic geometry gave us a measure
in t∗ and the holomorphic structure gave a representation of T and
consequently a multiplicity diagram in ℓ∗ ⊆ t∗ . These two pictures in
t∗ looked very similar. Our goal is to explain this relationship between
the symplectic data and the holomorphic data in a greater generality.
Any finite dimensional complex representation V of T splits into
one dimensional representations, V = Cα1 ⊕ · · · ⊕ Cαd where αi ∈ ℓ∗ ;
see section 1.2. Let mα = mult (α) = the number of times that α
occurs among α1 , . . . , αd . The representation V can be visualized by
its multiplicity diagram in t∗ in which we associate to each weight α its
multiplicity mα .
3. NEXT EXAMPLES: CP1 AND CPn
39
Every λ ∈ T acts on V by the matrix
 α1

λ
0
..
.
ρ(λ) = 
.
0
λαd
The character of the representation is the P
function χ : T −→ C defined
α1
αd
χ
by (λ) = trace ρ(λ) = λ + . . . + λ = α∈ℓ∗ mα λα .
If V is the space of holomorphic
C and S 1 acts as
P∞ −kfunctions on
before then we have χ(λ) = k=0 λ for λ ∈ S 1 . Note that in this
example V is infinite dimensional. We could make sense of the weight
spaces and their multiplicities, but V is not the direct sum of its weight
spaces. When considering compact manifolds, our V ’s will always be
finite dimensional, so this will not be a problem.
3. Next Examples: CP1 and CPn
The simplest compact examples are when M = CP1 . Instead of
looking at the space of holomorphic functions, we will now look at the
space of holomorphic sections of a complex line bundle.
3.1. Line bundles. We have a principal C× -bundle, (C2 r 0) −→
CP1 , where C× acts on the right by (z, w) · a = (za, wa). For every
integer k we get a one dimensional C× -module, C−k , by letting C×
act on C by λ · v = λ−k v. We can then take the associated bundle.
This is the complex line bundle over CP1 obtained as the quotient of
(C2 r 0) × C by C× acting on the right;
(z, w, v) · a = (za, wa, ak v).
(3.1)
Its Chern number is k. We denote this bundle by Lk = (C2 r0)×C× C−k
and the projection map by π : Lk −→ CP1 .
The space of holomorphic sections is Γhol (Lk ) := {s̃ : CP1 −→ Lk | s̃
holomorphic, π ◦ s̃ =identity}. We can identify this with the space of
holomorphic functions f : (C2 r 0) −→ C which are homogeneous of
degree k, by setting s([z, w]) = [z, w, f (z, w)].
Lemma 3.2. Every holomorphic function on (C2 r 0) which is homogeneous of degree k ≥ 0 is a polynomial. There are no holomorphic
functions on (C2 r 0) which are homogeneous of degree k < 0.
Proof. By Hartogs’ theorem, f extends to a holomorphic function
defined on all of C2 . By homogeneity, the Taylor expansion of f can
only contain terms of order k.
40 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
Corollary 3.3. If k ≥ 0 then the space Γhol (Lk ) can be identified
with the space
P of homogeneous polynomials of degree k in two variables;
f (z, w) = ki=0 z i w k−i . If k < 0 then Γhol (Lk ) = 0.
3.2. Representations of SU (2). Consider the standard left action of SU(2) on C2 . This induces an action on CP1 which lifts to
the bundle Lk by g · [z, w, v] = [g(z, w), v]. We obtain a representation of SU(2) on Γhol (Lk ) by (g · s)(p) = g(s(g −1p)) for p ∈ CP1 .
Equivalently, the action on the homogeneous polynomials is given by
(g · f )(z, w) = f (g −1(z, w)). This construction gives exactly all the
irreducible representations of SU(2). This motivates the study of the
spaces of holomorphic sections.
If we restrict the action to the maximal torus of SU(2);
λ 0
sends [z, w] to [λz, λ−1 w] for λ ∈ S 1 .
0 λ−1
(3.4)
This action is not effective because λ = −1 acts trivially. We generally
consider effective actions, but by careful bookkeeping we can also deal
with actions with finite kernels.
The space Γhol (Lk ) breaks into one dimensional weight spaces spanned
by the monomials. The monomial z i w k−i has a weight k −2i. For k = 3
the weight diagram looks like
−5
−4
−3
−2
−1
0
1
2
3
4
5
(3.5)
and all the four weights have multiplicity 1.
3.3. The symplectic form. Recall, a connection on a line bundle
π
L −→ M can be given by a 1-form θ on the total space of L, which
is invariant with respect to the action of C× and whose restriction to
1
da/a, where a ∈ C is a coordinate on the fiber; [Kt].
each fiber is 2πi
We can write dθ = π ∗ ω for a 2-form ω on M, called the curvature of
the connection. The cohomology class of ω is independent of the choice
of connection and is called the Chern class of L.
Consider the bundle Lk over CP1 , for k ≥ 0. Let us work in the
coordinate patch U = {[z, w] | z 6= 0} with the complex coordinate
ξ = w/z. The homogeneous function f (z) = z k defines a trivializing
section over U; let a be the corresponding coordinate on the fiber. We
can choose the connection form to be
1
da
kξdξ
θ=
.
−
2πi a
1 + ξξ
3. NEXT EXAMPLES: CP1 AND CPn
41
The curvature is then
ω = dθ =
kdξ ∧ dξ
.
2πi(1 + ξξ)2
The integral of ω over CP1 is equal to the Chern number, k, where in
the integration we use the following convention.
∂
∂
Remark 3.6. We choose the orientation of C such that { ∂y
, ∂x
}
is an oriented basis. In general, on a complex manifold with local
coordinates z1 , . . . , zn we take { ∂y∂ 1 , ∂x∂ 1 , . . . , ∂y∂n , ∂x∂n } to be an oriented
basis when zj = xj + iyj . If we had taken the standard orientation on a
complex manifold, then Theorem 3.1 would be mult (α) = (−1)n ρ(α).
Our funny convention for the orientation is such that the integral of
the curvature of the tautological line bundle is −1. We note that our
definition of the curvature ω is consistant with [Kt]. The curvature
defined by algebraic geometers is Ω = 2πiω. The Chern class is then
obtained by integrating 2πi Ω with respect to the standard orientation
of a complex manifold.
It is sometimes easier to work with polar coordinates, ξ = re2πiϕ .
Further, we can identify CP1 with the unit sphere S 2 such that [z, w] 7→
ξ = w/z is the stereographic projection.
h
r
On S 2 we have
coordinates; for (x1 , x2 , x3 ) ∈ S 2 we write
√ cylindrical
x1 + ix2 = 1 − h2 e2πiϕ and x3 = h. In terms of (h, ϕ) we then have
k
dϕ ∧ dh.
2
We note that the standard area form on S 2 with total area 4π is equal
to 2πdϕ ∧ dh. The tangent bundle has a Chern number k = 2, and the
standard area form is 2π times its curvature.
ω=
3.4. The moment map. The circle action (3.4) is generated by
∂
the vector field ξM = 2 ∂ϕ
. To compute the moment map, take
ι(ξ)ω =
k ∂
ι( )dϕ ∧ dh = kdh
2 ∂ϕ
thus the moment map is Φ(ϕ, h) = kh. This is just a multiple of the
height function on the sphere. For k = 3, the image of the moment
42 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
map is
−3
3
Notice its similarity with the weight diagram (3.5).
3.5. The case of CPn . The above arguments generalize to the
case of CPn . For n = 2 we write CP2 = (C3 r 0)/C× . For every integer
k we have a line bundle Lk = (C3 r 0) ×C× C−k . The maximal torus
of SU(3) is T = {(λ0 , λ1 , λ2 ) | |λi | = 1, λ0 λ1 λ2 = 1}. Its Lie algebra
is a subspace of R3 and the dual space is the quotient t∗ = R3 /(the
diagonal). The group SU(3) acts on C3 and thus on CP2 and on Lk . Its
maximal torus acts on CP2 by (λ0 , λ1 , λ2 )·[z0 , z1 , z2 ] = [λ0 z0 , λ1 z1 , λ2 z2 ].
For k ≥ 0 and for Γhol (Lk ) defined as before, the weight vectors are the
homogeneous monomials of degree k, i.e., z0j0 z1j1 z2j2 with j0 +j1 +j2 = k.
This monomial has a weight (−j0 , −j1 , −j2 ). For k = 3, whose weights
form the following picture in t∗ .
k
2
2
2
The moment map in this case is Φ([z0 , z1 , z2 ]) = kzk
2 (|z0 | , |z1 | , |z2 | )
where kzk2 = |z0 |2 + |z1 |2 + |z2 |2 . For k = 3, its image is the triangle
{(−x1 , −x2 , −x3 ) | x1 + x2 + x3 = 3, x1 , x2 , x3 ≥ 0}.
3.6. The index for CP1 . In the above examples for CP1 we
related the space of holomorphic sections of Lk to the image of the
moment map. This relation breaks up when k < 0, in that case
there are no nontrivial holomorphic sections. We fix this by replacing the space Γhol (Lk ) by the index of a certain differential operator.
For CP1 , this index will be the formal difference (as representations
4. THE INDEX
43
of S 1 ) of H 0 (CP1 , Lk ) and H 1 (CP1 , Lk ). These can be defined either as sheaf cohomology or, as we will do later, by Dolbeault cohomology. We have H 0 (CP1 , Lk ) = Γhol (Lk ) and, by Serre duality,
H 1 (CP1 , Lk ) ∼
= H 0 (CP1 , L−k−2 ). It follows that for k < 0, if S 1 acts as
in (3.4), then the index is given by
−k
−(k−2)
k−2
k
(3.7)
where all the multiplicities are now −1.
In the symplectic context, instead of considering the image of the
moment map, we now consider the push-forward of Liouville measure,
as described in section 1.7. If k < 0 then this is a negative measure,
because ω is no longer compatible with the orientation on CP1 . The
push-forward measure is now minus Lebesgue measure, supported on
the interval [−k, k]:
−k
k
The density function takes values −1 and 0. It coincides with the
multiplicity function (3.7) outside the boundary of the interval.
Remark 3.8. In this example we normalize Lebesgue measure so
that the volume of (R/2Z) is 1. The lattice 2Z consists of those x for
which the action (3.4) of e2πix is trivial. In general, for effective actions,
we will normalize Lebesgue measure on t∗ such that volume(t∗/ℓ∗ ) = 1
where ℓ∗ is the integral weight lattice.
4. The index
In general we will be considering the equivariant index of certain
elliptic operators over (M, T ).
The index of a linear operator D : H+ −→ H− , is defined to be the
formal difference of vector spaces ker(D) − coker(D). It is determined
by its dimension; dim(ker(D)) − dim(coker(D)). In our case, H± will
be spaces of sections of certain vector bundles over the manifold M.
These are infinite dimensional but the kernel and cokernel of our D will
be finite dimensional.
If T acts on H± and commutes with D then T acts on the kernel
and cokernel. Thus the index becomes a virtual T -representation, i.e.,
a formal difference of two representations; ρker −ρcoker . It is determined
by its virtual character , this is the function from T to C which is defined
by
χD (λ) = trace (ρker (λ)) − trace (ρcoker (λ)) for λ ∈ T.
44 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
We can write
χD (λ) =
X
mα λα
α∈ℓ∗
where the multiplicities mα ∈ Z might be negative. We will consider
the multiplicity function mult : ℓ∗ −→ Z, which sends α 7→ mα .
5. Statement of the Theorem
In sections 2 and 3 we saw a relationship between the image of the
moment map and certain multiplicity diagrams associated to (M, T ).
To generalize this relationship we need the following data.
• An even dimensional compact manifold, M 2n
• Certain extra structure on M 2n .
• An n dimensional torus T n acting effectively on M 2n and on the
extra structure.
By extra structure on M we mean one of three possible structures;
(i) A complex structure on M and a holomorphic line bundle L −→
M.
(ii) An almost complex structure on M and a smooth line bundle
L −→ M.
(iii) A spinc structure on M.
For each one of these structures we will construct a function ρ :
t∗ −→ R, defined almost everywhere, and a function mult : ℓ∗ −→ Z.
We give the precise definitions in sections 7 and 8.
Theorem 3.1. Let α ∈ ℓ∗ , then
mult (α) = ρ(α)
(5.1)
whenever the right term is defined. In case (iii) the right term is always
defined.
The function ρ will be the density function for a D-H measure.
The function mult will essentially be the equivariant index of a certain
elliptic differential operator. Theorem 3.1 combined with Theorem 2.1
gives a description of the equivariant index of D in terms of winding
numbers of a map Φ : M/T −→ t∗ .
Remark 5.2. The structure (i) is a special case of (ii). Moreover,
an almost complex structure and a smooth line bundle on M determine
a spinc structure. The corresponding functions mult will coincide. The
functions ρ and ρ̃ which correspond to the structures (ii) and (iii) will
differ slightly. More specifically, ρ(α) will be undefined for some α ∈ ℓ∗ .
The hyperplanes on which ρ̃(α) is not defined are small shifts of the
6. OUTLINE OF THE PROOF
45
hyperplanes on which ρ(α) is not defined, and they avoid the weight
lattice, so that ρ̃(α) is defined for all α ∈ ℓ∗ .
From the above data we will get a presymplectic form ω on (M, T ).
In cases (i) and (ii) ω will be the curvature of L with respect to some
connection. In case (iii), ω will be half the curvature of a line bundle
L̃ which comes with the spinc structure. We can then consider the
associated D-H measure on t∗ . The function ρ : t∗ −→ R is its density
function.
6. Outline of the proof
We will apply localization formulas to describe the left and right
terms of (5.1) as certain sums, and the summands will coincide. Here
are a few more details.
In Theorem 2.2 we described a formula of Guillemin, Lerman and
Sternberg which expresses the D-H measure as a sum:
X
ρ(α) =
ρp (α) for α ∈ t∗ .
p
We sum over the fixed points p in M. The function ρp is determined
by the linear isotropy action of t∗ at p and by the value of the moment
map at p. It is supported on a polygonal cone in t∗ , where it is equal
to either 1 or −1.
On the other hand, for the character of the equivariant index we
can use a formula of Atiyah and Bott, which we describe in section 9:
X
χD (λ) =
νp (λ) for λ ∈ T.
p
Again we sum over the fixed points in M. Each νp depends on the linear
isotropy action of T at p and on the lifting of the T action. One can
translate this formula into a formula for the multiplicity function by
imitating the definition of the ρp ’s above. This will be done in section
10. It gives
X
mult (α) =
µp (α) for α ∈ ℓ∗ .
p
where µp is a function on ℓ∗ . In examining the above formulas we get
that for every fixed point p and every α ∈ ℓ∗ we have ρp (α) = µp (α) if
ρp (α) is defined.
46 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
7. The elliptic operators
In this section we describe the structures (i), (ii) and (iii) which
were listed in section 5. For each of these, we describe the elliptic differential operator D : H+ −→ H− whose index we will later compute.
In subsections 7.1 and 7.2 we describe the Dolbeault operator for a
complex manifold with a holomorphic line bundle, and more generally,
for an almost complex manifold with a smooth line bundle. This description follows closely the thesis of M. Grossberg [G]. In subsection
7.3 we describe the Dirac spinc operator for a spinc manifold.
7.1. Almost complex structures and the Dolbeault operator. Let M be a complex manifold. Then every tangent space Tp M
has a complex structure, i.e., a linear map Jp : Tp M −→ Tp M such that
(Jp )2 = −Identity. These fit into a smooth bundle map J : T M −→
T M with J 2 = −Identity. An almost complex structure on M is defined to be such a J. This means that each Tp M has the structure of
a complex vector space, but there might not exist local holomorphic
coordinates on M.
The complexification of T M then splits into the ±i eigenspaces of
J;
T M ⊗R C ∼
= (T M)1,0 ⊕ (T M)0,1 .
Dually, we get a splitting of T ∗ M ⊗R C. This induces a bigrading on
the wedge products of this bundle;
^
M ^
r
p,q ∗
(T ∗ M ⊗ C) =
T M
(7.1)
p+q=r
Vp,q
Vp
Vq
where
T ∗ M :=
(T ∗ M)1,0 ⊗C (T ∗ M)0,1 . Taking the spaces of
smooth sections, we get a decomposition of the complex-valued differential forms;
M
Ωr (M, C) =
Ωp,q (M).
p+q=r
If M is a complex manifold and z1 , . . . , P
zn are local complex coordinates, then Ωp,q (M) consists of the forms aI,J dzi1 ∧ . . . ∧ dzip ∧ dz j1 ∧
. . . ∧ dz jq where aI,J are smooth, complex-valued functions on M.
The Dolbeault operator ∂¯ : Ωp,q (M) −→ Ωp,q+1(M) is defined to be
the exterior differentiation d followed by the projection Ωp+q+1 (M, C) −→
Ωp,q+1 (M). Consider the sequence
∂¯
∂¯
∂¯
∂¯
. . . −→ Ω0,q−1 (M) −→ Ω0,q (M) −→ Ω0,q+1 (M) −→ . . . .
(7.2)
We would like this to be an elliptic complex; the definition of this is
explained nicely in [AB1, AB2].
7. THE ELLIPTIC OPERATORS
47
Indeed, (7.2) is a sequence of differential operators and its symbol
sequence is exact. It might not be a complex, though. Being a complex
means that ∂¯2 = 0. By [NN] this happens if and only if the almost
complex structure is integrable, i.e., comes from an honest complex
structure. In that case, (7.2) is indeed an elliptic complex, called the
Dolbeault complex.
Being an elliptic complex over a compact manifold implies that the
¯ are finite dimensional. One can then
cohomology groups H q (M, ∂)
P
¯ We
form the index, which is the alternating sum q (−1)q H q (M, ∂).
think of this asPa formal difference of vector spaces. It is determined
¯
by the integer q (−1)q dim H q (M, ∂).
If the almost complex structure is not integrable then we can still
can define a Dolbeault complex by “rolling up the operator”, into
¯ ∂¯∗
∂+
0 −→ Ω0,even (M) −→ Ω0,odd (M) −→ 0
(7.3)
where ∂¯∗ is the adjoint of ∂¯ with respect to some Hermitian metric.
¯ ∂¯∗ . This is an elliptic differential operator, regardless of
Denote D = ∂+
whether the almost complex structure is integrable. It follows that this
operator is Fredholm, i.e., its kernel and cokernel are finite dimensional.
So we can form its index; kerD−cokerD. In the complex case, this index
coincides with that of (7.2). In the almost complex case, this index
provides us with the desired “cohomology”. This technique was applied
in [G] to compute a certain Dolbault index by deforming the complex
structure into an almost complex structure, and then computing the
index of the rolled-up operator for the almost complex structure.
Strictly speaking, one should work with appropriate completions of
Ω0,q (M) which are Hilbert spaces. Nevertheless, the kernel and cokernel
will always be represented by smooth differential forms. We shall not
discuss the relevant analytic details.
By a theorem of J. Dieudonnee (1943), the index is a locally constant function from the space of Fredholm operators to Z. If follows
that the index of D is independent of the choice of Hermitian metric
used in defining this operator, because any two metrics can be joined
by an interval of metrics.
7.2. The twisted Dolbeault operator. We can get more interesting information if we twist the Dolbeaux operator by a line bundle.
First assume that M is a complex manifold and L −→ M is a holomorphic line bundle. Consider the differential forms with coefficients in L.
These split into a direct sum of the spaces Ωp,q (M, L) = Ωp,q (M)⊗Γ(L).
Here Γ(L) denotes the smooth sections of L and the tensor product is
48 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
formed over the ring of smooth functions. Locally we can take a holomorphic trivializing section for L. Then we can define an operator ∂¯
as before, and these fit together to an operator ∂¯L which is well defined
globally. We then form the twisted complex
∂¯
∂¯
∂¯
∂¯
L
L
L
L
. . . −→
Ω0,q−1 (M, L) −→
Ω0,q (M, L) −→
Ω0,q+1 (M, L) −→
...
and denote its cohomology groups by H q (M, ∂¯L ). This is also equal to
the sheaf cohomology H q (M, OL ) where OL is the sheaf of holomorphic sections of L. We denote either of these cohomology groups by
H q (M, L).
If we have an almost complex structure which is not integrable
then we cannot talk about holomorphic line bundles. In this case we
take L to be a smooth line bundle over M. We can still define an
operator ∂¯L , which now involves the choice of a connection on L. A
connection is a map ∇L : Γ(L) −→ Γ(T ∗ M ⊗L). It decomposes as ∇L =
∇′L + ∇′′L , where ∇′L and ∇′′L take values in Ω1,0 (M, L) and Ω0,1 (M, L)
respectively. We then define ∂¯L : Ωp,q (M, L) −→ Ωp,q+1 (M, L) by
¯ ⊗ s + (−1)q ϕ ∧ ∇′′ s.
∂¯L (ϕ ⊗ s) = (∂ϕ)
L
In the complex case, if the connection on L is chosen to be compatible
with the holomorphic structure then this coincides with the previous
definition.
In the almost complex case we must again roll up the complex into
one operator,
DL : Ω0,even (M, L) −→ Ω0,odd (M, L)
where DL = (∂¯ + ∂¯∗ )⊗1+1⊗∇′′L . As before, the operator DL is elliptic,
its index is independent of the choices of a connection and a Hermitian
metric, and in the holomorphic case this index coincides with the index
of ∂¯L .
c
7.3. Spin structure and the Dirac operator. As a reference
to the material in this section, see [LM].
7.3.1. Algebra. We begin by describing the group Spinc (m). We are
interested in the case m = 2n although most of what follows applies to
odd m as well.
The group Spin(2n) is a double covering of SO(2n);
p
Spin(2n) −→ SO(2n)
(7.4)
Spinc (2n) = Spin(2n) ×Z2 U(1)
(7.5)
with kernel = Z2 . For 2n > 2 this is the universal covering. The group
Spinc (2n) is a central extension of Spin(2n) given by
7. THE ELLIPTIC OPERATORS
49
where U(1) are the complex numbers of norm 1 and Z2 is the subgroup
{1, −1}. The map [A, λ] 7→ [p(A), λ2 ] defines a double covering
Spinc (2n) −→ SO(2n) × SO(2).
(7.6)
We identify both U(1) and SO(2) with the complex numbers of norm
1. We think of U(1) as a double covering of SO(2); this notation will
be convenient later.
We are interested in two real representations of Spin(2n), called
the spin representations and denoted ∆+ and ∆− . The center Z2 acts
nontrivially in these representations. These fit together with the scalar
±
action of U(1) to give representations of Spinc (2n) on ∆±
C = ∆ ⊗ C.
The group Spin(2n) acts on R2n via the action of SO(2n). There
exists a natural map
σ : R2n ⊗ ∆+ −→ ∆−
which is equivariant with respect to the actions of Spin(2n).
One way to obtain the above representations and maps is the following. Let T (2n) denote the full tensor algebra of R2n . Let I denote
the ideal generated by the elements v ⊗ v + ||v||2 · 1 for v ∈ R2n , where
|| · || is the standard norm. The Clifford algebra Cl(2n) is the quotient
T (2n)/I. The group Spin(2n) can be realized as a subgroup of the
group of units in Cl(2n). It acts on Cl(2n) on the left by the Clifford multiplication. Under this action, Cl(2n) splits into 2n copies of
∆+ ⊕ ∆− . The vector space R2n embeds into the Cl(2n), and the map
−
σ : R2n ⊗ ∆+
C −→ ∆C is obtained from the Clifford multiplication.
7.3.2. Topology. Now take a smooth manifold M of dimension 2n.
Take a Riemann metric and an orientation on M. The bundle of oriented orthogonal frames is a principal SO(2n)-bundle over M, denote
it PSO(2n) −→ M. A spin structure on M is a principal bundle PSpin(2n)
and a double covering PSpin(2n) −→ PSO(2n) which is equivariant with
respect to the homomorphism (7.4).
Many oriented manifolds do not admit spin structures. For example, CP2 is not spin. But many manifolds admit a spinc structure,
which we now define.
Take a smooth manifold M of dimension 2n. Fix a Riemann metric
and an orientation on M, and fix a complex line bundle L̃ −→ M with
a Hermitian metric. This data determines a principal bundle over M
with structure group SO(2n) × SO(2), denote it by PSO(2n)×SO(2) . A
spinc structure on M is the above data, together with a principal Spinc bundle, PSpinc (2n) , and a double covering PSpinc (2n) −→ PSO(2n)×SO(2)
which is equivariant with respect to the homomorphism (7.6).
50 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
c
Remark 7.7.
1. Any complex manifold has a spin
Vn structure,
for which L̃ is the determinant bundle; L̃p = C Tp M [LM,
appendix D].
2. Given a line bundle L −→ M and a spinc structure on M, we
can twist the structure by L to produce a new spinc structure,
in which L̃ is replaced by L̃ ⊗ L⊗2 .
3. Every spin manifold is also spinc , with L̃ being the trivial bundle.
The spinor bundles over M are the associated bundles; SC± = P ×G
where P = PSpinc (2n) and G = Spinc (2n). The Dirac spinc operator is an elliptic differential operator ∂\c : Γ(SC+ ) −→ Γ(SC− ) which is
constructed in the following way.
Recall that we have a map σ : R2n ⊗ ∆+ −→ ∆− . This gives for
every p ∈ M a map σp : Tp∗ M ⊗ (SC+ )p −→ (SC− )p . On the space of
sections we get a map σ̃ : Γ(T ∗ M ⊗ SC+ ) −→ Γ(SC− ). A connection on
PSpinc (2n) induces a connection on the associated bundle SC+ and thus a
∆±
C
∇
map Γ(SC+ ) −→ Γ(T ∗ M ⊗ SC+ ). The Dirac operator is the composition,
∇
σ̃
Γ(SC+ ) −→ Γ(T ∗ M ⊗ SC+ ) −→ Γ(SC− ).
8. Lifting actions
In section 7 we described an elliptic differential operator D : H+ −→
H− for each one of the extra structures on M which were listed in
section 5. In §8.1 we explain what we mean by a lifting of the torus
action to the extra structure. In §8.2 we explain how such a lifting
gives rise to actions of T on the spaces H+ and H− . The operator D
will be equivariant.
8.1. Definitions of our liftings. Take a manifold M of dimension 2n and take a torus T of dimension n which acts effectively on
M.
First, suppose that we have a principal bundle P −→ M with
structure group G. A lifting of the torus action to P is defined to be a
smooth left action of T on P which commutes with the principal right
action of G and such that the bundle map P −→ M is T -equivariant.
Second, a lifting of the torus action to a vector bundle E −→ M
is defined to be a smooth action of T on E which sends each fiber
linearly onto another fiber and such that the bundle map E −→ M is
T -equivariant.
Now suppose that M is a complex manifold of dimension n and
L −→ M is a holomorphic line bundle. A torus action on this structure is a holomorphic torus action on M together with a lifting to a
holomorphic action on L.
8. LIFTING ACTIONS
51
Next, suppose that M has an almost complex structure and that we
have a smooth line bundle L −→ M. A torus action on this structure
is a smooth torus action on M which preserves the almost complex
structure, and a smooth lifting of the action to L.
Finally, suppose that M has a spinc structure. We now follow
closely the notation of §7.3.2. A torus action on the spinc structure
consists of a lifting of the torus action on M to an action on the bundle
PSpinc (2n) with the following properties. Note that such a lifting induces
actions of T on PSO(2n) and on L̃. We require the T -action on M to
preserve the Riemann metric and the action on PSO(2n) to be that which
is induced from the action on M; and we require the action on L̃ to
preserve the Hermitian metric.
8.2. Our equivariant index. We now describe how the above
torus actions induce an action of the torus on the vector spaces H± .
First assume that we have an almost complex structure on M and
a smooth line bundle L. The case of a holomorphic line bundle over a
complex manifold is a special case of this, as was explained in sections
7.1 and 7.2. We then have H+ = Ω0,even (M, L) and H− = Ω0,odd (M, L).
The T -action on M induces an action on the space of smooth functions
by
(λ · f )(p) = f (λ−1 · p).
(8.1)
Similarly, λ ∈ T acts on differential forms by pulling-back via the
action of λ−1 . This preserves the spaces Ωp,q (M) because the T -action
preserves the almost complex structure on M. Next, the T -action on
L induces an action on the space of sections of L by
(λ · s)(p) = λ(s(λ−1 p)).
(8.2)
These two actions fit together to an action on the space of twisted
forms, i.e., on Ωp,q (M, L) = Ωp,q (M) ⊗ Γ(L), and thus to actions of T
on the spaces H± .
In constructing the operator DL we used a Riemann metric on M,
a Hermitian structure on L and a connection on L. If these are chosen
to be T -equivariant (which we can alway arrange by averaging) then
the operator DL is equivariant.
Now assume that we are given a spinc structure on M on which
T acts, as in §8.1. The action of T on PSpinc (2n) induces an action
on every associated bundle. In particular we get actions of T on the
spinor bundles SC± and on their spaces of sections, H± . To construct
the operator ∂\c we used a connection on PSpinc (2n) . If this connection is
chosen to be T -equivariant then the operator ∂\c will be T -equivariant.
52 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
9. The Lefschetz formula of Atiyah and Bott
In section 7 we described an elliptic differential operator D : H+ −→
H− for each one of the extra structures on M which were listed in
section 5. In section 8 we described actions of T on the spaces H+
and H− . The operator D is T -equivariant, so its kernel and cokernel
are finite dimensional representations of T . We can then form the
equivariant index of D as was defined in section 4. In this section
we compute its character using the localization formula of Atiyah and
Bott.
9.1. Statement of the formula. Let E + , E − be complex vector
bundles over M. Let H± = Γ(E ± ) be the spaces of smooth sections.
D
Let H+ −→ H− be an elliptic differential operator. Let f : M −→ M
be a diffeomorphism and let f + : E + −→ E + and f − : E − −→ E −
be liftings of f . This means that they map the fiber over p ∈ M
isomorphically to the fiber over f (p) ∈ M. Then we get mappings
on the spaces of sections; f˜+ : H+ −→ H+ is given by (f˜+ · s)(p) =
f + (s(f −1 (p))), and similarly for f˜− . We assume that the operator D is
compatible with the liftings of f ; this means that D◦ f˜+ = f˜− ◦D. Then
f˜± induce maps A+ : kerD −→ kerD and A− : cokerD −→ cokerD.
The Lefschetz number of f (and of the lifting) is defined to be
L(f ) = trace (A+ ) − trace (A− ).
The formula of Atiyah and Bott computes this complex number.
Since f is a diffeomorphism it has an inverse, f −1 . At every fixed
point p of f , the map 1−(dfp )−1 sends Tp M to itself, so its determinant
is well defined. We assume that the graph of f is transversal to the
diagonal in M × M, then det(1 − (dfp )−1 ) is nonzero. Moreover, fp±
sends the fiber Ep± to itself, thus its trace is well defined.
Theorem 3.2 (Atiyah and Bott). Given a diffeomorphism f and
a lifting as described above, its Lefschetz number is given by the formula
X
L(f ) =
ν(p)
p
where we sum over the fixed points p of f in M and where
ν(p) =
trace C fp+ − trace C fp−
.
| det(1 − dfp−1 )|
(9.1)
Remark 9.2. The original theorem in [AB1] applies in a more
general setting, where f is not required to have an inverse. The formula
(9.1) is a special case.
9. THE LEFSCHETZ FORMULA OF ATIYAH AND BOTT
53
Suppose that T acts on M and lifts to E ± . Then for every λ ∈ T
we get a diffeomorphism f = fλ . The character of the equivariant
index is χD (λ) = L(fλ ). If T acts with isolated fixed points, then for
almost every λ, the graph of fλ is transversal to the diagonal in M ×M.
Theorem 3.2 gives
X
χD (λ) =
ν(p)(λ).
p
It is left to compute the terms ν(p) for our particular differential
operators; the Dolbeault operator D = DL and the Dirac operator
D = ∂\c . These are computations in linear algebra which we will now
describe.
9.2. Computation of ν(p) for the Dolbeault operator. Assume that M is a manifold with an almost complex structure J. Let
f : M −→ M be a diffeomorphism which preserves J and whose graph
is transversal to the diagonal in M × M. Suppose that we have a line
bundle L −→ M and a lifting f L : L −→ L of f . We can form the
twisted Dolbeault operator DL as in §7.2. Let H± be the even/odd
differential forms in Ω0,q (M, L). The lifting of f to L induces maps
f˜± : H± −→ H± , as explained in §8.2. We can then form the Lefschetz
number L(f ) and apply Theorem 3.2.
Proposition 9.3 (Atiyah and Bott, [AB2]). The contribution of
the fixed point p is
ν(p) =
fpL
.
detC (1 − dfp−1 )
(9.4)
Remark 9.5. Since p is a fixed point, f L acts on the fiber Lp as
multiplication by a complex number; this is the numerator fpL . The
denominator makes sense because the almost complex structure makes
Tp M into a complex vector space.
Proof. First recall that we have a decomposition
T ∗ M ⊗ C = T ′ M ⊕ T ′′ M
(9.6)
∗
where T ′ M =V(T ∗ M)1,0 and T ′′ M = (TV
M)0,1 . We have H± = Γ(E ± )
where E + = even T ′′ M ⊗ L and E − = odd T ′′ M ⊗ L.
The lifting of f to T ∗ M is (df −1 )∗ : T ∗ M −→ T ∗ M. Because f
preserves J, the map (df −1)∗ ⊗ 1 preserves the decomposition (9.6). So
we
(df −1 )∗ ⊗ 1 =V(df −1 )′ + (df −1 )′′ and we can form the maps
Vq can−1write
V
(df )′′ : q T ′′ M −→ q T ′′ M. The lifting of f to E + is given by
^
f + = even (df −1)′′ ⊗ f L ,
54 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
and similarly for f − .
The numerator of ν(p) for the rolled-up complex is trace C fp+ −
P
V
trace C fp− , which in our case is equal to q (−1)q trace ( q (dfp−1)′′ ⊗fpL ).
To obtain (9.4) we will need the following observations from linear
algebra .
Denote A = dfp−1. This is a linear operator on a complex vector
space V = Tp M. On V ⊗R C = V ′ ⊕ V ′′ we have A ⊗ 1 = A′ ⊕ A′′ .
V
V
′′
The operator
operators k A′′ on k V ′′ . These satisfy the
P A qinduces V
equality q (−1) trace C ( q A′′ ) = detC (I − A′′ ), this can be seen by
putting A in a triangular form. Moreover, detR (I − A) = detC ((I −
A) ⊗ 1) = detC (I − A′ ) · detC (I − A′′ ). Also, detC (I − A′ ) = detC (I − A)
when in the right
vector space. Finally,
Vterm we view V as a complex
V
note that trace ( q (dfp−1)′′ ⊗ fpL ) = trace ( q (dfp−1 )′′ ) · fpL . Putting all
these facts together, we get the desired expression for ν(p).
Corollary 9.7. Suppose that we have a torus action on M and
on the extra structure (i) or (ii) as defined in section 8. Then for each
λ ∈ T we get a diffeomorphism fλ of M and a lifting fλL . Suppose that
p is a fixed point, suppose that T acts on Tp M (viewed as a complex
vector space) with weights α1 , . . . , αn and on the fiber Lp with a weight
µ. Then as a function of λ, the contribution of ν(p) is given by
λµ
.
−αi )
i=1 (1 − λ
ν(p)(λ) = Qn
(9.8)
Proof. This follows from (9.4) when we write d(fλ )p as a diagonal
matrix with entries λα1 , . . . , λαn and when we write fλL = λµ .
9.3. Computation of ν(p) for the Dirac operator. We first
discuss the maximal tori in the relevant groups, and their weight lattices. Denote the standard maximal tori in the groups Spinc (2n),
Spin(2n) and SO(2n) by TSpinc (2n) , TSpin(2n) and TSO(2n) respectively.
The map (7.4) restricts to a double covering TSpin(2n) −→ TSO(2n) . From
(7.5) we get TSpinc (2n) = TSpin(2n) ×Z2 U(1), and (7.6) restricts to a double
covering TSpinc (2n) −→ TSO(2n) × SO(2).
Write TSO(2n) = Rn /Zn and let x1 , . . . , xn be the standard basis for
its weight lattice. The double covering TSpin(2n) −→ TSO(2n) induces an
∼
=
isomorphism t∗SO(2n) −→ t∗Spin(2n) . With this identification, the weight
lattice of Spin(2n) is generated by x1 , . . . , xn and 12 (x1 + . . . + xn ).
Let y be the standard generator for the weight lattice of SO(2) and
remember that the map U(1) −→ SO(2) is a double covering, then
the weight lattice of U(1) is generated by y/2. Finally, the weight
9. THE LEFSCHETZ FORMULA OF ATIYAH AND BOTT
55
lattice of TSpinc (2n) = TSpin(2n) ×Z2 U(1) is generated by x1 , . . . , xn , y,
and 12 (x1 + . . . + xn + y).
Remember, we have H± = Γ(E ± ) where E ± = PSpinc (2n) ×Spinc (2n)
±
c
∆±
C , and ∆C are the spin representations of Spin (2n).
Let p be a fixed point in M. Fix an identification of the fiber of
PSpinc (2n) over p with the group Spinc (2n). This determines an identification of the fiber Ep± with the space ∆±
C and an identification of
2n
±
Tp M with R . The action of T on Ep is then given by an inclusion
ip : T ֒→ Spinc (2n) followed by the left action of Spinc (2n) on ∆±
C.
Similarly, the isotropy action of T on Tp M is given by an inclusion
T ֒→ SO(2n) followed by the left action of SO(2n) on R2n . We can
assume that the images of T lie in the standard maximal tori TSpinc (2n)
and TSO(2n) . Also, the action of T on L̃p defines a map T −→ SO(2).
All these maps fit into a commuting diagram,
ip
T ֒→
m
T ֒→
TSpinc (2n)
֒→
↓ double cover
TSO(2n) × SO(2) ֒→
Spinc (2n)
↓ double cover
SO(2n) × SO(2)
Let ∆±
C = W1 ⊕ . . . ⊕ WM be the weight decomposition with respect
to the actions of TSpin(2n) . Then M = 2n−1 and the weights are αj =
1
(±x1 ± x2 ± . . . ± xn ). For the Wj ’s occurring in ∆+
C there is an even
2
numbers of plus signs in αj , and for those occurring in ∆−
C there is an
odd numbers of plus signs.
The standard action of U(1) on C is given by the weight y/2. Thus
TSpinc (2n) acts on Wj with a weight αj + y/2. Let λ ∈ T and let λp =
ip (λ) be its image in TSpinc (2n) . Then the numerator of ν(p)(λ) is
X
=
1
λp2
(±x1 ±x2 ±...±xn +y)
even number of
plus signs
n
Y
y/2
i /2
(λ−x
λp
p
i=1
−
X
1
λp2
(±x1 ±x2 ±...±xn +y)
odd number of
plus signs
− λxp i /2 ).
On the other hand, the action on Tp M ⊗ C splits into 2n weight
spaces with weights ±xi . Let A = 1 − dfλ−1 . Since detR (A)|
QnTp M =
detC (A ⊗ 1)|Tp M ⊗C , we get that the denominator of ν(p)(λ) is i=1 (1 −
i
λxp i )(1 − λ−x
p ).
Now, suppose that T acts on Tp M with weights ±α1 , . . . , ±αn and
that T acts on the fiber L̃ with a weight µ, so that αi = i∗p xi and
56 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
µ = i∗p y. Then
λµ/2
ν(p)(λ) = Qn
Qn
i=1
−αi /2
i=1 (λ
(1 − λαi )(1
− λαi /2 )
.
− λ−αi )
(9.9)
Note that this formula only depends on the pointwise actions of T on
Tp M and on L̃p . Also, note that the αi ’s are only determined up to
sign. The orientation of M restricts this ambiguity to only allow an
even number of simultaneous sign changes. The formula (9.9) does not
change under such sign changes.
10. Expanding the terms
P
In this section we expand χD (λ) = p ν(p)(λ) as a Laurent polynomial in λ. The exponents of this polynomials are weights in t∗ , and
their coefficients will give our desired multiplicity function for the index.
The problematic thing is that although the index is a polynomial,
each ν(p) is not a polynomial, and can be written as an infinite power
series in more than one way. We want to expand (9.8) or (9.9) into
power series in a consistent manner for the various fixed points p so that
the sum over p will give the polynomial χD (λ). In the following subsections we describe a recipe for the expansion. This recipe is directly
adapted from the formula of Guillemin-Lerman-Sternberg described in
Theorem 2.2.
10.1. Examples.
Example 10.1. For n = 1, we can expand equation (9.8) as
λµ
= λµ (1 + λ−α + λ−2α + . . . )
1 − λ−α
and obtain the multiplicity diagram
µ−2α
µ−α
µ
µ+α
µ+2α
(10.2)
µ+3α
where the circles denote multiplicity 1. But we could also write
α
λµ
µ −λ
=
λ
(
) = λµ (−λα − λ2α − λ3α − . . . )
−α
α
1−λ
1−λ
and obtain
µ−2α
µ−α
µ
µ+α
µ+2α
µ+3α
(10.3)
10. EXPANDING THE TERMS
57
where the squares denote multiplicity −1. Note that in the second
picture, the ray flips direction and the boundary weight µ is erased.
Example 10.4. Let us consider the action of S 1 on M = CP1 by
rotations, λ · [z, w] = [z, λw], and let L be the tangent bundle of CP1 ;
this is a complex line bundle to which the action lifts naturally. There
are two fixed points; the North pole pN = [0, 1] and the South pole
pS = [1, 0]. The isotropy weights at the poles are αN = 1 and αS = −1
respectively. These are also the weights for the action on Lp because
Lp = Tp M, so we have µN = 1 and µS = −1. Thus ν(pN )(λ) = 1−λλ −1
λ−1
. If we expand ν(pN ) as in (10.2) and ν(pS ) as in
and ν(pS )(λ) = 1−λ
(10.3) then we get the two diagrams:
−3
−3
−2
−2
−1
−1
0
0
1
1
2
2
where circles denote multiplicity 1 and squares denote multiplicity −1.
These combine to give the multiplicity diagram for the index,
−3
−2
−1
0
1
2
(10.5)
We could also expand ν(pN ) as in (10.3) and ν(pS ) as in (10.2) and get
the same final answer. But any other choice of expansions would give
an infinite power series which doesn’t converge anywhere.
Example 10.6. Again let us consider the action of S 1 on M = CP1
with the line bundle L = T M as above. This time let us think of M
as a spinc manifold. The line bundle L̃ which V
is associated to the spinc
structure is then L̃ = det ⊗L⊗2 where det = nT M. For CP1 we get
L̃ = (T M)⊗3 . The circle acts on the fiber L̃pN with a weight µN = 3
and on the fiber L̃pS with a weight µS = −3. The isotropy weights are
as before; αN = 1 and αS = −1. From (9.9) we get
χ(λ) = ν(pN )(λ) + ν(pS )(λ)
λ3/2 (λ−1/2 − λ1/2 ) λ−3/2 (λ1/2 − λ−1/2 )
=
+
(1 − λ)(1 − λ−1 )
(1 − λ−1 )(1 − λ)
= λ + 1 + λ−1 .
The exponents which occur in this polynomial form the same interval
as in (10.5).
58 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
Now let us look at the symplectic picture. Let ω̃ be the curvature
of L̃ and let Φ̃ be the corresponding moment map, which is determined
by the lifting of the action to L̃ as in section 1.5.
Let ω = 12 ω̃ and let Φ = 21 Φ̃. Consider the density function ρ(α)
for the D-H measure corresponding to ω. The formula in Theorem 2.2
expresses it as a sum of two terms, corresponding to the fixed points
p = pN and p = pS . Each term will be a ray with vertex Φ(p). From
section 1.5 we get that at a fixed point p, the value of the moment map
Φ̃ is equal to the weight µp . For ω itself we get half of that weight, so
Φ(pN ) = 3/2 and Φ(pS ) = −3/2. Thus the symplectic picture is
−2
−1
0
1
2
(10.7)
The weights in (10.5) are exactly those integral weights which occur in
(10.7). There is no question on whether or not we should include the
boundary weights; the boundary of the interval (10.7) does not hit any
weight.
10.2. The case of an almost complex structure. By (9.8) we
have
n
Y
1
µ
ν(p)(λ) = λ
.
1 − λ−αi
i=1
We can expand each factor either as in (10.2) or as in (10.3). We pick
a choice of signs, ǫi = ±1 for i = 1, . . . , n, and expand the ith factor
in the direction of αi′ = −ǫi αi . If ǫi = 1 for all i then we get
n
Y
ν(p)(λ) = λ
(1 + λ−αi + λ−2αi + . . . ).
µ
i=1
In general, the exponents which occur form a polyhedral cone in ℓ∗ ,
spanned
rays in the directions of αi′ , and the coefficients are equal
Qby
n
to ǫ = i=1 ǫi = ±1. P
Denote Cp = {µ + i ri αi′ | ri ≥ 0}. For α ∈ ℓ∗ let mα be the
coefficient of λα in the expansion, then mα = ǫ for α ∈ interior(Cp )
and mα = 0 for α 6∈ Cp . Among the weights on the boundary of Cp ,
some occur in the expansion and some don’t.
As illustrated in Example 10.4, for different p’s we should expand all
the ν(p)’s in the same direction. Denote by α1p , . . . , αnp the isotropy
weights at p. As in Theorem 2.2, we can choose β ∈ t such that
hβ, αip i =
6 0 for all i and p. Then, in ν(p) we expand the ith factor in
′
′
′
the direction of αip
where αip
= ±αip is chosen such that hαip
, βi > 0.
10. EXPANDING THE TERMS
59
It is left to prove that this recipe gives the character of the in′
dex. Indeed, if λ = exp(iβ) ∈ TC and hβ, αip
i > 0 for all i and p
α′i
2πihiβ,α′i i
then all our power series will
;
P converge at λ (because λ = e
see section 12). Therefore p ν(p)(λ) will converge for every such λ.
χ
The character
P D , being a polynomial, clearly converges for such λ.
χD (λ) for all such λ. Since this holds for
Therefore
p ν(p)(λ) =
a nonempty open set of λ’s, it follows that we also have an equality
P
χD (λ) as formal power series in λ. In particular, the
p ν(p)(λ) =
multiplicity function for the index is given by the sum over p of the
multiplicity functions for ν(p). This is what we wanted.
10.3. The case of a spinc structure. If n = 1 then, by (9.9),
we have
λµ/2 (λ−α/2 − λα/2 )
ν(p)(λ) =
(1 − λα )(1 − λ−α )
λ(µ−α)/2 (1 − λα )
= λ(µ−α)/2 (1 + λ−α + λ−2α + .(10.8)
..)
=
α
−α
(1 − λ )(1 − λ )
λ(µ+α)/2 (λ−α − 1)
= λ(µ+α)/2 (−1 − λα − λ2α − . .(10.9)
.)
=
(1 − λα )(1 − λ−α )
For n > 1 we choose ǫi,p = ±1 for all i, p as in section 10.2, and we
expand the ith factor of ν(p) by (10.8) or (10.9) according to whether
ǫi,p = 1 or = −1. The same argument as in section 10.2 shows that the
χ
sum over p of the power series is equal
Q to the polynomial D (λ).
′
Let αi = P
−ǫi,p αi,p and let ǫ = i ǫi,p as in section 10.2. Denote
Cp = {µ/2 + i ri αi′ | ri ≥ 0}. For α ∈ ℓ∗ , let mα be the coefficient of
λα in the expansion of ν(p). Again, mα = ǫ for α ∈ Cp and mα = 0 for
α 6∈ Cp . We claim that the boundary of Cp contains no integral points.
Remembering that αi′ = ±αi , this claim will follow from:
P
Lemma 10.10. α = µ/2 + ri αi is in ℓ∗ if and only if every ri is
equal to ni + 12 for some integer ni .
Proof. Recall, the weight lattice of TSpinc (2n) is generated over Z by
x1 , . . . , xn , y, and 21 (x1 +. . .+xn +y). Also recall that µ, α1 , . . . , αn were
the pull-backs of y, x1, . . . , xn under an inclusion ip : T ֒→ TSpinc (2n) . It
follows that the weight lattice of T is generated over Z by α1 , . . . , αn ,
µ, and 21 (α1 + . . . + αn + µ).
We note that the αi ’s are only determined up to sign. The orientation of M reduces this ambiguity so that the αi ’s are determined up
to a sign change of an even number of them. Note that ν(p) flips sign
when we flip the sign of an αi , so that ν(p) is well defined.
60 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
10.4. Proof of Theorem 3.1. Now consider the symplectic picture. In the spinc case, let µp be the weight for the action of T on
the fiber L̃p over a fixed point p. Let ω be half the curvature of L̃,
then the moment map sends p to Φ(p) = µp /2. In the almost complex
case with a line bundle L, let µp be the weight for the action of T on
Lp . Let ω be the curvature of L, then the moment map sends p to
Φ(p) = µp . In both cases, the polyhedral cone Cp on which the weights
are supported has a vertex Φ(p) and rays generated by αi′ . This is the
very same cone as that which occurs in the formula in Theorem 2.2.
(The recipe which determines the αi′ s in the formula for the index was
directly copied from the one in Theorem 2.2). Additionally, all the
weights of this cone occur with a multiplicity ǫ = ±1, according to the
parity of the number of i’s for which αi′ = −αi . This number ǫ is equal
to the density function on the p’th cone in Theorem 2.2. Therefore, as
we sum over the fixed p’s, we get that mult (α) = ρ(α) for every weight
α ∈ ℓ∗ , where mult (α) is the coefficient of λα in the polynomial χD (λ),
and when α does not lie on the boundary of the cones Cp .
11. Examples
11.1. Toric manifolds. In [KT], Sue Tolman and myself proved
Theorem 3.1 for toric manifolds, i.e., smooth toric varieties. These are
complex manifolds which have holomorphic line bundles. Our proof
was different than the proof given in this chapter. In particular, the
computation of the equivariant index was direct and did not use the
Atiyah-Bott formula (9.1).
See [KT] for the basic definition and construction of toric manifolds.
We worked with a construction that we had learned from M. Audin,
which is in the context of complex manifolds, is different from the
standard algebraic construction, and is in the spirit of a construction
of T. Delzant.
Toric manifolds are the only examples that I know for simply connected compact complex manifolds with a holomorphic line bundle to
which the torus action lifts. It is known that every smooth complex
algebraic variety of complex dimension n with an effective action of
an n dimensional complex torus is a toric variety. I suspect that this
should be true in the context of complex manifolds, and not just in the
algebraic context, although I do not know a proof.
11.2. Completely integrable spaces with almost complex
structures. The only other examples which I know for compact manifolds with a completely integrable torus action and with fixed points
11. EXAMPLES
61
are constructed from toric manifolds by means of equivariant connected
sum, as described in example 2.6.7.
Recall, given two toric varieties (Mi , T ), i = 1, 2 of complex dimension n, we get a space M = M1 ♥M2 . For n = 2, M. Grossberg and I
have shown that the resulting (M, T ) admits an invariant almost complex structure. Thus this provides us with an example of a manifold
with the extra structure (ii) which is not a toric manifold. We now
decribe this construction.
To construct an invariant almost complex structure on such M, we
need to define a linear operator Jp : Tp M −→ Tp M for every p ∈ M,
which makes Tp M into a complex vector space, i.e., Jp2 = −identity.
J is required to depend smoothly on p but there might not be local
holomorphic coordinate systems.
Fix an invariant Riemann metric and an orientation on M, then
we have a principal SO(4)-bundle, P , whose fiber over p ∈ M is the
set of orthonormal, oriented bases of Tp M. The subgroup U(2) ⊆
SO(4) acts on the fibers on the right by the restriction of the principal
action. The choice of an almost complex structure amounts to choosing
a global section of Q = P/U(2). This is a bundle over M with a fiber
SO(4)/U(2) and T acts on it on the left. Choosing an invariant almost
complex structure on M amounts to choosing a section of Q/T over
M/T . Over the smooth points of M/T , the quotient Q/T is a bundle
with fiber SO(4)/U(2).
Recall, M was obtained from M1 and M2 by removing neighborhoods U1 , U2 of a free orbit in each of M1 and M2 and then gluing the
manifolds M1 r U1 and M2 r U2 along their boundary.
The ordinary complex structure on CP 2 gives invariant sections of
Q over M1 rU1 and M2 rU2 . In M, a neighborhood of the gluing locus
is a tube of the form T × S 1 × I where I is an interval. So the complex
structure on M gives a section of the bundle Q/T which is defined on
a neighborhood of the boundary of the tube S 1 × I in M/T .
All we need is to extend this section across the rest of the tube in
M/T . This is a question of a homotopical nature. Over the tube, Q/T
is a trivial bundle with fiber SO(4)/U(2), which is topologically S 2 . (A
trivialization comes, for example, from viewing this tube as punctured
disc in M1 /T , whereas the bundle Q/T is defined and is trivial over the
whole disc). The boundaries of the tube are closed loops and the fiber
of Q/T is topologically S 2 . Because every loop in S 2 is homotopically
trivial, it follows that the sections can indeed be extended.
11.3. Completely integrable manifolds with spinc structures. M. Grossberg and I have shown that if (Mi , T ) have equivariant
62 3. INDEX THEORY FOR COMPLETELY INTEGRABLE SPINc MANIFOLDS
spinc structures then M1 ♥M2 also has an equivariant spinc structure.
For n = 3, if M1 and M2 are complex manifolds, we showed that the
ordinary complex structures do not extend to a T -invariant almost
complex structure on (M, T ). We suspect that such (M, T ) might not
admit any invariant almost complex structure. If this is true, then this
would provide an example for a manifold which admits an equivariant
spinc -structure but not an equivariant almost complex structure.
CHAPTER 4
Periodic Hamiltonian flows on four dimensional
manifolds
1. Introduction
Let (M, ω) be a compact four dimensional symplectic manifold and
let Φ : M −→ R be a smooth function with isolated critical points.
The corresponding Hamiltonian flow is generated by the vector field
ξM such that
dΦ = −ι(ξM )ω.
(1.1)
We assume that this flow is periodic, i.e., that ξM generates a circle
action on M. In the notation of Chapter 1 we then have a Hamiltonian
space (M, S 1 , ω, Φ) and Φ is the moment map. The action of S 1 has
isolated fixed points.
M. Audin [Au1],[Au2, Ch.IV] and, independently, K. Ahara and
A. Hattori [AH] have described a small number of minimal models such that for every compact four dimensional Hamiltonian space
(M, S 1 , ω, Φ), the underlying S 1 -space (M, S 1 ) is obtained from a minimal model by a sequence of blow-ups. One may further ask, which
blow-ups give rise to the same manifold, and which forms ω can we put
on a given space (M, S 1 ).
In this chapter we extend their result to a classification of the
Hamiltonian spaces (M, S 1 , ω, Φ) up to isomorphism, i.e., up to an
equivariant symplectomorphism which preserves Φ. This is part of a
paper in preparation [K], which will also account for the cases of nonisolated fixed points. For the rest of this chapter we assume that S 1
acts with isolated fixed points.
Our main result is that (M, S 1 , ω) is obtained from a symplectic toric manifold (M, T 2 , ω) by restricting the action to a sub-circle
S 1 ⊂ T 2 . In particular this means that (M, S 1 , ω) admits a compatible
Kähler structure and the S 1 action extends to a T 2 action. Additionally, (M, S 1 , ω) is determined up to isomorphism by (M, S 1 , [ω]), where
[ω] is the cohomology class of ω in H 2 (M, R).
Many of the ideas presented here are taken from [Au1], [Au2,
Ch.IV] and [AH]. We give more detailed credits within the text.
63
644. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
This work is motivated by the desire to understand Hamiltonian
torus actions; (M, T, ω, Φ). If the action is effective then dim T ≤
1/2 dim M. Delzant [De] gave a complete classification of the Hamiltonian spaces in which dim T = 1/2 dim M. All of these are symplectic
toric varieties. They are parametrized by a certain collection of convex
polyhedra in Rn , the bijection being by associating to (M, T, ω, Φ) the
image of the moment map Φ. If dim T < 1/2 dim M then many questions are left unanswered. We assume that T acts effectively and with
isolated fixed points. One question is; when does the T action extend
to an action of a torus T̃ with dim T̃ = 1/2 dim M? In some examples
one can extend the action after performing some blow-ups of M. Is this
true in general? Another question is: does (M, ω) admit a compatible
Kähler structure which is T -invariant? The author is not aware of any
counterexamples.
In section 2 we construct a labeled graph from the data (M, S 1 , Φ).
In section 4 we show that every such graph coincides with a graph
which is associated to a smooth toric variety. In section 5 we show
that the labeled graph determines (M, S 1 , Φ) up to an equivariant
diffeomorphism that preserves Φ. In section 6 we show that the labeled graph determines (M, S 1 , ω) up to an equivariant symplectomorphism. This together with section 4 implies that every Hamiltonian
space (M, S 1 , ω, Φ) with isolated fixed points is isomorphic to a toric
variety. In section 7 we state the precise classification of Hamiltonian
S 1 -spaces with isolated fixed points.
Acknowledgement. I wish to thank the following people for various useful discussions; M. Audin, V. Ginzburg, M. Grossberg, V.
Guillemin, D. McDuff, R. Sjamaar, S. Sternberg, S. Tolman. I wish
to thank the Weizmann Institute of science for their hospitality and
support during the summer of 1992.
2. Constructing a graph
We start with a Hamiltonian space (M, S 1 , ω, Φ) as in the introduction. In this section we will construct a graph with integer and
real labels. The integer labels will depend on the manifold and action.
The real labels are the values of Φ at the fixed points. Up to a global
constant, these depend only on the cohomology class [ω] in H 2 (M, R).
Our graph is essentially the same as that of M Audin [Au1, 2.1],
[Au2, ch.IV, prop.5.3.1]. Our construction is different in that it does
not use a Riemann metric, the roles of the vertices and the edges are
reversed, and we add extra real labels. Our integer labels are different
but then contain the same information as Audin’s.
2. CONSTRUCTING A GRAPH
65
Let p be a fixed point. By the local normal form [GS2, §4] (see
Examples 1.2.1(i) and 1.9.15(i)), there are complex coordinates (z, w) :
M −→ C2 defined near p such that
(i) The action is λ · (z, w) = (λm z, λn w) for (z, w) ∈ C2 , λ ∈ C, |λ| = 1.
(ii) The symplectic form is ω = 2i1 (dz ∧ dz + dw ∧ dw)
(iii) The moment map is Φ(z, w) = Φ(p) − m2 |z|2 − n2 |w|2.
(2.1)
The integers m, n are called the isotropy weights at p. They are relatively prime because the action is effective (on M, and thus effective
on a neighborhood of p).
w
n
m
z
The fixed points of the action are exactly the critical points of Φ
by (1.1) and the nondegeneracy of ω. There are exactly one maximum
pmax and one minimum pmin for Φ, and no other local extrema; see
[GS2, lemma 5.1] or [A, theorem 1]. By (iii), the isotropy weights
are negative at the maximum and are positive at the minimum; at a
nonextremal fixed point there is one positive and one negative weight.
Such a point will be called an interior fixed point.
Now take a point p ∈ M which is not fixed by the action. Then its
stabilizer is a cyclic group of order k; denote it Zk = {λ ∈ C | λk =
1} ⊆ S 1 . Then an invariant neighborhood of p is isomorphic to S 1 ×Zk
(I × D 2 ) where D 2 is a disc in C and I is an interval around zero. Let
a be in S 1 , let h be a coordinate on I, and z be the complex coordinate
on D 2 . In this model we have [aλ, h, z] = [a, h, λl z] for all λ ∈ Zk ; for
some 0 ≤ l < k. See Examples 1.2.1(ii) and 1.9.15(ii).
(i) The action of S 1 is λ · [a, h, z] = [λa, h, z].
(ii) The symplectic form is ω = dh ∧ dθ + 2i1 dz ∧ dz where a = e2πiθ .
(iii) The moment map is Φ[a, h, z] = Φ(p) + h.
(2.2)
664. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
Definition 2.3. A Zk -sphere is the closure of a component of the
fixed point set of Zk in M, for k ≥ 2.
Lemma 2.4. Let k ≥ 2 and let N be a component of the fixed point
set of Zk in M. Then
1. There are two fixed points p, q ∈ M such that the closure of N is
N = N ∪ {p, q}.
This is a Zk -sphere by the above definition. We denote it by
Npq .
2. (Npq , ω, S 1) is isomorphic to a sphere S 2 with S 1 acting by rotations k times, ω is a constant multiple of the area form, and p, q
are the North and South poles. Φ is linear function of the height.
k is an isotropy weight at p and −k is an isotropy weight at q.
p
k times
q
Proof of (1). The key ingredient is the local normal form (2.1)
and (2.2). If s is a fixed point then by (2.1), its neighborhood intersects
N only if one of the isotropy weights is k. Then, in a neighborhood of
s, N = N ∪ {s} is a coordinate plane.
If the stabilizer of s is ZK then by (2.2), a neighborhood of s =
[1, 0, 0] intersects N only if K = k, and then N = N = {z = 0} near s.
From the above two models we see that the boundary of N consists
of isolated fixed points and that N /S 1 is a one dimensional manifold
whose boundary are those fixed points. By connectedness and compactness, N/S 1 is a closed interval, so it has exactly two boundary
points, say, p and q. Then N = N ∪ {p, q} as desired.
Proof of (2). Let I be the closed interval [Φ(q), Φ(p)]. There is
a curve γ : I −→ N such that Φ ◦ γ = id, this can be constructed with
the help of (2.1) and (2.2). Let S 2 be the unit sphere in R3 and let
S 1 act on S 2 by rotation k times. Let L : [−1, 1] −→ I be an affine
bijection, then the desired isomorphism of Npq with S 2 is given by
√
√
e2πiθ · γ(L(t)) 7→ ( 1 − t2 cos(2πkθ), 1 − t2 sin(2πkθ), t).
2. CONSTRUCTING A GRAPH
67
2.5. The graph. We now associate with (M, S 1 , ω) a labeled graph.
To each fixed point p we assign a vertex hpi, labeled by the real number
Φ(p). To each Zk -sphere Npq we assign an edge hpqi, labeled by the
integer k.
We add the following edges, labeled by the integer 1. Let pmax , pmin
be the extrema of Φ. Let p be a fixed point with isotropy weights
{m, −n} where m, n > 0. If m = 1 then we draw an edge hpmax pi; if
n = 1 then we draw an edge hp pmini. In §3.6 we will see that these
edges also correspond to spheres inside M, which depend on the choice
of a Riemann metric.
Remark 2.6. Up to a global shift, the Φ(p)’s only depend on the
cohomology class of ω. Indeed, let p, q be fixed points. Let I be the
interval 0 ≤ t ≤ 1 and let γ : I −→ M be a smooth curve with γ(0) = q
2πis
and γ(1) = p. Define γ̃ : I × I −→
· γ(t), then
R M∗ by γ̃(s, t) = e
one can check that Φ(p) − Φ(q) = I×I γ̃ ω, which only depends on the
cohomology class of ω.
R In particular, if Npq is a Zk -sphere, then this
integral is equal to k Npq ω.
If p is a non-extremal fixed point p then hpi lies on exactly two
edges, hqpi and hpq ′ i. Thus the graph consists of several branches,
each starting from the top vertex and ending at the bottom.
Example 2.7. Let M = S 2 ×S 2 and let S 1 act diagonally, rotating
the second factor twice as fast as the first factor. Take the symplectic
structure to be the sum of the area forms on the two factors, normalized
so that the area of S 2 is 2. Denote by N and S the North and South
poles of S 2 . Let h : S 2 −→ R be the height function, with h(N) = 1 and
h(S) = −1. Then the moment map is Φ = h◦π1 +h◦π2 where π1 , π2 are
the projections of S 2 × S 2 to the first and second components. There
are four fixed points; (N, N), (S, N), (N, S), and (S, S); and there are
two Z2 -spheres; {N} × S 2 and {S} × S 2 . The corresponding graph is
1
2
2
1
2.8. Delzant Spaces. A Delzant space is a Hamiltonian space
(M, T, ω, Φ) with M compact and dim T = 21 dim M = n. It is determined up to isomorphism by the polytope image (Φ) in t∗ . The
polytopes which occur in this way are called Delzant polytopes. Every Delzant space admits an invariant complex structure for which ω
684. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
is Kähler. The Delzant spaces are exactly the compact, smooth toric
varieties with an invariant symplectic form.
For n = 2, the Delzant polytopes are the convex polygons in R2
which satisfy the following property. If Ei−1 , Ei are two consecutive
edges of the polygon then there exist integer vectors ui−1 , ui ∈ Z2
which are perpendicular to Ei−1 , Ei and such that det(ui−1 , ui ) = 1;
see [De].
i
If we restrict the action to a subcircle S 1 ֒→ T then the corresponding moment map is i∗ ◦ Φ. We can assume that t∗ = R2 , that
i∗ (x, y) = y and that the integer weight lattices in R2 and R are Z2
and Z respectively.
For i = 1, 2, . . . , let pi = (xi , yi ) be the vertices of the polygon,
ordered cyclically. Let Ei = pi−1 , pi be an edge, then Si = Φ−1 (Ei ) is a
2-sphere in M. If the normal of Ei is ui = (ki , bi ) then we can write
yi − yi−1
ki
=− .
xi − xi−1
bi
If ki and bi are relatively prime integers and |ki | =
6 1 then Si is a Zk sphere for k = ki . If |ki | = 1 then the action on Si is free. If ki = 0
then the action fixes Si , so we then have non-isolated fixed points. We
assume that ki 6= 0 for all i.
The graph of such a space (M, S 1 , ω, i∗ ◦ Φ) consists precisely of the
edges of the polytope; the edge Ei is labeled by the integer |ki | and the
vertex hpi i is labeled by the real number yi . There are two exceptions
–
1. If the endpoints of Ei are the extrema of i∗ and also ki = 1 then
there is no corresponding edge in the graph.
2. If ki = 1 but neither endpoints of Ei are extrema of i∗ then Ei
does not appear as an edge in the graph. Instead, we have two
edges in the graph, which connect each of hpi−1 i and hpi i to an
extremal vertex.
Example 2.9. The manifold in example 2.7 corresponds to the
polygon
Example 2.10. The polygons
2. CONSTRUCTING A GRAPH
69
correspond to the graphs
1
2
1
1
1
1
1
2
1
1
1
1
1
1
In constructing the graph for the polygon on the right, both of the
above exceptions (1) and (2) apply.
2.11. Push-forward measure. Let (M, S 1 , ω, Φ) be a Hamiltonian S 1 -space with isolated fixed points. Liouville measure on M is
given by the volume form mω = ω 2 /2!. Its push-forward via Φ is a
measure on R of the form
Φ∗ mω = ρ(y)|dy|
where |dy| denotes Lebesgue measure. Let (n, n′ ) be the isotropy
weights at pmin, let (−m, −m′ ) be the isotropy weights at pmax , let
(mp , −np ) be the isotropy weights at any other fixed point p. Let
ymin = Φ(pmin ), ymax = Φ(pmax ), yp = Φ(p) at other p’s.
Define
(
x if x ≥ 0
Θ(x) =
(2.12)
0 if x < 0
Lemma 2.13. The push-forward measure mDH = Φ∗ mω is determined from the graph by
X 1
1
1
ρ(y) =
Θ(y − yp ) +
Θ(y
−
y
)
−
Θ(y − ymax ).
min
′
′
nn
m
n
mm
p
p
p
(2.14)
This is a special case of a formula of Guillemin-Lerman-Sternberg
[GLS1].
Proof. The density function ρ(y) is characterized by the following
properties. It is supported on y ≥ ymin , it is linear on each component I
of R r{yp }, and its slope on I is equal to the Chern class of Φ−1 (y) −→
Φ−1 (y)/S 1 as a circle bundle over an orbifold [DH]. This Chern class is
1
for y = ymin +0 [Au2], and as y crosses yp , the Euler class decreases
nn′
by mp1np [MD, lemma 3],[Au2, ch.IV lemma 1.2.1].
704. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
Remark 2.15. The function ρ(y) above is convex on the interval
ymin < y < ymax because it is continuous, piecewise linear, and has a
decreasing slope.
Corollary 2.16.
X 1
1
1
+
=
mm′ nn′
mp np
p
where we sum over the interior fixed points.
Proof. The push-forward measure is supported on ymin ≤ y ≤
ymax , thus ρ(y)
P =1 0 for 1all y > ymax . The coefficient of y in (2.14) is
1
then nn′ − p mp np + mm′ = 0.
Example 2.17. Consider the Delzant spaces which correspond to
the polygons in Example 2.10. By the theorem of Duistermaat and
Heckman [DH], the push-forward of Liouville measure to t∗ = R2 is
Lebesgue measure on the polygon. If we let S 1 act as in §2.8, then the
push-forward measure on Lie(S 1 )∗ = R is given by projecting the polygon to R by (x, y) 7→ y and computing the push-forward of Lebesgue
measure. For all the manifolds in Example 2.10 we get the same density
function on R;
2
1
The second and third manifolds are isomorphic as S 1 -spaces (although
not as T 2 -spaces), because they hcorrespond to the same graph; this
will follow from section 6. But the first manifold is different; in fact,
its Betti numbers are different than those of the second and third manifolds; this can be read from the graph – see section 6. This gives
a negative answer to a question of V. Guillemin, whether the pushforward measure Φ∗ ω 2 determines (M, S 1 , ω).
3. Gradient spheres
In this section we define gradient spheres. The important point is
that all the edges in our graph correspond to gradient spheres in the
manifold. The idea of looking at gradient spheres, which is the key
method in this paper, is directly stolen from [Au2, Au1] and [AH].
In this section, Lemma 3.3 and part 3.6 are perhaps new. I learned
the argument in §3.1 from V. Ginzburg. I learned Example 3.8 from S.
Tolman.
3. GRADIENT SPHERES
71
3.1. Compatible metrics. Let (M, S 1 , ω) be a symplectic S 1 manifold. A compatible metric is a Riemann metric on M which is
S 1 -invariant and is compatible with ω in the following sense; hu, vi =
ω(Ju, v) where J : T M −→ T M satisfies J 2 = −id, i.e., J is an almost
complex structure on M.
The space of compatible metrics is nonempty and is path connected
by the following argument. The space of all metrics clearly has these
properties because a convex combination of two Riemann metrics is
again a Riemann metric. The space of all metrics projects continuously
onto the space of compatible metrics in the following way. First average
a metric with respect to the S 1 action, to get an invariant metric h , i′ .
Then in each vector space V = Tp M define the metric h , i in the
following way. There is a unique linear operator A : V −→ V such
that hu, vi′ = ω(Au,
. −A2 is symmetric and positive
√ v) for all u, v ∈ V −1
definite. Let P = −A2 and J = AP . Then take hu, vi = ω(Ju, v).
Suppose that we are given a compatible metric on a neighborhood
of a compact subset V ⊆ M. Then we can extend it to a compatible
metric on M which coincides with the given metric on V ; simply extend
it smoothly to M and apply the above procedure. We shall use this
fact implicitly in the subsequent sections.
3.2. Gradient spheres. Fix a compatible metric on (M, S 1 , ω).
Consider the gradient flow of the moment map Φ. It is generated by
the vector field JξM where ξM is the vector field which generates the
S 1 -action. The S 1 -action and the gradient flow combine into an action
of C× ∼
= S 1 × R on M.
Recall, the critical points of Φ are the fixed points of the action.
Following [AH], let G be a C× -orbit G in M. Then all the gradient
trajectories in G approach the same fixed points p and q at times ∞
and −∞ respectively.
The closure of G is then G = G ∪ {p, q} and is homeomorphic to a
2-sphere with p and q the North and South poles. We call it a gradient
sphere between p and q; see [Au2] and [AH].
If there is a unique gradient sphere between p and q then we denote
it Gpq . This condition fails if and only if p = pmax and q = pmin.
Let p be a fixed point with isotropy weights m, n. Consider a local
normal form (2.1) near p, let J be the standard complex structure and
take the standard metric. Let G, G′ be gradient spheres through p
with stabilizers of orders |m|, |n| respectively, which are ascending or
descending from p according to the signs of m and n. For example, we
could take G, G′ to be the coordinate planes near p, but sometimes we
could take different G, G′ .
724. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
Lemma 3.3. There are complex coordinates z ′ , w ′ near p in which
the action is λ · (z ′ , w ′ ) = (λm z ′ , λn w ′ ), the moment map is Φ(z ′ , w ′) =
− m2 |z ′ |2 − n2 |w ′ |2 and for which G and G′ are the coordinate planes.
Proof. The action of C× on C2 is given by u·(z, w) = (um z, un w), u ∈
C . Recall, G and G′ are closures of C× -orbits.
If m and n have opposite signs then we can take (z ′ , w ′) = (z, w)
because the coordinate planes are the unique C× -orbits whose closures
contain p = (0, 0).
Suppose m and n are positive. If m ≥ 2 and n ≥ 2 then again we
can take (z ′ , w ′) = (z, w) because the coordinate planes are exactly the
fixed-point-sets for the subgroups Zm and Zm of S 1 .
Suppose that m = n = 1. Then the C× orbits are complex lines,
and we get the coordinates (z ′ , w ′) by composing (z, w) with a complex
linear transformation of C2 .
Finally, assume that m ≥ 2 and n = 1. The only C× -orbit with
a stabilizer of order m is G = C × {0}. A C× -orbit with a nontrivial
stabilizer is of the form G′ = {(um α, uβ) | u ∈ C}, with β 6= 0, [AH,
lemma 4.9]. We will set z ′ = z − βαm w m and w ′ = f w where f is a positive S 1 -invariant function of z and w which will soon be determined. In
these new coordinates, the action of S 1 remains λ·(z ′ , w ′) = (λm z ′ , λw ′)
and we have G = {w ′ = 0} and G′ = {z ′ = 0}. We choose f such that
Φ will have the desired form, i.e., we require
2
m α m 1
m
1
− z − m w − f 2 |w|2 = − |z|2 − |w|2.
2
β
2
2
2
×
This amounts to
f 2 = 1 − m|
α 2 2m−2
α wm
z
|
|w|
+
2mℜ(
).
βm
β m |w|2
The right hand side avoids zero if z and w are very small, because
m ≥ 2. Thus the square root f is a smooth function of z and w.
We note that the symplectic form is no longer standard in the new
coordinates z ′ , w ′ .
Lemma 3.4.
1. Every Zk -sphere is a gradient sphere.
2. Every gradient sphere whose stabilizer is nontrivial is a Zk -sphere.
Recall, a Zk -sphere Npq is the closure of a component N of the fixed
point set of Zk ⊂ S 1 for some k ≥ 2. Also recall that a gradient sphere
Gpq is the closure of a nontrivial C× -orbit.
Proof of (1). Fix N as above. The gradient flow commutes with
the S 1 -action, thus N is C× -invariant. Since dim N = 2 and N is
3. GRADIENT SPHERES
73
connected and ξM 6≡ 0 on N, N must consist of exactly one C× -orbit
G, so then Npq = Gpq .
Proof of (2). Fix a C× -orbit G with a nontrivial stabilizer Zk .
Then G is contained in some N. In (1) we have shown that N consists
of exactly one C× -orbit, thus G = N and so Gpq = Npq .
Corollary 3.5 (Rigidity). Suppose that the stabilizer of Gpq is
nontrivial. Then Gpq ⊂ M is determined by the S 1 -action and not by
the symplectic structure or by the choice of the metric.
3.6. Non-rigidity. Let Gpq be a gradient sphere with a trivial
stabilizer. Assume that p 6= pmax and q 6= pmin . Denote by p− and
q + the gradient manifolds which descend from p and ascend from q
respectively. Then their closures are both equal to Gpq . We will show
that we can then perturb the metric such that p− and q + become
disjoint;
p
q
Therefore, for a generic metric, every gradient sphere Gpq with a trivial
stabilizer reaches an extremum of Φ, i.e., has p = pmax or q = pmin . It
follows that all the edges in the graph of (M, S 1 , ω), even those which
are labeled by 1 (see §2.5), correspond to gradient spheres in M.
Let Gpq be a gradient sphere with a trivial stabilizer. Pick s ∈
Gpq r {p, q} and choose normal coordinates in a neighborhood U of s,
as in (2.2) with k = 1;
(e2πiθ , h, z) : U −→ S 1 × I × D 2 .
∂
The gradient flow is generated by the vector field J( ∂θ
) where J is
the almost complex structure. We can choose the normal form such
∂
∂
) is a positive multiple of ∂h
; thus Gpq ∩ U =
that if z = 0 then J( ∂θ
{z = 0}. Let z = x + iy, then we can write
J(
∂
∂
∂
∂
∂
)=a +b
+c
+d .
∂θ
∂θ
∂h
∂x
∂y
(3.7)
By S 1 invariance, a, b, c, d are functions of h, z. Additionally, at z = 0
we have a = c = d = 0. We normalize the gradient flow to obtain
a flow for which dh
= 1, i.e., we divide (3.7) by the positive function
dt
b. Restricting to level sets of h and dividing by the S 1 action, we get
a time dependent flow on D 2 , where h is the time parameter. It is
744. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
generated by
ξ = b1
∂
∂
+ b2 ,
∂x
∂y
where b1 = c/b and b2 = d/b are functions of h, z. We have b1 (h, 0) =
b2 (h, 0) = 0 for all h, so by choosing D 2 small enough we will assume
that b21 + b22 < 1 everywhere. Let Fh : D 2 −→ D 2 be the corresponding
flow, such that dFh /dh = ξ for −δ < h < δ, and F−δ = id. Of course,
Fh might not be defined on all of D 2 ; however, Fh (0) = 0 for all h, so
Fh is defined near z = 0.
We will construct a new almost complex structure Jˆ which coincides
with J outside U but for which the corresponding flow Fˆh satisfies
F̂δ (0) 6= 0. The gradient manifolds p− and q + corresponding to Jˆ will
be disjoint, because p− enters U at (δ, 0) and q + enters U at (−δ, 0)
ˆ
and Fˆh keeps track of the gradient flow of J.
ˆ
We first construct the flow Fh . Take a smooth function ρ : I ×
2
D −→ R+ supported on a small neighborhood of (0, 0) and with
ρ(0, 0) > 0. Let Gh : D 2 −→ D 2 be the time dependent flow gen∂
erated by ρ(h, z) ∂x
for which G−δ = id; then Gδ (0) 6= 0. The flow
ˆ
Fh = Gh ◦ Fh satisfies the required properties. Moreover, if we choose
ρ small enough then its generating vector field
∂
∂
ξˆ = b̂1
+ b̂2
∂x
∂y
2
2
still satisfies bˆ1 + bˆ2 < 1.
In the remainder of this section we will construct an almost complex
ˆ b̂ = b̂2 .
ˆ ∂ ) = b̂ ∂ + ĉ ∂ + dˆ ∂ with ĉ/b̂ = b̂1 , d/
structure Jˆ for which J(
∂θ
∂h
∂x
∂y
We will do this in an open subset V whose closure is contained in U,
and such that ξˆ = ξ on U rV , so we can extend Jˆ to an almost complex
structure on M which coincides with J outside U.
∂
∂
∂
∂
In the basis { ∂θ
, ∂h
, ∂x
, ∂y
}, an almost complex structure is defined
by a 4 × 4 matrix J˜ such that J˜2 = −I. The symplectic structure is
defined by the matrix
J2 0
0 1
0 0
ω̃ =
where J2 =
and 0 =
.
0 J2
−1 0
0 0
The compatibility conditions is J˜t ω̃ J˜ = ω̃, and ω̃ J˜ should be positive
˜ Then an invariant
definite. Denote by J the set of such matrices J.
1
almost complex structure on U = S × I × D 2 is given by a map
I × D 2 −→ J .
3. GRADIENT SPHERES
75
The coefficients of our vector field ξˆ define a map
(b̂1 ,b̂2 )
I × D 2 −→ D 2 (1).
We get our Jˆ by composing
(b̂1 ,b̂2 )
M
I × D 2 −→ D 2 (1) −→ J
where M is defined by
−b̂J2
B
,
B
−b̂J2
p
ĉ dˆ
1 + ĉ2 + dˆ2 . Then
where J2 is as before, B =
,
and
b̂
=
dˆ −ĉ
ˆ ∂ ) = b̂ ∂ + ĉ ∂ + dˆ ∂ . We choose ĉ, dˆ in such a way that bˆ1 = ĉ/b̂
J(
∂θ
∂h
∂x
∂y
2
2
ˆ
ˆ
and b2 = d/b̂. To do this, given bˆ1 , bˆ2 , define N 2 = bˆ1 + bˆ2 . Then set
ˆ = √ 1 (bˆ1 , bˆ2 ). The reader is invited to check
b̂ = √ 1
and (ĉ, d)
M(bˆ1 , bˆ2 ) =
1−N 2
1−N 2
that this gives an almost complex structure which is compatible with
ω and which has the desired properties.
Example 3.8. Consider the toric varieties which correspond to the
second and third polygons in Examples 2.10; with the S 1 -actions as
described in §2.8. If we take the standard Kähler metrics then the
arrangements of gradient spheres for these metrics are given by the
edges of the polygons. These arrangements can be described by the
following graphs:
4.0
1
4.0
1
3.0
3.0
1
1
1
1.0
1
1.0
0.0
0.0
1
In the left graph we have an edge labeled 1 which does not reach the
top or bottom vertex. This means that the metric was not generic.
For a generic metric, the graph would be identical to the graph on the
right; thus the corresponding two Delzant spaces become isomorphic
once we restrict the T 2 -action to S 1 .
Topologically, the right polygon gives S 2 × S 2 and the left polygon
gives an S 2 bundle over S 2 which can be written as M = P ×S 1 S 2 where
P −→ S 2 is a principal circle bundle with Chern class 2. S. Tolman
764. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
has described an explicit S 1 -equivariant mapping between these two
4-manifolds.
4. Combinatorics
In this section we prove that every graph which was constructed in
§2.5 actually comes from a Delzant polygon as in §2.8.
I learned the trick in the proof of Lemma 4.6 from a talk of A.
Khovanskii. Lemma 4.3, Corollary 4.5 and §4.11 already appear in
[Au1, §3.3]. I learned the result of §4.10 from [AH, §6]; the method
of proof here is different.
4.1. Disc bundles over S 2 . Let S 1 act on S 2 by rotating it k
times, fixing the North and South poles as in Lemma 2.4. Let E −→ S 2
be a complex line bundle and consider a lifting of the action to E. The
fiber over the North [South] pole is mapped to itself by the action
t : z 7→ tmN z [tmS z]. We then have mN − mS = ek where e is the
Euler class of E. A proof of this can be found in [AH, lemma 4.3] and
it can be seen directly by decomposing S 2 into the upper and lower
hemispheres. I learned this from M. Grossberg.
In particular, consider a chain of gradient spheres; G1 , . . . , Gl where
Gi = Gpi ,pi−1 and p0 , pl are the minimum and maximum for Φ. Let ki
be the order of the stabilizer of Gi . Then the isotropy weights at pi for
1 ≤ i ≤ l − 1 are {ki , −ki+1 }. We draw this situation as:
pl
pi+1
ki+1
ki
pi
pi−1
p0
Since the action is effective, gcd(ki , ki+1) = 1. If 2 ≤ i ≤ l − 1 then
a neighborhood of Gi in M is equivariantly diffeomorphic to a discbundle of the normal bundle of Gi in M. This can be made into a
complex line bundle, and by the discussion above, −ki+1 − ki−1 = ei ki
where ei is the Euler class. Therefore
ei = −(ki−1 + ki+1 )/ki is a negative integer
gcd(ki , ki+1 ) = 1.
(4.2)
Lemma 4.3. Let k1 , . . . , kl be integers which satisfy the conditions
(4.2). Then there are vectors u1 , . . . , ul ∈ Z2 with det(ui , ui+1) = 1
and π(ui ) = ki , with π : Z2 −→ Z the projection to the first coordinate.
4. COMBINATORICS
77
Proof. Write ui = (ki , bi ), then det(ui , ui+1) = ki bi+1 − bi ki+1 .
Write
and subtract, then
ki−1 bi − bi−1 ki = 1
ki bi+1 − bi ki+1 = 1
ki+1 + ki−1
bi .
(4.4)
ki
We can choose b1 , b2 ∈ Z such that k1 b2 −b1 k2 = 1 because gcd(k1 , k2 ) =
1. Construct the other bi ’s by (4.4).
bi+1 = −bi−1 +
Corollary 4.5. Denote by Ci the cone generated by ui and ui+1 .
Then the cones C1 , . . . , Cl−1 fit together to form a smooth fan in Z2 ,
as in the following picture:
u4
u3
u1
C3
u2
C2
C1
Indeed, since det(ui , ui+1 ) > 0, the vectors u1 , u2 , . . . are arranged
in a counter-clockwise order. Our Ci ’s don’t overlap because ki > 0
for all i, so the ui ’s are on the right half-plane. Smooth means that
det(ui , ui+1 ) = 1.
In the following subsections we will show that there are just two
branches in our graph, and that the corresponding two fans fit together
to a complete fan in Z2 .
Lemma 4.6. Let k1 , . . . , kl be positive integers that satisfy the conditions (4.2). Then
1
1
d
+ ...+
=
k1 k2
kl−1 kl
k1 kl
where d is a positive integer.
Proof.
We use the notation of Lemma 4.3. One can easily show
R
that Ci e−x dx ∧ dy converges and is equal to ki k1i+1 det(ui ui+1) = ki k1i+1 .
R
Similarly, Let C = C1 ∪ C2 ∪ · · · ∪ Cl−1 = R+ u1 + R+ ul , then C e−x dx ∧
dy = k11kl det(u1 ul ). The desired equality follows from additivity of the
integral with d = det(u1ul ).
784. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
4.7. Denote by r the number of branches in our graph. Let
{m1 , . . . , mr } be the integer labels of the top edge in each branch.
Let {−m, −m′ } be the isotropy weights at the maximum of Φ. Then
mj ∈ {1, m, m′ } for all j, and if m ≥ 2 [or m′ ≥ 2] then m [or m′ ] occurs at most once among {m1 , . . . , mr }. A similar statement holds for
the bottom labels {n1 , . . . , nr } where {n, n′ } are the isotropy weights
at the minimum.
Let r be the number of branches of length ≥ 2; suppose that these
branches are the ones indexed by 1 ≤ j ≤ r.
Lemma 4.8.
1
d1
dr
1
+ ′ =
+ ...+
′
mm
nn
m1 n1
mr nr
where dj are positive integers.
Proof. By Corollary 2.16,
r
X
1
1
1
1
).
( j j + ...+ j
+ ′ =
′
mm
nn
k1 k2
klj −1 kljj
j=1
(4.9)
where k1j , . . . , kljj are the integer labels in the jth branch. By Lemma
P
4.6, we can write i kj 1 kj = mj1nj dj for all j.
i−1 i
4.10. The graph has two branches. We will show that Lemma
4.8, together with the restriction on the mj ’s and nj ’s in §4.7, imply
that r ≤ 2. The reader is invited to check that the cases listed below
exhaust all possibilities.
Case 1:
r ≥ 2, m1 = m, n1 = n, m3 = n3 = . . . = mr = nr = 1.
This is illustrated as
m
n
1
1
1
1
By Lemma 4.8, mm
′ + nn′ = mn d1 + m n d2 + d3 + . . . + dr . This can
2 2
be rewritten as
1
1 1
1
1
d2
1
− ′ ′ ) + d3 + . . . + dr .
(d1 − 1) + (
( − ′ )( ′ − ) =
m n m
n
mn
m2 n2 m n
The left term is < 1. The first two summands on the right are ≥ 0
because d1 ≥ 1, d2 ≥ 1, m2 ∈ {1, m′ } and n2 ∈ {1, n′} so that m2 n2 ≤
m′ n′ . Therefore, the right term is at least r − 2. It follows that r = 2.
So we have two possibilities, illustrated below.
4. COMBINATORICS
m
m′
79
m 1
m′ =n′ ≥2
n
n 1
n′
1
1
The second picture is impossible; it would imply that mm
′ + nn′ =
1
1
d + 1·1
d2 in which the left term is ≤ 1 and the right term is > 1.
mn 1
r ≥ 2, m1 = 1, n2 = 1;
Case 2:
1
m2
n1 1
By Lemma 4.8,
re-written as
1
mm′
+
1
nn′
=
1
d
n1 1
+
1
d
m2 2
+
Pr
1
j=3 mj nj dj .
This can be
r
0=(
X 1
1
1
1
1
d1 − ′ ) + ( d2 −
)
+
dj .
n1
nn
m2
mm′
m
n
j
j
j=3
The first summand is ≥ 0 because n1 ∈ {1, n, n′ }, the second summand
is ≥ 0 because m2 ∈ {1, m, m′ }, and the third summand is ≥ 0 as well.
It follows that r = 2, d1 = 1, {n, n′ } = {1, n1 }, d2 = 1, {m, m′ } =
{1, m2 }. This implies that r = 2.
Case 3:
r = 1; r = 1, 2, or 3.
m
m
m=n
≥2
m′ =n′
≥2
m′ =n′ =1
n
1
n
m′ =n′
≥2
1
d1
1
1
In the third case, by Lemma 4.8, mm
′ + mm′ = 1·1 . This is impossible
because the right term is a positive integer and the left term is < 1.
Case 4:
impossible.
If r = 0 then by Lemma 4.8,
1
mm′
+
1
nn′
= 0 which is
804. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
4.11. Constructing a fan. We have seen that the graph is of one
of the following three forms.
kl =m
m′ =kl′′
kl =m
kl =m
m′ =n′ =1
m′ =n′
≥2
k1 =n
n′ =k1′
k1 =n
k1 =n
Construct u1 , . . . , ul1 from k1 , . . . , kl as in Lemma 4.3. In a similar way, construct u′1 from u1 , n, −n′ ; and construct u′2 , . . . , u′l2 from
u′1 , −k1′ , . . . , −kl′′ . Then the cones don’t overlap because Cj are in the
left half-plane and Cj′ are in the right half-plane.
Claim: det(u′l′ , ul ) = 1. Why? because,
X
X
1
1
1
1
′
−
=0
,
u
)
+
det(u
−
+
′
l
l
′
mm′
mj mj+1 nn′
mj m′j+1
by Lemma 4.6. This plus Lemma 4.8 implies the claim.
Corollary 4.12. The graph of (M, S 1 , ω) corresponds to a complete smooth fan in Z2 . This is also shown in [Au1, §3.3].
Lemma 4.13. Let (M, S 1 , ω, Φ) be as in section 1, with isolated
fixed points. Consider its associated graph, as was constructed in §2.5.
Then there is a Delzant space to which is associated the same graph,
as in §2.8.
Proof. We need to find a Delzant polytope in R2 whose associated
graph, in the sense of §2.8, is our graph. Let ui = (ki , bi ) for 1 ≤ i ≤ l
and u′i = (−ki′ , b′i ) for 1 ≤ i ≤ l′ as in §4.11. Let y0 = y0′ = ymin ,
′
y1 , . . . , yl−1 , y1′ , . . . , yl−1
and yl = yl′ = ymax be the real labels of the
vertices. Define x0 = ymin and by induction,
bi
xi = xi−1 − (yi − yi−1 )
ki
b′
′
x′i = x′i−1 + i′ (yi′ − yi−1
).
ki
Construct two piecewise linear paths (x(t), t) and (x′ (t), t) by setting
x(yi ) = xi , making x(t) linear for yi−1 ≤ t ≤ yi , and similarly for x′ (t).
To get a polytope, we need to show that the paths x(t), x′ (t) do not
cross each other for ymin < t < ymax and that
x(t) = x′ (t) for t = ymax .
(4.14)
The reader is invited to check that once these conditions are satisfied,
the paths x(t) and x′ (t) for a Delzant polytope as desired.
5. UNIQUENESS – I
81
In Lemma 4.15 below we work out an expression for the difference
x (t) − x(t), which is exactly the expression for the push-forward of
Liouville measure given in Lemma 2.13. Since ρ(t) ≥ 0 for all t and
ρ(ymax ) = 0, it follows that x′ (t) ≥ x(t) for all t and x′ (ymax ) = x(ymax ),
which is what we needed.
′
Lemma 4.15.
x′ (t) − x(t) =
1
Θ(t − ymin) −
nn′
Pl′
− i=1 k′ 1 k′ Θ(t
i−1 i
Pl
−
1
i=1 ki−1 ki Θ(t − yi )
1
yi′ ) + mm
′ Θ(t − ymax )
where Θ was defined in (2.12).
Proof. For yi−1 < t < yi ,
bi
dx
=− .
dt
ki
−bi+1 ki
1
) − (− kbii ) = bi ki+1
= − ki+1
(the last equality follows
Since (− kbi+1
ki+1 ki
ki
i+1
from Lemma 4.3), we have that
X 1
b1
x(t) = ymin − Θ(t − ymin) −
Θ(t − yi ).
k1
k
k
i
i+1
i
b′
We have a similar expression for the curve x′ (t). Since k1′ + kb11 = nn1 ′
1
(see [Au2]), we get that
X 1
X 1
1
Θ(t−y
)
+
Θ(t−yi′ ),
x′ (t) −x(t) =
Θ(t−y
)
+
i
min
′ ′
nn′
k
k
k
k
i i+1
i i+1
i
i
i.e., x′ (t) − x(t) = ρ(t).
5. Uniqueness – I
Assume that (M1 , S 1 , ω1 , Φ1 ) and (M2 , S 1 , ω2 , Φ2 ) correspond to the
same graph. In this section we construct a diffeomorphism F : M1 −→
M2 which is S 1 -invariant and such that Φ2 ◦ F = Φ1 . We don’t require
F to respect the symplectic forms. We will first define F on neighborhoods of the fixed points, then extend it to neighborhoods of the
gradient spheres, then extend it to be defined everywhere.
We note that M. Audin explains that the graph determines the
manifold and circle action by observing that a neighborhood of the
gradient spheres is obtained by plumbing of disk-bundles over S 2 and
the parameters that determine the plumbing are labels of the graph
and by the fact that plumbing is uniquely defined. In this way she
obtains an equivariant diffeomorphism between neighborhoods of the
824. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
gradients spheres, below the maximum of Φ. We chose to describe
such an equivariant diffeomorphism more explicitly, with the hope that
similar descriptions might work in higher dimensions.
From now on until the end of this section, isomorphism means
equivariant diffeomorphism which preserves Φ.
Near each fixed point in M1 and M2 we take the standard metric
with respect to a local normal form (2.1). We then extend these to
metrics on M1 and on M2 which are compatible with the symplectic
structures and the S 1 -actions, and which are generic in the sense of
§3.6.
Until the end of this section, by gradient sphere we will only refer to
those gradient spheres which correspond to edges of our graph; i.e., we
ignore those infinitely many spheres which stretch all the way betwween
pmin and pmax and whose stabilizer is trivial.
5.1. Let p be a fixed point in M1 or in M2 . There are at most two
gradient spheres through p and the orders of their stabilizers are the
isotropy weights at p, this follows from §4.10 by the correspondence
between gradient spheres and edges of the graph which was established
in section 3. Lemma 3.3 gives a normal form for a neighborhood of p
such that the gradient spheres are the coordinate planes. This normal
form only depends on the isotropy weights at p, and if p1 and p2 are
points on M1 , M2 which correspond to the same vertex on the graph,
then they have the same isotropy weights. Therefore they have neighborhoods U1 , U2 and there is an isomorphism F : U1 −→ U2 which
sends the gradient spheres through p1 to the gradient spheres through
p2 .
5.2. Take a gradient sphere G1 = Gpq in M1 . Denote by Iǫ the
open interval (Φ(q) + ǫ, Φ(p) − ǫ). Choose ǫ small enough so that
Gǫ1 := Gpq ∩ Φ−1 (Iǫ ) intersects the neighborhoods of p and q on which
F has already been defined.
Let U1 and U2 be small neighborhoods of Gǫ1 and Gǫ2 respectively.
For x ∈ Iǫ which is close to the endpoints of Iǫ , the map F restricts
−1
to an equivariant diffeomorphism Fx from Φ−1
1 (x) ∩ U1 to Φ2 (x) ∩ U2 .
We can extend these Fx ’s to be defined for all x ∈ Iǫ and be smooth in
x, perhaps after shrinking U1 and U2 , using the following observations.
By the local normal form (2.2), Φi (x)∩Ui is isomorphic to S 1 ×Zm Vi
where Vi are small neighborhoods of the origin in C. Next, any Fx :
S 1 ×Zm V1 −→ S 1 ×Zm V2 can be equivariantly deformed to a map which
is linear on V1 , i.e., is given by an element of S 1 × Gl(2); perhaps by
shrinking V1 and V2 . Finally, the group S 1 × Gl(2) is connected.
5. UNIQUENESS – I
83
5.3. We now have an isomorphism F : U1 −→ U2 where U1 , U2
are open subsets of M1 and M2 which contain the fixed points and
the gradient spheres. Let m be the maximal value of the moment
map on M1 (and thus on M2 too). Fix ǫ > 0 small enough so that
Φ−1
1 (m − ǫ, m) is contained in U1 , and such that the interval [m − ǫ, m)
contains no critical values of φ1 (and thus of Φ2 ). We will now extend
−1
F to an isomorphism from Φ−1
1 (−∞, m − ǫ) to Φ2 (−∞, m − ǫ), by
“sweeping” along the gradient flows of the moment maps. We do this
in the following way.
For i = 1, 2 denote by Tti : Mi −→ Mi the gradient flow of Φi , so
we have T0i = idMi and dtd (Tti (q)) = J(ξMi )|q for q ∈ Mi . For t ∈ R
we define Ft to be the conjugation of F by the gradient flow. Let
Uti = Tti (Ui ), then the following diagram commutes:
F
Ut1 −−−t→ Ut2
x
x


1
2
Tt 
Tt 
F
U1 −−−→ U2
Ft is S 1 -equivariant but it will generally not preserve the moment maps.
1
We fix this by considering a new function; Gt = Tτ2 ◦ F ◦ T−t
, where
τ = τ (q, t) is chosen in such a way that Φ2 ◦ Gt = Φ1 . We get the
function τ from the implicit function theorem if we define f (q, t, τ ) =
1
Φ(Tτ2 (F (T−t
(q)))) and note that the set {(q, t, t̃) | f (q, t, t̃) = Φ(q)} ⊆
1
Ut × R × R is transverse to {q = const, t = const}. For this set to
project onto all (q, t) we need to use the fact that the original map
F sent gradient spheres to gradient spheres. Note that τ (q, t) is then
smooth in q and in t and is monotonous in t.
Now fix ǫ > 0 and pick t sufficiently large so that Ut1 contains
−1
Φ1 (−∞, m − ǫ). Then the restriction of Gt to Φ−1
1 (−∞, m − ǫ) gives
the desired isomorphism.
5.4.
We have constructed an isomorphism
−1
F − : Φ−1
1 (−∞, m − ǫ) −→ Φ2 (−∞, m − ǫ).
Recall, m was the maximal value of Φ1 (and of Φ2 ) and ǫ > 0 was very
small, so that we also have an isomorphism
−1
F + : Φ−1
1 (m − ǫ/2, ∞) −→ Φ2 (m − ǫ/2, ∞)
given by the restriction of F , where F was constructed earlier. It
remains to extend F − and F + across the set Φ−1
1 (I) where I is the
−1
∼
interval (m − ǫ, m − ǫ/2). We have Φ−1
(I)
Φ
= 2 (I) ∼
= P × I where
1
P is a level set of Φ1 (or of Φ2 ). By arguing as in §5.2, it is sufficient
844. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
to show that the space of equivariant diffeomorphisms of P is path
connected.
As is explained in [Au2], P is a Seifert fibered space whose exceptional fibers come from the Zk -sheres in M. By §4.10, P has at most
two exceptional orbits, so B := P/S 1 is a two dimensional orbifold with
at most two singular points. Let f : P −→ P be an equivariant diffeomorphism, then it descends to a diffeomorphism f of B. As illustrated
in [AH, §10], one can adapt to this situation a theorem of Smale [S]
to get a diffeotopy ft : B −→ B with f0 = id and f1 = f . Although
P −→ B is not a circle bundle in the usual sense, one can still think
of it as a circle bundle over an orbifold and choose a connection on it.
As in [De, §1], one can use this connection to lift ft to an equivariant
diffeotopy ft : P −→ P with f1 = f . Finally, the space of bundle maps
of P is isomorphic to the space of smooth functions B −→ S 1 . This
space it is path connected, so f0 can be connected to the identity map.
6. Uniqueness – II
In section 5 we showed that the labeled graph determines (M, S 1 , Φ)
up to an equivariant diffeomorphism that preserves Φ. In this section
we show that (M, S 1 , Φ) determines (M, S 1 , ω) up to an equivariant
symplectomorphism. This was conjectured by Delzant, as I heard from
V. Ginzburg. As a corollary, the labeled graph determines the Hamiltonian space (M, S 1 , ω, Φ) up to isomorphism.
This section is influenced by the techniques of Delzant [De, §1].
First, a standard argument shows that (M, S 1 , Φ) determines the
class [ω] in H 2 (M). Indeed, Φ is a perfect Morse function on M [GS2,
§5] whose critical points correspond to vertices of our graph. The
gradient manifolds Gpq form completing cycles for the critical points
p, therefore they form a basis Rof H∗ (M, Z).
Thus the class [ω] is
determined by the real numbers Gpq ω, where p 6= pmax . These numbers
R
are determined by the graph because if Gpq is a Zk -sphere then Gpq ω =
1
(Φ(p) − Φ(q)); see Remark 2.6. If the action on Gpq is free then the
m
same holds with m = 1.
We now wish to show that (M, S 1 , Φ) determines (M, S 1 , ω). Let
ω0 , ω1 be two S 1 -invariant symplectic forms for which Φ is a moment
map. We want to find an equivariant diffeomorphism F : M −→ M
such that F ∗ ω1 = ω0 . We have already shown that [ω0 ] = [ω1 ] in
H 2 (M, R). By Moser’s method [W1, lecture 5] it is sufficient to find a
path of symplectic forms ωt , 0 ≤ t ≤ 1 such that [ωt ] = [ω0 ] = [ω1 ] for
all t and such that each ωt is S 1 -invariant. We define ωt = (1−t)ω0 +tω1 ,
we will show that it is nondegenerate for all 0 ≤ t ≤ 1, at every p ∈ M.
6. UNIQUENESS – II
85
Case 1:
Suppose that p is a fixed point. In Lemma 6.1 we will
show that ω|Tp M is determined by dΦ|Tp M and by the isotropy weights
at p, thus ω0 |Tp M = ω1 |Tp M = ωt |Tp M .
Case 2:
Suppose that p has a discrete stabilizer. Choose v ∈ Tp M
such that Φ∗ v > 0. Let ξM be the vector field on M that generates the
circle action. Then ωt (ξM |p , v) > 0 for all 0 ≤ t ≤ 1, so the rank of ωt
at p is at least 2.
It is sufficient to show that ωt is nondegenerate on the reduced
space, Vp = ker(dΦ|p )/span(ξM |p ). If p has a trivial stabilizer then Vp
is the tangent at [p] to the reduced space, Mα = Φ−1 (α)/S 1 , where
α = Φ(p). Each ωt induces a 2-form, ωt , on Mα . Since ω0 and ω1
are area forms which define the same orientation on Mα , the ωt ’s are
nondegenerate. Note that this argument fails in higher dimensions.
The same argument works if p has a finite stabilizer; in this case Mα
is an orbifold and we then need to pass to a finite covering of Mα in a
neighborhood of [p].
Lemma 6.1. Let p ∈ M be a fixed point for the circle action. Then
the linear symplectic form ω|Tp M is determined by the isotropy weights
at p and by Φ.
Proof. We may apply the local normal form (2.1). The generator
∂
∂
∂
for the circle action is the vector field ξM = im(z ∂z
− z ∂z
) + in(w ∂w
−
∂
w ∂w ). Any real 2-form can be written as
i
ω = (Adz∧dz+Bdz∧dw−Bdz∧dw+Cdz∧dw−Cdz∧dw+Ddw∧dw)
2
where A, D ∈ R and B, C ∈ C. The S 1 -invariance amounts to the
conditions
λm+n B = B
and λm−n C = C
for all λ ∈ C, |λ| = 1.
Case 1:
If m 6= ±n then B = C = 0, so ω = 2i (Adz ∧ dz +
Ddw ∧ dw). The moment map is Φ(z, w) = − 21 (mA|z|2 + nD|w|2 ), so
A = − m2 Φ(1, 0) and D = − n2 Φ(0, 1).
Case 2:
If m = −n then C = 0, so ω = 2i (Adz ∧ dz + Bdz ∧ dw −
Bdz ∧ dw + Ddw ∧ dw). The moment map is Φ(z, w) = − 12 (mA|z|2 +
2mℜ(Bzw) − mD|w|2 ), so A = − m2 Φ(1, 0), D = m2 Φ(0, 1), ℜB =
− m1 Φ(1, 1) − 21 A + 21 D and ℑB = − m1 Φ(−i, 1) − 12 A + 21 D.
Case 3:
If m = n then B = 0, so ω = 2i (Adz ∧ dz + Cdz ∧ dw −
Cdz ∧ dw + Ddw ∧ dw). The moment map is Φ(z, w) = − 12 (mA|z|2 +
mD|w|2 + 2mℜ(Czw)), so A = − m2 Φ(1, 0), D = − m2 Φ(0, 1), ℜC =
− m1 Φ(1, 1) − 21 A − 12 D and ℑC = − m1 Φ(−i, 1) − 12 A − 12 D.
864. PERIODIC HAMILTONIAN FLOWS ON FOUR DIMENSIONAL MANIFOLDS
In all three cases, ω is determined by Φ, m, n.
7. Classification
From §2.8, lemma 4.13 and sections 5, 6 we learn that the Hamiltonian S 1 -spaces are determined by certain graphs and that every such
space is isomorphic to a Delzant space, whose graph is determined by
a corresponding Delzant polygon. We now determine which polygons
give isomorphic spaces.
We follow the notation of §2.8. Let ymin, ymax denote the extrema
of the y-coordinate on a polygon.
Theorem 4.1. The Hamiltonian S 1 spaces with isolated fixed points
are classified by a set of polygons in R2 , modulo an equivalence relation.
The set consists of the Delzant polygons (defined in §2.8) with the
following two additional properties.
(i) Each of ymin, ymax is obtained on a unique vertex; hpmini, hpmax i.
(ii) Let E be an edge of the polytope which is perpendicular to the
vector (1, b) with b ∈ Z. Then E reaches either hpmini or hpmax i.
The equivalence relation is the following. Two polygons are equivalent
if they differ by an affine transformation of R2 of the form (x, y) 7→
(a + x + my, y) or of the form (x, y) 7→ (a − x + my, y) with m ∈ Z.
Proof. In §2.8 we described how, given a Delzant polytope, we
get a periodic Hamiltonian flow; and how to read the corresponding
graph from the given polytope. From Lemma 4.13 it follows that we
get all the possible graphs in this way, from Delzant polytopes which
satisfy the properties (i) and (ii). From the uniqueness (section 6) it
follows that all Hamiltonian periodic flows are obtained in this way.
By examining the proof of Lemma 4.13 it follows that two Delzant
polytopes give the same graph if and only if they differ by an affine
transformation of the above form.
CHAPTER 5
Poisson Duality
1. Introduction
Let (M, ω) be a compact symplectic manifold and let G be a compact, connected Lie group. Assume that G acts on M by symplectic
diffeomorphisms and that we have a moment map Φ : M −→ g∗ . Recall that C ∞ (M) is a Poisson algebra. We will be interested in the
following Poisson subalgebras:
1. C ∞ (M)G are the G-invariant smooth functions on M.
2. Φ∗ C ∞ (g∗ ) are the collective Hamiltonians; these are the pullbacks f ◦ Φ where f is a smooth function on g∗ .
3. C ∞ (M)Φ are the smooth functions on M which are constant
along the level sets of Φ.
We have {f, g} = 0 whenever f is invariant and g is a collective
Hamiltonian. In fact, the centralizer of the collective Hamiltonians
consists exactly of the invariant functions. Here the centralizer of a
subset R ⊆ C∞ (M) is defined to be the following Poisson subalgebra
R{} = {f ∈ C ∞ (M) | {f, h} = 0 for all h ∈ R}.
The inverse might fail to hold; the inclusion
{}
C ∞ (M)G ⊃ Φ∗ C ∞ (g∗ )
(1.1)
could be strict; see [L].
Definition 1.2. Two Poisson subalgebras A and B of C ∞ (M) are
Poisson-duals if A{} = B and B{} = A.
Thus the invariants and the collective Hamiltonians are Poissonduals if and only if we have equality in (1.1); if and only if the algebra of
collective Hamiltonians is equal to its double centralizer in the Poisson
algebra C ∞ (M).
In this chapter we will prove that if one replaces the algebra of
collective Hamiltonians by the algebra C ∞ (M)Φ , then
Theorem 5.1. The Poisson algebras C ∞ (M)G and C ∞ (M)Φ are
Poisson-duals in C ∞ (M).
87
88
5. POISSON DUALITY
Notice that C ∞ (M)Φ contains the collective Hamiltonians, but it
can be larger: There could be a function on g∗ which is not smooth,
but whose pullback is a smooth function on M; see [L].
Theorem 1 will consist of two parts, proved in sections 2 and 3
respectively.
Part I:
Define (ker dΦ)ann = {f ∈ C ∞ (M) | ζf = 0 for any ζ ∈
ker dΦ}. Then (ker dΦ)ann and C ∞ (M)G are Poisson-duals in C ∞ (M).
Part II:
(ker dΦ)ann = C ∞ (M)Φ .
A main ingredient in the proof of Part I is Proposition 2.6, showing
that if a smooth vector field on M is tangent to the regular G-orbits
then it is tangent to all the G-orbits. The proof of Part II uses two main
ingredients. First, the connectedness of the level sets of the moment
map, which follows from a theorem of F. Kirwan. Secondly, Guillemin,
Sternberg and Marle’s local normal form for the moment map; Theorem
1.3.
Now suppose the action of G is not Hamiltonian, then we can
still define a moment map Φ locally, if we do not require it to be
G-equivariant. (ker dΦ)ann still makes sense, and the proof of Part I
still works. In particular, this implies the following
Corollary 1.3. Let G be a compact Lie group which acts on a
symplectic manifold (M, ω) and preserves the symplectic form ω. Then
the algebra of invariant functions is equal to its own double centralizer
in the Poisson algebra C ∞ (M).
We now describe some background to this problem. Collective
Hamiltonians are related to the complete integrability of various physical systems. They arise in many examples in physics and are discussed
by Kazhdan, Kostant and Sternberg in [KKS] and by Guillemin and
Sternberg in [GS1].
In certain interesting cases we have Hamiltonian actions of two Lie
groups G, G′ such that the corresponding two algebras of invariant
functions are Poisson-duals. This gives rise to a bijection between the
coadjoint orbits of G and those of G′ which occur in the images of
the corresponding moment maps; this is explained in [KKS] in the
case that the G orbits form a foliation. This phenomenon is a classical
analogue of the notion of Howe duality in representation theory. The
Φ
Φ′
moment maps g∗ ← M −→ g′∗ then form a dual pair.
Dual pairs were studied in a greater generality by A. Weinstein in
J
J2
[W2]. These are pairs of Poisson mappings P1 ←1 M −→
P2 where M is
symplectic and P1 , P2 are Poisson manifolds; such that the two algebras
Ji∗ C ∞ (Pi ) are Poisson-dual in C ∞ (M). Weinstein concentrates on the
2. PROOF OF THEOREM ?? – PART I
89
case that Ji are submersions, (or more generally, are of constant rank).
One then gets a bijection between the symplectic leaves in P1 and
those in P2 . One important case is when J1 : M −→ g∗ is the moment
map and J2 : M −→ M/G is the quotient map, assuming that the
G-orbits form a fibration of M. Our results in this chapter deal with
the more general case, when we allow singular G-orbits in M, i.e., we
don’t require the moment map to have a constant rank.
In [GS4], Guillemin and Sternberg conjectured that the algebra of
collective Hamiltonians should be equal to its double centralizer in the
Poisson algebra C ∞ (M); equivalently, that the algebra of G-invariants
and the algebra of collective Hamiltonians are Poisson-duals. The have
shown that this is the case locally, if we restrict our functions to regions
in M where Φ has constant rank (i.e., where the orbits of G form a
foliation). They have also shown that if G is a torus then the double
centralizer of the collective Hamiltonians consists of functions which
are constant on the level sets of Φ.
E. Lerman in [L] has proved the conjecture of Guillemin and Sternberg in the case that the image of the moment map avoids the walls of
the Weyl chambers in g∗ ; i.e., when the stabilizers under the coadjoint
action of the points in the image of the moment map are all tori. He
gave a counterexample, where this condition is not satisfied, and where
there is a strict inclusion in (1.1).
2. Proof of Theorem 5.1 – Part I
Let G be a compact Lie group which acts on the manifold M by
smooth diffeomorphisms. As in §1.1 we denote by g · q the tangent
space to the G-orbit through q. Similarly for any subspace m ⊆ g we
denote m · q = {ξ · q | ξ ∈ m} ⊆ Tq M where ξ · q = ξM |q .
Consider an orbit O = G · p in M. Denote H = Stab(p) and
W = Tp M/Tp O, then H acts on W linearly. Recall (§1.1); the normal
bundle of O in M is N = G ×H W where G acts by left multiplication.
Recall (Theorem 1.1); there is a G-equivariant diffeomorphism between
a neighborhood of the zero section in N and a neighborhood of O in
M, which maps the zero section of N onto O in the obvious manner.
Let q ∈ N be in the fiber above p. Then the tangent to N at q is
the (nondirect) sum:
Tq N = W + g · q.
(2.1)
W is the tangent to the fiber and g · q is the tangent to the G-orbit
at q. The intersection of these two spaces is the tangent space to the
90
5. POISSON DUALITY
H-orbit:
W ∩ g · q = h · q.
(2.2)
Moreover, for x ∈ g we have x · q ∈ W if and only if x ∈ h.
Now suppose that we have a splitting
g=h⊕m
where m is invariant under the adjoint action of H. Such a splitting
determines a G-invariant connection on the principal bundle G −→
G/H and also on the associated bundle N . The decomposition of Tq N
into a vertical and a horizontal part is given by
Tq N = W ⊕ m · q.
Let Vert : Tq N −→ W be the projection onto the vertical component.
Then Vert(v) = v − x · q for some x ∈ m, therefore:
Corollary 2.3. Vert(v) ∈ g · q if and only if v ∈ g · q.
Recall; the regular points of M are those whose orbit has the maximal dimension. These form an open dense subset of M (Proposition
1.1.5). Therefore, a point is regular if and only if it is regular in a small
invariant neighborhood of its orbit. This allows us to replace M by N .
Lemma 2.4. Let q0 be a point in N = G ×H W which lies in the
fiber W over p. Then q0 is a regular point for the G-action if and only
if it is a regular point for the linear action of H on W .
Proof. Let q be a point in the fiber over p. By (2.1) and (2.2),
dim W + dim(G · q) = dim N + dim(H · q)
(2.5)
Therefore, if dim(G · q0 ) is maximal in N then dim(H · q0 ) is maximal
in W .
Conversely, assume that dim(H · q0 ) is maximal in W . The set of
regular points in W is open, therefore there is a neighborhood U of
q0 in W on which all H-orbits have the same dimension. By (2.5),
dim(G · q) is constant for q ∈ U. The set G · U is a neighborhood of q0
in N . The set of regular points in N is dense so it must intersect G · U.
But dim(G · q) is constant on G · U, because dim(G · q) does not change
along a G-orbit. It follows that all the points in G · U are regular, and
in particular that q0 is regular with respect to the G-action on N .
Proposition 2.6. Let G be a compact Lie group, acting smoothly
on M. Let ζ be a continuous vector field on M such that ζp ∈ g · p for
all regular p. Then ζp ∈ g · p for all p ∈ M.
2. PROOF OF THEOREM ?? – PART I
91
Proof. Let us fix p ∈ M and show that ζp ∈ g · p. By Theorem
1.1 we can view ζ as a vector field on the normal bundle N = G ×H W ,
with p = [e, 0, 0]. Fix an Ad(H)-invariant splitting g = h ⊕ m as
above. This determines a connection on G ×H W as described above.
By Corollary 2.3, it is enough to prove our theorem for the vector field
Vert(ζ), obtained from our ζ by pointwise projection into the vertical
component. So let us assume that ζ was a vertical vector field to begin
with, and we will show that ζp = 0. Lemma 2.4 reduces our story to
the linear situation. We are left to prove:
Lemma 2.7. Let W be a representation space for the compact group
H. Let ζ be a continuous vector field on W with the property that ζq is
tangent to H · q for all regular q. Then ζ0 = 0
Proof. Fix an H-invariant inner product (, ) on W . By Proposition 1.1.5, the set of regular points is open and dense. It follows that
we can approach zero from all directions in W by regular points, i.e.,
for any vector v ∈ W of norm 1 there is a sequence of regular points
pn ∈ W converging to zero, such that limn−→∞ |ppnn| = v,
We have (ζpn , pn ) = 0, because ζpn is tangent to the H-orbit at
pn , and this orbit is contained in the sphere through pn . Therefore
(ζpn , |ppnn | ) = 0, and in the limit we have
(ζ0 , v) = 0
(2.8)
by the continuity of ζ. Finally, since (2.8) holds for any v ∈ W of norm
1, it follows that ζ0 = 0.
This ends the proof of Proposition 2.6.
We now assume that the G-action on M preserves a symplectic
form ω and that we have a moment map Φ : M −→ g∗ . Recall that
C ∞ (M)G denotes the G-invariant smooth functions on M. We denote
by (ker dΦ)ann the set of smooth functions f on M which satisfy
ζf = 0 for any ζ ∈ ker dp Φ
(2.9)
at any p ∈ M.
From the definition of the moment map (1.5.3) it follows that for
all p ∈ M,
ker dp Φ = (g · p)ω
(2.10)
g · p = (ker dp Φ)ω .
(2.11)
as subspaces of Tp M; where ·ω denotes ω-orthogonal complement. Since
ω is nondegenerate, (2.10) is equivalent to:
92
5. POISSON DUALITY
Recall (§1.4) that to every function f ∈ C ∞ (M) we associated a
vector field Xf by
df = −ι(Xf )ω
(2.12)
{f, g} = Xg f.
(2.13)
and that the Poisson bracket was defined by
We now give critera for the function f to belong to the family C ∞ (M)
or to the family (ker dΦ)ann , in terms of the vector field Xf .
Lemma 2.14.
1. f ∈ C ∞ (M)G if and only if Xf |p ∈ kerdp Φ for
all p ∈ M.
2. f ∈ (ker dΦ)ann if and only if Xf |p ∈ g · p for all p ∈ M.
Proof. Since G is connected, f ∈ C ∞ (M)G if and only if ζf = 0
for every ζ ∈ g · p, at any p ∈ M. By (2.12), this is equivalent to
ω(Xf , ζ) = 0 for every ζ ∈ g · p. By (2.10), this is equivalent to
Xf ∈ ker dp Φ. This proves part (1).
The proof of part (2) is similar: We have ζf = −ω(Xf , ζ) so that
ζf = 0 for all ζ ∈ kerdp Φ if and only if Xf ∈ (kerdp Φ)ω , which is equal
to g · p by (2.11).
0.
Corollary 2.15. If f ∈ C ∞ (M)G and g ∈ (ker dΦ)ann then {f, g} =
Proof. By (2.13), {f, g} = Xg f . By part (2) of Lemma 2.14, Xg |p
is tangent to the G-orbit at every p ∈ M. Since f is G-invariant, its
derivative in the direction of an orbit is zero.
In particular it follows that
((ker dΦ)ann ){} ⊇ C ∞ (M)G
{}
C ∞ (M)G
Lemma 2.17. For any p ∈ M,
⊇ (ker dΦ)ann
(2.16)
{Xf |p | f ∈ (ker dΦ)ann } = g · p.
Proof. Direction ⊆ follows from Lemma 2.14, part (2). For the
direction ⊇, fix a vector in g · p, say, ξ · p with ξ ∈ g. Take f = Φξ
to the the ξth coordinate of the moment map. From the definition of
the moment map and from (2.12) it follows that Xf |p = ξ · p. The
function f = Φξ is in (ker dΦ)ann , because ζ ∈ ker dΦ implies ζΦξ =
h dΦ(ζ), ξi = 0.
Lemma 2.18. For any p ∈ M,
{Xf |p | f ∈ C ∞ (M)G } ⊆ ker dp Φ.
If p is regular then we have equality in (2.19).
(2.19)
2. PROOF OF THEOREM ?? – PART I
93
Proof. To prove the inclusion, take f ∈ C ∞ (M)G . By G-invariance,
ζf = 0 for any ζ ∈ g·p. By (2.12), this is just saying that Xf |p ∈ (g·p)ω .
And by (2.10), Xf |p ∈ ker dp Φ.
Now suppose that p is a regular point. By Theorem 1.1, a neighborhood of G · p looks just like a neighborhood of the zero section in a
homogeneous bundle, G×H W . By Lemma 2.4, p is also a regular point
for the linear action of H on W . It follows that H acts on W trivially.
So the bundle is simply a product G/H × W , where G acts on the first
factor. The G-invariant functions are the pullbacks of functions on W .
The tangent vectors that kill them are those vectors which are tangent
to the G-orbit, and no other vectors. So we have:
{ζ ∈ Tp M | ζf = 0 for all f ∈ C ∞ (M)G } = g · p
(2.20)
for every regular p. (We always have the inclusion ⊇, which may be
a strict inclusion if p is not regular). By (2.12), ζf = −ω(Xf , ζ), so
the set on the left of (2.20) is just the ω-orthogonal complement to
{Xf | f ∈ C ∞ (M)G }. Moreover, the right hand side of (2.20) is just
(ker dp Φ)ω , by (2.11). So passing to the ω-orthogonal complements of
(2.20) gives us (2.19).
We can now prove the first ingredient for of Theorem 5.1:
Proposition 2.21. C ∞ (M)G and (ker dΦ)ann are Howe duals in
the Poisson algebra C ∞ (M).
We need to show:
1. ((ker dΦ)ann ){} = C ∞ (M)G
{}
2. C ∞ (M)G = (ker dΦ)ann
Proof of (1). Take g ∈ ((ker dΦ)ann ){} . This means that {g, f } =
0 for any f ∈ (ker dΦ)ann . By (2.13), this is equivalent to:
Xg |p ∈ {Xf |p | f ∈ (ker dΦ)ann .}ω
(2.22)
for any p ∈ M. By Theorem 2.17, the right term in (2.22) is equal to
(g · p)ω which, by (2.10), is just ker dp Φ. Therefore, (2.22) is equivalent
to
Xg |p ∈ ker dp Φ
at any p ∈ M. This last condition, for g ∈ C ∞ (M), is equivalent to
g ∈ C ∞ (M)G by part 1 of Lemma 2.14.
{}
Proof of (2). By (2.16) we have C ∞ (M)G ⊇ (ker dΦ)ann . Take
{}
g in C ∞ (M)G . By (2.13),
Xg |p ∈ {Xf |p | f ∈ C ∞ (M)G }ω
(2.23)
94
5. POISSON DUALITY
for any p ∈ M. By Lemma 2.18, the right term in 2.23 is equal to
(ker dp Φ)ω whenever p is a regular point for the action of G on M. By
(2.11), this implies Xg |p ∈ g · p for any regular p ∈ M. By Theorem
2.6: Xg |p ∈ g · p for all points p ∈ M. Finally, by part 2 of Lemma
2.14, g ∈ (ker dΦ)ann .
3. Proof of Theorem 1 - Part II.
Let (M, ω) be a symplectic manifold, and let the compact group G
act on it in a Hamiltonian fashion, with a moment map:
Φ : M −→ g∗ .
Recall our notation: C ∞ (M)Φ is the set of smooth functions on M
which are constant on the level sets of Φ. (ker dΦ)ann is the set of
smooth functions f on M which satisfy
ζf = 0 for any ζ ∈ ker dp Φ
(3.1)
at any p ∈ M. Let us assume that M is compact. The following
proposition, together with Theorem 2.21, will imply Theorem 5.1.
Proposition 3.2. (ker dΦ)ann = C ∞ (M)Φ .
Proof. Let f ∈ (ker dΦ)ann , and fix α ∈ g∗ . Let γt , t ∈ R be
a path in the level set Φ−1 (α). The tangent vectors to γt are all in
ker dΦ. So by (3.1), they all annihilate f . Thus dtd f (γt ) = 0 at every
t, and f is constant along γt . This implies that f is constant on the
connected components of Φ−1 (α). The level set Φ−1 (α) is connected,
this follows from the proof of a theorem of F. Kirwan [Ki], also see
[CDM]. Therefore, f ∈ C ∞ (M)Φ , so we have proved the direction ⊆.
For the direction ⊇, fix f ∈ C ∞ (M)Φ . Fix p ∈ M, and let us show
that ζf = 0 for any ζ ∈ ker dp Φ. By the local normal form for the
moment map (Theorem 1.3) we can pretend that f is a function on
G ×H (N ∗ ⊕ V ), where H, N and V are as in the notation of Theorem
1.3. Let π : G × (N ∗ ⊕ V ) −→ G ×H (N ∗ ⊕ V ) be the quotient modulo
H, and let us denote the composition Φ ◦ π by Φ̂. Let F = π ∗ f be
the pullback of f to a function on G × (N ∗ ⊕ V ). We will prove that
ζ̂F = 0 for any ζ̂ ∈ ker dΦ̂|(e,0,0) . This will imply that ζf = ζ̂F = 0 for
ζ = dπ(ζ̂). Since dπ is surjective, the images dπ(ζ̂) for ζ̂ ∈ ker dΦ̂ are
exactly kerdΦ, so this will imply that f ∈ (ker dΦ)ann .
Let us first calculate ker dΦ̂|(e,0,0) . By Theorem 1.3,
Φ̂(g, ν, v) = Ad∗ (g)(α + ν + ΦV (v))
where ΦV : V −→ h∗ is the moment map for the linear symplectic
action of H on V . It is given by the formula: hΦV (v), ξi = 21 ωV (Aξ v, v)
3. PROOF OF THEOREM 1 - PART II.
95
for ξ ∈ h, where Aξ is the matrix in sp(V ) corresponding to ξ, and
where ωV is the symplectic form on V ; see §1.6.3. Any tangent vector
to G ×H (N ∗ ⊕ V ) at (e, 0, 0) can be written as ζ̂ = (ξ, ν, v) where
ξ ∈ g, ν ∈ N ∗ and v ∈ V .
d
|t=0 Φ̂(etξ , tν, tv)
dt
d
|t=0 (Ad∗ (etξ )(α + tν + ΦV (tv)))
=
dt
= ad∗ (ξ)(α) + ν
dΦ̂(ζ̂) =
where ad∗ denotes the infinitesimal action of G on g∗ . (The differential
of ΦV (tv) is zero at 0, because ΦV (tv) = t2 Φ(v)). So ker dΦ̂ consists of
the vectors (ξ, ν, v) for which
ad∗ (ξ)(α) + ν = 0.
(3.3)
The first summand lies in the tangent to the coadjoint orbit; [g, α] (in
the notation of §1,6). The second summand lies in the space h∗ , which
is embedded in g∗ according to the splitting g = h ⊕ m. In Lemma 3.5
we will prove that [g, α] = (g/gα )∗ = the annihilator of gα in g∗ , where
gα is the Lie algebra of the stabilizer of α. This space intersects h∗
trivially, since h ⊆ gα . Therefore, condition (3.3) is equivalent to the
vanishing of both ad∗ (ξ)(α) and ν. It follows that
ker dΦ̂ = gα × {0} × V.
(3.4)
Lemma 3.5. [g, α] = (g/gα )∗ .
Proof. Again, we follow the notation of §1.6. We have the following equalities:
h[ξ, α], ζi = hα, −[ξ, ζ]i = −h[ζ, α], ξi.
The last term vanishes for any ζ ∈ gα . In the first term, [ξ, α] run over
all elements of [g, α] as ξ runs over g. So we get that the elements of
[g, α] annihilate gα . By counting dimensions, these are all the elements
of g∗ that annihilate gα , thus [g, α] = (g/gα )∗ .
Now, let f ∈ C ∞ (M)Φ and let F = π ∗ f : G × (N ∗ ⊕ V ) −→ R be
its pullback. Then
F (g, ν, v) = h(Ad∗ (g)(α + ν + ΦV (v)))
(3.6)
for some continuous h : g∗ −→ R. We want to show that ζ̂F |(e,0,0) = 0
for any ζ̂ ∈ ker dΦ̂. By (3.4), it is enough to take ζ̂ ∈ gα or ζ̂ ∈ V . By
(3.6), F (g, 0, 0) = h(Ad∗ (g)(α)) = h(α) for any g ∈ Gα , thus ζ̂F = 0
96
5. POISSON DUALITY
for any ζ̂ ∈ gα . Also, F (e, 0, v) = F (e, 0, −v) (since ΦV (v) = ΦV (−v)),
so ζ̂F = 0 for any ζ̂ ∈ V .
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