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āā’ Ļ(z), ei = Ī»i , āi ā I
ā’Ļ|zi |2 + Ī»i = Ī»i , āi ā I
āā’
āā’ āi ā I .
zi = 0 ,

ā¢ The Td -action on Cd preserves Ļ, so the Td -action takes Z = Ļā’1 (ā )
onto itself, so Td acts on Z.

Exercise. The stratiļ¬cation of Z is just the stratiļ¬cation of Z into Td
orbit types. More speciļ¬cally, if z ā Z and Ļ(z) ā FI then the stabilizer
of z in Td is (Td )I where

I = (i1 , . . . , ir ) ,
185
29.3 Conclusion of the Delzant Construction

FI = {y ā ā | y, ei = Ī»i , āi ā I} ,
and
(Td )I = {(e2Ļit1 , . . . , e2Ļitd ) | e2Ļits = 1, ās ā I}
/

Suppose that z = (z1 , . . . , zd ) ā Cd . Then
Hint:
(e2Ļit1 z1 , . . . , e2Ļitd zd ) = (z1 , . . . , zd )
if and only if e2Ļits = 1 whenever zs = 0.

ā™¦

In order to show that N acts freely on Z, consider the worst case scenario of
points z ā Z whose stabilizer under the action of Td is a large as possible. Now
(Td )I is largest when FI = {y} is a vertex of ā . Then y satisļ¬es n equations

i ā I = {i1 , . . . , in } .
y, ei = Ī»i ,

Lemma 29.3 Let z ā Z be such that Ļ(z) is a vertex of ā . Let (Td )I be the
stabilizer of z. Then the map Ļ : Td ā’ Tn maps (Td )I bijectively onto Tn .

Since N = ker Ļ, this lemma shows that in the worst case, the stabilizer of z
intersects N in the trivial group. It will follow that N acts freely at this point
and hence on Z.
Proof of the lemma. Suppose that Ļ(z) = y is a vertex of ā . Renumber
the indices so that
I = (1, 2, . . . , n) .
Then
(Td )I = {(e2Ļit1 , . . . , e2Ļitn , 1, . . . , 1) | ti ā R} .
The hyperplanes meeting at y are

y , ei = Ī» i , i = 1, . . . , n .

By deļ¬nition of Delzant polytope, the set Ļ(e1 ), . . . , Ļ(en ) is a basis of Zn . Thus,
Ļ : (Td )I ā’ Tn is bijective.
This proves the theorem in the worst case scenario, and hence in general.

29.3 Conclusion of the Delzant Construction

We continue the construction of (Mā , Ļā ) from ā. We already have that

Mā = Z/N

is a compact 2n-dimensional manifold. Let Ļā be the reduced symplectic form.
186 29 DELZANT CONSTRUCTION

Claim. The manifold (Mā , Ļā ) is a hamiltonian Tn -space with a moment map
Āµ having image Āµ(Mā ) = ā.
Suppose that z ā Z. The stabilizer of z with respect to the Td -action is
(Td )I , and
(Td )I ā© N = {e} .
In the worst case scenario, FI is a vertex of ā and (Td )I is an n-dimensional
subgroup of Td . In any case, there is a right inverse map Ļ ā’1 : Tn ā’ (Td )I .
Thus, the exact sequence

0 ā’ā’ N ā’ā’ Td ā’ā’ Tn ā’ā’ 0

splits, and Td = N Ć— Tn .
Apply the results on reduction for product groups (Section 24.3) to our
situation of Td = N Ć— Tn acting on (Mā , Ļā ). The moment map is

Ļ : Cd ā’ā’ (Rd )ā— = nā— ā• (Rn )ā— .

Let j : Z ā’ Cd be the inclusion map, and let

pr1 : (Rd )ā— ā’ā’ nā— pr2 : (Rd )ā— ā’ā’ (Rn )ā—
and

be the projection maps. The map

pr2 ā—¦ Ļ ā—¦ j : Z ā’ā’ (Rn )ā—

is constant on N -orbits. Thus there exists a map

Āµ : Mā ā’ā’ (Rn )ā—

such that
Āµ ā—¦ p = pr2 ā—¦ Ļ ā—¦ j .
The image of Āµ is equal to the image of pr2 ā—¦ Ļ ā—¦ j. We showed earlier that
Ļ(Z) = ā . Thus

Image of Āµ = pr2 (ā ) = pr2 ā—¦ Ļ ā— (ā) = ā .
id

Thus (Mā , Ļā ) is the required toric manifold corresponding to ā.

29.4 Idea Behind the Delzant Construction

We use the idea that Rd is āuniversalā in the sense that any n-dimensional
polytope ā with d facets can be obtained by intersecting the negative orthant
Rd with an aļ¬ne plane A. Given ā, to construct A ļ¬rst write ā as:
ā’

ā = {x ā Rn | x, vi ā¤ Ī»i , i = 1, . . . , d} .
187
29.4 Idea Behind the Delzant Construction

Deļ¬ne
Ļ ā— : Rn
Ļ : Rd ā’ā’ Rn ā’ā’ Rd .
with dual map
ā’ā’ vi
ei

Then
Ļ ā— ā’ Ī» : Rn ā’ā’ Rd
is an aļ¬ne map, where Ī» = (Ī»1 , . . . , Ī»d ). Let A be the image of Ļ ā— ā’ Ī». Then
A is an n-dimensional aļ¬ne plane.
Claim. We have the equality (Ļ ā— ā’ Ī»)(ā) = Rd ā© A.
ā’

Proof. Let x ā Rn . Then

(Ļ ā— ā’ Ī»)(x) ā Rd Ļ ā— (x) ā’ Ī», ei ā¤ 0, āi
āā’
ā’
āā’ x, Ļ(ei ) ā’ Ī»i ā¤ 0, āi
āā’ x, vi ā¤ Ī»i , āi
āā’ xāā.

We conclude that ā Rd ā© A. Now Rd is the image of the moment map
ā’ ā’
for the standard hamiltonian action of T on Cd
d

Ļ : Cd ā’ā’ Rd
ā’ā’ ā’Ļ(|z1 |2 , . . . , |zd |2 ) .
(z1 , . . . , zd )

Facts.

ā¢ The set Ļā’1 (A) ā‚ Cd is a compact submanifold. Let i : Ļ ā’ Cd denote
inclusion. Then iā— Ļ0 is a closed 2-form which is degenerate. Its kernel is
an integrable distribution. The corresponding foliation is called the null
foliation.
ā¢ The null foliation of iā— Ļ0 is a principal ļ¬bration, so we take the quotient:

ā’ Ļā’1 (A)
N
ā“
= Ļā’1 (A)/N
Mā

Let Ļā be the reduced symplectic form.
ā¢ The (non-eļ¬ective) action of Td = N Ć—Tn on Ļā’1 (A) has a āmoment mapā
with image Ļ(Ļā’1 (A)) = ā. (By āmoment mapā we mean a map satisfying
the usual deļ¬nition even though the closed 2-form is not symplectic.)

Theorem 29.4 For any x ā ā, we have that Āµā’1 (x) is a single Tn -orbit.
188 29 DELZANT CONSTRUCTION

Proof. Exercise.
First consider the standard Td -action on Cd with moment map Ļ : Cd ā’ Rd .
Show that Ļā’1 (y) is a single Td -orbit for any y ā Ļ(Cd ). Now observe that

y ā ā = Ļ ā— (ā) āā’ Ļā’1 (y) ā Z .

Suppose that y = Ļ ā— (x). Show that Āµā’1 (x) = Ļā’1 (y)/N . But Ļā’1 (y) is a single
Td -orbit where Td = N Ć— Tn , hence Āµā’1 (x) is a single Tn -orbit.
Therefore, for toric manifolds, ā is the orbit space.
Now ā is a manifold with corners. At every point p in a face F , the tangent
space Tp ā is the subspace of Rn tangent to F . We can visualize (Mā , Ļā , Tn , Āµ)
from ā as follows. First take the product Tn Ć— ā. Let p lie in the interior of
Tn Ć— ā. The tangent space at p is Rn Ć— (Rn )ā— . Deļ¬ne Ļp by:

Ļp (v, Ī¾) = Ī¾(v) = ā’Ļp (Ī¾, v) and Ļp (v, v ) = Ļ(Ī¾, Ī¾ ) = 0 .

for all v, v ā Rn and Ī¾, Ī¾ ā (Rn )ā— . Then Ļ is a closed nondegenerate 2-form
on the interior of Tn Ć— ā. At the corner there are directions missing in (Rn )ā— ,
so Ļ is a degenerate pairing. Hence, we need to eliminate the corresponding
directions in Rn . To do this, we collapse the orbits corresponding to subgroups
of Tn generated by directions orthogonal to the annihilator of that face.
Example. Consider

(S 2 , Ļ = dĪø ā§ dh, S 1 , Āµ = h) ,

where S 1 acts on S 2 by rotation. The image of Āµ is the line segment I = [ā’1, 1].
The product S 1 Ć— I is an open-ended cylinder. By collapsing each end of the
ā™¦
cylinder to a point, we recover the 2-sphere.

Exercise. Build CP2 from T2 Ć— ā where ā is a right-angled isosceles triangle.
ā™¦
Finally, Tn acts on Tn Ć— ā by multiplication on the Tn factor. The moment
map for this action is projection onto the ā factor.
Homework 22: Delzant Theorem

1. (a) Consider the standard (S 1 )3 -action on CP3 :

(eiĪø1 , eiĪø2 , eiĪø3 ) Ā· [z0 , z1 , z2 , z3 ] = [z0 , eiĪø1 z1 , eiĪø2 z2 , eiĪø3 z3 ] .

Exhibit explicitly the subsets of CP3 for which the stabilizer under
this action is {1}, S 1 , (S 1 )2 and (S 1 )3 . Show that the images of these
subsets under the moment map are the interior, the facets, the edges
and the vertices, respectively.
(b) Classify all 2-dimensional Delzant polytopes with 4 vertices, up to
translation and the action of SL(2; Z).
Hint: By a linear transformation in SL(2; Z), you can make one of the angles
in the polytope into a square angle. Check that automatically another angle
also becomes 90o .
(c) What are all the 4-dimensional symplectic toric manifolds that have
four ļ¬xed points?

2. Take a Delzant polytope in Rn with a vertex p and with primitive (inward-
pointing) edge vectors u1 , . . . , un at p. Chop oļ¬ the corner to obtain a
new polytope with the same vertices except p, and with p replaced by n
new vertices:
p + Īµuj , j = 1, . . . , n ,
where Īµ is a small positive real number. Show that this new polytope
is also Delzant. The corresponding toric manifold is the Īµ-symplectic
blowup of the original one.

p

22 Ā˜
222 Ā Ā˜
Ā˜
d Ā˜
d
d
d

189
190 HOMEWORK 22

3. The toric 4-manifold Hn corresponding to the polygon with vertices (0, 0),
(n + 1, 0), (0, 1) and (1, 1), for n a nonnegative integer, is called a Hirze-
bruch surfaces.

rr
d
rr
d
rr
d
rr
d

(a) What is the manifold H0 ? What is the manifold H1 ?
Hint:

d
d
d
d

(b) Construct the manifold Hn by symplectic reduction of C4 with re-
spect to an action of (S 1 )2 .
(c) Exhibit Hn as a CP1 -bundle over CP1 .

4. Which 2n-dimensional toric manifolds have exactly n + 1 ļ¬xed points?
30 Duistermaat-Heckman Theorems
30.1 Duistermaat-Heckman Polynomial
Ļn
Let (M 2n , Ļ) be a symplectic manifold. Then is the symplectic volume form.
n!

Deļ¬nition 30.1 The Liouville measure (or symplectic measure) of a
Borel subset 17 U of M is
Ļn
mĻ (U) = .
U n!

Let G be a torus. Suppose that (M, Ļ, G, Āµ) is a hamiltonian G-space, and
that the moment map Āµ is proper.
Deļ¬nition 30.2 The Duistermaat-Heckman measure, mDH , on gā— is the
push-forward of mĻ by Āµ : M ā’ gā— . That is,
Ļn
mDH (U ) = (Āµā— mĻ )(U ) =
n!
Āµā’1 (U )

for any Borel subset U of gā— .
For a compactly-supported function h ā C ā (gā— ), we deļ¬ne its integral with
respect to the Duistermaat-Heckman measure to be
Ļn
(h ā—¦ Āµ)
h dmDH = .
n!
M
gā—

On gā— regarded as a vector space, say Rn , there is also the Lebesgue (or
euclidean) measure, m0 . The relation between mDH and m0 is governed by the
Radon-Nikodym derivative, denoted by dmDH , which is a generalized function
dm0
satisfying
dmDH
h dmDH = h dm0 .
dm0
gā— gā—

Theorem 30.3 (Duistermaat-Heckman, 1982 [31]) The Duistermaat-
Heckman measure is a piecewise polynomial multiple of Lebesgue (or euclidean)
measure m0 on gā— Rn , that is, the Radon-Nikodym derivative
dmDH
f=
dm0
is piecewise polynomial. More speciļ¬cally, for any Borel subset U of g ā— ,

mDH (U ) = f (x) dx ,
U

where dx = dm0 is the Lebesgue volume form on U and f : gā— Rn ā’ R is
polynomial on any region consisting of regular values of Āµ.
17 Theset B of Borel subsets is the Ļ-ring generated by the set of compact subsets, i.e., if
A, B ā B, then A \ B ā B, and if Ai ā B, i = 1, 2, . . ., then āŖā Ai ā B.
i=1

191
192 30 DUISTERMAAT-HECKMAN THEOREMS

The proof of Theorem 30.3 for the case G = S 1 is in Section 30.3. The
proof for the general case, which follows along similar lines, can be found in, for
instance, [53], besides the original articles.

The Radon-Nikodym derivative f is called the Duistermaat-Heckman
polynomial. In the case of a toric manifold, the Duistermaat-Heckman poly-
nomial is a universal constant equal to (2Ļ)n when ā is n-dimensional. Thus
the symplectic volume of (Mā , Ļā ) is (2Ļ)n times the euclidean volume of ā.
Example. Consider (S 2 , Ļ = dĪø ā§ dh, S 1 , Āµ = h). The image of Āµ is the interval
[ā’1, 1]. The Lebesgue measure of [a, b] ā [ā’1, 1] is
m0 ([a, b]) = b ā’ a .
The Duistermaat-Heckman measure of [a, b] is

dĪø dh = 2Ļ(b ā’ a) .
mDH ([a, b]) =
{(Īø,h)āS 2 |aā¤hā¤b}

Consequently, the spherical area between two horizontal circles depends only on
the vertical distance between them, a result which was known to Archimedes
around 230 BC.
Corollary 30.4 For the standard hamiltonian action of S 1 on (S 2 , Ļ), we have
mDH = 2Ļ m0 .
ā™¦

30.2 Local Form for Reduced Spaces
Let (M, Ļ, G, Āµ) be a hamiltonian G-space, where G is an n-torus.18 Assume
that Āµ is proper. If G acts freely on Āµā’1 (0), it also acts freely on nearby levels
Āµā’1 (t), t ā gā— and t ā 0. Consider the reduced spaces
Mred = Āµā’1 (0)/G Mt = Āµā’1 (t)/G
and
with reduced symplectic forms Ļred and Ļt . What is the relation between these
reduced spaces as symplectic manifolds?
For simplicity, we will assume G to be the circle S 1 . Let Z = Āµā’1 (0) and let
i : Z ā’ M be the inclusion map. We ļ¬x a connection form Ī± ā ā„¦1 (Z) for the
principal bundle
EZ
S1 ā‚

Ļ
c
Mred
18 The discussion in this section may be extended to hamiltonian actions of other compact
Lie groups, not necessarily tori; see [53, Exercises 2.1-2.10].
193
30.2 Local Form for Reduced Spaces

that is, LX # Ī± = 0 and Ä±X # Ī± = 1, where X # is the inļ¬nitesimal generator for
the S 1 -action. From Ī± we construct a 2-form on the product manifold ZĆ—(ā’Īµ, Īµ)
by the recipe
Ļ = Ļ ā— Ļred ā’ d(xĪ±) ,
gā— . (By abuse of
x being a linear coordinate on the interval (ā’Īµ, Īµ) ā‚ R
notation, we shorten the symbols for forms on ZĆ—(ā’Īµ, Īµ) which arise by pullback
via projection onto each factor.)
Lemma 30.5 The 2-form Ļ is symplectic for Īµ small enough.

Proof. The form Ļ is clearly closed. At points where x = 0, we have

Ļ|x=0 = Ļ ā— Ļred + Ī± ā§ dx ,
which satisļ¬es
ā‚
Ļ|x=0 X # , =1,
ā‚x
so Ļ is nondegenerate along Z Ć— {0}. Since nondegeneracy is an open condition,
we conclude that Ļ is nondegenerate for x in a suļ¬ciently small neighborhood
of 0.
Notice that Ļ is invariant with respect to the S 1 -action on the ļ¬rst factor of
Z Ć— (ā’Īµ, Īµ). In fact, this S 1 -action is hamiltonian with moment map given by
projection onto the second factor,
x : Z Ć— (ā’Īµ, Īµ) ā’ā’ (ā’Īµ, Īµ) ,

as is easily veriļ¬ed:
Ä±X # Ļ = ā’Ä±X # d(xĪ±) = ā’ LX # (xĪ±) +d Ä±X # (xĪ±) = dx .
0 x

Lemma 30.6 There exists an equivariant symplectomorphism between a neigh-
borhood of Z in M and a neighborhood of Z Ć— {0} in Z Ć— (ā’Īµ, Īµ), intertwining
the two moment maps, for Īµ small enough.

Proof. The inclusion i0 : Z ā’ Z Ć— (ā’Īµ, Īµ) as Z Ć— {0} and the natural inclusion
i : Z ā’ M are S 1 -equivariant coisotropic embeddings. Moreover, they satisfy
iā— Ļ = iā— Ļ since both sides are equal to Ļ ā— Ļred , and the moment maps coincide on
0
Z because iā— x = 0 = iā— Āµ. Replacing Īµ by a smaller positive number if necessary,
0
the result follows from the equivariant version of the coisotropic embedding
theorem stated in Section 8.3.19
19 The equivariant version of Theorem 8.6 needed for this purpose may be phrased as follows:
Let (M0 , Ļ0 ), (M1 , Ļ1 ) be symplectic manifolds of dimension 2n, G a compact Lie group acting
on (Mi , Ļi ), i = 0, 1, in a hamiltonian way with moment maps Āµ0 and Āµ1 , respectively, Z
a manifold of dimension k ā„ n with a G-action, and Ī¹i : Z ā’ Mi , i = 0, 1, G-equivariant
coisotropic embeddings. Suppose that Ī¹ā— Ļ0 = Ī¹ā— Ļ1 and Ī¹ā— Āµ0 = Ī¹ā— Āµ1 . Then there exist G-
0 1 0 1
invariant neighborhoods U0 and U1 of Ī¹0 (Z) and Ī¹1 (Z) in M0 and M1 , respectively, and a
G-equivariant symplectomorphism Ļ• : U0 ā’ U1 such that Ļ• ā—¦ Ī¹0 = Ī¹1 and Āµ0 = Ļ•ā— Āµ1 .
194 30 DUISTERMAAT-HECKMAN THEOREMS

Therefore, in order to compare the reduced spaces

Mt = Āµā’1 (t)/S 1 , tā0,

we can work in Z Ć— (ā’Īµ, Īµ) and compare instead the reduced spaces

xā’1 (t)/S 1 , tā0.

Proposition 30.7 The reduced space (Mt , Ļt ) is symplectomorphic to

(Mred , Ļred ā’ tĪ²) ,

where Ī² is the curvature form of the connection Ī±.

Proof. By Lemma 30.6, (Mt , Ļt ) is symplectomorphic to the reduced space at
level t for the hamiltonian space (Z Ć— (ā’Īµ, Īµ), Ļ, S 1 , x). Since xā’1 (t) = Z Ć— {t},
where S 1 acts on the ļ¬rst factor, all the manifolds xā’1 (t)/S 1 are diļ¬eomorphic
to Z/S 1 = Mred. As for the symplectic forms, let Ī¹t : Z Ć— {t} ā’ Z Ć— (ā’Īµ, Īµ) be
the inclusion map. The restriction of Ļ to Z Ć— {t} is

Ī¹ā— Ļ = Ļ ā— Ļred ā’ tdĪ± .
t

By deļ¬nition of curvature, dĪ± = Ļ ā— Ī². Hence, the reduced symplectic form on
xā’1 (t)/S 1 is
Ļred ā’ tĪ² .

In loose terms, Proposition 30.7 says that the reduced forms Ļt vary linearly
in t, for t close enough to 0. However, the identiļ¬cation of Mt with Mred as
abstract manifolds is not natural. Nonetheless, any two such identiļ¬cations are
isotopic. By the homotopy invariance of de Rham classes, we obtain:

Theorem 30.8 (Duistermaat-Heckman, 1982 [31]) The cohomology class
of the reduced symplectic form [Ļt ] varies linearly in t. More speciļ¬cally,

[Ļt ] = [Ļred ] + tc ,

where c = [ā’Ī²] ā HdeRham (Mred ) is the ļ¬rst Chern class of the S 1 -bundle
2

Z ā’ Mred.

Remark on conventions. Connections on principal bundles are Lie algebra-
valued 1-forms; cf. Section 25.2. Often the Lie algebra of S 1 is identiļ¬ed with
2ĻiR under the exponential map exp : g 2ĻiR ā’ S 1 , Ī¾ ā’ eĪ¾ . Given a prin-
cipal S 1 -bundle, by this identiļ¬cation the inļ¬nitesimal action maps the gener-
ator 2Ļi of 2ĻiR to the generating vector ļ¬eld X # . A connection form A is
then an imaginary-valued 1-form on the total space satisfying LX # A = 0 and
Ä±X # A = 2Ļi. Its curvature form B is an imaginary-valued 2-form on the base
195
30.3 Variation of the Symplectic Volume

satisfying Ļ ā— B = dA. By the Chern-Weil isomorphism, the ļ¬rst Chern class
i
of the principal S 1 -bundle is c = [ 2Ļ B].
In this lecture, we identify the Lie algebra of S 1 with R and implicitly use
R ā’ S 1 , t ā’ e2Ļit . Hence, given a principal
the exponential map exp : g
S 1 -bundle, the inļ¬nitesimal action maps the generator 1 of R to X # , and here a
connection form Ī± is an ordinary 1-form on the total space satisfying LX # Ī± = 0
and Ä±X # Ī± = 1. The curvature form Ī² is an ordinary 2-form on the base satisfying
Ļ ā— Ī² = dĪ±. Consequently, we have A = 2ĻiĪ±, B = 2ĻiĪ² and the ļ¬rst Chern
ā™¦
class is given by c = [ā’Ī²].

30.3 Variation of the Symplectic Volume

Let (M, Ļ, S 1 , Āµ) be a hamiltonian S 1 -space of dimension 2n and let (Mx , Ļx )
be its reduced space at level x. Proposition 30.7 or Theorem 30.8 imply that,
for x in a suļ¬ciently narrow neighborhood of 0, the symplectic volume of Mx ,
nā’1
(Ļred ā’ xĪ²)nā’1
Ļx
vol(Mx ) = = ,
(n ā’ 1)! (n ā’ 1)!
Mx Mred

is a polynomial in x of degree n ā’ 1. This volume can be also expressed as

Ļ ā— (Ļred ā’ xĪ²)nā’1
ā§Ī± .
vol(Mx ) =
(n ā’ 1)!
Z

Recall that Ī± is a chosen connection form for the S 1 -bundle Z ā’ Mred and Ī² is
its curvature form.
Now we go back to the computation of the Duistermaat-Heckman measure.
For a Borel subset U of (ā’Īµ, Īµ), the Duistermaat-Heckman measure is, by deļ¬-
nition,
Ļn
mDH (U ) = .
Āµā’1 (U ) n!

Using the fact that (Āµā’1 (ā’Īµ, Īµ), Ļ) is symplectomorphic to (Z Ć— (ā’Īµ, Īµ), Ļ) and,
moreover, they are isomorphic as hamiltonian S 1 -spaces, we obtain

Ļn
mDH (U ) = .
n!
ZĆ—U

Since Ļ = Ļ ā— Ļred ā’ d(xĪ±), its power is

Ļ n = n(Ļ ā— Ļred ā’ xdĪ±)nā’1 ā§ Ī± ā§ dx .

By the Fubini theorem, we then have

Ļ ā— (Ļred ā’ xĪ²)nā’1
ā§ Ī± ā§ dx .
mDH (U ) =
(n ā’ 1)!
U Z
196 30 DUISTERMAAT-HECKMAN THEOREMS

Therefore, the Radon-Nikodym derivative of mDH with respect to the Lebesgue
measure, dx, is

Ļ ā— (Ļred ā’ xĪ²)nā’1
ā§ Ī± = vol(Mx ) .
f (x) =
(n ā’ 1)!
Z

The previous discussion proves that, for x ā 0, f (x) is a polynomial in x.
The same holds for a neighborhood of any other regular value of Āµ, because we
may change the moment map Āµ by an arbitrary additive constant.
Homework 23: S 1 -Equivariant Cohomology
1. Let M be a manifold with a circle action and X # the vector ļ¬eld on M
generated by S 1 . The algebra of S 1 -equivariant forms on M is the
algebra of S 1 -invariant forms on M tensored with complex polynomials in
x,
1
ā„¦ā¢ 1 (M ) := (ā„¦ā¢ (M ))S ā—R C[x] .
S
The product ā§ on ā„¦ā¢ 1 (M ) combines the wedge product on ā„¦ā¢ (M ) with
S
the product of polynomials on C[x].
S
on C[x] where the monomial x has degree 2. Check that (ā„¦ā¢ 1 (M ), ā§)
S
is then a supercommutative graded algebra, i.e.,
Ī± ā§ Ī² = (ā’1)deg Ī±Ā·deg Ī² Ī² ā§ Ī±
for elements of pure degree Ī±, Ī² ā ā„¦ā¢ 1 (M ).
S
ā¢
(b) On ā„¦S 1 (M ) we deļ¬ne an operator
dS 1 := d ā— 1 ā’ Ä±X # ā— x .
In other words, for an elementary form Ī± = Ī± ā— p(x),
dS 1 Ī± = dĪ± ā— p(x) ā’ Ä±X # Ī± ā— xp(x) .
The operator dS 1 is called the Cartan diļ¬erentiation. Show that
dS 1 is a superderivation of degree 1, i.e., check that it increases degree
by 1 and that it satisļ¬es the super Leibniz rule:
dS 1 (Ī± ā§ Ī²) = (dS 1 Ī±) ā§ Ī² + (ā’1)deg Ī± Ī± ā§ dS 1 Ī² .

(c) Show that d2 1 = 0.
S
Cartan magic formula.
Hint:

2. The previous exercise shows that the sequence
d d d
1 1 1
0 ā’ā’ ā„¦0 1 (M ) ā’ā’ ā„¦1 1 (M ) ā’ā’ ā„¦2 1 (M ) ā’ā’ . . .
S S S
S S S

forms a graded complex whose cohomology is called the equivariant co-
homology20 of M for the given action of S 1 . The kth equivariant coho-
mology group of M is
ker dS 1 : ā„¦k 1 ā’ā’ ā„¦k+1
S1
k S
HS1 (M ) := .
kā’1
im dS 1 : ā„¦S 1 ā’ā’ ā„¦k 1
S
20 The equivariant cohomology of a topological space M endowed with a continuous
action of a topological group G is, by deļ¬nition, the cohomology of the diagonal quotient
(M Ć— EG)/G, where EG is the universal bundle of G, i.e., EG is a contractible space where
G acts freely. H. Cartan [21, 58] showed that, for the action of a compact Lie group G on a
manifold M , the de Rham model (ā„¦ā¢ (M ), dG ) computes the equivariant cohomology, where
G
ā„¦ā¢ (M ) are the G-equivariant forms on M . [8, 9, 29, 53] explain equivariant cohomology in
G
the symplectic context and [58] discusses equivariant de Rham theory and many applications.

197
198 HOMEWORK 23

(a) What is the equivariant cohomology of a point?
(b) What is the equivariant cohomology of S 1 with its multiplication
action on itself?
(c) Show that the equivariant cohomology of a manifold M with a free
S 1 -action is isomorphic to the ordinary cohomology of the quotient
space M/S 1 .
Let Ļ : M ā’ M/S 1 be projection. Show that
Hint:
Ļā— : H ā¢ (M/S 1 ) ā¢
ā’ā’ HS 1 (M )
ā— Ī± ā— 1]
[Ī±] ā’ā’ [Ļ
is a well-deļ¬ned isomorphism. It helps to choose a connection on the principal
S 1 -bundle M ā’ M/S 1 , that is, a 1-form Īø on M such that LX # Īø = 0 and
Ä±X # Īø = 1. Keep in mind that a form Ī² on M is of type Ļ ā— Ī± for some Ī± if and
only if it is basic, that is LX # Ī² = 0 and Ä±X # Ī² = 0.

3. Suppose that (M, Ļ) is a symplectic manifold with an S 1 -action. Let
Āµ ā C ā (M ) be a real function. Consider the equivariant form
Ļ := Ļ ā— 1 + Āµ ā— x .
Show that Ļ is equivariantly closed, i.e., dS 1 Ļ = 0 if and only if Āµ
is a moment map. The equivariant form Ļ is called the equivariant
symplectic form.
4. Let M 2n be a compact oriented manifold, not necessarily symplectic, acted
1
upon by S 1 . Suppose that the set M S of ļ¬xed points for this action is
ļ¬nite. Let Ī±(2n) be an S 1 -invariant form which is the top degree part of
1
an equivariantly closed form of even degree, that is, Ī±(2n) ā ā„¦2n (M )S is
such that there exists Ī± ā ā„¦ā¢ 1 (M ) with
S

Ī± = Ī±(2n) + Ī±(2nā’2) + . . . + Ī±(0)
1
where Ī±(2k) ā (ā„¦2k (M ))S ā— C[x] and dS 1 Ī± = 0.
1
(a) Show that the restriction of Ī±(2n) to M \ M S is exact.
1
The generator X # of the S 1 -action does not vanish on M \ M S .
Hint:
Y,X #
1
Hence, we can deļ¬ne a connection on M S by Īø(Y ) = , where Ā·, Ā· is
X # ,X #
1
some S 1 -invariant metric on M . Use Īø ā ā„¦1 (M \ M S ) to chase the primitive
of Ī±(2n) all the way up from Ī±(0) .
(b) Compute the integral of Ī±(2n) over M .
Stokesā™ theorem allows to localize the answer near the ļ¬xed points.
Hint:

This exercise is a very special case of the Atiyah-Bott-Berline-Vergne lo-
calization theorem for equivariant cohomology [8, 14].
5. What is the integral of the symplectic form Ļ on a surface with a hamil-
tonian S 1 -action, knowing that the S 1 -action is free outside a ļ¬nite set of
ļ¬xed points?
Exercises 3 and 4.
Hint:
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Index
action Atiyah-Guillemin-Sternberg the-
orem, 170
moduli space, 158
Yang-Mills theory, 155
coordinates, 111
deļ¬nition, 128
Banyaga theorem, 92
eļ¬ective, 171
base, 155
free, 135
basis
gauge group, 158
for skew-symmetric bilinear maps,
hamiltonian, 129, 130, 133, 164
3
inļ¬nitesimal, 156, 164
Beltrami
locally free, 135
Laplace-Beltrami operator, 98
minimizing, 115, 120
Betti number, 100
of a path, 114, 115, 119
biholomorphic map, 83
principle of least action, 114
bilinear map, see skew-symmetric bi-
smooth, 129
linear map
symplectic, 129
billiards, 30
transitive, 135
Birkhoļ¬
action-angle coordinates, 111
PoincarĀ“-Birkhoļ¬ theorem, 33
e
blowup, 189
Borel subset, 191
action, 131, 137
Bott
representation, 130, 131
moduli space, 158
almost complex manifold, 70
Morse-Bott function, 175
almost complex structure
Yang-Mills theory, 155
compatibility, 70 bracket
contractibility, 77 Lie, 108
deļ¬nition, 70 Poisson, 108, 109, 134, 164
integrability, 75, 82
three geometries, 67 C 1 -topology, 53, 54
almost complex submanifold, 76 canonical
almost symplectic manifold, 74 symplectic form on a coadjoint
angle coordinates, 110 orbit, 139, 150, 162
angular momentum, 137, 138 symplectomorphism, 12
(J-)anti-holomorphic tangent vectors, canonical form on T ā— X
78 coordinate deļ¬nition, 9, 10
antisymmetry, 108 intrinsic deļ¬nition, 10
arc-length, 25 naturality, 11
Archimedes, 192 Cartan
Arnold diļ¬erentiation, 197
Arnold-Liouville theorem, 110 magic formula, 36, 40, 44
conjecture, 33, 55, 56 Cauchy-Riemann equations, 84
Atiyah characteristic distribution, 53

207
208 INDEX

chart compatibility, 68, 77
complex, 83 on a vector space, 68
Darboux, 7 polar decomposition, 69
Chern complex surface, 103
ļ¬rst Chern class, 194, 195 complex torus, 103
Chevalley cohomology, 165 complex vector space, 68
Christoļ¬el complex-antilinear cotangent vectors,
equations, 120 79
symbols, 120 complex-linear cotangent vectors, 79
circle bundle, 159 complex-valued form, 79
classical mechanics, 107 conehead orbifold, 150
action, 131, 137, 162 conjecture
orbit, 139, 162 Arnold, 33, 55, 56
representation, 130, 131 Hodge, 101
codiļ¬erential, 98 Seifert, 65
cohomology Weinstein, 65, 66
S 1 -equivariant, 197 conjugation, 131
Chevalley, 165 connectedness, 170, 175, 176
de Rham, 13, 39 connection
Dolbeault, 81 ļ¬‚at, 159
equivariant, 197 form, 156
Lie algebra, 165 moduli space, 159
coindex, 175, 176 on a principal bundle, 155
coisotropic space, 158
embedding, 49, 53 conormal
subspace, 8 bundle, 18
commutator ideal, 165 space, 18
comoment map, 133, 134, 164 conservative system, 113
compatible constrained system, 114
almost complex structure, 70, constraint set, 114
74 contact
contact structure on S 2nā’1 , 64
complex structure, 68
linear structures, 72 dynamics, 63
triple, 70, 75 element, 57, 61, 62
complete vector ļ¬eld, 129 example of contact structure, 58
completely integrable system, 110 local contact form, 57
complex local normal form, 59
atlas, 89 locally deļ¬ning 1-form, 57
chart, 83 manifold, 57
diļ¬erentials, 86, 87 point, 57
Hodge theory, 99 structure, 57
manifold, 83 contactomorphism, 63
projective space, 89, 95, 96, 103, contractibility, 77
136, 169, 181 convexity, 170
complex structure cotangent bundle
209
INDEX

canonical symplectomorphism, connection, 156
S 4 is not an almost complex man-
11, 12
conormal bundle, 18 ifold, 76
coordinates, 9 embeddiing
is a symplectic manifold, 9 closed, 15
lagrangian submanifold, 16ā“18 deļ¬nition, 15
projectivization, 59 embedding
sphere bundle, 59 coisotropic, 49, 53
zero section, 16 isotropic, 53
critical set, 175 lagrangian, 51
curvature form, 157 energy
classical mechanics, 107
Dā™Alembert energy-momentum map, 153
variational principle, 114 kinetic, 112, 113
Darboux potential, 112, 113
chart, 7 equations
theorem, 7, 45, 46 Christoļ¬el, 120
theorem for contact manifolds,
Euler-Lagrange, 105, 120, 123
59
Hamilton, 123, 148
theorem in dimension two, 50
Hamilton-Jacobi, 105
de Rham cohomology, 13, 39
of motion, 113
deformation equivalence, 42
equivariant
deformation retract, 40
cohomology, 197
Delzant
coisotropic embedding, 193
construction, 183, 185, 186
form, 197
example of Delzant polytope, 177
moment map, 134
example of non-Delzant poly-
symplectic form, 198
tope, 178
tubular neighborhood theorem,
polytope, 177, 189
143
theorem, 179, 189
euclidean
Dolbeault
distance, 24, 25
cohomology, 81
inner product, 24, 25
theorem, 88
measure, 191
theory, 78
norm, 25
dual function, 122, 126
space, 24
Duistermaat-Heckman
Euler
measure, 191
Euler-Lagrange equations, 105,
polynomial, 191, 192
116, 120, 123
theorem, 191, 194
variational principle, 114
dunce cap orbifold, 150
evaluation map, 129
dynamical system, 33
exactly homotopic to the identity,
56
eļ¬ective
example
action, 171
2-sphere, 97
moment map, 176
210 INDEX

complex projective space, 89, 103, Taubes, 103
181 toric manifold, 172
complex submanifold of a KĀØh-a weighted projective space, 151
ler manifold, 103 exponential map, 35
complex torus, 103
facet, 178
Delzant construction, 181
FernĀ“ndez-Gotay-Gray example, 102
a
FernĀ“ndez-Gotay-Gray, 102
a
ļ¬rst Chern class, 194, 195
Gompf, 103
ļ¬rst integral, 109
hermitian matrices, 132
ļ¬xed point, 29, 33, 55
Hirzebruch surfaces, 178, 190
ļ¬‚at connection, 159
Hopf surface, 102
ļ¬‚ow, 35
Kodaira-Thurston, 102
form
McDuļ¬, 43
area, 50
non-singular projective variety,
canonical, 9, 10
95
complex-valued, 79
of almost complex manifold, 76
connection, 156
of compact complex manifold,
curvature, 157
101
de Rham, 6
of compact KĀØhler manifold, 96,
a
Fubini-Study, 96, 169
101
harmonic, 98, 99
of compact symplectic manifold,
101 KĀØhler, 90, 98
a
of complex manifold, 89 Killing, 158
of contact manifold, 62 Liouville, 13
of contact structure, 58 on a complex manifold, 85
of Delzant polytope, 177 positive, 92
of hamiltonian actions, 129, 130 symplectic, 6
of inļ¬nite-dimensional symplec- tautological, 9, 10, 20
tic manifold, 158 type, 79
of KĀØhler submanifold, 95
a free action, 135
of lagrangian submanifold, 16 Fubini theorem, 195
of mechanical system, 113 Fubini-Study form, 96, 169
of non-almost-complex manifold, function
76 biholomorphic, 89
of non-Delzant polytope, 178 dual, 122, 126
of reduced system, 153 generating, 29
of symplectic manifold, 6, 9 hamiltonian, 106, 134
of symplectomorphism, 22 J-holomorphic, 82
oriented surfaces, 50 Morse-Bott, 175
product of KĀØhler manifolds, 103
a stable, 121, 125
quotient topology, 135 strictly convex, 121, 125
reduction, 169
G-space, 134
Riemann surface, 103
gauge
simple pendulum, 112
group, 158, 159
spherical pendulum, 152
Stein manifold, 103 theory, 155
211
INDEX

transformation, 159 vector ļ¬eld, 105, 106
Gauss lemma, 28 harmonic form, 98, 99
generating function, 17, 22, 23, 29 Hausdorļ¬ quotient, 136
geodesic Heckman, see Duistermaat-Heckman
curve, 25 hermitian matrix, 132
ļ¬‚ow, 26, 27 hessian, 121, 125, 175
Hirzebruch surface, 178, 190
geodesically convex, 25
Hodge
minimizing, 25, 119, 120
Gompf construction, 103 complex Hodge theory, 99
Gotay conjecture, 101
coisotropic embedding, 53 decomposition, 98, 99
FernĀ“ndez-Gotay-Gray, 102
a diamond, 101
gradient vector ļ¬eld, 107 number, 100
ā—-operator, 98
gravitational potential, 113
gravity, 112, 152 theorem, 98ā“100
Gray theory, 98
FernĀ“ndez-Gotay-Gray (A. Gray),
a (J-)holomorphic tangent vectors, 78
102 homotopy
theorem (J. Gray), 59 deļ¬nition, 40
Gromov formula, 39, 40
pseudo-holomorphic curve, 67, invariance, 39
82 operator, 40
group Hopf
gauge, 158, 159 ļ¬bration, 64, 156
S 4 is not almost complex, 76
Lie, 128
of symplectomorphisms, 12, 53 surface, 102
one-parameter group of diļ¬eo- vector ļ¬eld, 64
morphisms, 127, 128
immersion, 15
product, 149
index, 175, 176
semisimple, 158
inļ¬nitesimal action, 156, 164
structure, 155
integrable
Guillemin
almost complex structure, 75,
Atiyah-Guillemin-Sternberg the-
82
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