<< стр. 8(всего 9)СОДЕРЖАНИЕ >>
в‡ђв‡’ П†(z), ei = О»i , в€Ђi в€€ I
в€’ПЂ|zi |2 + О»i = О»i , в€Ђi в€€ I
в‡ђв‡’
в‡ђв‡’ в€Ђi в€€ I .
zi = 0 ,

вЂў The Td -action on Cd preserves П†, so the Td -action takes Z = П†в€’1 (в€† )
onto itself, so Td acts on Z.

Exercise. The stratiп¬Ѓcation of Z is just the stratiп¬Ѓcation of Z into Td
orbit types. More speciп¬Ѓcally, if z в€€ Z and П†(z) в€€ 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 в€€ в€† | y, ei = О»i , в€Ђi в€€ I} ,
and
(Td )I = {(e2ПЂit1 , . . . , e2ПЂitd ) | e2ПЂits = 1, в€Ђs в€€ I}
/

Suppose that z = (z1 , . . . , zd ) в€€ Cd . Then
Hint:
(e2ПЂit1 z1 , . . . , e2ПЂitd zd ) = (z1 , . . . , zd )
if and only if e2ПЂits = 1 whenever zs = 0.

в™¦

In order to show that N acts freely on Z, consider the worst case scenario of
points z в€€ 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 в€† . Then y satisп¬Ѓes n equations

i в€€ I = {i1 , . . . , in } .
y, ei = О»i ,

Lemma 29.3 Let z в€€ Z be such that П†(z) is a vertex of в€† . Let (Td )I be the
stabilizer of z. Then the map ПЂ : Td в†’ Tn maps (Td )I bijectively onto Tn .

Since N = ker ПЂ, 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 П†(z) = y is a vertex of в€† . Renumber
the indices so that
I = (1, 2, . . . , n) .
Then
(Td )I = {(e2ПЂit1 , . . . , e2ПЂitn , 1, . . . , 1) | ti в€€ R} .
The hyperplanes meeting at y are

y , ei = О» i , i = 1, . . . , n .

By deп¬Ѓnition of Delzant polytope, the set ПЂ(e1 ), . . . , ПЂ(en ) is a basis of Zn . Thus,
ПЂ : (Td )I в†’ 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 (Mв€† , П‰в€† ) from в€†. We already have that

Mв€† = Z/N

is a compact 2n-dimensional manifold. Let П‰в€† be the reduced symplectic form.
186 29 DELZANT CONSTRUCTION

Claim. The manifold (Mв€† , П‰в€† ) is a hamiltonian Tn -space with a moment map
Вµ having image Вµ(Mв€† ) = в€†.
Suppose that z в€€ Z. The stabilizer of z with respect to the Td -action is
(Td )I , and
(Td )I в€© N = {e} .
In the worst case scenario, FI is a vertex of в€† and (Td )I is an n-dimensional
subgroup of Td . In any case, there is a right inverse map ПЂ в€’1 : Tn в†’ (Td )I .
Thus, the exact sequence

0 в€’в†’ N в€’в†’ Td в€’в†’ Tn в€’в†’ 0

splits, and Td = N Г— Tn .
Apply the results on reduction for product groups (Section 24.3) to our
situation of Td = N Г— Tn acting on (Mв€† , П‰в€† ). The moment map is

П† : Cd в€’в†’ (Rd )в€— = nв€— вЉ• (Rn )в€— .

Let j : Z в†’ Cd be the inclusion map, and let

pr1 : (Rd )в€— в€’в†’ nв€— pr2 : (Rd )в€— в€’в†’ (Rn )в€—
and

be the projection maps. The map

pr2 в—¦ П† в—¦ j : Z в€’в†’ (Rn )в€—

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

Вµ : Mв€† в€’в†’ (Rn )в€—

such that
Вµ в—¦ p = pr2 в—¦ П† в—¦ j .
The image of Вµ is equal to the image of pr2 в—¦ П† в—¦ j. We showed earlier that
П†(Z) = в€† . Thus

Image of Вµ = pr2 (в€† ) = pr2 в—¦ ПЂ в€— (в€†) = в€† .
id

Thus (Mв€† , П‰в€† ) is the required toric manifold corresponding to в€†.

29.4 Idea Behind the Delzant Construction

We use the idea that Rd is вЂњuniversalвЂќ in the sense that any n-dimensional
polytope в€† with d facets can be obtained by intersecting the negative orthant
Rd with an aп¬ѓne plane A. Given в€†, to construct A п¬Ѓrst write в€† as:
в€’

в€† = {x в€€ Rn | x, vi в‰¤ О»i , i = 1, . . . , d} .
187
29.4 Idea Behind the Delzant Construction

Deп¬Ѓne
ПЂ в€— : Rn
ПЂ : Rd в€’в†’ Rn в€’в†’ Rd .
with dual map
в€’в†’ vi
ei

Then
ПЂ в€— в€’ О» : Rn в€’в†’ Rd
is an aп¬ѓne map, where О» = (О»1 , . . . , О»d ). Let A be the image of ПЂ в€— в€’ О». Then
A is an n-dimensional aп¬ѓne plane.
Claim. We have the equality (ПЂ в€— в€’ О»)(в€†) = Rd в€© A.
в€’

Proof. Let x в€€ Rn . Then

(ПЂ в€— в€’ О»)(x) в€€ Rd ПЂ в€— (x) в€’ О», ei в‰¤ 0, в€Ђi
в‡ђв‡’
в€’
в‡ђв‡’ x, ПЂ(ei ) в€’ О»i в‰¤ 0, в€Ђi
в‡ђв‡’ x, vi в‰¤ О»i , в€Ђi
в‡ђв‡’ xв€€в€†.

We conclude that в€† Rd в€© A. Now Rd is the image of the moment map
в€’ в€’
for the standard hamiltonian action of T on Cd
d

П† : Cd в€’в†’ Rd
в€’в†’ в€’ПЂ(|z1 |2 , . . . , |zd |2 ) .
(z1 , . . . , zd )

Facts.

вЂў The set П†в€’1 (A) вЉ‚ Cd is a compact submanifold. Let i : П† в†’ Cd denote
inclusion. Then iв€— П‰0 is a closed 2-form which is degenerate. Its kernel is
an integrable distribution. The corresponding foliation is called the null
foliation.
вЂў The null foliation of iв€— П‰0 is a principal п¬Ѓbration, so we take the quotient:

в†’ П†в€’1 (A)
N
в†“
= П†в€’1 (A)/N
Mв€†

Let П‰в€† be the reduced symplectic form.
вЂў The (non-eп¬Ђective) action of Td = N Г—Tn on П†в€’1 (A) has a вЂњmoment mapвЂќ
with image П†(П†в€’1 (A)) = в€†. (By вЂњmoment mapвЂќ we mean a map satisfying
the usual deп¬Ѓnition even though the closed 2-form is not symplectic.)

Theorem 29.4 For any x в€€ в€†, we have that Вµв€’1 (x) is a single Tn -orbit.
188 29 DELZANT CONSTRUCTION

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

y в€€ в€† = ПЂ в€— (в€†) в‡ђв‡’ П†в€’1 (y) вЉ† Z .

Suppose that y = ПЂ в€— (x). Show that Вµв€’1 (x) = П†в€’1 (y)/N . But П†в€’1 (y) is a single
Td -orbit where Td = N Г— Tn , hence Вµв€’1 (x) is a single Tn -orbit.
Therefore, for toric manifolds, в€† is the orbit space.
Now в€† is a manifold with corners. At every point p in a face F , the tangent
space Tp в€† is the subspace of Rn tangent to F . We can visualize (Mв€† , П‰в€† , Tn , Вµ)
from в€† as follows. First take the product Tn Г— в€†. Let p lie in the interior of
Tn Г— в€†. The tangent space at p is Rn Г— (Rn )в€— . Deп¬Ѓne П‰p by:

П‰p (v, Оѕ) = Оѕ(v) = в€’П‰p (Оѕ, v) and П‰p (v, v ) = П‰(Оѕ, Оѕ ) = 0 .

for all v, v в€€ Rn and Оѕ, Оѕ в€€ (Rn )в€— . Then П‰ is a closed nondegenerate 2-form
on the interior of Tn Г— в€†. At the corner there are directions missing in (Rn )в€— ,
so П‰ 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 , П‰ = dОё в€§ dh, S 1 , Вµ = h) ,

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

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

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

(eiОё1 , eiОё2 , eiОё3 ) В· [z0 , z1 , z2 , z3 ] = [z0 , eiОё1 z1 , eiОё2 z2 , eiОё3 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 п¬Ѓxed points?

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

p

22 В˜
222 В В˜
В˜
d В˜
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 п¬Ѓxed points?
30 Duistermaat-Heckman Theorems
30.1 Duistermaat-Heckman Polynomial
П‰n
Let (M 2n , П‰) be a symplectic manifold. Then is the symplectic volume form.
n!

Deп¬Ѓnition 30.1 The Liouville measure (or symplectic measure) of a
Borel subset 17 U of M is
П‰n
mП‰ (U) = .
U n!

Let G be a torus. Suppose that (M, П‰, G, Вµ) is a hamiltonian G-space, and
that the moment map Вµ is proper.
Deп¬Ѓnition 30.2 The Duistermaat-Heckman measure, mDH , on gв€— is the
push-forward of mП‰ by Вµ : M в†’ gв€— . That is,
П‰n
mDH (U ) = (Вµв€— mП‰ )(U ) =
n!
Вµв€’1 (U )

for any Borel subset U of gв€— .
For a compactly-supported function h в€€ C в€ћ (gв€— ), we deп¬Ѓne its integral with
respect to the Duistermaat-Heckman measure to be
П‰n
(h в—¦ Вµ)
h dmDH = .
n!
M
gв€—

On gв€— 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
gв€— gв€—

Theorem 30.3 (Duistermaat-Heckman, 1982 ) The Duistermaat-
Heckman measure is a piecewise polynomial multiple of Lebesgue (or euclidean)
measure m0 on gв€— Rn , that is, the Radon-Nikodym derivative
dmDH
f=
dm0
is piecewise polynomial. More speciп¬Ѓcally, for any Borel subset U of g в€— ,

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

where dx = dm0 is the Lebesgue volume form on U and f : gв€— Rn в†’ R is
polynomial on any region consisting of regular values of Вµ.
17 Theset B of Borel subsets is the Пѓ-ring generated by the set of compact subsets, i.e., if
A, B в€€ B, then A \ B в€€ B, and if Ai в€€ B, i = 1, 2, . . ., then в€Єв€ћ Ai в€€ 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, , 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 (2ПЂ)n when в€† is n-dimensional. Thus
the symplectic volume of (Mв€† , П‰в€† ) is (2ПЂ)n times the euclidean volume of в€†.
Example. Consider (S 2 , П‰ = dОё в€§ dh, S 1 , Вµ = h). The image of Вµ is the interval
[в€’1, 1]. The Lebesgue measure of [a, b] вЉ† [в€’1, 1] is
m0 ([a, b]) = b в€’ a .
The Duistermaat-Heckman measure of [a, b] is

dОё dh = 2ПЂ(b в€’ a) .
mDH ([a, b]) =
{(Оё,h)в€€S 2 |aв‰¤hв‰¤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 , П‰), we have
mDH = 2ПЂ m0 .
в™¦

30.2 Local Form for Reduced Spaces
Let (M, П‰, G, Вµ) be a hamiltonian G-space, where G is an n-torus.18 Assume
that Вµ is proper. If G acts freely on Вµв€’1 (0), it also acts freely on nearby levels
Вµв€’1 (t), t в€€ gв€— and t в‰€ 0. Consider the reduced spaces
Mred = Вµв€’1 (0)/G Mt = Вµв€’1 (t)/G
and
with reduced symplectic forms П‰red and П‰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 = Вµв€’1 (0) and let
i : Z в†’ M be the inclusion map. We п¬Ѓx a connection form О± в€€ в„¦1 (Z) for the
principal bundle
EZ
S1 вЉ‚

ПЂ
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 # О± = 0 and Д±X # О± = 1, where X # is the inп¬Ѓnitesimal generator for
the S 1 -action. From О± we construct a 2-form on the product manifold ZГ—(в€’Оµ, Оµ)
by the recipe
Пѓ = ПЂ в€— П‰red в€’ d(xО±) ,
gв€— . (By abuse of
x being a linear coordinate on the interval (в€’Оµ, Оµ) вЉ‚ R
notation, we shorten the symbols for forms on ZГ—(в€’Оµ, Оµ) which arise by pullback
via projection onto each factor.)
Lemma 30.5 The 2-form Пѓ is symplectic for Оµ small enough.

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

Пѓ|x=0 = ПЂ в€— П‰red + О± в€§ dx ,
which satisп¬Ѓes
в€‚
Пѓ|x=0 X # , =1,
в€‚x
so Пѓ is nondegenerate along Z Г— {0}. Since nondegeneracy is an open condition,
we conclude that Пѓ is nondegenerate for x in a suп¬ѓciently small neighborhood
of 0.
Notice that Пѓ is invariant with respect to the S 1 -action on the п¬Ѓrst factor of
Z Г— (в€’Оµ, Оµ). In fact, this S 1 -action is hamiltonian with moment map given by
projection onto the second factor,
x : Z Г— (в€’Оµ, Оµ) в€’в†’ (в€’Оµ, Оµ) ,

as is easily veriп¬Ѓed:
Д±X # Пѓ = в€’Д±X # d(xО±) = в€’ LX # (xО±) +d Д±X # (xО±) = dx .
0 x

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

Proof. The inclusion i0 : Z в†’ Z Г— (в€’Оµ, Оµ) as Z Г— {0} and the natural inclusion
i : Z в†’ M are S 1 -equivariant coisotropic embeddings. Moreover, they satisfy
iв€— Пѓ = iв€— П‰ since both sides are equal to ПЂ в€— П‰red , and the moment maps coincide on
0
Z because iв€— x = 0 = iв€— Вµ. Replacing Оµ 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 , П‰0 ), (M1 , П‰1 ) be symplectic manifolds of dimension 2n, G a compact Lie group acting
on (Mi , П‰i ), i = 0, 1, in a hamiltonian way with moment maps Вµ0 and Вµ1 , respectively, Z
a manifold of dimension k в‰Ґ n with a G-action, and О№i : Z в†’ Mi , i = 0, 1, G-equivariant
coisotropic embeddings. Suppose that О№в€— П‰0 = О№в€— П‰1 and О№в€— Вµ0 = О№в€— Вµ1 . Then there exist G-
0 1 0 1
invariant neighborhoods U0 and U1 of О№0 (Z) and О№1 (Z) in M0 and M1 , respectively, and a
G-equivariant symplectomorphism П• : U0 в†’ U1 such that П• в—¦ О№0 = О№1 and Вµ0 = П•в€— Вµ1 .
194 30 DUISTERMAAT-HECKMAN THEOREMS

Therefore, in order to compare the reduced spaces

Mt = Вµв€’1 (t)/S 1 , tв‰€0,

we can work in Z Г— (в€’Оµ, Оµ) and compare instead the reduced spaces

xв€’1 (t)/S 1 , tв‰€0.

Proposition 30.7 The reduced space (Mt , П‰t ) is symplectomorphic to

(Mred , П‰red в€’ tОІ) ,

where ОІ is the curvature form of the connection О±.

Proof. By Lemma 30.6, (Mt , П‰t ) is symplectomorphic to the reduced space at
level t for the hamiltonian space (Z Г— (в€’Оµ, Оµ), Пѓ, S 1 , x). Since xв€’1 (t) = Z Г— {t},
where S 1 acts on the п¬Ѓrst factor, all the manifolds xв€’1 (t)/S 1 are diп¬Ђeomorphic
to Z/S 1 = Mred. As for the symplectic forms, let О№t : Z Г— {t} в†’ Z Г— (в€’Оµ, Оµ) be
the inclusion map. The restriction of Пѓ to Z Г— {t} is

О№в€— Пѓ = ПЂ в€— П‰red в€’ tdО± .
t

By deп¬Ѓnition of curvature, dО± = ПЂ в€— ОІ. Hence, the reduced symplectic form on
xв€’1 (t)/S 1 is
П‰red в€’ tОІ .

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

Theorem 30.8 (Duistermaat-Heckman, 1982 ) The cohomology class
of the reduced symplectic form [П‰t ] varies linearly in t. More speciп¬Ѓcally,

[П‰t ] = [П‰red ] + tc ,

where c = [в€’ОІ] в€€ HdeRham (Mred ) is the п¬Ѓrst Chern class of the S 1 -bundle
2

Z в†’ 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 identiп¬Ѓed with
2ПЂiR under the exponential map exp : g 2ПЂiR в†’ S 1 , Оѕ в†’ eОѕ . Given a prin-
cipal S 1 -bundle, by this identiп¬Ѓcation the inп¬Ѓnitesimal action maps the gener-
ator 2ПЂi of 2ПЂiR to the generating vector п¬Ѓeld X # . A connection form A is
then an imaginary-valued 1-form on the total space satisfying LX # A = 0 and
Д±X # A = 2ПЂi. Its curvature form B is an imaginary-valued 2-form on the base
195
30.3 Variation of the Symplectic Volume

satisfying ПЂ в€— B = dA. By the Chern-Weil isomorphism, the п¬Ѓrst Chern class
i
of the principal S 1 -bundle is c = [ 2ПЂ B].
In this lecture, we identify the Lie algebra of S 1 with R and implicitly use
R в†’ S 1 , t в†’ e2ПЂit . Hence, given a principal
the exponential map exp : g
S 1 -bundle, the inп¬Ѓnitesimal action maps the generator 1 of R to X # , and here a
connection form О± is an ordinary 1-form on the total space satisfying LX # О± = 0
and Д±X # О± = 1. The curvature form ОІ is an ordinary 2-form on the base satisfying
ПЂ в€— ОІ = dО±. Consequently, we have A = 2ПЂiО±, B = 2ПЂiОІ and the п¬Ѓrst Chern
в™¦
class is given by c = [в€’ОІ].

30.3 Variation of the Symplectic Volume

Let (M, П‰, S 1 , Вµ) be a hamiltonian S 1 -space of dimension 2n and let (Mx , П‰x )
be its reduced space at level x. Proposition 30.7 or Theorem 30.8 imply that,
for x in a suп¬ѓciently narrow neighborhood of 0, the symplectic volume of Mx ,
nв€’1
(П‰red в€’ xОІ)nв€’1
П‰x
vol(Mx ) = = ,
(n в€’ 1)! (n в€’ 1)!
Mx Mred

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

ПЂ в€— (П‰red в€’ xОІ)nв€’1
в€§О± .
vol(Mx ) =
(n в€’ 1)!
Z

Recall that О± is a chosen connection form for the S 1 -bundle Z в†’ Mred and ОІ is
its curvature form.
Now we go back to the computation of the Duistermaat-Heckman measure.
For a Borel subset U of (в€’Оµ, Оµ), the Duistermaat-Heckman measure is, by deп¬Ѓ-
nition,
П‰n
mDH (U ) = .
Вµв€’1 (U ) n!

Using the fact that (Вµв€’1 (в€’Оµ, Оµ), П‰) is symplectomorphic to (Z Г— (в€’Оµ, Оµ), Пѓ) and,
moreover, they are isomorphic as hamiltonian S 1 -spaces, we obtain

Пѓn
mDH (U ) = .
n!
ZГ—U

Since Пѓ = ПЂ в€— П‰red в€’ d(xО±), its power is

Пѓ n = n(ПЂ в€— П‰red в€’ xdО±)nв€’1 в€§ О± в€§ dx .

By the Fubini theorem, we then have

ПЂ в€— (П‰red в€’ xОІ)nв€’1
в€§ О± в€§ dx .
mDH (U ) =
(n в€’ 1)!
U Z
196 30 DUISTERMAAT-HECKMAN THEOREMS

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

ПЂ в€— (П‰red в€’ xОІ)nв€’1
в€§ О± = vol(Mx ) .
f (x) =
(n в€’ 1)!
Z

The previous discussion proves that, for x в‰€ 0, f (x) is a polynomial in x.
The same holds for a neighborhood of any other regular value of Вµ, because we
may change the moment map Вµ 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 п¬Ѓeld 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
в„¦вЂў 1 (M ) := (в„¦вЂў (M ))S вЉ—R C[x] .
S
The product в€§ on в„¦вЂў 1 (M ) combines the wedge product on в„¦вЂў (M ) with
S
the product of polynomials on C[x].
S
on C[x] where the monomial x has degree 2. Check that (в„¦вЂў 1 (M ), в€§)
S
is then a supercommutative graded algebra, i.e.,
О± в€§ ОІ = (в€’1)deg О±В·deg ОІ ОІ в€§ О±
for elements of pure degree О±, ОІ в€€ в„¦вЂў 1 (M ).
S
вЂў
(b) On в„¦S 1 (M ) we deп¬Ѓne an operator
dS 1 := d вЉ— 1 в€’ Д±X # вЉ— x .
In other words, for an elementary form О± = О± вЉ— p(x),
dS 1 О± = dО± вЉ— p(x) в€’ Д±X # О± вЉ— xp(x) .
The operator dS 1 is called the Cartan diп¬Ђerentiation. Show that
dS 1 is a superderivation of degree 1, i.e., check that it increases degree
by 1 and that it satisп¬Ѓes the super Leibniz rule:
dS 1 (О± в€§ ОІ) = (dS 1 О±) в€§ ОІ + (в€’1)deg О± О± в€§ dS 1 ОІ .

(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 в€’в†’ в„¦0 1 (M ) в€’в†’ в„¦1 1 (M ) в€’в†’ в„¦2 1 (M ) в€’в†’ . . .
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 : в„¦k 1 в€’в†’ в„¦k+1
S1
k S
HS1 (M ) := .
kв€’1
im dS 1 : в„¦S 1 в€’в†’ в„¦k 1
S
20 The equivariant cohomology of a topological space M endowed with a continuous
action of a topological group G is, by deп¬Ѓnition, the cohomology of the diagonal quotient
(M Г— 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 (в„¦вЂў (M ), dG ) computes the equivariant cohomology, where
G
в„¦вЂў (M ) are the G-equivariant forms on M . [8, 9, 29, 53] explain equivariant cohomology in
G
the symplectic context and  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 ПЂ : M в†’ M/S 1 be projection. Show that
Hint:
ПЂв€— : H вЂў (M/S 1 ) вЂў
в€’в†’ HS 1 (M )
в€— О± вЉ— 1]
[О±] в€’в†’ [ПЂ
is a well-deп¬Ѓned isomorphism. It helps to choose a connection on the principal
S 1 -bundle M в†’ M/S 1 , that is, a 1-form Оё on M such that LX # Оё = 0 and
Д±X # Оё = 1. Keep in mind that a form ОІ on M is of type ПЂ в€— О± for some О± if and
only if it is basic, that is LX # ОІ = 0 and Д±X # ОІ = 0.

3. Suppose that (M, П‰) is a symplectic manifold with an S 1 -action. Let
Вµ в€€ C в€ћ (M ) be a real function. Consider the equivariant form
П‰ := П‰ вЉ— 1 + Вµ вЉ— x .
Show that П‰ is equivariantly closed, i.e., dS 1 П‰ = 0 if and only if Вµ
is a moment map. The equivariant form П‰ 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 п¬Ѓxed points for this action is
п¬Ѓnite. Let О±(2n) be an S 1 -invariant form which is the top degree part of
1
an equivariantly closed form of even degree, that is, О±(2n) в€€ в„¦2n (M )S is
such that there exists О± в€€ в„¦вЂў 1 (M ) with
S

О± = О±(2n) + О±(2nв€’2) + . . . + О±(0)
1
where О±(2k) в€€ (в„¦2k (M ))S вЉ— C[x] and dS 1 О± = 0.
1
(a) Show that the restriction of О±(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 deп¬Ѓne a connection on M S by Оё(Y ) = , where В·, В· is
X # ,X #
1
some S 1 -invariant metric on M . Use Оё в€€ в„¦1 (M \ M S ) to chase the primitive
of О±(2n) all the way up from О±(0) .
(b) Compute the integral of О±(2n) over M .
StokesвЂ™ theorem allows to localize the answer near the п¬Ѓxed 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 П‰ on a surface with a hamil-
tonian S 1 -action, knowing that the S 1 -action is free outside a п¬Ѓnite set of
п¬Ѓxed points?
Exercises 3 and 4.
Hint:
References
 Abraham, R., Marsden, J. E., Foundations of Mechanics, second edition,
 Aebischer, B., Borer, M., KВЁlin, M., Leuenberger, Ch., Reimann, H.M.,
a
Symplectic Geometry. An Introduction Based on the Seminar in Bern,
1992, Progress in Mathematics 124, BirkhВЁuser Verlag, Basel, 1994.
a
 Arnold, V., Mathematical Methods of Classical Mechanics, Graduate Texts
in Math. 60, Springer-Verlag, New York, 1978.
 Arnold, V., First steps of symplectic topology, VIIIth International
Congress on Mathematical Physics (Marseille, 1986), 1-16, World Sci. Pub-
lishing, Singapore, 1987.
 Arnold, V., Givental, A., Symplectic geometry, Dynamical Systems IV,
Symplectic Geometry and its Applications, edited by Arnold, V. and
Novikov, S., Encyclopaedia of Mathematical Sciences 4, Springer-Verlag,
Berlin-New York, 1990.
 Atiyah, M., Convexity and commuting Hamiltonians, Bull. London Math.
Soc. 14 (1982), 1-15.
 Atiyah, M., Bott, R., The Yang-Mills equations over Riemann surfaces,
Topology 23 (1984), 1-28. Philos. Trans. Roy. Soc. London 308 (1983),
523-615.
 Atiyah, M., Bott, R., The moment map and equivariant cohomology, Topol-
ogy 23 (1984), 1-28.
 Audin, M., The Topology of Torus Actions on Symplectic Manifolds,
Progress in Mathematics 93, BirkhВЁuser Verlag, Basel, 1991.
a
 Audin, M., Spinning Tops. A Course on Integrable Systems, Cambridge
Studies in Advanced Mathematics 51, Cambridge University Press, Cam-
bridge, 1996.
 Audin, M., Lafontaine, J., Eds., Holomorphic Curves in Symplectic Geom-
etry, Progress in Mathematics 117, BirkhВЁuser Verlag, Basel, 1994.
a
 Auroux, D., Asymptotically holomorphic families of symplectic submani-
folds, Geom. Funct. Anal. 7 (1997), 971-995.
 Berline, N., Getzler, E., Vergne, M., Heat Kernels and Dirac Operators,
Grundlehren der Mathematischen Wissenschaften 298, Springer-Verlag,
Berlin, 1992.
 Berline, N., Vergne, M., Classes caractВґristiques Вґquivariantes, formule de
e e
localisation en cohomologie Вґquivariante, C. R. Acad. Sci. Paris SВґr. I
e e
Math. 295 (1982), 539-541.

199
200 REFERENCES

 Berline, N., Vergne, M., ZВґros dвЂ™un champ de vecteurs et classes caractВґris-
e e
tiques Вґquivariantes, Duke Math. J. 50 (1983), 539-549.
e
 Biran, P., A stability property of symplectic packing, Invent. Math. 136
(1999), 123-155.
 Bredon, G., Introduction to Compact Transformation Groups, Pure and
Applied Mathematics 46, Academic Press, New York-London, 1972.
 Bott, R., Tu, L., Diп¬Ђerential Forms in Algebraic Topology, Graduate Texts
in Mathematics 82, Springer-Verlag, New York-Berlin, 1982.
 Cannas da Silva, A., Guillemin, V., Woodward, C., On the unfolding of
folded symplectic structures, Math. Res. Lett. 7 (2000), 35-53.
 Cannas da Silva, A., Weinstein, A., Geometric Models for Noncommutative
Algebras, Berkeley Mathematics Lecture Notes series, Amer. Math. Soc.,
Providence, 1999.
 Cartan, H., La transgression dans un groupe de Lie et dans un espace п¬ЃbrВґe
principal, Colloque de Topologie (Espaces FibrВґs), Bruxelles, 1950, 57-71,
e
Masson et Cie., Paris, 1951.
 Chern, S.S., Complex Manifolds Without Potential Theory, with an ap-
pendix on the geometry of characteristic classes, second edition, Universi-
text, Springer-Verlag, New York-Heidelberg, 1979.
 Delzant, T., Hamiltoniens pВґriodiques et images convexes de lвЂ™application
e
moment, Bull. Soc. Math. France 116 (1988), 315-339.
 Donaldson, S., Symplectic submanifolds and almost-complex geometry, J.
Diп¬Ђerential Geom. 44 (1996), 666-705.
 Donaldson, S., Lefschetz п¬Ѓbrations in symplectic geometry, Proceedings of
the International Congress of Mathematicians, vol. II (Berlin, 1998), Doc.
Math. 1998, extra vol. II, 309-314.
 Donaldson, S., Lefschetz pencils on symplectic manifolds, J. Diп¬Ђerential
Geom. 53 (1999), 205-236.
 Donaldson, S., Kronheimer, P., The Geometry of Four-Manifolds, Oxford
Mathematical Monographs, The Clarendon Press, Oxford University Press,
New York, 1990.
 Duistermaat, J.J., On global action-angle coordinates, Comm. Pure Appl.
Math. 33 (1980), 687-706.
 Duistermaat, J.J., Equivariant cohomology and stationary phase, Symplec-
tic Geometry and Quantization (Sanda and Yokohama, 1993), edited by
Maeda, Y., Omori, H. and Weinstein, A., 45-62, Contemp. Math. 179,
Amer. Math. Soc., Providence, 1994.
201
REFERENCES

 Duistermaat, J.J., The Heat Kernel Lefschetz Fixed Point Formula for the
Spin-c Dirac Operator, Progress in Nonlinear Diп¬Ђerential Equations and
their Applications 18, BirkhВЁuser Boston, Inc., Boston, 1996.
a

 Duistermaat, J.J., Heckman, G., On the variation in the cohomology of
the symplectic form of the reduced phase space, Invent. Math. 69 (1982),
259-268; Addendum, Invent. Math. 72 (1983), 153-158.

 Eliashberg, Y., Classiп¬Ѓcation of overtwisted contact structures on 3-
manifolds, Invent. Math. 98 (1989), 623-637.

 Eliashberg, Y., Contact 3-manifolds twenty years since J. MartinetвЂ™s work,
Ann. Inst. Fourier (Grenoble) 42 (1992), 165-192.

 Eliashberg, Y., Gromov, M., Lagrangian intersection theory: п¬Ѓnite-
dimensional approach, Geometry of Diп¬Ђerential Equations, 27-118, Amer.
Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, 1998.

 Eliashberg, Y., Thurston, W., Confoliations, University Lecture Series 13,
Amer. Math. Soc., Providence, 1998.

 Eliashberg, Y., Traynor, L., Eds., Symplectic Geometry and Topology, lec-
tures from the Graduate Summer School held in Park City, June 29-July 19,
1997, IAS/Park City Mathematics Series 7, Amer. Math. Soc., Providence,
1999.

 FernВґndez, M., Gotay, M., Gray, A., Compact parallelizable four-
a
dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc.
103 (1988), 1209-1212.

 Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies
131, Princeton University Press, Princeton, 1993.

 Geiges, H., Applications of contact surgery, Topology 36 (1997), 1193-1220.

 Geiges, H., Gonzalo, J., Contact geometry and complex surfaces, Invent.
Math. 121 (1995), 147-209.

 Ginzburg, V., Guillemin, V., Karshon, Y., Cobordism theory and localiza-
tion formulas for Hamiltonian group actions, Internat. Math. Res. Notices
1996, 221-234.

 Ginzburg, V., Guillemin, V., Karshon, Y., The relation between compact
and non-compact equivariant cobordisms, Tel Aviv Topology Conference:
Rothenberg Festschrift (1998), 99-112, Contemp. Math., 231, Amer. Math.
Soc., Providence, 1999.

 Giroux, E., Topologie de contact en dimension 3 (autour des travaux de
Yakov Eliashberg), SВґminaire Bourbaki 1992/93, AstВґrisque 216 (1993),
e e
7-33.
202 REFERENCES

 Giroux, E., Une structure de contact, mЛ†me tendue, est plus ou moins
e
Вґ
tordue, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), 697-705.
 Givental, A., Periodic mappings in symplectic topology (Russian), Funkt-
sional. Anal. i Prilozhen 23 (1989), 37-52, translation in Funct. Anal. Appl.
23 (1989), 287-300.
 Gompf, R., A new construction of symplectic manifolds, Ann. of Math.
142 (1995), 527-595.
 Gotay, M., On coisotropic imbeddings of presymplectic manifolds, Proc.
Amer. Math. Soc. 84 (1982), 111-114.
 Griп¬ѓths, P., Harris, J., Principles of Algebraic Geometry, Chapter 0,
reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons,
Inc., New York, 1994.
 Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent.
Math. 82 (1985), 307-347.
 Gromov, M., Partial Diп¬Ђerential Relations, Springer-Verlag, Berlin-New
York, 1986.
 Gromov, M., Soft and hard symplectic geometry, Proceedings of the In-
ternational Congress of Mathematicians 1 (Berkeley, Calif., 1986), 81-98,
Amer. Math. Soc., Providence, 1987.
 Guillemin, V., Course 18.966 вЂ“ Geometry of Manifolds, M.I.T., Spring of
1992.
 Guillemin, V., Moment Maps and Combinatorial Invariants of Hamiltonian
T n -spaces, Progress in Mathematics 122, BirkhВЁuser, Boston, 1994.
a
 Guillemin, V., Pollack, A., Diп¬Ђerential Topology, Prentice-Hall, Inc., En-
glewood Cliп¬Ђs, N.J., 1974.
 Guillemin, V., Sternberg, S., Geometric Asymptotics, Math. Surveys and
Monographs 14, Amer. Math. Soc., Providence, 1977.
 Guillemin, V., Sternberg, S., Convexity properties of the moment mapping,
Invent. Math. 67 (1982), 491-513.
 Guillemin, V., Sternberg, S., Symplectic Techniques in Physics, second edi-
tion, Cambridge University Press, Cambridge, 1990.
 Guillemin, V., Sternberg, S., Supersymmetry and Equivariant de Rham
Theory, with an appendix containing two reprints by H. Cartan, Mathe-
matics Past and Present, Springer-Verlag, Berlin, 1999.
 Hausmann, J.-C., Knutson, A., The cohomology ring of polygon spaces,
Ann. Inst. Fourier (Grenoble) 48 (1998), 281-321.
203
REFERENCES

 Hausmann, J.-C., Knutson, A., Cohomology rings of symplectic cuts, Dif-
ferential Geom. Appl. 11 (1999), 197-203.

 Hitchin, N., Segal, G., Ward, R., Integrable Systems. Twistors, Loop groups,
and Riemann Surfaces Oxford Graduate Texts in Mathematics 4, The
Clarendon Press, Oxford University Press, New York, 1999.

 Hofer, H., Pseudoholomorphic curves in symplectizations with applications
to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993),
515-563.

 Hofer, H. Viterbo, C., The Weinstein conjecture in the presence of holo-
morphic spheres, Comm. Pure Appl. Math. 45 (1992), 583-622.

 Hofer, H., Zehnder, E., Symplectic Invariants and Hamiltonian Dynamics,
BirkhВЁuser Advanced Texts: Basler LehrbВЁcher, BirkhВЁuser Verlag, Basel,
a u a
1994.

 HВЁrmander, L., An Intorduction to Complex Analysis in Several Variables,
o
third edition, North-Holland Mathematical Library 7, North-Holland Pub-
lishing Co., Amsterdam-New York, 1990.

 Jacobson, N., Lie Algebras, republication of the 1962 original, Dover Pub-
lications, Inc., New York, 1979.

 Jeп¬Ђrey, L., Kirwan, F., Localization for nonabelian group actions, Topology
34 (1995), 291-327.

 Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geome-
try, Mathematical Notes 31, Princeton University Press, Princeton, 1984.

 Kodaira, K., On the structure of compact complex analytic surfaces, I,
Amer. J. Math. 86 (1964), 751-798.

 Kronheimer, P., Developments in symplectic topology, Current Develop-
ments in Mathematics (Cambridge, 1998), 83-104, Int. Press, Somerville,
1999.

 Kronheimer, P., Mrowka, T., Monopoles and contact structures, Invent.
Math. 130 (1997), 209-255.

 Lalonde, F., McDuп¬Ђ, D., The geometry of symplectic energy, Ann. of Math.
(2) 141 (1995), 349-371.

 Lalonde, F., Polterovich, L., Symplectic diп¬Ђeomorphisms as isometries of
HoferвЂ™s norm, Topology 36 (1997), 711-727.

 Lerman, E., Meinrenken, E., Tolman, S., Woodward, C., Nonabelian con-
vexity by symplectic cuts, Topology 37 (1998), 245-259.
204 REFERENCES

 Marsden, J., Ratiu, T., Introduction to Mechanics and Symmetry. A Basic
Exposition of Classical Mechanical Systems, Texts in Applied Mathematics
17, Springer-Verlag, New York, 1994.

 Marsden, J., Weinstein, A., Reduction of symplectic manifolds with sym-
metry, Rep. Mathematical Phys. 5 (1974), 121-130.

 Martin, S., Symplectic quotients by a nonabelian group and by its maximal
torus, to appear in Annals of Math.

 Martin, S., Transversality theory, cobordisms, and invariants of symplectic
quotients, to appear in Annals of Math.

 Martinet, J., Formes de contact sur les variВґtВґs de dimension 3, Proceed-
ee
ings of Liverpool Singularities Symposium II (1969/1970), 142-163, Lecture
Notes in Math. 209, Springer, Berlin, 1971.

 McDuп¬Ђ, D., Examples of simply-connected symplectic non-KВЁhlerian man-
a
ifolds, J. Diп¬Ђerential Geom. 20 (1984), 267-277.

 McDuп¬Ђ, D., The local behaviour of holomorphic curves in almost complex
4-manifolds, J. Diп¬Ђerential Geom. 34 (1991), 143-164.

 McDuп¬Ђ, D., Salamon, D., Introduction to Symplectic Topology, Oxford
Mathematical Monographs, Oxford University Press, New York, 1995.

 Meinrenken, E., Woodward, C., Hamiltonian loop group actions and Ver-
linde factorization, J. Diп¬Ђerential Geom. 50 (1998), 417-469.

 Meyer, K., Symmetries and integrals in mechanics, Dynamical Systems
New York, 1973.

 Milnor, J., Morse Theory, based on lecture notes by M. Spivak and R. Wells,
Annals of Mathematics Studies 51, Princeton University Press, Princeton,
1963.

 Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc.
120 (1965), 286-294.

 Mumford, D., Fogarty, J., Kirwan, F., Geometric Invariant Theory, Ergeb-
nisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, Berlin,
1994.

 Newlander, A., Nirenberg, L., Complex analytic coordinates in almost com-
plex manifolds, Ann. of Math. 65 (1957), 391-404.

 Salamon, D., Morse theory, the Conley index and Floer homology, Bull.
London Math. Soc. 22 (1990), 113-140.
205
REFERENCES

 Satake, I., On a generalization of the notion of manifold, Proc. Nat. Acad.
Sci. U.S.A. 42 (1956), 359-363.

 Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15
(1983), 401-487.

 Seidel, P., Lagrangian two-spheres can be symplectically knotted, J. Dif-
ferential Geom. 52 (1999), 145-171.

 Sjamaar, R., Lerman, E., Stratiп¬Ѓed symplectic spaces and reduction, Ann.
of Math. 134 (1991), 375-422.

 Spivak, M., A Comprehensive Introduction to Diп¬Ђerential Geometry, Vol.
I, second edition, Publish or Perish, Inc., Wilmington, 1979.

 Taubes, C., The Seiberg-Witten invariants and symplectic forms, Math.
Res. Lett. 1 (1994), 809-822.

 Taubes, C., More constraints on symplectic forms from Seiberg-Witten
invariants, Math. Res. Lett. 2 (1995), 9-13.

 Taubes, C., The Seiberg-Witten and Gromov invariants, Math. Res. Lett.
2 (1995), 221-238.

 Thomas, C., Eliashberg, Y., Giroux, E., 3-dimensional contact geometry,
Contact and Symplectic Geometry (Cambridge, 1994), 48-65, Publ. Newton
Inst. 8, Cambridge University Press, Cambridge, 1996.

 Thurston, W., Some simple examples of symplectic manifolds, Proc. Amer.
Math. Soc. 55 (1976), 467-468.

 Tolman, S., Weitsman, J., On semifree symplectic circle actions with iso-
lated п¬Ѓxed points, Topology 39 (2000), 299-309.

 Viterbo, C., A proof of WeinsteinвЂ™s conjecture in R2n , Ann. Inst. H.
PoincarВґ Anal. Non LinВґaire 4 (1987), 337-356.
e e

 Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds,
Advances in Math. 6 (1971), 329-346.

 Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Se-
ries in Mathematics 29, Amer. Math. Soc., Providence, 1977.

 Weinstein, A., On the hypotheses of RabinowitzвЂ™ periodic orbit theorems,
J. Diп¬Ђerential Equations 33 (1979), 353-358.

 Weinstein, A., Neighborhood classiп¬Ѓcation of isotropic embeddings, J. Dif-
ferential Geom. 16 (1981), 125-128.

 Weinstein, A., Symplectic geometry, Bull. Amer. Math. Soc. (N.S.) 5
(1981), 1-13.
206 REFERENCES

 Wells, R.O., Diп¬Ђerential Analysis on Complex Manifolds, second edition,
Graduate Texts in Mathematics 65, Springer-Verlag, New York-Berlin,
1980.
 Weyl, H., The Classical Groups. Their Invariants and Representations,
Princeton Landmarks in Mathematics, Princeton University Press, Prince-
ton, 1997.
 Witten, E., Two-dimensional gauge theories revisited, J. Geom. Phys. 9
(1992), 303-368.
 Woodward, C., Multiplicity-free Hamiltonian actions need not be KВЁhler,
a
Invent. Math. 131 (1998), 311-319.
 Woodward, C., Spherical varieties and existence of invariant KВЁhler struc-
a
tures, Duke Math. J. 93 (1998), 345-377.
Index
action Atiyah-Guillemin-Sternberg the-
orem, 170
moduli space, 158
Yang-Mills theory, 155
coordinates, 111
deп¬Ѓnition, 128
Banyaga theorem, 92
eп¬Ђective, 171
base, 155
free, 135
basis
gauge group, 158
for skew-symmetric bilinear maps,
hamiltonian, 129, 130, 133, 164
3
inп¬Ѓnitesimal, 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
Birkhoп¬Ђ
action-angle coordinates, 111
PoincarВґ-Birkhoп¬Ђ 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
deп¬Ѓnition, 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 в€— X
78 coordinate deп¬Ѓnition, 9, 10
antisymmetry, 108 intrinsic deп¬Ѓnition, 10
arc-length, 25 naturality, 11
Archimedes, 192 Cartan
Arnold diп¬Ђerentiation, 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
п¬Ѓrst Chern class, 194, 195 complex torus, 103
Chevalley cohomology, 165 complex vector space, 68
Christoп¬Ђel 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
codiп¬Ђerential, 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 п¬‚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 2nв€’1 , 64
complex structure, 68
linear structures, 72 dynamics, 63
triple, 70, 75 element, 57, 61, 62
complete vector п¬Ѓeld, 129 example of contact structure, 58
completely integrable system, 110 local contact form, 57
complex local normal form, 59
atlas, 89 locally deп¬Ѓning 1-form, 57
chart, 83 manifold, 57
diп¬Ђerentials, 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, 16вЂ“18 deп¬Ѓnition, 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
DвЂ™Alembert energy-momentum map, 153
variational principle, 114 kinetic, 112, 113
Darboux potential, 112, 113
chart, 7 equations
theorem, 7, 45, 46 Christoп¬Ђel, 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
eп¬Ђective
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 KВЁh-a weighted projective space, 151
ler manifold, 103 exponential map, 35
complex torus, 103
facet, 178
Delzant construction, 181
FernВґndez-Gotay-Gray example, 102
a
FernВґndez-Gotay-Gray, 102
a
п¬Ѓrst Chern class, 194, 195
Gompf, 103
п¬Ѓrst integral, 109
hermitian matrices, 132
п¬Ѓxed point, 29, 33, 55
Hirzebruch surfaces, 178, 190
п¬‚at connection, 159
Hopf surface, 102
п¬‚ow, 35
Kodaira-Thurston, 102
form
McDuп¬Ђ, 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 KВЁhler manifold, 96,
a
Fubini-Study, 96, 169
101
harmonic, 98, 99
of compact symplectic manifold,
101 KВЁhler, 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 inп¬Ѓnite-dimensional symplec- tautological, 9, 10, 20
tic manifold, 158 type, 79
of KВЁhler 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 KВЁhler 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 п¬Ѓeld, 105, 106
Gauss lemma, 28 harmonic form, 98, 99
generating function, 17, 22, 23, 29 Hausdorп¬Ђ quotient, 136
geodesic Heckman, see Duistermaat-Heckman
curve, 25 hermitian matrix, 132
п¬‚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
FernВґndez-Gotay-Gray, 102
a diamond, 101
gradient vector п¬Ѓeld, 107 number, 100
в€—-operator, 98
gravitational potential, 113
gravity, 112, 152 theorem, 98вЂ“100
Gray theory, 98
FernВґndez-Gotay-Gray (A. Gray),
a (J-)holomorphic tangent vectors, 78
102 homotopy
theorem (J. Gray), 59 deп¬Ѓnition, 40
Gromov formula, 39, 40
pseudo-holomorphic curve, 67, invariance, 39
82 operator, 40
group Hopf
gauge, 158, 159 п¬Ѓbration, 64, 156
S 4 is not almost complex, 76
Lie, 128
of symplectomorphisms, 12, 53 surface, 102
one-parameter group of diп¬Ђeo- vector п¬Ѓeld, 64
morphisms, 127, 128
immersion, 15
product, 149
index, 175, 176
semisimple, 158
inп¬Ѓnitesimal action, 156, 164
structure, 155
integrable
Guillemin
almost complex structure, 75,
Atiyah-Guillemin-Sternberg the-
82
 << стр. 8(всего 9)СОДЕРЖАНИЕ >>