ńņš. 8 |

ā’Ļ|zi |2 + Ī»i = Ī»i , āi ā I

āā’

āā’ āi ā I .

zi = 0 ,

ā¢ The Td -action on Cd preserves Ļ, so the Td -action takes Z = Ļā’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 , āi ā I} ,

and

(Td )I = {(e2Ļit1 , . . . , e2Ļitd ) | e2Ļits = 1, ās ā I}

/

Suppose that z = (z1 , . . . , zd ) ā Cd . Then

Hint:

(e2Ļit1 z1 , . . . , e2Ļitd zd ) = (z1 , . . . , zd )

if and only if e2Ļits = 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 = {(e2Ļit1 , . . . , e2Ļitn , 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 , Ļā ) from ā. We already have that

Mā = Z/N

is a compact 2n-dimensional manifold. Let Ļā be the reduced symplectic form.

186 29 DELZANT CONSTRUCTION

Claim. The manifold (Mā , Ļā ) 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 , Ļā ). 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 , Ļā ) is the required toric manifold corresponding to ā.

29.4 Idea Behind the Delzant Construction

We use the idea that Rd is āuniversalā 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, āi

āā’

ā’

āā’ x, Ļ(ei ) ā’ Ī»i ā¤ 0, āi

āā’ x, vi ā¤ Ī»i , āi

āā’ xāā.

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 Ļā’1 (A) ā‚ Cd is a compact submanifold. Let i : Ļ ā’ Cd denote

inclusion. Then iā— Ļ0 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— Ļ0 is a principal ļ¬bration, so we take the quotient:

ā’ Ļā’1 (A)

N

ā“

= Ļā’1 (A)/N

Mā

Let Ļā be the reduced symplectic form.

ā¢ The (non-eļ¬ective) action of Td = N Ć—Tn on Ļā’1 (A) has a āmoment mapā

with image Ļ(Ļā’1 (A)) = ā. (By āmoment 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 Ļā’1 (y) is a single Td -orbit for any y ā Ļ(Cd ). Now observe that

y ā ā = Ļ ā— (ā) āā’ Ļā’1 (y) ā Z .

Suppose that y = Ļ ā— (x). Show that Āµā’1 (x) = Ļā’1 (y)/N . But Ļā’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 , Ļā , 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 Ļp by:

Ļp (v, Ī¾) = Ī¾(v) = ā’Ļp (Ī¾, v) and Ļp (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

Ļn

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

Ļn

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,

Ļn

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

Ļn

(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 āŖā 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 , Ļā ) 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)āS 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 Ļred and Ļt . 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— Ļred 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— Ļred + 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— Ļred , 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 , Ļ0 ), (M1 , Ļ1 ) be symplectic manifolds of dimension 2n, G a compact Lie group acting

on (Mi , Ļi ), 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— Ļ0 = Ī¹ā— Ļ1 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 , tā0,

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

xā’1 (t)/S 1 , tā0.

Proposition 30.7 The reduced space (Mt , Ļt ) is symplectomorphic to

(Mred , Ļred ā’ tĪ²) ,

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

Proof. By Lemma 30.6, (Mt , Ļt ) 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— Ļred ā’ tdĪ± .

t

By deļ¬nition of curvature, dĪ± = Ļ ā— Ī². Hence, the reduced symplectic form on

xā’1 (t)/S 1 is

Ļred ā’ tĪ² .

In loose terms, Proposition 30.7 says that the reduced forms Ļt 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 [Ļt ] varies linearly in t. More speciļ¬cally,

[Ļt ] = [Ļred ] + 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

2ĻiR under the exponential map exp : g 2ĻiR 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 2Ļi of 2ĻiR 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 = 2Ļi. 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’ e2Ļit . 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 = 2ĻiĪ±, B = 2ĻiĪ² 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 , Ļx )

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

(Ļred ā’ xĪ²)nā’1

Ļx

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— (Ļred 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,

Ļn

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

Ļn

mDH (U ) = .

n!

ZĆ—U

Since Ļ = Ļ ā— Ļred ā’ d(xĪ±), its power is

Ļ n = n(Ļ ā— Ļred ā’ xdĪ±)nā’1 ā§ Ī± ā§ dx .

By the Fubini theorem, we then have

Ļ ā— (Ļred 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— (Ļred 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].

(a) We grade ā„¦ā¢ 1 (M ) by adding the usual grading on ā„¦ā¢ (M ) to a grading

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:

Ļā— : 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:

References

[1] Abraham, R., Marsden, J. E., Foundations of Mechanics, second edition,

Addison-Wesley, Reading, 1978.

[2] Aebischer, B., Borer, M., KĀØlin, M., Leuenberger, Ch., Reimann, H.M.,

a

Symplectic Geometry. An Introduction Based on the Seminar in Bern,

1992, Progress in Mathematics 124, BirkhĀØuser Verlag, Basel, 1994.

a

[3] Arnold, V., Mathematical Methods of Classical Mechanics, Graduate Texts

in Math. 60, Springer-Verlag, New York, 1978.

[4] Arnold, V., First steps of symplectic topology, VIIIth International

Congress on Mathematical Physics (Marseille, 1986), 1-16, World Sci. Pub-

lishing, Singapore, 1987.

[5] 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.

[6] Atiyah, M., Convexity and commuting Hamiltonians, Bull. London Math.

Soc. 14 (1982), 1-15.

[7] 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.

[8] Atiyah, M., Bott, R., The moment map and equivariant cohomology, Topol-

ogy 23 (1984), 1-28.

[9] Audin, M., The Topology of Torus Actions on Symplectic Manifolds,

Progress in Mathematics 93, BirkhĀØuser Verlag, Basel, 1991.

a

[10] Audin, M., Spinning Tops. A Course on Integrable Systems, Cambridge

Studies in Advanced Mathematics 51, Cambridge University Press, Cam-

bridge, 1996.

[11] Audin, M., Lafontaine, J., Eds., Holomorphic Curves in Symplectic Geom-

etry, Progress in Mathematics 117, BirkhĀØuser Verlag, Basel, 1994.

a

[12] Auroux, D., Asymptotically holomorphic families of symplectic submani-

folds, Geom. Funct. Anal. 7 (1997), 971-995.

[13] Berline, N., Getzler, E., Vergne, M., Heat Kernels and Dirac Operators,

Grundlehren der Mathematischen Wissenschaften 298, Springer-Verlag,

Berlin, 1992.

[14] Berline, N., Vergne, M., Classes caractĀ“ristiques Ā“quivariantes, formule de

e e

localisation en cohomologie Ā“quivariante, C. R. Acad. Sci. Paris SĀ“r. I

e e

Math. 295 (1982), 539-541.

199

200 REFERENCES

[15] Berline, N., Vergne, M., ZĀ“ros dā™un champ de vecteurs et classes caractĀ“ris-

e e

tiques Ā“quivariantes, Duke Math. J. 50 (1983), 539-549.

e

[16] Biran, P., A stability property of symplectic packing, Invent. Math. 136

(1999), 123-155.

[17] Bredon, G., Introduction to Compact Transformation Groups, Pure and

Applied Mathematics 46, Academic Press, New York-London, 1972.

[18] Bott, R., Tu, L., Diļ¬erential Forms in Algebraic Topology, Graduate Texts

in Mathematics 82, Springer-Verlag, New York-Berlin, 1982.

[19] Cannas da Silva, A., Guillemin, V., Woodward, C., On the unfolding of

folded symplectic structures, Math. Res. Lett. 7 (2000), 35-53.

[20] Cannas da Silva, A., Weinstein, A., Geometric Models for Noncommutative

Algebras, Berkeley Mathematics Lecture Notes series, Amer. Math. Soc.,

Providence, 1999.

[21] Cartan, H., La transgression dans un groupe de Lie et dans un espace ļ¬brĀ“e

principal, Colloque de Topologie (Espaces FibrĀ“s), Bruxelles, 1950, 57-71,

e

Masson et Cie., Paris, 1951.

[22] 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.

[23] Delzant, T., Hamiltoniens pĀ“riodiques et images convexes de lā™application

e

moment, Bull. Soc. Math. France 116 (1988), 315-339.

[24] Donaldson, S., Symplectic submanifolds and almost-complex geometry, J.

Diļ¬erential Geom. 44 (1996), 666-705.

[25] Donaldson, S., Lefschetz ļ¬brations in symplectic geometry, Proceedings of

the International Congress of Mathematicians, vol. II (Berlin, 1998), Doc.

Math. 1998, extra vol. II, 309-314.

[26] Donaldson, S., Lefschetz pencils on symplectic manifolds, J. Diļ¬erential

Geom. 53 (1999), 205-236.

[27] Donaldson, S., Kronheimer, P., The Geometry of Four-Manifolds, Oxford

Mathematical Monographs, The Clarendon Press, Oxford University Press,

New York, 1990.

[28] Duistermaat, J.J., On global action-angle coordinates, Comm. Pure Appl.

Math. 33 (1980), 687-706.

[29] 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

[30] Duistermaat, J.J., The Heat Kernel Lefschetz Fixed Point Formula for the

Spin-c Dirac Operator, Progress in Nonlinear Diļ¬erential Equations and

their Applications 18, BirkhĀØuser Boston, Inc., Boston, 1996.

a

[31] 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.

[32] Eliashberg, Y., Classiļ¬cation of overtwisted contact structures on 3-

manifolds, Invent. Math. 98 (1989), 623-637.

[33] Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinetā™s work,

Ann. Inst. Fourier (Grenoble) 42 (1992), 165-192.

[34] Eliashberg, Y., Gromov, M., Lagrangian intersection theory: ļ¬nite-

dimensional approach, Geometry of Diļ¬erential Equations, 27-118, Amer.

Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, 1998.

[35] Eliashberg, Y., Thurston, W., Confoliations, University Lecture Series 13,

Amer. Math. Soc., Providence, 1998.

[36] 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.

[37] FernĀ“ndez, M., Gotay, M., Gray, A., Compact parallelizable four-

a

dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc.

103 (1988), 1209-1212.

[38] Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies

131, Princeton University Press, Princeton, 1993.

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

[40] Geiges, H., Gonzalo, J., Contact geometry and complex surfaces, Invent.

Math. 121 (1995), 147-209.

[41] Ginzburg, V., Guillemin, V., Karshon, Y., Cobordism theory and localiza-

tion formulas for Hamiltonian group actions, Internat. Math. Res. Notices

1996, 221-234.

[42] 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.

[43] Giroux, E., Topologie de contact en dimension 3 (autour des travaux de

Yakov Eliashberg), SĀ“minaire Bourbaki 1992/93, AstĀ“risque 216 (1993),

e e

7-33.

202 REFERENCES

[44] Giroux, E., Une structure de contact, mĖme tendue, est plus ou moins

e

Ā“

tordue, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), 697-705.

[45] 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.

[46] Gompf, R., A new construction of symplectic manifolds, Ann. of Math.

142 (1995), 527-595.

[47] Gotay, M., On coisotropic imbeddings of presymplectic manifolds, Proc.

Amer. Math. Soc. 84 (1982), 111-114.

[48] Griļ¬ths, P., Harris, J., Principles of Algebraic Geometry, Chapter 0,

reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons,

Inc., New York, 1994.

[49] Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent.

Math. 82 (1985), 307-347.

[50] Gromov, M., Partial Diļ¬erential Relations, Springer-Verlag, Berlin-New

York, 1986.

[51] 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.

[52] Guillemin, V., Course 18.966 ā“ Geometry of Manifolds, M.I.T., Spring of

1992.

[53] Guillemin, V., Moment Maps and Combinatorial Invariants of Hamiltonian

T n -spaces, Progress in Mathematics 122, BirkhĀØuser, Boston, 1994.

a

[54] Guillemin, V., Pollack, A., Diļ¬erential Topology, Prentice-Hall, Inc., En-

glewood Cliļ¬s, N.J., 1974.

[55] Guillemin, V., Sternberg, S., Geometric Asymptotics, Math. Surveys and

Monographs 14, Amer. Math. Soc., Providence, 1977.

[56] Guillemin, V., Sternberg, S., Convexity properties of the moment mapping,

Invent. Math. 67 (1982), 491-513.

[57] Guillemin, V., Sternberg, S., Symplectic Techniques in Physics, second edi-

tion, Cambridge University Press, Cambridge, 1990.

[58] 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.

[59] Hausmann, J.-C., Knutson, A., The cohomology ring of polygon spaces,

Ann. Inst. Fourier (Grenoble) 48 (1998), 281-321.

203

REFERENCES

[60] Hausmann, J.-C., Knutson, A., Cohomology rings of symplectic cuts, Dif-

ferential Geom. Appl. 11 (1999), 197-203.

[61] 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.

[62] Hofer, H., Pseudoholomorphic curves in symplectizations with applications

to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993),

515-563.

[63] Hofer, H. Viterbo, C., The Weinstein conjecture in the presence of holo-

morphic spheres, Comm. Pure Appl. Math. 45 (1992), 583-622.

[64] Hofer, H., Zehnder, E., Symplectic Invariants and Hamiltonian Dynamics,

BirkhĀØuser Advanced Texts: Basler LehrbĀØcher, BirkhĀØuser Verlag, Basel,

a u a

1994.

[65] HĀØrmander, 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.

[66] Jacobson, N., Lie Algebras, republication of the 1962 original, Dover Pub-

lications, Inc., New York, 1979.

[67] Jeļ¬rey, L., Kirwan, F., Localization for nonabelian group actions, Topology

34 (1995), 291-327.

[68] Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geome-

try, Mathematical Notes 31, Princeton University Press, Princeton, 1984.

[69] Kodaira, K., On the structure of compact complex analytic surfaces, I,

Amer. J. Math. 86 (1964), 751-798.

[70] Kronheimer, P., Developments in symplectic topology, Current Develop-

ments in Mathematics (Cambridge, 1998), 83-104, Int. Press, Somerville,

1999.

[71] Kronheimer, P., Mrowka, T., Monopoles and contact structures, Invent.

Math. 130 (1997), 209-255.

[72] Lalonde, F., McDuļ¬, D., The geometry of symplectic energy, Ann. of Math.

(2) 141 (1995), 349-371.

[73] Lalonde, F., Polterovich, L., Symplectic diļ¬eomorphisms as isometries of

Hoferā™s norm, Topology 36 (1997), 711-727.

[74] Lerman, E., Meinrenken, E., Tolman, S., Woodward, C., Nonabelian con-

vexity by symplectic cuts, Topology 37 (1998), 245-259.

204 REFERENCES

[75] 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.

[76] Marsden, J., Weinstein, A., Reduction of symplectic manifolds with sym-

metry, Rep. Mathematical Phys. 5 (1974), 121-130.

[77] Martin, S., Symplectic quotients by a nonabelian group and by its maximal

torus, to appear in Annals of Math.

[78] Martin, S., Transversality theory, cobordisms, and invariants of symplectic

quotients, to appear in Annals of Math.

[79] Martinet, J., Formes de contact sur les variĀ“tĀ“s de dimension 3, Proceed-

ee

ings of Liverpool Singularities Symposium II (1969/1970), 142-163, Lecture

Notes in Math. 209, Springer, Berlin, 1971.

[80] McDuļ¬, D., Examples of simply-connected symplectic non-KĀØhlerian man-

a

ifolds, J. Diļ¬erential Geom. 20 (1984), 267-277.

[81] McDuļ¬, D., The local behaviour of holomorphic curves in almost complex

4-manifolds, J. Diļ¬erential Geom. 34 (1991), 143-164.

[82] McDuļ¬, D., Salamon, D., Introduction to Symplectic Topology, Oxford

Mathematical Monographs, Oxford University Press, New York, 1995.

[83] Meinrenken, E., Woodward, C., Hamiltonian loop group actions and Ver-

linde factorization, J. Diļ¬erential Geom. 50 (1998), 417-469.

[84] Meyer, K., Symmetries and integrals in mechanics, Dynamical Systems

(Proc. Sympos., Univ. Bahia, Salvador, 1971), 259-272. Academic Press,

New York, 1973.

[85] Milnor, J., Morse Theory, based on lecture notes by M. Spivak and R. Wells,

Annals of Mathematics Studies 51, Princeton University Press, Princeton,

1963.

[86] Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc.

120 (1965), 286-294.

[87] Mumford, D., Fogarty, J., Kirwan, F., Geometric Invariant Theory, Ergeb-

nisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, Berlin,

1994.

[88] Newlander, A., Nirenberg, L., Complex analytic coordinates in almost com-

plex manifolds, Ann. of Math. 65 (1957), 391-404.

[89] Salamon, D., Morse theory, the Conley index and Floer homology, Bull.

London Math. Soc. 22 (1990), 113-140.

205

REFERENCES

[90] Satake, I., On a generalization of the notion of manifold, Proc. Nat. Acad.

Sci. U.S.A. 42 (1956), 359-363.

[91] Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15

(1983), 401-487.

[92] Seidel, P., Lagrangian two-spheres can be symplectically knotted, J. Dif-

ferential Geom. 52 (1999), 145-171.

[93] Sjamaar, R., Lerman, E., Stratiļ¬ed symplectic spaces and reduction, Ann.

of Math. 134 (1991), 375-422.

[94] Spivak, M., A Comprehensive Introduction to Diļ¬erential Geometry, Vol.

I, second edition, Publish or Perish, Inc., Wilmington, 1979.

[95] Taubes, C., The Seiberg-Witten invariants and symplectic forms, Math.

Res. Lett. 1 (1994), 809-822.

[96] Taubes, C., More constraints on symplectic forms from Seiberg-Witten

invariants, Math. Res. Lett. 2 (1995), 9-13.

[97] Taubes, C., The Seiberg-Witten and Gromov invariants, Math. Res. Lett.

2 (1995), 221-238.

[98] 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.

[99] Thurston, W., Some simple examples of symplectic manifolds, Proc. Amer.

Math. Soc. 55 (1976), 467-468.

[100] Tolman, S., Weitsman, J., On semifree symplectic circle actions with iso-

lated ļ¬xed points, Topology 39 (2000), 299-309.

[101] Viterbo, C., A proof of Weinsteinā™s conjecture in R2n , Ann. Inst. H.

PoincarĀ“ Anal. Non LinĀ“aire 4 (1987), 337-356.

e e

[102] Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds,

Advances in Math. 6 (1971), 329-346.

[103] Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Se-

ries in Mathematics 29, Amer. Math. Soc., Providence, 1977.

[104] Weinstein, A., On the hypotheses of Rabinowitzā™ periodic orbit theorems,

J. Diļ¬erential Equations 33 (1979), 353-358.

[105] Weinstein, A., Neighborhood classiļ¬cation of isotropic embeddings, J. Dif-

ferential Geom. 16 (1981), 125-128.

[106] Weinstein, A., Symplectic geometry, Bull. Amer. Math. Soc. (N.S.) 5

(1981), 1-13.

206 REFERENCES

[107] Wells, R.O., Diļ¬erential Analysis on Complex Manifolds, second edition,

Graduate Texts in Mathematics 65, Springer-Verlag, New York-Berlin,

1980.

[108] Weyl, H., The Classical Groups. Their Invariants and Representations,

Princeton Landmarks in Mathematics, Princeton University Press, Prince-

ton, 1997.

[109] Witten, E., Two-dimensional gauge theories revisited, J. Geom. Phys. 9

(1992), 303-368.

[110] Woodward, C., Multiplicity-free Hamiltonian actions need not be KĀØhler,

a

Invent. Math. 131 (1998), 311-319.

[111] Woodward, C., Spherical varieties and existence of invariant KĀØhler struc-

a

tures, Duke Math. J. 93 (1998), 345-377.

Index

action Atiyah-Guillemin-Sternberg the-

orem, 170

adjoint, 131, 137

moduli space, 158

coadjoint, 131, 137

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

adapted coordinates, 18

blowup, 189

adjoint

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

coadjoint conļ¬guration space, 107, 113

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

Ehresmann coadjoint orbits, 137, 139

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

ńņš. 8 |