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V . The bundle H is called the horizontal bundle.

25.2 Connection and Curvature Forms

A connection on a principal bundle P may be equivalently described in terms
of 1-forms.

Deп¬Ѓnition 25.3 A connection form on a principal bundle P is a Lie-algebra-
valued 1-form
k
в€€ в„¦1 (P ) вЉ— g
Ai вЉ— X i
A=
i=1

such that:
157
25.2 Connection and Curvature Forms

(a) A is G-invariant, with respect to the product action of G on в„¦1 (P ) (in-
duced by the action on P ) and on g (the adjoint representation), and
(b) A is vertical, in the sense that Д±X # A = X for any X в€€ g.

Exercise. Show that a connection T P = V вЉ• H determines a connection form
A and vice-versa by the formula
H = ker A = {v в€€ T P | Д±v A = 0} .
в™¦
Given a connection on P , the splitting T P = V вЉ• H induces the following
splittings for bundles:
T в€—P = V в€— вЉ• Hв€—

в€§2 T в€— P = (в€§2 V в€— ) вЉ• (V в€— в€§ H в€— ) вЉ• (в€§2 H в€— )
.
.
.
and for their sections:
в„¦1 (P ) = в„¦1 (P ) вЉ• в„¦1 (P )
vert horiz

в„¦2 (P ) = в„¦2 (P ) вЉ• в„¦2 (P ) вЉ• в„¦2 (P )
vert mix horiz
.
.
.
The corresponding connection form A is in в„¦1 вЉ— g. Its exterior derivative dA
vert
is in
в„¦2 (P ) вЉ— g = в„¦2 вЉ• в„¦2 вЉ• в„¦2 horiz вЉ— g ,
vert mix
and thus decomposes into three components,
dA = (dA)vert + (dA)mix + (dA)horiz .

Exercise. Check that:
1
ci m A в€§ Am вЉ— Xi , where
(a) (dA)vert (X, Y ) = [X, Y ], i.e., (dA)vert = 2
i, ,m
i
the c m вЂ™s are the structure constants of the Lie algebra with respect to
c i m Xi ;
the chosen basis, and deп¬Ѓned by [X , Xm ] =
i, ,m

(b) (dA)mix = 0.
в™¦
According to the previous exercise, the relevance of dA may come only from
its horizontal component.
Deп¬Ѓnition 25.4 The curvature form of a connection is the horizontal com-
ponent of its connection form. I.e., if A is the connection form, then
в€€ в„¦2
horiz вЉ— g .
curv A = (dA)horiz
Deп¬Ѓnition 25.5 A connection is called п¬‚at if its curvature is zero.
158 25 MOMENT MAP IN GAUGE THEORY

25.3 Symplectic Structure on the Space of Connections

Let P be a principal G-bundle over B. If A is a connection form on P , and if
a в€€ в„¦1horiz вЉ— g is G-invariant for the product action, then it is easy to check that
A + a is also a connection form on P . Reciprocally, any two connection forms on
P diп¬Ђer by an a в€€ (в„¦1 G
horiz вЉ— g) . We conclude that the set A of all connections
on the principal G-bundle P is an aп¬ѓne space modeled on the linear space

a = (в„¦1 G
horiz вЉ— g) .

Now let P be a principal G-bundle over a compact oriented 2-dimensional
riemannian manifold B (for instance, B is a Riemann surface). Suppose that
the group G is compact or semisimple. Atiyah and Bott  noticed that the cor-
responding space A of all connections may be treated as an inп¬Ѓnite-dimensional
symplectic manifold. This will require choosing a G-invariant inner product
В·, В· on g, which always exists, either by averaging any inner product when G
is compact, or by using the Killing form on semisimple groups.
Since A is an aп¬ѓne space, its tangent space at any point A is identiп¬Ѓed with
the model linear space a. With respect to a basis X1 , . . . , Xk for the Lie algebra
g, elements a, b в€€ a14 are written

ai вЉ— Xi bi вЉ— X i .
a= and b=

If we wedge a and b, and then integrate over B using the riemannian volume,
we obtain a real number:
G
в„¦2 (P ) в„¦2 (B)
a Г— a в€’в†’ в€’в†’ R
П‰: horiz

(a, b) в€’в†’ a i в€§ b j X i , Xj в€’в†’ a i в€§ b j X i , Xj .
i,j B i,j

We have used that the pullback ПЂ в€— : в„¦2 (B) в†’ в„¦2 (P ) is an isomorphism onto
G
its image в„¦2 (P ) .
horiz

Exercise. Show that if w(a, b) = 0 for all b в€€ a, then a must be zero. в™¦
The map П‰ is nondegenerate, skew-symmetric, bilinear and constant in the
sense that it does not depend on the base point A. Therefore, it has the right to
be called a symplectic form on A, so the pair (A, П‰) is an inп¬Ѓnite-dimensional
symplectic manifold.

25.4 Action of the Gauge Group

Let P be a principal G-bundle over B. A diп¬Ђeomorphism f : P в†’ P commuting
with the G-action determines a diп¬Ђeomorphism fbasic : B в†’ B by projection.
14 The choice of symbols is in honor of Atiyah and Bott!
159
25.5 Case of Circle Bundles

Deп¬Ѓnition 25.6 A diп¬Ђeomorphism f : P в†’ P commuting with the G-action
is a gauge transformation if the induced fbasic is the identity. The gauge
group of P is the group G of all gauge transformations of P .
The derivative of an f в€€ G takes a connection T P = V вЉ• H to another
connection T P = V вЉ• Hf , and thus induces an action of G in the space A of
all connections. Recall that A has a symplectic form П‰. Atiyah and Bott 
noticed that the action of G on (A, П‰) is hamiltonian, where the moment map
(appropriately interpreted) is the map
G
в„¦2 (P ) вЉ— g
Вµ: A в€’в†’

в€’в†’ curv A ,
A
i.e., the moment map вЂњisвЂќ the curvature! We will describe this construction in
detail for the case of circle bundles in the next section.
Remark. The reduced space at level zero
M = Вµв€’1 (0)/G
is the space of п¬‚at connections modulo gauge equivalence, known as the mod-
uli space of п¬‚at connections. It turns out that M is a п¬Ѓnite-dimensional
в™¦
symplectic orbifold.

25.5 Case of Circle Bundles
What does the Atiyah-Bott construction of the previous section look like for the
case when G = S 1 ?
EP
S1 вЉ‚

ПЂ
c
B
1
Let v be the generator of the S -action on P , corresponding to the basis 1 of
g R. A connection form on P is a usual 1-form A в€€ в„¦1 (P ) such that
Lv A = 0 and Д±v A = 1 .
If we п¬Ѓx one particular connection A0 , then any other connection is of the form
G
A = A0 + a for some a в€€ a = в„¦1 (P ) = в„¦1 (B). The symplectic form on
horiz
a = в„¦1 (B) is simply
П‰ : a Г— a в€’в†’ R

(a, b) в€’в†’ aв€§b .
B
в€€в„¦2 (B)
160 25 MOMENT MAP IN GAUGE THEORY

The gauge group is G = Maps(B, S 1 ), because a gauge transformation is multi-
plication by some element of S 1 over each point in B:

G в€’в†’ Diп¬Ђ(P )
П€:

h : B в†’ S1 П€h : P в†’ P
в€’в†’
p в†’ h(ПЂ(p)) В· p

The Lie algebra of G is

Lie G = Maps(B, R) = C в€ћ (B) .

Its dual space is
в€—
(Lie G) = в„¦2 (B) ,
where the duality is provided by integration over B

C в€ћ (B) Г— в„¦2 (B) в€’в†’ R

(h, ОІ) в€’в†’ hОІ .
B

(it is topological or smooth duality, as opposed to algebraic duality) .
The gauge group acts on the space of all connections by

G в€’в†’ Diп¬Ђ(A)

в€’в†’ (A в†’ A в€’ПЂ в€— dОё)
h(x) = eiОё(x)
в€€a

Exercise. Check the previous assertion about the action on connections.
Hint: First deal with the case where P = S 1 Г— B is a trivial bundle, in which
case h в€€ G acts on P by
П€h : (t, x) в€’в†’ (t + Оё(x), x) ,
and where every connection can be written A = dt + ОІ, with ОІ в€€ в„¦1 (B). A
gauge transformation h в€€ G acts on A by
в€—
A в€’в†’ П€hв€’1 (A) .
в™¦
The inп¬Ѓnitesimal action of G on A is
dП€ : Lie G в€’в†’ П‡(A)

в€’в†’ X # = vector п¬Ѓeld described by the transformation
X
(A в†’ A в€’dX )
в€€в„¦1 (B)=a

so that X # = в€’dX.
161
25.5 Case of Circle Bundles

Finally, we will check that

Вµ : A в€’в†’ (Lie G)в€— = в„¦2 (B)

A в€’в†’ curv A

is indeed a moment map for the action of the gauge group on A.
Exercise. Check that in this case:
G
в„¦2 (P ) = в„¦2 (B) ,
в€€
(a) curv A = dA horiz

(b) Вµ is G-invariant.
в™¦
The previous exercise takes care of the equivariance condition, since the
action of G on в„¦2 (B) is trivial.
Take any X в€€ Lie G = C в€ћ (B). We need to check that

dВµX (a) = П‰(X # , a) , в€Ђa в€€ в„¦1 (B) . ()

As for the left-hand side of ( ), the map ВµX ,

ВµX : A в€’в†’ R
A в€’в†’ X В· dA ,
X , dA =
B
в€€C в€ћ (B) в€€в„¦2 (B)

is linear in A. Consequently,

dВµX : a в€’в†’ R
a в€’в†’ X В· da .
B

As for the right-hand side of ( ), by deп¬Ѓnition of П‰, we have

П‰(X # , a) = X# В· a = в€’ dX В· a .
B B

But, by Stokes theorem,, the last integral is

в€’ dX В· a = X В· da ,
B B

so we are done in proving that Вµ is the moment map.
Homework 19: Examples of Moment Maps

1. Suppose that a Lie group G acts in a hamiltonian way on two symplectic
manifolds (Mj , П‰j ), j = 1, 2, with moment maps Вµ : Mj в†’ gв€— . Prove that
the diagonal action of G on M1 Г— M2 is hamiltonian with moment map
Вµ : M1 Г— M2 в†’ gв€— given by

Вµ(p1 , p2 ) = Вµ1 (p1 ) + Вµ2 (p2 ) , for pj в€€ Mj .

2. Let Tn = {(t1 , . . . , tn ) в€€ Cn : |tj | = 1, for all j } be a torus acting on Cn
by
(t1 , . . . , tn ) В· (z1 , . . . , zn ) = (tk1 z1 , . . . , tkn zn ) ,
1 n

where k1 , . . . , kn в€€ Z are п¬Ѓxed. Show that this action is hamiltonian with
moment map Вµ : Cn в†’ (tn )в€— Rn given by
1
Вµ(z1 , . . . , zn ) = в€’ 2 (k1 |z1 |2 , . . . , kn |zn |2 ) ( + constant ) .

3. The vector п¬Ѓeld X # generated by X в€€ g for the coadjoint representation
of a Lie group G on gв€— satisп¬Ѓes XОѕ , Y = Оѕ, [Y, X] , for any Y в€€ g.
#

Equip the coadjoint orbits with the canonical symplectic forms. Show
that, for each Оѕ в€€ gв€— , the coadjoint action on the orbit G В· Оѕ is hamiltonian
with moment map the inclusion map:

Вµ : G В· Оѕ в†’ gв€— .

4. Consider the natural action of U(n) on (Cn , П‰0 ). Show that this action is
hamiltonian with moment map Вµ : Cn в†’ u(n) given by
i
Вµ(z) = 2 zz в€— ,

where we identify the Lie algebra u(n) with its dual via the inner product
(A, B) = trace(Aв€— B).
Denote the elements of U(n) in terms of real and imaginary parts
Hint:
h в€’k
g = h+i k. Then g acts on R2n by the linear symplectomorphism .
k h
The Lie algebra u(n) is the set of skew-hermitian matrices X = V + i W where
V = в€’V t в€€ RnГ—n and W = W t в€€ RnГ—n . Show that the inп¬Ѓnitesimal action
is generated by the hamiltonian functions
ВµX (z) = в€’ 2 (x, W x) + (y, V x) в€’ 1 (y, W y)
1
2

where z = x + i y, x, y в€€ Rn and (В·, В·) is the standard inner product. Show that
1 1
ВµX (z) = i z в€— Xz i trace(zz в€— X)
= .
2 2

Check that Вµ is equivariant.

162
163
HOMEWORK 19

5. Consider the natural action of U(k) on the space (CkГ—n , П‰0 ) of complex
(k Г— n)-matrices. Identify the Lie algebra u(k) with its dual via the inner
product (A, B) = trace(Aв€— B). Prove that a moment map for this action
is given by
Вµ(A) = 2 AAв€— + Id , for A в€€ CkГ—n .
i
2i
Id
(The choice of the constant is for convenience in Homework 20.)
2i

Exercises 1 and 4.
Hint:

2
6. Consider the U(n)-action by conjugation on the space (Cn , П‰0 ) of complex
(n Г— n)-matrices. Show that a moment map for this action is given by

Вµ(A) = 2 [A, Aв€— ] .
i

Previous exercise and its вЂњtransposeвЂќ version.
Hint:
26 Existence and Uniqueness of Moment Maps

26.1 Lie Algebras of Vector Fields
Let (M, П‰) be a symplectic manifold and v в€€ П‡(M ) a vector п¬Ѓeld on M .
в‡ђв‡’
v is symplectic Д±v П‰ is closed ,
в‡ђв‡’
v is hamiltonian Д±v П‰ is exact .
The spaces
П‡sympl (M ) = symplectic vector п¬Ѓelds on M ,
П‡ham (M ) = hamiltonian vector п¬Ѓelds on M .
are Lie algebras for the Lie bracket of vector п¬Ѓelds. C в€ћ (M ) is a Lie algebra for
the Poisson bracket, {f, g} = П‰(vf , vg ). H 1 (M ; R) and R are regarded as Lie
algebras for the trivial bracket. We have two exact sequences of Lie algebras:
0 в€’в†’ П‡ham (M ) в†’ П‡sympl (M ) в€’в†’ H 1 (M ; R) в€’в†’ 0
v в€’в†’ [Д±v П‰]

C в€ћ (M ) в€’в†’ П‡ham (M )
0 в€’в†’ в†’ в€’в†’ 0
R
f в€’в†’ vf .

In particular, if H 1 (M ; R) = 0, then П‡ham (M ) = П‡sympl (M ).
Let G be a connected Lie group. A symplectic action П€ : G в†’ Sympl(M, П‰)
induces an inп¬Ѓnitesimal action
g в€’в†’ П‡sympl (M )
dП€ :
X в€’в†’ X # = vector п¬Ѓeld generated by the
one-parameter group {exp tX(e) | t в€€ R} .

в™¦
Exercise. Check that the map dП€ is a Lie algebra anti-homomorphism.
The action П€ is hamiltonian if and only if there is a Lie algebra homo-
morphism Вµв€— : g в†’ C в€ћ (M ) lifting dП€, i.e., making the following diagram
commute.
E П‡sympl (M )
C в€ћ (M )
d
s В

d В
d В
Вµв€— d В  dП€
d В
g
в€—
The map Вµ is then called a comoment map (deп¬Ѓned in Lecture 22).
Existence of Вµв€— в‡ђв‡’ Existence of Вµ
comoment map moment map

в†ђв†’
Lie algebra homomorphism equivariance

164
165
26.2 Lie Algebra Cohomology

26.2 Lie Algebra Cohomology

Let g be a Lie algebra, and

:= О›k gв€— = k-cochains on g
Ck
= alternating k-linear maps g Г— . . . Г— g в€’в†’ R .
k

Deп¬Ѓne a linear operator Оґ : C k в†’ C k+1 by

(в€’1)i+j c([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ) .
Оґc(X0 , . . . , Xk ) =
i<j

Exercise. Check that Оґ 2 = 0. в™¦
The Lie algebra cohomology groups (or Chevalley cohomology groups)
Оґ Оґ Оґ
of g are the cohomology groups of the complex 0 в†’ C 0 в†’ C 1 в†’ . . .:

ker Оґ : C k в€’в†’ C k+1
H k (g; R) := .
im Оґ : C kв€’1 в€’в†’ C k
Theorem 26.1 If g is the Lie algebra of a compact connected Lie group G,
then
H k (g; R) = HdeRham (G) .
k

Proof. Exercise. Hint: by averaging show that the de Rham cohomology can
be computed from the subcomplex of G-invariant forms.
Meaning of H 1 (g; R) and H 2 (g; R):

вЂў An element of C 1 = gв€— is a linear functional on g. If c в€€ gв€— , then
Оґc(X0 , X1 ) = в€’c([X0 , X1 ]). The commutator ideal of g is

[g, g] := {linear combinations of [X, Y ] for any X, Y в€€ g} .

Since Оґc = 0 if and only if c vanishes on [g, g], we conclude that

H 1 (g; R) = [g, g]0

where [g, g]0 вЉ† gв€— is the annihilator of [g, g].

вЂў An element of C 2 is an alternating bilinear map c : g Г— g в†’ R.

Оґc(X0 , X1 , X2 ) = в€’c([X0 , X1 ], X2 ) + c([X0 , X2 ], X1 ) в€’ c([X1 , X2 ], X0 ) .

If c = Оґb for some b в€€ C 1 , then

c(X0 , X1 ) = (Оґb)(X0 , X1 ) = в€’b([X0 , X1 ] ).
166 26 EXISTENCE AND UNIQUENESS OF MOMENT MAPS

26.3 Existence of Moment Maps

Theorem 26.2 If H 1 (g; R) = H 2 (g, R) = 0, then any symplectic G-action is
hamiltonian.

Proof. Let П€ : G в†’ Sympl(M, П‰) be a symplectic action of G on a symplectic
manifold (M, П‰). Since

H 1 (g; R) = 0 в‡ђв‡’ [g, g] = g

and since commutators of symplectic vector п¬Ѓelds are hamiltonian, we have

dП€ : g = [g, g] в€’в†’ П‡ham (M ).

The action П€ is hamiltonian if and only if there is a Lie algebra homomorphism
Вµв€— : g в†’ C в€ћ (M ) such that the following diagram commutes.
E C в€ћ (M ) E П‡ham (M )
R
d
s В

d В
d В
?d В  dП€
d В
g
We п¬Ѓrst take an arbitrary vector space lift П„ : g в†’ C в€ћ (M ) making the diagram
commute, i.e., for each basis vector X в€€ g, we choose

П„ (X) = П„ X в€€ C в€ћ (M ) such that v(П„ X ) = dП€(X) .

The map X в†’ П„ X may not be a Lie algebra homomorphism. By construction,
П„ [X,Y ] is a hamiltonian function for [X, Y ]# , and (as computed in Lecture 16)
{П„ X , П„ Y } is a hamiltonian function for в€’[X #, Y # ]. Since [X, Y ]# = в€’[X # , Y # ],
the corresponding hamiltonian functions must diп¬Ђer by a constant:

П„ [X,Y ] в€’ {П„ X , П„ Y } = c(X, Y ) в€€ R .

By the Jacobi identity, Оґc = 0. Since H 2 (g; R) = 0, there is b в€€ gв€— satisfying
c = Оґb, c(X, Y ) = в€’b([X, Y ]). We deп¬Ѓne

Вµв€— : g в€’в†’ C в€ћ (M )
X в€’в†’ Вµв€— (X) = П„ X + b(X) = ВµX .

Now Вµв€— is a Lie algebra homomorphism:

Вµв€— ([X, Y ]) = П„ [X,Y ] + b([X, Y ]) = {П„ X , П„ Y } = {ВµX , ВµY } .

So when is H 1 (g; R) = H 2 (g; R) = 0?
167
26.4 Uniqueness of Moment Maps

A compact Lie group G is semisimple if g = [g, g].
Examples. The unitary group U(n) is not semisimple because the multiples of
the identity, S 1 В· Id, form a nontrivial center; at the level of the Lie algebra, this
corresponds to the 1-dimensional subspace R В· Id of constant matrices which are
not commutators since they are not traceless.
Any direct product of the other compact classical groups SU(n), SO(n) and
Sp(n) is semisimple (n > 1). Any commutative Lie group is not semisimple. в™¦

Theorem 26.3 (Whitehead Lemmas) Let G be a compact Lie group.
H 1 (g; R) = H 2 (g; R) = 0 .
в‡ђв‡’
G is semisimple
A proof can be found in [66, pages 93-95].
Corollary 26.4 If G is semisimple, then any symplectic G-action is hamilto-
nian.

26.4 Uniqueness of Moment Maps
Let G be a compact Lie group.
Theorem 26.5 If H 1 (g; R) = 0, then moment maps for hamiltonian G-actions
are unique.

Proof. Suppose that Вµв€— and Вµв€— are two comoment maps for an action П€:
1 2

E П‡ham (M )
C в€ћ (M )
d
s В

d Вµв€— В
d2 В
Вµв€— d В  dП€
1
d В
g
For each X в€€ g, ВµX and ВµX are both hamiltonian functions for X # , thus
1 2
Вµ1 в€’ Вµ2 = c(X) is locally constant. This deп¬Ѓnes c в€€ gв€— , X в†’ c(X).
X X

Since Вµв€— , Вµв€— are Lie algebra homomorphisms, we have c([X, Y ]) = 0, в€ЂX, Y в€€
1 2
g, i.e., c в€€ [g, g]0 = {0}. Hence, Вµв€— = Вµв€— .
1 2

Corollary of this proof. In general, if Вµ : M в†’ gв€— is a moment map, then
given any c в€€ [g, g]0 , Вµ1 = Вµ + c is another moment map.
In other words, moment maps are unique up to elements of the dual of the
Lie algebra which annihilate the commutator ideal.
The two extreme cases are:
G semisimple: any symplectic action is hamiltonian ,
moment maps are unique .
G commutative: symplectic actions may not be hamiltonian ,
moment maps are unique up to any constant c в€€ gв€— .
168 26 EXISTENCE AND UNIQUENESS OF MOMENT MAPS

Example. The circle action on (T2 , П‰ = dОё1 в€§ dОё2 ) by rotations in the Оё1
в€‚
direction has vector п¬Ѓeld X # = в€‚Оё1 ; this is a symplectic action but is not
в™¦
hamiltonian.
Homework 20: Examples of Reduction

1. For the natural action of U(k) on CkГ—n with moment map computed in
exercise 5 of Homework 19, we have Вµв€’1 (0) = {A в€€ CkГ—n | AAв€— = Id}.
Show that the quotient

Вµв€’1 (0)/U(k) = G(k, n)

is the grassmannian of k-planes in Cn .

2. Consider the S 1 -action on (R2n+2 , П‰0 ) which, under the usual identiп¬Ѓca-
tion of R2n+2 with Cn+1 , corresponds to multiplication by eit . This action
is hamiltonian with a moment map Вµ : Cn+1 в†’ R given by

Вµ(z) = в€’ 1 |z|2 + 1
.
2 2

Prove that the reduction Вµв€’1 (0)/S 1 is CPn with the Fubini-Study sym-
plectic form П‰red = П‰FS .
Let pr : Cn+1 \ {0} в†’ CPn denote the standard projection. Check
Hint:
that
iВЇ
prв€— П‰FS = в€‚ в€‚ log(|z|2 ) .
2
Prove that this form has the same restriction to S 2n+1 as П‰red .

3. Show that the natural actions of Tn+1 and U(n + 1) on (CPn , П‰FS ) are
hamiltonian, and п¬Ѓnd formulas for their moment maps.

Previous exercise and exercises 2 and 4 of Homework 19.
Hint:

169
27 Convexity

27.1 Convexity Theorem

From now on, we will concentrate on actions of a torus G = Tm = Rm /Zm .

Theorem 27.1 (Atiyah , Guillemin-Sternberg )
Let (M, П‰) be a compact connected symplectic manifold, and let Tm be an
m-torus. Suppose that П€ : Tm в†’ Sympl(M, П‰) is a hamiltonian action with
moment map Вµ : M в†’ Rm . Then:

1. the levels of Вµ are connected;

2. the image of Вµ is convex;

3. the image of Вµ is the convex hull of the images of the п¬Ѓxed points of the
action.

The image Вµ(M ) of the moment map is hence called the moment polytope.
Proof. This proof (due to Atiyah) involves induction over m = dim Tm . Con-
sider the statements:

Am : вЂњthe levels of Вµ are connected, for any Tm -action;вЂќ

Bm : вЂњthe image of Вµ is convex, for any Tm -action.вЂќ

Then
в‡ђв‡’
(1) Am holds for all m ,
в‡ђв‡’
(2) Bm holds for all m .

вЂў A1 is a non-trivial result in Morse theory.

вЂў Amв€’1 =в‡’ Am (induction step) is in Homework 21.

вЂў B1 is trivial because in R connectedness is convexity.

вЂў Amв€’1 =в‡’ Bm is proved below.

Choose an injective matrix A в€€ ZmГ—(mв€’1) . Consider the action of an (mв€’1)-
subtorus
П€A : Tmв€’1 в€’в†’ Sympl(M, П‰)
Оё в€’в†’ П€AОё .

Exercise. The action П€A is hamiltonian with moment map ВµA = At Вµ : M в†’
Rmв€’1 . в™¦
Given any p0 в€€ Вµв€’1 (Оѕ),
A

p в€€ Вµв€’1 (Оѕ) в‡ђв‡’ At Вµ(p) = Оѕ = At Вµ(p0 )
A

170
171
27.2 Eп¬Ђective Actions

so that
Вµв€’1 (Оѕ) = {p в€€ M | Вµ(p) в€’ Вµ(p0 ) в€€ ker At } .
A

By the п¬Ѓrst part (statement Amв€’1 ), Вµв€’1 (Оѕ) is connected. Therefore, if we
A
в€’1
connect p0 to p1 by a path pt in ВµA (Оѕ), we obtain a path Вµ(pt ) в€’ Вµ(p0 ) in
ker At . But ker At is 1-dimensional. Hence, Вµ(pt ) must go through any convex
combination of Вµ(p0 ) and Вµ(p1 ), which shows that any point on the line segment
from Вµ(p0 ) to Вµ(p1 ) must be in Вµ(M ):

(1 в€’ t)Вµ(p0 ) + tВµ(p1 ) в€€ Вµ(M ) , 0в‰¤tв‰¤1.

Any p0 , p1 в€€ M can be approximated arbitrarily closely by points p0 and p1
with Вµ(p1 ) в€’ Вµ(p0 ) в€€ ker At for some injective matrix A в€€ ZmГ—(mв€’1) . Taking
limits p0 в†’ p0 , p1 в†’ p1 , we obtain that Вµ(M ) is convex.15
To prove part 3, consider the п¬Ѓxed point set C of П€. Homework 21 shows
that C is a п¬Ѓnite union of connected symplectic submanifolds, C = C1 в€Є. . .в€ЄCN .
The moment map is constant on each Cj , Вµ(Cj ) = О·j в€€ Rm , j = 1, . . . , N . By
the second part, the convex hull of {О·1 , . . . , О·N } is contained in Вµ(M ).
For the converse, suppose that Оѕ в€€ Rm and Оѕ в€€ convex hull of {О·1 , . . . , О·N }.
/
m
Choose X в€€ R with rationally independent components and satisfying

Оѕ, X > О·j , X , for all j .

By the irrationality of X, the set {exp tX(e) | t в€€ R} is dense in Tm , hence the
zeros of the vector п¬Ѓeld X # on M are the п¬Ѓxed points of the Tm -action. Since
ВµX = Вµ, X attains its maximum on one of the sets Cj , this implies

Оѕ, X > sup Вµ(p), X ,
pв€€M

hence Оѕ в€€ Вµ(M ). Therefore,
/

Вµ(M ) = convex hull of {О·1 , . . . , О·N } .

27.2 Eп¬Ђective Actions
An action of a group G on a manifold M is called eп¬Ђective if each group element
g = e moves at least one p в€€ M , that is,

Gp = {e} ,
pв€€M

where Gp = {g в€€ G | g В· p = p} is the stabilizer of p.

Corollary 27.2 Under the conditions of the convexity theorem, if the Tm -
action is eп¬Ђective, then there must be at least m + 1 п¬Ѓxed points.
15 Clearly Вµ(M ) is closed because it is compact.
172 27 CONVEXITY

Proof. If the Tm -action is eп¬Ђective, there must be a point p where the moment
map is a submersion, i.e., (dВµ1 )p , . . . , (dВµm )p are linearly independent. Hence,
Вµ(p) is an interior point of Вµ(M ), and Вµ(M ) is a nondegenerate convex polytope.
Any nondegenerate convex polytope in Rm must have at least m + 1 vertices.
The vertices of Вµ(M ) are images of п¬Ѓxed points.

Theorem 27.3 Let (M, П‰, Tm , Вµ) be a hamiltonian Tm -space. If the Tm -action
is eп¬Ђective, then dim M в‰Ґ 2m.

Proof. Fact: If П€ : Tm в†’ Diп¬Ђ(M ) is an eп¬Ђective action, then it has orbits of
dimension m; a proof may be found in .
On an m-dimensional orbit O, the moment map Вµ(O) = Оѕ is constant. For
p в€€ O, the exterior derivative

dВµp : Tp M в€’в†’ gв€—

maps Tp O to 0. Thus
Tp O вЉ† ker dВµp = (Tp O)П‰ ,
which shows that orbits O of a hamiltonian torus action are always isotropic
submanifolds of M . In particular, dim O = m в‰¤ 1 dim M .
2

Deп¬Ѓnition 27.4 A (symplectic) toric manifold16 is a compact connected
symplectic manifold (M, П‰) equipped with an eп¬Ђective hamiltonian action of a
torus T of dimension equal to half the dimension of the manifold:

1
dim T = dim M
2
and with a choice of a corresponding moment map Вµ.

Exercise. Show that an eп¬Ђective hamiltonian action of a torus Tn on a 2n-
dimensional symplectic manifold gives rise to an integrable system.
Hint: The coordinates of the moment map are commuting integrals of motion.

в™¦

27.3 Examples

1. The circle S1 acts on the 2-sphere (S 2 , П‰standard = dОё в€§ dh) by rota-
tions with moment map Вµ = h equal to the height function and moment
polytope [в€’1, 1].
16 In these notes, a toric manifold is always a symplectic toric manifold.
173
27.3 Examples

t t1
'\$
Вµ=h
E
t tв€’1
&%

1.вЂ™ The circle S1 acts on CP1 = C2 в€’ 0/ в€ј with the Fubini-Study form
П‰FS = 1 П‰standard , by eiОё В· [z0 , z1 ] = [z0 , eiОё z1 ]. This is hamiltonian with
4
|z1 |2
1 1
moment map Вµ[z0 , z1 ] = в€’ 2 В· and moment polytope в€’ 2 , 0 .
|z0 |2 +|z1 |2 ,

2. The T2 -action on CP2 by

(eiОё1 , eiОё2 ) В· [z0 , z1 , z2 ] = [z0 , eiОё1 z1 , eiОё2 z2 ]

has moment map

|z1 |2 |z2 |2
1
Вµ[z0 , z1 , z2 ] = в€’ , .
|z0 |2 + |z1 |2 + |z2 |2 |z0 |2 + |z1 |2 + |z2 |2
2

The п¬Ѓxed points get mapped as

[1, 0, 0] в€’в†’ (0, 0)
1
[0, 1, 0] в€’в†’ в€’ 2 , 0
1
[0, 0, 1] в€’в†’ 0, в€’ 2

Notice that the stabilizer of a preimage of the edges is S 1 , while the action
is free at preimages of interior points of the moment polytope.
174 27 CONVEXITY

T

1
(в€’ 2 , 0)
t(0, 0) E
t
dВ В В В В
В
dВ В В В
В
dВ В В
В
dВ В
В
dВ
dt 1
(0, в€’ 2 )

Exercise. What is the moment polytope for the T3 -action on CP3 as

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

в™¦

Exercise. What is the moment polytope for the T2 -action on CP1 Г— CP1 as

(eiОё , eiО· ) В· ([z0 , z1 ], [w0 , w1 ]) = ([z0 , eiОё z1 ], [w0 , eiО· w1 ]) ?

в™¦
Homework 21: Connectedness
Consider a hamiltonian action П€ : Tm в†’ Sympl (M, П‰), Оё в†’ П€Оё , of an m-
dimensional torus on a 2n-dimensional compact connected symplectic manifold
(M, П‰). If we identify the Lie algebra of Tm with Rm by viewing Tm = Rm /Zm ,
and we identify the Lie algebra with its dual via the standard inner product,
then the moment map for П€ is Вµ : M в†’ Rm .
1. Show that there exists a compatible almost complex structure J on (M, П‰)
в€— в€—
which is invariant under the Tm -action, that is, П€Оё J = JП€Оё , for all Оё в€€ Tm .
Hint: We cannot average almost complex structures, but we can average
riemannian metrics (why?). Given a riemannian metric g0 on M , its Tm -
в€—
average g = Tm П€Оё g0 dОё is Tm -invariant.

2. Show that, for any subgroup G вЉ† Tm , the п¬Ѓxed-point set for G ,

Fix (G) = Fix (П€Оё ) ,
Оёв€€G

is a symplectic submanifold of M .
For each p в€€ Fix (G) and each Оё в€€ G, the diп¬Ђerential of П€Оё at p,
Hint:
dП€Оё (p) : Tp M в€’в†’ Tp M ,
preserves the complex structure Jp on Tp M . Consider the exponential map
expp : Tp M в†’ M with respect to the invariant riemannian metric g(В·, В·) =
П‰(В·, JВ·). Show that, by uniqueness of geodesics, exp p is equivariant, i.e.,
expp (dП€Оё (p)v) = П€Оё (exp p v)
for any Оё в€€ G, v в€€ Tp M . Conclude that the п¬Ѓxed points of П€Оё near p correspond
to the п¬Ѓxed points of dП€Оё (p) on Tp M , that is

Tp Fix (G) = ker(Id в€’ dП€Оё (p)) .
Оёв€€G

Since dП€Оё (p) в—¦ Jp = Jp в—¦ dП€Оё (p), the eigenspace with eigenvalue 1 is invariant
under Jp , and is therefore a symplectic subspace.

3. A smooth function f : M в†’ R on a compact riemannian manifold M
is called a Morse-Bott function if its critical set Crit (f ) = {p в€€
M | df (p) = 0} is a submanifold of M and for every p в€€ Crit (f ), Tp Crit (f ) =
ker 2 f (p) where 2 f (p) : Tp M в†’ Tp M denotes the linear operator ob-
tained from the hessian via the riemannian metric. This is the natural
generalization of the notion of Morse function to the case where the crit-
ical set is not just isolated points. If f is a Morse-Bott function, then
Crit (f ) decomposes into п¬Ѓnitely many connected critical manifolds C.
The tangent space Tp M at p в€€ C decomposes as a direct sum
+ в€’
Tp M = T p C вЉ• E p вЉ• E p
+ в€’
where Ep and Ep are spanned by the positive and negative eigenspaces of
f (p). The index of a connected critical submanifold C is nв€’ = dim Ep ,
2 в€’
C
for any p в€€ C, whereas the coindex of C is n+ = dim Ep . +
C

175
176 HOMEWORK 21

For each X в€€ Rm , let ВµX = Вµ, X : M в†’ R be the component of Вµ along
X. Show that ВµX is a Morse-Bott function with even-dimensional critical
manifolds of even index. Moreover, show that the critical set

Crit (ВµX ) = Fix (П€Оё )
Оёв€€TX

is a symplectic manifold, where TX is the closure of the subgroup of Tm
generated by X.
Hint: Assume п¬Ѓrst that X has components independent over Q, so that TX =
Tm and Crit (ВµX ) = Fix (Tm ). Apply exercise 2. To prove that Tp Crit (ВµX ) =
ker 2 ВµX (p), show that ker 2 ВµX (p) = в€©Оёв€€Tm ker(Id в€’ dП€Оё (p)). To see
this, notice that the 1-parameter group of matrices (dП€exp tX )p coincides with
exp(tvp ), where vp = в€’Jp 2 ВµX (p) : Tp M в†’ Tp M is a vector п¬Ѓeld on Tp M .
The kernel of 2 ВµX (p) corresponds to the п¬Ѓxed points of dП€tX (p), and since X
has rationally independent components, these are the common п¬Ѓxed points of
all dП€Оё (p), Оё в€€ Tm . The eigenspaces of 2 ВµX (p) are even-dimensional because
they are invariant under Jp .

4. The moment map Вµ = (Вµ1 , . . . , Вµm ) is called eп¬Ђective if the 1-forms
dВµ1 , . . . , dВµm of its components are linearly independent. Show that, if Вµ
is not eп¬Ђective, then the action reduces to that of an (m в€’ 1)-subtorus.
Hint: If Вµ is not eп¬Ђective, then the function ВµX = Вµ, X is constant for
some nonzero X в€€ Rm . Show that we can neglect the direction of X.

5. Prove that the level set Вµв€’1 (Оѕ) is connected for every regular value Оѕ в€€ Rm .
Hint: Prove by induction over m = dim Tm . For the case m = 1, use the
lemma that all level sets f в€’1 (c) of a Morse-Bott function f : M в†’ R on a com-
pact manifold M are necessarily connected, if the critical manifolds all have
index and coindex = 1 (see [82, p.178-179]). For the induction step, you can as-
sume that П€ is eп¬Ђective. Then, for every 0 = X в€€ Rm , the function ВµX : M в†’
R is not constant. Show that C := в€ЄX=0 Crit ВµX = в€Є0=Xв€€Zm Crit ВµX where
each Crit ВµX is an even-dimensional proper submanifold, so the complement
M \ C must be dense in M . Show that M \ C is open. Hence, by continuity, to
show that Вµв€’1 (Оѕ) is connected for every regular value Оѕ = (Оѕ1 , . . . , Оѕm ) в€€ Rm , it
suп¬ѓces to show that Вµв€’1 (Оѕ) is connected whenever (Оѕ1 , . . . , Оѕmв€’1 ) is a regular
value for a reduced moment map (Вµ1 , . . . , Вµmв€’1 ). By the induction hypoth-
esis, the manifold Q = в€©j=1 Вµв€’1 (Оѕj ) is connected whenever (Оѕ1 , . . . , Оѕmв€’1 )
mв€’1
j
is a regular value for (Вµ1 , . . . , Вµmв€’1 ). It suп¬ѓces to show that the function
Вµm : Q в†’ R has only critical manifolds of even index and coindex (see [82,
p.183]), because then, by the lemma, the level sets Вµв€’1 (Оѕ) = Q в€© Вµв€’1 (Оѕm ) are
m
connected for every Оѕm .
Part XI
Symplectic Toric Manifolds
Native to algebraic geometry, toric manifolds have been studied by symplec-
tic geometers as examples of extremely symmetric hamiltonian spaces, and as
guinea pigs for new theorems. Delzant showed that symplectic toric manifolds
are classiп¬Ѓed (as hamiltonian spaces) by a set of special polytopes.

28 Classiп¬Ѓcation of Symplectic Toric Manifolds
28.1 Delzant Polytopes
A 2n-dimensional (symplectic) toric manifold is a compact connected sym-
plectic manifold (M 2n , П‰) equipped with an eп¬Ђective hamiltonian action of an
n-torus Tn and with a corresponding moment map Вµ : M в†’ Rn .
Deп¬Ѓnition 28.1 A Delzant polytope в€† in Rn is a convex polytope satisfying:
вЂў it is simple, i.e., there are n edges meeting at each vertex;
вЂў it is rational, i.e., the edges meeting at the vertex p are rational in the
sense that each edge is of the form p + tui , 0 в‰¤ t < в€ћ, where ui в€€ Zn ;
вЂў it is smooth, i.e., these u1 , . . . , un can be chosen to be a basis of Zn .

Remark. The Delzant polytopes are the simple rational smooth polytopes.
These are closely related to the Newton polytopes (which are the nonsingular
n-valent polytopes), except that the vertices of a Newton polytope are required
в™¦
to lie on the integer lattice and for a Delzant polytope they are not.

Examples of Delzant polytopes:

d
d
d
d

Вўd
d Вў d
d Вў d
d Вў d

d Вў 
Вў


177
178 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

The dotted vertical line in the trapezoildal example means nothing, except that
itвЂ™s a picture of a rectangle plus an isosceles triangle. For вЂњtallerвЂќ triangles,
smoothness would be violated. вЂњWiderвЂќ triangles (with integral slope) may
still be Delzant. The family of the Delzant trapezoids of this type, starting
with the rectangle, correspond, under the Delzant construction, to Hirzebruch
surfaces; see Homework 22.

Examples of polytopes which are not Delzant:

В„
d
r
rr В„d
rr В„d
rr В„d
r В„В

The picture on the left fails the smoothness condition, whereas the picture
on the right fails the simplicity condition.
Algebraic description of Delzant polytopes:
A facet of a polytope is a (n в€’ 1)-dimensional face.
Let в€† be a Delzant polytope with n = dim в€† and d = number of facets.
A lattice vector v в€€ Zn is primitive if it cannot be written as v = ku with
u в€€ Zn , k в€€ Z and |k| > 1; for instance, (1, 1), (4, 3), (1, 0) are primitive, but
(2, 2), (4, 6) are not.
Let vi в€€ Zn , i = 1, . . . , d, be the primitive outward-pointing normal vectors
to the facets.

r
ВЎ rr !
ВЎ
ВЎ rr ВЎ n=2
В‰
r
rr ВЎ rr ВЎ
rВЎ rВЎ d=3
r
ВЎ rr
ВЎ rr
ВЎ rr
ВЎ r
c

Then we can describe в€† as an intersection of halfspaces

в€† = {x в€€ (Rn )в€— | x, vi в‰¤ О»i , i = 1, . . . , d} for some О»i в€€ R .
179
28.2 Delzant Theorem

Example. For the picture below, we have

в€† = {x в€€ (R2 )в€— | x1 в‰Ґ 0, x2 в‰Ґ 0, x1 + x2 в‰¤ 1}
= {x в€€ (R2 )в€— | x, (в€’1, 0) в‰¤ 0 , x, (0, в€’1) в‰¤ 0 , x, (1, 1) в‰¤ 1} .

В
В
В
(0, 1)
В
r
d В
d В
d В
' dВ
d
d
d
dr
r
(0, 0) (1, 0)

c

в™¦

28.2 Delzant Theorem

We do not have a classiп¬Ѓcation of symplectic manifolds, but we do have a clas-
siп¬Ѓcation of toric manifolds in terms of combinatorial data. This is the content
of the Delzant theorem.

Theorem 28.2 (Delzant ) Toric manifolds are classiп¬Ѓed by Delzant
polytopes. More speciп¬Ѓcally, there is the following one-to-one correspondence
1в€’1
{toric manifolds} в€’в†’ {Delzant polytopes}
(M 2n , П‰, Tn , Вµ) в€’в†’ Вµ(M ).

We will prove the existence part (or surjectivity) in the Delzant theorem
following . Given a Delzant polytope, what is the corresponding toric man-
ifold?
?
(Mв€† , П‰в€† , Tn , Вµ) в†ђв€’ в€†n
180 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

28.3 Sketch of Delzant Construction

Let в€† be a Delzant polytope with d facets. Let vi в€€ Zn , i = 1, . . . , d, be the
primitive outward-pointing normal vectors to the facets. For some О»i в€€ R,

в€† = {x в€€ (Rn )в€— | x, vi в‰¤ О»i , i = 1, . . . , d} .

Let e1 = (1, 0, . . . , 0), . . . , ed = (0, . . . , 0, 1) be the standard basis of Rd . Con-
sider
ПЂ : Rd в€’в†’ Rn
ei в€’в†’ vi .

Claim. The map ПЂ is onto and maps Zd onto Zn .

Proof. The set {e1 , . . . , ed } is a basis of Zd . The set {v1 , . . . , vd } spans Zn for
the following reason. At a vertex p, the edge vectors u1 , . . . , un в€€ (Rn )в€— , form a
basis for (Zn )в€— which, without loss of generality, we may assume is the standard
basis. Then the corresponding primitive normal vectors to the facets meeting at
p are symmetric (in the sense of multiplication by в€’1) to the ui вЂ™s, hence form
a basis of Zn .
Therefore, ПЂ induces a surjective map, still called ПЂ, between tori:
ПЂ
Rd /Zd в€’в†’ Rn /Zn

Td Tn
в€’в†’ в€’в†’ 0 .

Let
(N is a Lie subgroup of Td )
N = kernel of ПЂ
n = Lie algebra of N
Rd = Lie algebra of Td
Rn = Lie algebra of Tn .
The exact sequence of tori
i ПЂ
0 в€’в†’ N в€’в†’ Td в€’в†’ Tn в€’в†’ 0

induces an exact sequence of Lie algebras
i ПЂ
0 в€’в†’ n в€’в†’ Rd в€’в†’ Rn в€’в†’ 0

with dual exact sequence
ПЂв€— iв€—
0 в€’в†’ (Rn )в€— в€’в†’ (Rd )в€— в€’в†’ nв€— в€’в†’ 0 .
i
Now consider Cd with symplectic form П‰0 = dzk в€§ dВЇk , and standard
z
2
hamiltonian action of Td

(e2ПЂit1 , . . . , e2ПЂitd ) В· (z1 , . . . , zd ) = (e2ПЂit1 z1 , . . . , e2ПЂitd zd ) .
181
28.3 Sketch of Delzant Construction

The moment map is П† : Cd в€’в†’ (Rd )в€—
П†(z1 , . . . , zd ) = в€’ПЂ(|z1 |2 , . . . , |zd |2 ) + constant ,
where we choose the constant to be (О»1 , . . . , О»d ). What is the moment map for
the action restricted to the subgroup N ?
Exercise. Let G be any compact Lie group and H a closed subgroup of G,
with g and h the respective Lie algebras. The inclusion i : h в†’ g is dual to the
projection iв€— : gв€— в†’ hв€— . Suppose that (M, П‰, G, П†) is a hamiltonian G-space.
Show that the restriction of the G-action to H is hamiltonian with moment map
iв€— в—¦ П† : M в€’в†’ hв€— .
в™¦
The subtorus N acts on Cd in a hamiltonian way with moment map
iв€— в—¦ П† : Cd в€’в†’ nв€— .
Let Z = (iв€— в—¦ П†)в€’1 (0) be the zero-level set.
Claim. The set Z is compact and N acts freely on Z.
This claim will be proved in the next lecture.
By the п¬Ѓrst claim, 0 в€€ nв€— is a regular value of iв€— в—¦ П†. Hence, Z is a compact
submanifold of Cd of dimension
dimR Z = 2d в€’ (d в€’ n) = d + n .
dim nв€—

The orbit space Mв€† = Z/N is a compact manifold of dimension
dimR Mв€† = d + n в€’ (d в€’ n) = 2n .
dim N

The point-orbit map p : Z в†’ Mв€† is a principal N -bundle over Mв€† .
Consider the diagram
j
Cd
в†’
Z
pв†“
Mв€†
where j : Z в†’ Cd is inclusion. The Marsden-Weinstein-Meyer theorem guaran-
tees the existence of a symplectic form П‰в€† on Mв€† satisfying
p в€— П‰в€† = j в€— П‰0 .

Exercise. Work out all details in the following simple example.
Let в€† = [0, a] вЉ‚ Rв€— (n = 1, d = 2). Let v(= 1) be the standard basis vector
in R. Then
в€† : x, v1 в‰¤ 0 v1 = в€’v
x, v2 в‰¤ a v2 = v .
182 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

The projection
ПЂ
R2 в€’в†’ R
в€’в†’ в€’v
e1
в€’в†’ v
e2
has kernel equal to the span of (e1 + e2 ), so that N is the diagonal subgroup of
T2 = S 1 Г— S 1 . The exact sequences become
i ПЂ
T2 S1
0 в€’в†’ в€’в†’ в€’в†’ в€’в†’ 0
N
в€— в€—
ПЂ i
0 в€’в†’ Rв€— (R2 )в€— nв€—
в€’в†’ в€’в†’ в€’в†’ 0
в€’в†’
(x1 , x2 ) x 1 + x2 .

The action of the diagonal subgroup N = {(e2ПЂit , e2ПЂit ) в€€ S 1 Г— S 1 } on C2 ,

(e2ПЂit , e2ПЂit ) В· (z1 , z2 ) = (e2ПЂit z1 , e2ПЂit z2 ) ,

has moment map

(iв€— в—¦ П†)(z1 , z2 ) = в€’ПЂ(|z1 |2 + |z2 |2 ) + a ,

with zero-level set
a
(iв€— в—¦ П†)в€’1 (0) = {(z1 , z2 ) в€€ C2 : |z1 |2 + |z2 |2 = }.
ПЂ
Hence, the reduced space is

(iв€— в—¦ П†)в€’1 (0)/N = CP1 projective space!

в™¦
29 Delzant Construction

29.1 Algebraic Set-Up

Let в€† be a Delzant polytope with d facets. We can write в€† as

в€† = {x в€€ (Rn )в€— | x, vi в‰¤ О»i , i = 1, . . . , d} ,

for some О»i в€€ R. Recall the exact sequences from the previous lecture
i ПЂ
в€’в†’ Td в€’в†’ Tn
0 в€’в†’ N в€’в†’ 0
i ПЂ
в€’в†’ Rd в€’в†’ Rn
0 в€’в†’ n в€’в†’ 0
в€’в†’ vi
ei

and the dual sequence
ПЂв€— iв€—
0 в€’в†’ (Rn )в€— в€’в†’ (Rd )в€— в€’в†’ nв€— в€’в†’ 0 .

The standard hamiltonian action of Td on Cd

(e2ПЂit1 , . . . , e2ПЂitd ) В· (z1 , . . . , zd ) = (e2ПЂit1 z1 , . . . , e2ПЂitd zd )

has moment map П† : Cd в†’ (Rd )в€— given by

П†(z1 , . . . , zd ) = в€’ПЂ(|z1 |2 , . . . , |zd |2 ) + (О»1 , . . . , О»d ) .

The restriction of this action to N has moment map

iв€— в—¦ П† : Cd в€’в†’ nв€— .

29.2 The Zero-Level

Let Z = (iв€— в—¦ П†)в€’1 (0).

Theorem 29.1 The level Z is compact and N acts freely on Z.

Proof. Let в€† be the image of в€† by ПЂ в€— . We will show that П†(Z) = в€† . Since
П† is a proper map and в€† is compact, it will follow that Z is compact.

Lemma 29.2 Let y в€€ (Rd )в€— . Then:

yв€€в€† в‡ђв‡’ y is in the image of Z by П† .

Proof of the lemma. The value y is in the image of Z by П† if and only if
both of the following conditions hold:
1. y is in the image of П†;

183
184 29 DELZANT CONSTRUCTION

2. iв€— y = 0.
Using the expression for П† and the third exact sequence, we see that these
conditions are equivalent to:

1. y, ei в‰¤ О»i for i = 1, . . . , d.
2. y = ПЂ в€— (x) for some x в€€ (Rn )в€— .

Suppose that the second condition holds, so that y = ПЂ в€— (x). Then

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

Thus, y в€€ П†(z) в‡ђв‡’ y в€€ ПЂ в€— (в€†) = в€† .
Hence, we have a surjective proper map П† : Z в†’ в€† . Since в€† is compact,
we conclude that Z is compact. It remains to show that N acts freely on Z.
We deп¬Ѓne a stratiп¬Ѓcation of Z with three equivalent descriptions:

вЂў Deп¬Ѓne a stratiп¬Ѓcation on в€† whose ith stratum is the closure of the union
of the i-dimensional faces of в€† . Pull this stratiп¬Ѓcation back to Z by П†.
We can obtain a more explicit description of the stratiп¬Ѓcation on Z:
вЂў Let F be a face of в€† with dim F = n в€’ r. Then F is characterized (as a
subset of в€† ) by r equations

y, ei = О»i , i = i 1 , . . . , ir .

We write F = FI where I = (i1 , . . . , ir ) has 1 в‰¤ i1 < i2 . . . < ir в‰¤ d.
Let z = (z1 , . . . , zd ) в€€ Z.

z в€€ П†в€’1 (FI ) в‡ђв‡’ П†(z) в€€ FI
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