Atiyah-Guillemin-Sternberg convexity theorem.pdf

Atiyah-Guillemin-Sternberg convexity theorem
Jun Ikeda
July 1, 2025
Abstract
We review symplectic geometry and prove the Atiyah–Guillemin-Sternberg (AGS)
convexity theorem. After summarising the essential notions of moment maps, Hamilto-
nian torus actions, and their Morse–Bott properties, we outline a self-contained proof
that highlights ideas from equivariant Morse theory.
Contents
1 Introduction 1
2 Preliminaries 2
3 Proof of Main Theorem 7
1 Introduction
Hamiltonian torus actions on compact symplectic manifolds provide a rich class of exam-
ples in symplectic geometry and connect to representation theory, algebraic geometry, and
combinatorics through their moment polytopes. The Atiyah-Guillemin-Sternberg convexity
theorem asserts that the image of the moment map for such an action is a convex polytope
whose vertices are determined by the fixed points. The goals of this note are to give a
modern, streamlined proof of the theorem and to collect the auxiliary results on Morse–Bott
theory of moment maps that are used along the way. The basic definitions of this note are
based on [3].
Theorem 1.1 (Atiyah [1], Guillemin-Sternberg [2]). Let (M, ω) be a compact connected
symplectic manifold, and let Tm
be an m-torus. Suppose that ψ : Tm
→ Sympl(M, ω) is a
hamiltonian action with moment map µ : M → Rm
. Then:
1. The levels of µ are connected;
2. The image of µ is convex;
1

For example, consider the standard T2
-action on M = CP2
given by
(eit1
, eit2
) · [z0 : z1 : z2] = [z0 : eit1
z1 : eit2
z2].
The moment map is
µ [z0 : z1 : z2]

= −1
2

|z1|2
|z0|2+|z1|2+|z2|2 , |z2|2
|z0|2+|z1|2+|z2|2

.
The image µ(CP2
) is the triangle in R2
with vertices (0, 0), −1
2
, 0

, and 0, −1
2

. These
are exactly the images of the fixed points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1], respectively.
One can extends this to CPn
, in which case µ(CPn
) is the standard n-simplex.
Here is the roadmap toward this theorem.
Lemma 2.6

Theorem 2.3
))
Lemma 2.5

Lemma 2.7
vv
Theorem 1.1
2 Preliminaries
Definition 2.1. 2-form ω is a symplectic if ω is closed and nondegenerate. A pair (M, ω)
is a symplectic manifold if M is a manifold and ω is a symplectic form.
Definition 2.2. Let (M, ω) be a symplectic manifold and G be a Lie group with associated
Lie algebra g. A symplectic action of G on (M, ω) ψ : G → Sympl(M, ω) is hamiltonian if
there exists a moment map µ : M → g∗
satisfying:
1. For each X ∈ g, the function µX
: M → R defined by µX(m) = ⟨µ(m), X⟩ satisfies
dµX
= ιX# ω;
2. µ is equivariant with respect to the given action ψ of G on M and the coadjoint action
Ad∗
of G on g∗
: µ ◦ ψg = Ad∗
g ◦ µ, for all g ∈ G.
(M, ω, G, µ) is called a hamiltonian G-space and µ is called a moment map.
To prove the first part of 1.1, we first show that the connectedness of levels of the moment
map for T1
-action.
Theorem 2.3. Let (M, ω) be a compact connected symplectic manifold, and let T1
be an
1-torus. Suppose that ψ : T1
→ Sympl(M, ω) is a hamiltonian action with moment map
µ : M → R1
. The levels of µ are connected.
Proof. 1. µ is a Bott-Morse function. For any p ∈ M and v ∈ TpM, one has
dµp(v) = ωp Xp, v

.
2

Hence
dµp = 0 ⇐⇒ Xp = 0,
which is exactly the condition that p is fixed by the T1
-action. Therefore
Crit(µ) = MT1
.
Each connected component of the fixed-point set of a compact group action is a closed
embedded submanifold. At any fixed point p ∈ MT1
, the tangent space splits symplec-
tically as
TpM = TpMT1
⊕ νp,
where νp is the symplectic normal space. In complex Darboux coordinates (z1, . . . , zn)
on νp adapted to the T1
-weights wi ∈ Z{0}, the moment map admits a local expansion
µ p + z

= µ(p) + 1
2
n
X
i=1
wi|zi|2
+ (higher-order terms).
It follows that the Hessian of µ at p, restricted to the normal directions νp, is a non-
degenerate quadratic form (since each wi ̸= 0). Equivalently,
ker Hessp µ

= TpMT1
.
Thus µ is a Bott-Morse function: its critical set is a union of closed submanifolds, and
the Hessian is nondegenerate in the normal directions.
2. The Morse index of µ along a connected component is even. By the equivariant
Darboux (or Weinstein) theorem, at any p ∈ C there exist symplectic coordinates
{(xi, yi)}n
i=1 on the normal space νpC ∼
= R2n
in which the generator X of the action is
linearized as
Xp =
n
X
i=1
wi −yi∂xi
+ xi∂yi

,
with each integer weight wi ̸= 0.
In these coordinates the moment map admits the Taylor expansion
µ p + (x, y)

= µ(p) + 1
2
n
X
i=1
wi x2
i + y2
i

+ O(∥(x, y)∥3
).
Thus the Hessian of µ at p, restricted to νpC, is the quadratic form 1
2
X
i
wi(x2
i + y2
i ).
Since each summand wi(x2
i +y2
i ) contributes two real eigenvalues (one in the xi-direction
and one in the yi-direction), the spectrum of the Hessian consists of each weight wi
with multiplicity 2.
The Morse index is the number of negative eigenvalues, namely 2 {i : wi 0} , and
the coindex is 2 {i : wi 0} . In particular both are even integers.
3

3. Preservation of connectivity of level sets. Let
c0 c1 · · · cm
be the (finitely many) distinct critical values of µ. By (2), every critical submanifold
has both Morse index and coindex even, so no index jump of exactly one occurs.
Standard Bott-Morse theory then implies that for any regular value c ∈ (cj, cj+1), the
inclusion
µ−1
(c) ,→ µ−1
[cj, cj+1]

induces a bijection on path-components. Hence the number of connected components
of µ−1
(c) remains constant as c varies between critical levels.
4. Connectivity at the minimum. Since M is compact and connected and µ is continuous,
it attains a minimum value cmin. The corresponding level set
µ−1
(cmin) = Cmin
is a critical submanifold and is connected. By (3), all other level sets µ−1
(c) are
homeomorphic (in the sense of component count) to Cmin, and hence are connected as
well.
Intuitively, for a hamiltonian T1
-action the function
µ : M → R
is a Morse-Bott function whose critical manifolds are precisely the fixed-point sets of the circle
action. Morse-Bott function is a natural generalization of Morse function, which is a smooth
function on a manifold whose critical points form a submanifold with a non-degeneracy
condition in the normal direction.
Definition 2.4 ([3]). A smooth function f : M → R on a compact manifold M is called a
Morse-Bott function if its critical set Crit(f) = {p ∈ M|df(p) = 0} is a submanifold of M
and for every p ∈ Crit(f), Tp Crit(f) = ker ∇2
f(p) where ∇2
f(p) denotes the linear operator
obtained from the hessian via the riemannian metric.
By choosing an T1
-invariant compatible almost-complex structure J and the associated
metric
g(·, ·) = ω(·, J·),
the negative gradient flow of µ preserves the circle orbits and deforms sublevel sets without
ever disconnecting them, because all Morse-indices are even. Hence each level set µ−1
(c)
remains connected.
For induction, the following lemma is useful.
Lemma 2.5. Let (M, ω) be a compact connected symplectic manifold, and let Tm
be an
m-torus. Suppose that ψ : Tm
→ Sympl(M, ω) is a hamiltonian action with moment map
µ : M → Rm
. For an injective matrix A ∈ Zm×(m−1)
, the action of (m − 1)-torus given by
ψA : Tm−1
→ Sympl(M, ω) (1)
θ 7→ ψAθ (2)
is hamiltonian with moment map µA = AT
µ.
4

Proof. Let
ϕ : Tm−1
= Rm−1
/Zm−1
−→ Rm
/Zm
= Tm
be the homomorphism induced by the integer matrix A, namely ϕ([θ]) = [Aθ]. Define the
Tm−1
-action
ψA = ψ ◦ ϕ Tm−1
−→ Sympl(M, ω).
For each X ∈ Rm−1
we denote by XA
M the fundamental vector field of ψA and by (AX)M
that of ψ. Then for any p ∈ M,
XA
M (p) =
d
dt t=0
ψA exp(tX)

p =
d
dt t=0
ψ exp(tAX)

p = (AX)M (p).
Since ψ is Hamiltonian with moment map µ : M → (Rm
)∗
, we have
d µ, η = ιηM
ω, ∀η ∈ Rm
.
Define
µA M −→ (Rm−1
)∗
, µA(p) = AT
µ(p)

.
Then for each X ∈ Rm−1
,
d µA, X = d AT
µ, X = d µ, AX = ι(AX)M
ω = ιXA
M
ω.
Hence µA satisfies the defining relation of a moment map for ψA.
Lemma 2.6. For any subgroup G ⊆ Tm
, the set of fixed points of G Fix(G) =
S
θ∈G Fix(ψθ)
is a symplectic submanifold of M.
Proof. First, choose an almost complex structure J on M compatible with the symplectic
form ω, and define the Riemannian metric
g(u, v) = ω u, Jv

.
Since Tm
is compact, we may average g over the Tm
-action to obtain a Tm
-invariant metric
(and hence a Tm
-invariant almost complex structure) which remains compatible with ω.
Next, fix θ ∈ G and let p ∈ Fix(ψθ). Because ψθ is an isometry of (M, g), the exponential
map
expp : TpM −→ M
is ψθ-equivariant. It follows that in a neighborhood of p,
Fix(ψθ) = expp

ker Id −dψθ(p)

,
so Fix(ψθ) is a submanifold whose tangent space at p is
Tp Fix(ψθ) = ker Id −dψθ(p)

.
5

Moreover, dψθ(p) commutes with Jp, hence ker(Id −dψθ(p)) is a Jp-invariant subspace
of TpM. Since ω(u, v) = g(u, Jv), the restriction of ω to this subspace is nondegenerate.
Therefore each Fix(ψθ) is a symplectic submanifold.
Finally, as
Fix(G) =
[
θ∈G
Fix(ψθ)
is a (disjoint) union of symplectic submanifolds, it too is a (possibly disconnected) symplectic
submanifold of M.
Lemma 2.7. µX
: M → R is a Morse-Bott function with even dimensional critical subman-
ifold of even index. Crit(µX
) =
S
θ∈TX Fix(ψθ) is a symplectic manifold, where TX
is the
closure of the subgroup of Tm
generated by X.
Proof. Let ψ: Tm
× M → M denote the Hamiltonian action and write TX
⊂ Tm
for the
closure of the one-parameter subgroup generated by X. A point p ∈ M is critical for µX
if
and only if the vector field
XM (p) =
d
dt
ψexp(tX)(p)
t=0
vanishes, i.e. p ∈ Fix(ψexp(tX)) for all t. Equivalently,
Crit(µX
) =
[
θ∈TX
Fix(ψθ).
By the equivariant Darboux theorem, in a neighborhood of any p ∈ Fix(ψθ) one can
choose symplectic coordinates (z1, . . . , zk, w1, . . . , wk) ∈ Ck
so that the TX
-action is diagonal
with real weights wj ̸= 0. In these coordinates the Hamiltonian function µX
takes the form
µX
(z, w) = µX
(p) −
k
X
j=1
λj|zj|2
+
n
X
j=k+1
λj|wj|2
,
where λj ∈ R {0} are the infinitesimal weights of the action on the normal bundle to the
fixed submanifold (the tangent to the fixed locus corresponds to the zero-weight subspace)
It follows that Crit(µX
) near p is a smooth submanifold of codimension 2k, so each con-
nected component is even-dimensional and µX
is Bott-nondegenerate along it. Indeed, the
Hessian Hp(µX
) vanishes exactly on the tangent to the fixed submanifold and is nondegen-
erate on its symplectic normal bundle (with signature (k, k)), giving Morse index k, which
is even.
Finally, because the fixed locus of a symplectic torus action is a symplectic submanifold
(being the zero set of a collection of commuting Hamiltonian vector fields), each component
of Crit(µX
) inherits a nondegenerate restriction of ω and hence is symplectic and of even
dimension. This concludes the proof that µX
is a Morse-Bott function with even-dimensional,
symplectic critical manifolds all of even index.
6

3 Proof of Main Theorem
Proof of 1.1. Let An be the first statement for Tn
-action and let Bn be the second statement
for Tn
-action. Theorem 2.3 shows A1, and since a connected set is convex in R1
, B1 also
follows. We show that An−1 =⇒ Bn and An−1 =⇒ An.
1. An−1 =⇒ Bn.
Let p0 ∈ µ−1
A (ξ). Then for any p ∈ M,
p ∈ µ−1
A (ξ) ⇐⇒ At
µ(p) = ξ = At
µ(p0),
so that
µ−1
A (ξ) =

p ∈ M | µ(p) − µ(p0) ∈ ker At
.
By the first part (statement Am−1), µ−1
A (ξ) is connected. Hence if p0 and p1 lie in
µ−1
A (ξ), there is a path {pt}t∈[0,1] ⊂ µ−1
A (ξ) joining them. Applying µ and subtracting
µ(p0) gives a path µ(pt) − µ(p0) lying in the linear subspace ker At
. Since ker At
is
1-dimensional, this path must pass through every point of the line segment between
its endpoints. In particular,
(1 − t)µ(p0) + tµ(p1) ∈ µ(M), 0 ≤ t ≤ 1.
Finally, given arbitrary p0, p1 ∈ M, one can approximate them arbitrarily closely by
points p′
0, p′
1 for which µ(p′
1) − µ(p′
0) ∈ ker At
for some injective matrix A ∈ Zm×(m−1)
.
Taking limits p′
0 → p0, p′
1 → p1 shows that the straight segment between µ(p0) and
µ(p1) lies in µ(M). Therefore µ(M) is convex.
2. An−1 =⇒ An.
Let µ = (µ′
, µn): M → Rn−1
× R be the moment map, where µ′
= pr1,...,n−1 ◦µ is
the moment map for the Tn−1
-subaction obtained by omitting the last circle, and
µn = ⟨µ, en⟩ is the Hamiltonian for the residual S1
generated by the last coordinate
vector en.
Fix a regular value
ξ = (η, c) ∈ Rn−1
× R,
so
µ−1
(ξ) = µ′−1
(η) ∩ µ−1
n (c).
By Lemma 2.5 the Tn−1
-subaction is Hamiltonian; the inductive hypothesis An−1 there-
fore tells us that every level of µ′
is connected. In particular, µ′−1
(η) is connected.
Lemma 2.7 shows that µn : M → R is a Morse-Bott function whose critical submani-
folds all have even Morse index. The case n = 1 (already proved in Theorem 2.3) then
implies that every level of µn is connected; hence µ−1
n (c) is connected.
Because µn is Tn−1
-invariant, it is constant along the Tn−1
-orbits that foliate µ′−1
(η).
Thus the restriction
µn µ′−1(η)
: µ′−1
(η) → R
7

is a continuous map from a connected space to R, so each of its level sets is connected.
In particular, the fiber
µ−1
(ξ) = µ′−1
(η) ∩ µ−1
n (c)
is connected.
Therefore An holds.
References
[1] M. Atiyah. “Convexity and commuting Hamiltonians”. In: Bull. London Math. Soc. 14
(1982), pp. 1–15.
[2] V. Guillemin and S. Sternberg. “Convexity properties of the moment mapping”. In:
Invent. Math. 67 (1982), pp. 491–513.
[3] Ana Cannas Silva. Lectures on Symplectic Geometry. 1st ed. Vol. 1764. Lecture Notes in
Mathematics. Springer Berlin, Heidelberg, 2001, pp. XII, 220. isbn: 978-3-540-42195-5.
doi: 10.1007/978-3-540-45330-7.
8

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