Part III Symplectic Geometry
Based on lectures by A. R. Pires
Notes taken by Dexter Chua
Lent 2018
These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.
The first part of the course will be an overview of the basic structures of symplectic ge-
ometry, including symplectic linear algebra, symplectic manifolds, symplectomorphisms,
Darboux theorem, cotangent bundles, Lagrangian submanifolds, and Hamiltonian sys-
tems. The course will then go further into two topics. The first one is moment maps and
toric symplectic manifolds, and the second one is capacities and symplectic embedding
problems.
Pre-requisites
Some familiarity with basic notions from Differential Geometry and Algebraic Topology
will be assumed. The material covered in the respective Michaelmas Term courses
would be more than enough background.
Contents
1 Symplectic manifolds
1.1 Symplectic linear algebra
1.2 Symplectic manifolds
1.3 Symplectomorphisms and Lagrangians
1.4 Periodic points of symplectomorphisms
1.5 Lagrangian submanifolds and fixed points
2 Complex structures
2.1 Almost complex structures
2.2 Dolbeault theory
2.3 ahler manifolds
2.4 Hodge theory
3 Hamiltonian vector fields
3.1 Hamiltonian vector fields
3.2 Integrable systems
3.3 Classical mechanics
3.4 Hamiltonian actions
3.5 Symplectic reduction
3.6 The convexity theorem
3.7 Toric manifolds
4 Symplectic embeddings
1 Symplectic manifolds
1.1 Symplectic linear algebra
In symplectic geometry, we study symplectic manifolds. These are manifolds
equipped with a certain structure on the tangent bundle. In this section, we first
analyze the condition fiberwise.
Definition
(Symplectic vector space)
.
A symplectic vector space is a real vector
space
V
together with a non-degenerate skew-symmetric bilinear map Ω :
V ×
V R.
Recall that
Definition
(Non-degenerate bilinear map)
.
We say a bilinear map is non-
degenerate if the induced map
˜
Ω : V V
given by v 7→ Ω(v, ·) is bijective.
Even if we drop the non-degeneracy condition, there aren’t that many sym-
plectic vector spaces around up to isomorphism.
Theorem
(Standard form theorem)
.
Let
V
be a real vector space and a skew-
symmetric bilinear form. Then there is a basis
{u
1
, . . . , u
k
, e
1
, . . . , e
n
, f
1
, . . . , f
n
}
of V such that
(i) Ω(u
i
, v) = 0 for all v V
(ii) Ω(e
i
, e
j
) = Ω(f
i
, f
j
) = 0.
(iii) Ω(e
i
, f
j
) = δ
ij
.
The proof is a skew-symmetric version of Gram–Schmidt.
Proof. Let
U = {u V : Ω(u, v) = 0 for all v V },
and pick a basis u
1
, . . . , u
k
of this. Choose any W complementary to U .
First pick
e
1
W \{
0
}
arbitrarily. Since
e
1
6∈ U
, we can pick
f
1
such that
Ω(e
1
, f
1
) = 1. Then define W
1
= span{e
i
, f
i
}, and
W
1
= {w W : Ω(w, v) = 0 for all v W
1
}.
It is clear that
W
1
W
1
=
{
0
}
. Moreover,
W
=
W
1
W
1
. Indeed, if
v W
,
then
v = (Ω(v, f
1
)e
1
Ω(v, e
1
)f
1
) + (v (Ω(v, f
1
)e
1
Ω(v, e
1
)f
1
)),
Then we are done by induction on the dimension.
Here
k
=
dim U
and 2
n
=
dim V k
are invariants of (
V,
Ω). The number
2n is called the rank of Ω. Non-degeneracy is equivalent to k = 0.
Exercise. is non-degenerate iff
n
= Ω ··· 6= 0.
By definition, every symplectic vector space is canonically isomorphic to
its dual, given by the map
˜
. As in the above theorem, a symplectic basis
{e
1
, . . . , e
n
, f
1
, . . . , f
n
} of V is a basis such that
Ω(e
i
, e
j
) = Ω(f
i
, f
j
) = 0, Ω(e
i
, f
i
) = δ
ij
.
Every symplectic vector space has a symplectic basis, and in particular has even
dimension. In such a basis, the matrix representing is
0
=
0 I
I 0
.
We will need the following definitions:
Definition
(Symplectic subspace)
.
If (
V,
Ω) is a symplectic vector space, a
symplectic subspace is a subspace W V such that |
W ×W
is non-degenerate.
Definition
(Isotropic subspace)
.
If (
V,
Ω) is a symplectic vector space, an
isotropic subspace is a subspace W V such that |
W ×W
= 0.
Definition
(Lagrangian subspace)
.
If (
V,
Ω) is a symplectic vector space, an
Lagrangian subspace is an isotropic subspace W with dim W =
1
2
dim V .
Definition
(Symplectomorphism)
.
A symplectomorphism between symplectic
vector spaces (
V,
Ω)
,
(
V
0
,
0
) is an isomorphism
ϕ
:
V V
0
such that
ϕ
0
= Ω.
The standard form theorem says any two 2
n
-dimensional symplectic vector
space (V, Ω) are symplectomorphic to each other.
1.2 Symplectic manifolds
We are now ready to define symplectic manifolds.
Definition
(Symplectic manifold)
.
A symplectic manifold is a manifold
M
of dimension 2
n
equipped with a 2-form
ω
that is closed (i.e. d
ω
= 0) and
non-degenerate (i.e. ω
n
6= 0). We call ω the symplectic form.
The closedness condition is somewhat mysterious. There are some motivations
from classical mechanics, for examples, but they are not very compelling, and
the best explanation for the condition is “these manifolds happen to be quite
interesting”.
Observe that by assumption, the form
ω
n
is nowhere-vanishing, and so it is
a volume form. If our manifold is compact, then pairing the volume form with
the fundamental class (i.e. integrating it over the manifold) gives the volume,
and in particular is non-zero. Thus,
ω
n
6
= 0
H
2n
dR
(
M
). Since the wedge is
well-defined on cohomology, an immediate consequence of this is that
Proposition.
If a compact manifold
M
2n
is such that
H
2k
dR
(
M
) = 0 for some
k < n, then M does not admit a symplectic structure.
A more down-to-earth proof of this result would be that if
ω
2k
= d
α
for some
α, then
Z
ω
n
=
Z
ω
2k
ω
2(nk)
=
Z
d(α ω
2(nk)
) = 0
by Stokes’ theorem.
Example. S
n
does not admit a symplectic structure unless
n
= 2 (or
n
= 0, if
one insists).
On the other hand, S
2
is a symplectic manifold.
Example.
Take
M
=
S
2
. Take the standard embedding in
R
3
, and for
p S
2
,
the tangent space consists of the vectors perpendicular to
p
. We can take the
symplectic form to be
ω
p
(u, v) = p ·(u × v).
Anti-symmetry and non-degeneracy is IA Vectors and Matrices.
Definition
(Symplectomorphism)
.
Let (
X
1
, ω
1
) and (
X
2
, ω
2
) be symplectic
manifolds. A symplectomorphism is a diffeomorphism
f
:
X
1
X
2
such that
f
ω
2
= ω
1
.
If we have a single fixed manifold, and two symplectic structures on it, there
are other notions of “sameness” we might consider:
Definition
(Strongly isotopic)
.
Two symplectic structures on
M
are strongly
isotopic if there is an isotopy taking one to the other.
Definition
(Deformation equivalent)
.
Two symplectic structures
ω
0
, ω
1
on
M
are deformation equivalent if there is a family of symplectic forms
ω
t
that start
and end at ω
0
and ω
1
respectively.
Definition
(Isotopic)
.
Two symplectic structures
ω
0
, ω
1
on
M
are isotopic
if there is a family of symplectic forms
ω
t
that start and end at
ω
0
and
ω
1
respectively, and further the cohomology class [ω
t
] is independent of t.
A priori, it is clear that we have the following implications:
symplectomorphic strongly isotopic isotopic deformation equivalent.
It turns out when M is compact, isotopic implies strongly isotopic:
Theorem
(Moser)
.
If
M
is compact with a family
ω
t
of symplectic forms with
[ω
t
] constant, then there is an isotopy ρ
t
: M M with ρ
t
ω
t
= ω
0
.
The key idea is to express the condition
ρ
t
ω
t
=
ω
0
as a differential equation
for
v
t
=
d
dt
ρ
t
, and ODE theory guarantees the existence of a
v
t
satisfying the
equation. Then compactness allows us to integrate it up to get a genuine ρ
t
.
Proof.
We set
ρ
0
to be the identity. Then the equation
ρ
t
ω
t
=
ω
0
is equivalent
to ρ
t
ω
t
being constant. If v
t
is the associated vector field to ρ
t
, then we need
0 =
d
dt
(ρ
t
ω
t
) = ρ
t
L
v
t
ω
t
+
dω
t
dt
.
So we want to solve
L
v
t
ω
t
+
dω
t
dt
= 0.
To solve for this, since [
dω
t
dt
] = 0, it follows that there is a family
µ
t
of 1-forms
such that
dω
t
dt
= dµ
t
. Then our equation becomes
L
v
t
ω
t
+ dµ
t
= 0.
By assumption, dω
t
= 0. So by Cartan’s magic formula, we get
dι
v
t
ω
t
+ dµ
t
= 0.
To solve this equation, it suffices to solve Moser’s equation,
ι
v
t
ω
t
+ µ
t
= 0,
which can be solved since ω
t
is non-degenerate.
There is also a relative version of this.
Theorem
(Relative Moser)
.
Let
X M
be a compact manifold of a manifold
M
,
and
ω
0
, ω
1
symplectic forms on
M
agreeing on
X
. Then there are neighbourhoods
U
0
, U
1
of X and a diffeomorphism ϕ : U
0
U
1
fixing X such that ϕ
ω
1
= ω
0
.
Proof. We set
ω
t
= (1 t)ω
0
+
1
.
Then this is certainly closed, and since this is constantly
ω
0
on
X
, it is non-
degenerate on
X
, hence, by compactness, non-degenerate on a small tubular
neighbourhood U
0
of X (by compactness). Now
d
dt
ω
t
= ω
1
ω
0
,
and we know this vanishes on
X
. Since the inclusion of
X
into a tubular
neighbourhood is a homotopy equivalence, we know [
ω
1
ω
0
] = 0
H
1
dR
(
U
0
).
Thus, we can find some
µ
such that
ω
1
ω
0
= d
µ
, and by translation by a
constant, we may suppose
µ
vanishes on
X
. We then solve Moser’s equation,
and the resulting ρ will be constant on X since µ vanishes.
We previously had the standard form theorem for symplectic vector spaces,
which is not surprising — we have the same for Riemannian metrics, for example.
Perhaps more surprisingly, give a symplectic manifold, we can always pick
coordinate charts where the symplectic form looks like the standard symplectic
form throughout:
Theorem
(Darboux theorem)
.
If (
M, ω
) is a symplectic manifold, and
p M
,
then there is a chart (U, x
1
, . . . , x
n
, y
1
, . . . , y
n
) about p on which
ω =
X
dx
i
dy
i
.
Proof. ω
can certainly be written in this form at
p
. Then relative Moser with
X = {p} promotes this to hold in a neighbourhood of p.
Thus, symplectic geometry is a global business, unlike Riemannian geometry,
where it is interesting to talk about local properties such as curvature.
A canonical example of a symplectic manifold is the cotangent bundle of a
manifold
X
. In physics,
X
is called the configuration space, and
M
=
T
X
is
called the phase space.
On a coordinate chart (
U, x
1
, . . . , x
n
) for
X
, a generic 1-form can be written
as
ξ =
n
X
i=1
ξ
i
dx
i
.
Thus, we have a coordinate chart on
T
X
given by (
T
U, x
1
, . . . , x
n
, ξ
1
, . . . , ξ
n
).
On T
U, there is the tautological 1-form
α =
n
X
i=1
ξ
i
dx
i
.
Observe that this is independent of the chart chosen, since it can be characterized
in the following coordinate-independent way:
Proposition.
Let
π
:
M
=
T
X X
be the projection map, and
π
:
T
X
T
M the pullback map. Then for ξ M, we have α
ξ
= π
ξ.
Proof. On a chart, we have
π
ξ
x
j
= ξ
x
j
= ξ
j
= α
ξ
x
j
,
and similarly both vanish on
ξ
j
.
This tautological 1-form is constructed so that the following holds:
Proposition.
Let
µ
be a one-form of
X
, i.e. a section
s
µ
:
X T
X
. Then
s
µ
α = µ.
Proof. By definition,
α
ξ
= ξ dπ.
So for x X, we have
s
µ
α
µ(x)
= µ(x) dπ ds
µ
= µ(x) d(π s
µ
) = µ(x) d(id
X
) = µ(x).
Given this, we can set
ω = dα =
n
X
i=1
dx
i
dξ
i
,
the canonical symplectic form on
T
X
. It is clear that it is anti-symmetric,
closed, and non-degenerate, because it looks just like the canonical symplectic
form on R
2n
.
Example.
Take
X
=
S
1
, parametrized by
θ
, and
T
X
=
S
1
×R
with coordinates
(θ, ξ
θ
). Then
ω = dθ dξ
θ
.
We can see explicitly how this is independent of coordinate. Suppose we
parametrized the circle by
τ
= 2
θ
. Then d
τ
= 2d
θ
. However, since we de-
fined ξ
θ
by
ξ = ξ
θ
(ξ) dθ
for all ξ T
S
1
, we know that ξ
τ
=
1
2
ξ
θ
, and hence
dθ dξ
θ
= dτ dξ
τ
.
Of course, if we have a diffeomorphism
f
:
X Y
, then the pullback map
induces a symplectomorphism
f
:
T
Y T
X
. However, not all symplecto-
morphisms arise this way.
Example.
If
X
=
S
1
and
T
X
=
S
1
× R
is given the canonical symplectic
structure, then the vertical translation
g
(
θ, ξ
) = (
θ, ξ
+
c
) is a symplectomorphism
that is not a lift of a diffeomorphism S
1
S
1
.
So when does a symplectomorphism come from a diffeomorphism?
Exercise.
A symplectomorphism
y
:
T
X T
X
is a lift of a diffeomorphism
X X iff g
α = α.
1.3 Symplectomorphisms and Lagrangians
We now consider Lagrangian submanifolds. These are interesting for many
reasons, but in this section, we are going to use them to understand symplecto-
morphisms.
Definition
(Lagrangian submanifold)
.
Let (
M, ω
) be a symplectic manifold,
and
L M
a submanifold. Then
L
is a Lagrangian submanifold if for all
p L
,
T
p
L
is a Lagrangian subspace of
T
p
M
. Equivalently, the restriction of
ω
to
L
vanishes and dim L =
1
2
dim M.
Example.
If (
M, ω
) is a surface with area form
ω
, and
L
is any 1-dimensional
submanifold, then
L
is Lagrangian, since any 1-dimensional subspace of
T
p
M
is
Lagrangian.
What do Lagrangian submanifolds of
T
X
look like? Let
µ
be a 1-form on
X, i.e. a section of T
X, and let
X
µ
= {(x, µ
x
) : x X} T
X.
The map
s
µ
:
X T
X
is an embedding, and
dim X
µ
=
dim X
=
1
2
dim T
X
.
So this is a good candidate for a Lagrangian submanifold. By definition, X
µ
is
Lagrangian iff s
µ
ω = 0. But
s
µ
ω = s
µ
(dα) = d(s
µ
α) = dµ.
So
X
µ
is Lagrangian iff
µ
is closed. In particular, if
h C
(
X
), then
X
dh
is
a Lagrangian submanifold of
T
X
. We call
h
the generating function of the
Lagrangian submanifold.
There are other examples of Lagrangian submanifolds of the cotangent bundle.
Let
S
be a submanifold of
X
, and
x X
. We define the conormal space of
S
at
x to be
N
x
S = {ξ T
x
X : ξ|
T
x
S
= 0}.
The conormal bundle of S is the union of all N
x
S.
Example. If S = {x}, then N
S = T
x
X.
Example. If S = X, then N
S = S is the zero section.
Note that both of these are
n
-dimensional submanifolds of
T
X
, and this is
of course true in general.
So if we are looking for Lagrangian submanifolds, this is already a good start.
In the case of
S
=
X
, this is certainly a Lagrangian submanifold, and so is
S = {x}. Indeed,
Proposition.
Let
L
=
N
S
and
M
=
T
X
. Then
L M
is a Lagrangian
submanifold.
Proof.
If
L
is
k
-dimensional, then
S X
locally looks like
R
k
R
n
, and it is
clear in this case.
Our objective is to use Lagrangian submanifolds to understand the following
problem:
Problem.
Let (
M
1
, ω
1
) and (
M
2
, ω
2
) be 2
n
-dimensional symplectic manifolds.
If f : M
1
M
2
is a diffeomorphism, when is it a symplectomorphism?
Consider the 4
n
-dimensional manifold
M
1
× M
2
, with the product form
ω = pr
1
ω
1
+ pr
2
ω
2
. We have
dω = pr
1
dω
1
+ pr
2
dω
2
= 0,
and ω is non-degenerate since
ω
2n
=
2n
n
pr
1
ω
n
1
pr
2
ω
n
2
6= 0.
So
ω
is a symplectic form on
M
1
× M
2
. In fact, for
λ
1
, λ
2
R \ {
0
}
, the
combination
λ
1
pr
1
ω
1
+ λ
2
pr
2
ω
2
is also symplectic. The case we are particularly interested in is
λ
1
= 1 and
λ
2
= 1, where we get the twisted product form.
˜ω = pr
1
ω
1
pr
2
ω
2
.
Let T
f
be the graph of f, namely
T
f
= {(p
1
, f(p
1
)) : p
1
M
1
} M
1
× M
2
.
and let γ
f
: M
1
M
1
× M
2
be the natural embedding with image T
f
.
Proposition. f
is a symplectomorphism iff
T
f
is a Lagrangian submanifold of
(M
1
× M
2
, ˜ω).
Proof.
The first condition is
f
ω
2
=
ω
1
, and the second condition is
γ
f
˜ω
= 0,
and
γ
f
˜ω = γ
f
pr
1
ω
1
= γ
f
pr
2
ω
2
= ω
1
f
ω
2
.
We are particularly interested in the case where
M
i
are cotangent bundles.
If
X
1
, X
2
are
n
-dimensional manifolds and
M
i
=
T
X
i
, then we can naturally
identify
T
(X
1
× X
2
)
=
T
X
1
× T
X
2
.
The canonical symplectic form on the left agrees with the canonical symplectic
form on the right. However, we want the twisted symplectic form instead. So we
define an involution
σ : M
1
× M
2
M
1
× M
2
that fixes M
1
and sends (x, ξ) 7→ (x, ξ) on M
2
. Then σ
ω = ˜ω.
Then, to find a symplectomorphism
M
1
M
2
, we can carry out the following
procedure:
(i) Start with L a Lagrangian submanifold of (M
1
× M
2
, ω).
(ii) Twist it to get L
σ
= σ(L).
(iii) Check if L
σ
is the graph of a diffeomorphism.
This doesn’t sound like a very helpful specification, until we remember that
we have a rather systematic way of producing Lagrangian submanifolds of
(
M
1
, ×M
2
, ω
). We pick a generating function
f C
(
X
1
× X
2
). Then d
f
is a
closed form on X
1
× X
2
, and
L = X
df
= {(x, y,
x
f
(x,y)
,
y
f
(x,y)
) : (x, y) X
1
× X
2
}
is a Lagrangian submanifold of (M
1
× M
2
, ω).
The twist L
σ
is then
L
σ
= {(x, y, d
x
f
(x,y)
, d
y
f
(x,y)
) : (x, y) X
1
× X
2
}
We then have to check if
L
σ
is the graph of a diffeomorphism
ϕ
:
M
1
M
2
. If so,
we win, and we call this the symplectomorphism generated by
f
. Equivalently,
we say f is the generating function for ϕ.
To check if L
σ
is the graph of a diffeomorphism, for each (x, ξ) T
X
1
, we
need to find (y, η) T
X
2
such that
ξ =
x
f
(x,y)
, η =
y
f
(x,y)
.
In local coordinates {x
i
, ξ
i
} for T
X
1
and {y
i
, η
i
} for T
X
2
, we want to solve
ξ
i
=
f
x
i
(x, y)
η
i
=
f
y
i
(x, y)
The only equation that requires solving is the first equation for
y
, and then the
second equation tells us what we should pick η
i
to be.
For the first equation to have a unique smoothly-varying solution for
y
,
locally, we must require
det
y
i
f
x
j

i,j
6= 0.
This is a necessary condition for
f
to generate a symplectomorphism
ϕ
. If this
is satisfied, then we should think a bit harder to solve the global problem of
whether ξ
i
is uniquely defined.
We can in fact do this quite explicitly:
Example. Let X
1
= X
2
= R
n
, and f : R
n
× R
n
R given by
(x, y) 7→
|x y|
2
2
=
1
2
n
X
i=1
(x
i
y
i
)
2
.
What, if any, symplectomorphism is generated by f? We have to solve
ξ
i
=
f
x
i
= y
i
x
i
η
i
=
f
y
i
= y
i
x
i
So we find that
y
i
= x
i
+ ξ
i
,
η
i
= ξ
i
.
So we have
ϕ(x, ξ) = (x + ξ, ξ).
If we use the Euclidean inner product to identify
T
R
n
=
R
2n
, and view
ξ
as a
vector, then ϕ is a free translation in R
n
.
That seems boring, but in fact this generalizes quite well.
Example.
Let
X
1
=
X
2
=
X
be a connected manifold, and
g
a Riemannian
metric on
X
. We then have the Riemannian distance
d
:
X × X R
, and we
can define
f : X × X R
(x, y) 7→
1
2
d(x, y)
2
.
Using the Riemannian metric to identify
T
X
=
T X
, we can check that
ϕ
:
T X T X is given by
ϕ(x, v) = (γ(1), ˙γ(1)),
where
γ
is the geodesic with
γ
(0) =
x
and
˙γ
(0) =
v
. Thus,
ϕ
is the geodesic
flow.
1.4 Periodic points of symplectomorphisms
Symplectic geometers have an unreasonable obsession with periodic point prob-
lems.
Definition
(Periodic point)
.
Let (
M, ω
) be a symplectic manifold, and
ϕ
:
M M
a symplectomorphism. An
n
-periodic point of
ϕ
is an
x M
such that
ϕ
n
(x) = x. A periodic point is an n-periodic point for some n.
We are particularly interested in the case where
M
=
T
X
with the canonical
symplectic form on
M
, and
ϕ
is generated by a function
f
:
X R
. We begin
with 1-periodic points, namely fixed points of f.
If ϕ(x
0
, ξ
0
) = (x
0
, ξ
0
), this means we have
ξ
0
=
x
f
(x
0
,x
0
)
ξ
0
=
y
f
(x
0
,x
0
)
In other words, we need
(
x
+
y
)f
(x
0
,x
0
)
= 0.
Thus, if we define
ψ : X R
x 7→ f(x, x),
then we see that
Proposition.
The fixed point of
ϕ
are in one-to-one correspondence with the
critical points of ψ.
It is then natural to consider the same question for
ϕ
n
. One way to think
about this is to consider the symplectomorphism ˜ϕ : M
n
M
n
given by
˜ϕ(m
1
, m
2
, . . . , m
n
) = (ϕ(m
n
), ϕ(m
1
), . . . , ϕ(m
n1
)).
Then fixed points of ˜ϕ correspond exactly to the n-periodic points of ϕ.
We now have to find a generating function for this
˜ϕ
. By inspection, we see
that this works:
˜
f((x
1
, . . . , x
n
), (y
1
, . . . , y
n
)) = f(x
1
, y
2
) + f(x
2
, y
3
) + ··· + f(x
n
, y
1
).
Thus, we deduce that
Proposition.
The
n
-periodic points of
ϕ
are in one-to-one correspondence with
the critical points of
ψ
n
(x
1
, . . . , x
n
) = f(x
1
, x
2
) + f(x
2
, x
3
) + ··· + f(x
n
, x
1
).
Example.
We consider the problem of periodic billiard balls. Suppose we have
a bounded, convex region Y R
2
.
Y
We put a billiard ball in
Y
and set it in motion. We want to see if it has periodic
orbits.
Two subsequent collisions tend to look like this:
x
θ
We can parametrize this collision as (
x, v
=
cos θ
)
Y ×
(
1
,
1). We can think of
this as the unit disk bundle of cotangent bundle on
X
=
Y
=
S
1
with canonical
symplectic form d
x
d
v
, using a fixed parametrization
χ
:
S
1
=
R/Z X
. If
ϕ
(
x, v
) = (
y, w
), then
v
is the projection of the unit vector pointing from
x
to
y
onto the tangent line at x. Thus,
v =
χ(y) χ(x)
kχ(y) χ(x)k
·
dχ
ds
s=x
=
x
(−kχ(x) χ(y)k),
and a similar formula holds for
w
, using that the angle of incidence is equal to
the angle of reflection. Thus, we know that
f(x) = −kχ(x) χ(y)k
is a generating function for ϕ.
The conclusion is that the
N
-periodic points are given by the critical points
of
(x
1
, . . . , x
N
) 7→ −|χ(x
1
) χ(x
2
)| |χ(x
2
) χ(x
3
)| ··· |χ(x
N
) χ(x
1
)|.
Up to a sign, this is the length of the “generalized polygon” with vertices
(x
1
, . . . , x
N
).
In general, if
X
is a compact manifold, and
ϕ
:
T
X T
X
is a symplecto-
morphism generated by a function
f
, then the number of fixed points of
ϕ
are
the number of fixed points of
ψ
(
x
) =
f
(
x, x
). By compactness, there is at least
a minimum and a maximum, so ϕ has at least two fixed points.
In fact,
Theorem
(Poincar´e’s last geometric theorem (Birkhoff, 1925))
.
Let
ϕ
:
A
A
be an area-preserving diffeomorphism such that
ϕ
preserves the boundary
components, and twists them in opposite directions. Then
ϕ
has at least two
fixed points.
1.5 Lagrangian submanifolds and fixed points
Recall what we have got so far. Let (
M, ω
) be a symplectic manifold,
ϕ
:
M M
.
Then the graph of
ϕ
is a subset of
M ×M
. Again let
˜ω
be the twisted product form
on
M ×M
. Then we saw that a morphism
ϕ
:
M M
is a symplectomorphism
iff the graph of
ϕ
is Lagrangian in (
M ×M, ˜ω
). Moreover, the set of fixed points
is exactly graph ϕ, where ∆ = graph id
M
is the diagonal.
If
ϕ
is “close to” the identity, map, then its graph is close to ∆. Thus, we are
naturally led to understanding the neighbourhoods of the identity. An important
theorem is the following symplectic version of the tubular neighbourhood theorem:
Theorem
(Lagrangian neighbourhood theorem)
.
Let (
M, ω
) be a symplectic
manifold,
X
a compact Lagrangian submanifold, and
ω
0
the canonical symplectic
form on
T
X
. Then there exists neighbourhoods
U
0
of
X
in
T
X
and
U
of
X
in M and a symplectomorphism ϕ : U
0
U sending X to X.
An equivalent theorem is the following:
Theorem
(Weinstein)
.
Let
M
be a 2
n
-dimensional manifold,
X n
-dimensional
compact submanifold, and
i
:
X M
the inclusion, and symplectic forms
ω
0
, ω
1
on
M
such that
i
ω
0
=
i
ω
1
= 0, i.e.
X
is Lagrangian with respect to both
symplectic structures. Then there exists neighbourhoods
U
0
, U
1
of
X
in
M
such
that ρ|
X
= id
X
and ρ
ω
1
= ω
0
.
We first prove these are equivalent. This amounts to identifying the (dual of
the) cotangent bundle of X with the normal bundle of X.
Proof of equivalence.
If (
V,
Ω) is a symplectic vector space,
L
a Lagrangian
subspace, the bilinear form
Ω : V/L ×L R
([v], u) Ω(v, u).
is non-degenerate and gives a natural isomorphism
V/L
=
L
. Taking
V
=
T
p
M
and L = T
p
X, we get an isomorphism
NX = T M |
X
/T X
=
T
X.
Thus, by the standard tubular neighbourhood theorem, there is a neighbourhood
N
0
of
X
in
NX
and a neighbourhood
N
of
X
in
M
, and a diffeomorphism
ψ
:
N
0
N
. We now have two symplectic forms on
N
0
the one from the
cotangent bundle and the pullback of that from
M
. Then applying the second
theorem gives the first.
Conversely, if we know the first theorem, then applying this twice gives us
symplectomorphisms between neighbourhoods of
X
in
M
under each symplectic
structure with a neighbourhood in the cotangent bundle.
It now remains to prove the second theorem. This is essentially an application
of the relative Moser theorem, which is where the magic happens. The bulk of
the proof is to get ourselves into a situation where relative Moser applies.
Proof of second theorem.
For
p X
, we define
V
=
T
p
M
and
U
=
T
p
X
, and
W
any complement of
U
. By assumption,
U
is a Lagrangian subspace of both
(
V, ω
0
|
p
=
0
) and (
V, ω
1
|
p
=
1
). We apply the following linear-algebraic
lemma:
Lemma.
Let
V
be a 2
n
-dimensional vector space,
0
,
1
symplectic structures
on
V
. Suppose
U
is a subspace of
V
Lagrangian with respect to both
0
and
1
, and
W
is any complement of
V
. Then we can construct canonically a linear
isomorphism H : V V such that H|
U
= id
U
and H
1
= Ω
2
.
Note that the statement of the theorem doesn’t mention
W
, but the con-
struction of
H
requires a complement of
V
, so it is canonical only after we pick
a W .
By this lemma, we get canonically an isomorphism
H
p
:
T
p
M T
p
M
such
that
H
p
|
T
p
X
=
id
T
p
X
and
H
p
ω
1
|
p
=
ω
0
|
p
. The canonicity implies
H
p
varies
smoothly with p. We now apply the Whitney extension theorem
Theorem
(Whitney extension theorem)
.
Let
X
be a submanifold of
M
,
H
p
:
T
p
M T
p
M
smooth family of isomorphisms such that
H
p
|
T
p
X
=
id
T
p
X
. Then
there exists an neighbourhood
N
of
X
in
M
and an embedding
h
:
N M
such that h|
X
= id
X
and for all p X, dh
p
= H
p
.
So at
p X
, we have
h
ω
1
|
p
= (d
h
p
)
ω
1
|
P
=
h
p
ω
1
|
p
=
ω
0
|
p
. So we are done
by relative Moser.
Example.
We can use this result to understand the neighbourhood of the iden-
tity in the group
Symp
(
M, ω
) of symplectomorphisms of a symplectic manifold
(M, ω).
Suppose
ϕ, id Symp
(
M, ω
). Then the graphs Γ
ϕ
and = Γ
id
are La-
grangian submanifolds of (
M × M, ˜ω
). Then by our theorem, there is a neigh-
bourhood
U
of in (
M × M, ˜ω
) that is symplectomorphic to a neighbourhood
U
0
of the zero section of (T
M, ω
0
).
Suppose
ϕ
is sufficiently
C
0
-close to
id
. Then Γ
ϕ
U
. If
ϕ
is sufficiently
C
1
-close to the identity, then the image of Γ
ϕ
in
U
0
is a smooth section
X
µ
for
some 1-form µ.
Now
ϕ
is a symplectomorphism iff Γ
ϕ
is Lagrangian iff
X
µ
is Lagrangian iff
µ
is closed. So a small
C
1
-neighbourhood of
id
in
Sym
(
M, ω
) is “the same as” a
small
C
1
-neighbourhood of the zero-section in the space of closed 1-forms on
X
.
We can also use this to understand fixed points of symplectomorphisms
(again!).
Theorem.
Let (
M, ω
) be a compact symplectic manifold such that
H
1
dR
(
M
) = 0.
Then any symplectomorphism
ϕ
:
M M
sufficiently close to the identity has
at least two fixed points.
Proof.
The graph of
ϕ
corresponds to a closed 1-form on
M
. Since
µ
is closed
and
H
1
dR
(
M
) = 0, we know
µ
= d
h
for some
h C
. Since
M
is compact,
h
has at least two critical points (the global maximum and minimum). Since the
fixed points corresponding to the points where
µ
vanish (so that Γ
ϕ
intersects
∆), we are done.
Counting fixed points of symplectomorphisms is a rather popular topic in
symplectic geometry, and Arnold made some conjectures about these. A version
of this is
Theorem
(Arnold conjecture)
.
Let (
M, ω
) be a compact symplectic manifold
of dimension 2
n
, and
ϕ
:
M M
a symplectomorphism. Suppose
ϕ
is exactly
homotopic to the identity and non-degenerate. Then the number of fixed points
of ϕ is at least
P
2n
i=0
dim H
i
(M, R).
We should think of the sum
P
2n
i=0
dim H
i
(
M, R
) as the minimal number of
critical points of a function, as Morse theory tells us.
We ought to define the words we used in the theorem:
Definition
(Exactly homotopic)
.
We say
ϕ
is exactly homotopic to the identity
if there is isotopy
ρ
t
:
M M
such that
ρ
0
=
id
and
ρ
1
=
ϕ
, and further there
is some 1-periodic family of functions
h
t
such that
ρ
t
is generated by the vector
field v
t
defined by ι
v
t
ω = dh
t
.
The condition
ι
v
t
ω
d
h
t
says
v
t
is a Hamiltonian vector field, which we will
discuss soon after this.
Definition
(Non-degenerate function)
.
A endomorphism
ϕ
:
M M
is non-
degenerate iff all its fixed points are non-degenerate, i.e. if
p
is a fixed point,
then det(id dϕ
p
) 6= 0.
Example.
In the original statement of the Arnold conjecture, which is the case
(T
2
, dθ
1
dθ
2
), any symplectomorphism has at least 4 fixed points.
In the case where
h
t
is actually not time dependent, Arnold’s conjecture is
easy to prove.
2 Complex structures
2.1 Almost complex structures
Symplectic manifolds are very closely related to complex manifolds. A first
(weak) hint of this fact is that they both have to be even dimensional! In general,
symplectic manifolds will have almost complex structures, but this need not
actually come from a genuine complex structure. If it does, then we say it is
ahler, which is a very strong structure on the manifold.
We begin by explaining what almost complex structures are, again starting
from the linear algebraic side of the story. A complex vector space can be thought
of as a real vector space with a linear endomorphism that acts as “multiplication
by i”.
Definition
(Complex structure)
.
Let
V
be a vector space. A complex structure
is a linear J : V V with J
2
= 1.
Here we call it a complex structure. When we move on to manifolds, we will
call this “almost complex”, and a complex structure is a stronger condition. It
is clear that
Lemma.
There is a correspondence between real vector spaces with a complex
structure and complex vector spaces, where J acts as multiplication by i.
Our symplectic manifolds come with symplectic forms on the tangent space.
We require the following compatibility condition:
Definition
(Compatible complex structure)
.
Let (
V,
Ω) be a symplectic vector
space, and
J
a complex structure on
V
. We say
J
is compatible with if
G
J
(u, v) = Ω(u, Jv) is an inner product. In other words, we need
Ω(Ju, Jv) = Ω(u, v), Ω(v, Jv) 0
with equality iff v = 0.
Example. On the standard symplectic vector space (R
2n
, Ω), we set
J
0
(e
i
) = f
i
, J
0
(f
i
) = e
i
.
We can then check that this is compatible with the symplectic structure, and in
fact gives the standard inner product.
Proposition
(Polar decomposition)
.
Let (
V,
Ω) be a symplectic vector space,
and
G
an inner product on
V
. Then from
G
, we can canonically construct a
compatible complex structure
J
on (
V,
Ω). If
G
=
G
J
for some
J
, then this
process returns J.
Note that in general, G
J
(u, v) = Ω(u, Jv) 6= G(u, v).
Proof. Since , G are non-degenerate, we know
Ω(u, v) = G(u, Av)
for some
A
:
V V
. If
A
2
=
1, then we are done, and set
J
=
A
. In general,
we also know that
A
is skew-symmetric, i.e.
G
(
Au, v
) =
G
(
u, Av
), which is
clear since is anti-symmetric. Since
AA
t
is symmetric and positive definite, it
makes sense to write down
AA
T
(e.g. by diagonalizing), and we take
J =
AA
t
1
A =
p
A
2
1
A.
It is clear that
J
2
=
1, since
A
commutes with
AA
t
, so this is a complex
structure. We can write this as
A
=
AA
T
J
, and this is called the (left) polar
decomposition of A.
We now check that
J
is a compatible, i.e.
G
J
(
u, v
) = Ω(
u, Jv
) is symmetric
and positive definite. But
G
J
(u, v) = G(u,
AA
t
v),
and we are done since
AA
t
is positive and symmetric.
Notation.
Let (
V,
Ω) be a symplectic vector space. We write
J
(
V,
Ω) for the
space of all compatible complex structures on (V, Ω).
Proposition. J(V, Ω) is path-connected.
Proof.
Let
J
0
, J
1
J
(
V,
Ω). Then this induces inner products
G
J
0
, G
J
1
. Let
G
t
= (1
t
)
G
J
0
+
tG
J
1
be a smooth family of inner products on
V
. Then apply
polar decomposition to get a family of complex structures that start from
J
0
to
J
1
.
A quick adaptation of the proof shows it is in fact contractible.
We now move on to the case of manifolds.
Definition
(Almost complex structure)
.
An almost complex structure
J
on
a manifold is a smooth field of complex structures on the tangent space
J
p
:
T
p
M T
p
M, J
2
p
= 1.
Example.
Suppose
M
is a complex manifold with local complex coordinates
z
1
, . . . , z
n
on U M. We have real coordinates {x
i
, y
i
} given by
z
i
= x
i
+ iy
i
.
The tangent space is spanned by
x
i
,
y
i
. We define J by
J
p
x
j
=
y
j
, J
p
y
j
=
x
j
.
The Cauchy–Riemann equations imply this is globally well-defined. This
J
is
called the canonical almost-complex structure on the complex manifold M.
Definition
(Integrable almost complex structure)
.
An almost complex structure
on M is called integrable if it is induced by a complex structure.
Example.
It is a fact that
CP
2
#
CP
2
#
CP
2
has an almost complex but no
complex structure.
Definition
(Compatible almost complex structure)
.
An almost complex struc-
ture
J
on
M
is compatible with a symplectic structure
ω
if
J
p
is compatible with
ω
p
for all p M. In this case, (ω, g
J
, J) is called a compatible triple.
Any two of the structures determine the three, and this gives rise to the nice
fact that the intersection of any two of O(2n), Sp(2n, R) and GL(n, C) is U(n).
Performing polar decomposition pointwise gives
Proposition.
Let (
M, ω
) be a symplectic manifold, and
g
a metric on
M
. Then
from
g
we can canonically construct a compatible almost complex structure
J.
As before, in general, g
J
( ·, ·) 6= g( ·, ·).
Corollary.
Any symplectic manifold has a compatible almost complex structure.
The converse does not hold. For example,
S
6
is almost complex but not
symplectic.
Notation.
Let (
M, ω
) be a symplectic manifold. We write
J
(
V,
Ω) for the space
of all compatible almost complex structures on (M, ω).
The same proof as before shows that
Proposition. J(M, ω) is contractible.
Proposition.
Let
J
be an almost complex structure on
M
that is compatible
with ω
0
and ω
1
. Then ω
0
and ω
1
are deformation equivalent.
Proof.
Check that
ω
t
= (1
t
)
ω
0
+
1
works, which is non-degenerate since
ω
t
(
·, J ·
) is a positive linear combination of inner products, hence is non-
degenerate.
Proposition.
Let (
M, ω
) be a symplectic manifold,
J
a compatible almost
complex structure. If
X
is an almost complex submanifold of (
M, J
), i.e.
J(T X) = T X, then X is a symplectic submanifold of (M, ω).
Proof.
We only have to check
ω|
T X
is non-degenerate, but Ω(
·, J ·
) is a metric,
so is in particular non-degenerate.
We shall not prove the following theorem:
Theorem
(Gromov)
.
Let (
M, J
) be an almost complex manifold with
M
open,
i.e.
M
has no closed connected components. Then there exists a symplectic form
ω
in any even 2-cohomology class and such that
J
is homotopic to an almost
complex structure compatible with ω.
2.2 Dolbeault theory
An almost complex structure on a manifold allows us to discuss the notion of
holomorphicity, and this will in turn allow us to stratify our
k
-forms in terms of
“how holomorphic” they are.
Let (
M, J
) be an almost complex manifold. By complexifying
T M
to
T M C
and then extending J linearly, we can split T M C into its ±i eigenspace.
Notation.
We write
T
1,0
for the +
i
eigenspace of
J
and
T
1,0
for the
i
-
eigenspace of
J
. These are called the
J
-holomorphic tangent vectors and the
J-anti-holomorphic tangent vectors respectively.
We write
T
1,0
for the +
i
eigenspace of
J
, and
T
0,1
the
i
-eigenspace of
J
.
Then we have a splitting
T M C
T
1,0
T
0,1
.
We can explicitly write down the projection maps as
π
1,0
(w) =
1
2
(w iJw)
π
0,1
(w) =
1
2
(w + iJw).
Example.
On a complex manifold with local complex coordinates (
z
1
, . . . , z
n
),
the holomorphic tangent vectors
T
1,0
is spanned by the
z
j
, while
T
0,1
is spanned
by the
¯z
j
.
Similarly, we can complexify the cotangent bundle, and considering the
±i
eigenspaces of (the dual of) J gives a splitting
(π
1,0
, π
0,1
) : T
M C
T
1,0
T
0,1
.
These are the complex linear cotangent vectors and complex anti-linear cotangent
vectors.
Example.
In the complex case,
T
1,0
is spanned by the d
z
j
and
T
0,1
is spanned
by the d¯z
j
.
More generally, we can decompose
V
k
(T
M C) =
V
k
(T
1,0
T
0,1
) =
M
p+q=k
V
`
T
1,0
V
m
T
0,1
M
p+q=k
Λ
p,q
.
We write
k
(M, C) = sections of Λ
k
(T
M C)
p,q
(M, C) = sections of Λ
p,q
.
So
k
(M, C) =
M
p+q=k
p,q
(M, C).
The sections in
p,q
(M, C) are called forms of type (p, q).
Example. In the complex case, we have, locally,
Λ
p,q
p
= C{dz
I
dz
K
: |I| = , |K| = m}
and
p,q
=
X
|I|=p,|K|=q
b
I,K
dz
I
d¯z
K
: b
IK
C
(U, C)
.
As always, we have projections
π
p,q
:
V
k
(T
M C) Λ
p,q
.
Combining the exterior derivative d : Ω
k
(
M, C
)
k1
(
M, C
) with projections
yield the and
¯
operators
: Ω
p,q
p+1,q
¯
: Ω
p,q
p,q+1
.
Observe that for functions
f
, we have d
f
=
f
+
¯
f
, but this is not necessarily
true in general.
Definition
(
J
-holomorphic)
.
We say a function
f
is
J
-holomorphic if
¯
f
= 0,
and J-anti-holomorphic if f = 0.
It would be very nice if the sequence
p,q
p,q+1
p,q+2
···
¯
¯
¯
were a chain complex, i.e.
¯
2
= 0. This is not always true. However, this is
true in the case where
J
is integrable. Indeed, if
M
is a complex manifold and
β
k
(M, C), then in local complex coordinates, we can write
β =
X
p+q=k
X
|I|=p,|K|=q
b
I,K
dz
I
d¯z
K
.
So
dβ =
X
p+q=k
X
|I|=p,|K|=q
db
I,K
dz
I
d¯z
K
=
X
p+q=k
X
|I|=p,|K|=q
( +
¯
)b
I,K
dz
I
d¯z
K
= ( +
¯
)
X
p+q=k
X
|I|=p,|K|=q
b
I,K
dz
I
d¯z
K
Thus, on a complex manifold, d = +
¯
.
Thus, if β
p,q
, then
0 = d
2
β =
2
β + (
¯
+
¯
)β +
¯
2
β.
Since the three terms are of different types, it follows that
2
=
¯
2
=
¯
+
¯
= 0.
In fact, the converse of this computation is also true, which we will not prove:
Theorem (Newlander–Nirenberg). The following are equivalent:
¯
2
= 0
2
= 0
d = +
¯
J is integrable
N = 0
where N is the Nijenhuis torsion
N(X, Y ) = [JX, JY ] J[JX, Y ] J[X, JY ] [X, Y ].
When our manifold is complex, we can then consider the cohomology of the
chain complex.
Definition
(Dolbeault cohomology groups)
.
Let (
M, J
) be a manifold with an
integrable almost complex structure. The Dolbeault cohomology groups are the
cohomology groups of the cochain complex
p,q
p,q+1
p,q+2
···
¯
¯
¯
.
Explicitly,
H
p,q
Dolb
(M) =
ker(
¯
: Ω
p,q
p,q+1
)
im(
¯
: Ω
p,q1
p,q
)
.
2.3 ahler manifolds
In the best possible scenario, we would have a symplectic manifold with a
compatible, integrable almost complex structure. Such manifolds are called
ahler manifolds.
Definition
(K¨ahler manifold)
.
A ahler manifold is a symplectic manifold
equipped with a compatible integrable almost complex structure. Then
ω
is
called a ahler form.
Let (
M, ω, J
) be a ahler manifold. What can we say about the ahler form
ω? We can decompose it as
ω
2
(M)
2
(M, C) = Ω
2,0
1,1
2,2
.
We claim that
Lemma. ω
1,1
.
Proof. Since ω( ·, J ·) is symmetric, we have
J
ω(u, v) = ω(Ju, Jv) = ω(v, JJu) = ω(v, u) = ω(u, v).
So J
ω = ω.
On the other hand,
J
acts on holomorphic forms as multiplication by
i
and
anti-holomorphic forms by multiplication by
1 (by definition). So it acts on
2,0
and Ω
0,2
by multiplication by
1 (locally
2,0
is spanned by d
z
i
d
z
j
, etc.),
while it fixes
1,1
. So ω must lie in
1,1
.
We can explore what the other conditions on ω tell us. Closedness implies
0 = dω = ω +
¯
ω = 0.
So we have ω =
¯
ω = 0. So in particular ω H
1,1
Dolb
(M).
In local coordinates, we can write
ω =
i
2
X
j,k
h
j,k
dz
j
d¯z
k
for some
h
j,k
. The fact that
ω
is real valued, so that
¯ω
=
ω
gives some constraints
on what the h
jk
can be. We compute
¯ω =
i
2
X
j,k
h
jk
d¯z
j
dz
k
=
i
2
X
j,k
h
jk
dz
k
¯z
j
.
So we have
h
kj
= h
jk
.
The non-degeneracy condition ω
n
6= 0 is equivalent to det h
jk
6= 0, since
ω
n
=
i
2
n
n! det(h
jk
) dz
1
d¯z
1
··· dz
n
d¯z
n
.
So h
jk
is a non-singular Hermitian matrix.
Finally, we take into account the compatibility condition
ω
(
v, Jv
)
>
0. If we
write
v =
X
j
a
j
z
j
+ b
j
¯z
j
,
then we have
Jv = i
X
j
a
j
z
j
b
j
¯z
j
.
So we have
ω(v, Jv) =
i
2
X
h
jk
(ia
j
b
k
ia
j
b
k
) =
X
h
jk
a
j
b
k
> 0.
So the conclusion is that h
jk
is positive definite.
Thus, the conclusion is
Theorem.
A ahler form
ω
on a complex manifold
M
is a
- and
¯
-closed
form of type (1, 1) which on a local chart is given by
ω =
i
2
X
j,k
h
jk
dz
j
d¯z
k
where at each point, the matrix (h
jk
) is Hermitian and positive definite.
Often, we start with a complex manifold, and want to show that it has a
ahler form. How can we do so? First observe that we have the following
proposition:
Proposition.
Let (
M, ω
) be a complex K¨ahler manifold. If
X M
is a complex
submanifold, then (
X, i
ω
) is K¨ahler, and this is called a ahler submanifold.
In particular, if we can construct ahler forms on
C
n
and
CP
n
, then we
have ahler forms for a lot of our favorite complex manifolds, and in particular
complex projective varieties.
But we still have to construct some ahler forms to begin with. To do so,
we use so-called strictly plurisubharmonic functions.
Definition
(Strictly plurisubharmonic (spsh))
.
A function
ρ C
(
M, R
) is
strictly plurisubharmonic (spsh) if locally,
2
ρ
z
j
¯z
k
is positive definite.
Proposition.
Let
M
be a complex manifold, and
ρ C
(
M
;
R
) strictly
plurisubharmonic. Then
ω =
i
2
¯
ρ
is a ahler form.
We call ρ the ahler potential for ω.
Proof. ω
is indeed a 2-form of type (1, 1). Since
2
=
¯
2
= 0 and
¯
=
¯
,
we know ω =
¯
ω = 0. We also have
ω =
i
2
X
j,k
2
ρ
z
j
¯z
k
dz
j
d¯z
k
,
and the matrix is Hermitian positive definite by assumption.
Example. If M = C
n
=
R
2n
, we take
ρ(z) = |z|
2
=
X
z
k
¯z
k
.
Then we have
h
jk
=
2
ρ
z
j
¯z
k
= δ
jk
,
so this is strictly plurisubharmonic. Then
ω =
i
2
X
dz
j
d¯z
k
=
i
2
X
d(x
j
+ iy
j
) d(x
k
iy
k
)
=
X
dx
k
dy
k
,
which is the standard symplectic form. So (
C
n
, ω
) is ahler and
ρ
=
|z|
2
is a
(global) ahler potential for ω
0
.
There is a local converse to this result.
Proposition.
Let
M
be a complex manifold,
ω
a closed real-valued (1
,
1)-form
and
p M
, then there exists a neighbourhood
U
of
p
in
M
and a
ρ C
(
U, R
)
such that
ω = i∂
¯
ρ on U.
Proof. This uses the holomorphic version of the Poincar´e lemma.
When ρ is ahler, such a ρ is called a local ahler potential for ω.
Note that it is not possible to have a global ahler potential on a closed
ahler manifold, because if ω =
i
2
¯
ρ, then
ω = d
i
2
¯
ρ
is exact, and we know symplectic forms cannot be exact.
Example. Let M = C
n
and
ρ(z) = log(|z|
2
+ 1).
It is an exercise to check that ρ is strictly plurisubharmonic. Then
ω
F S
=
i
2
¯
(log(|z
2
| + 1))
is another ahler form on C
n
, called the Fubini–Study form.
The reason this is interesting is that it allows us to put a ahler structure
on CP
n
.
Example.
Let
M
=
CP
n
. Using homogeneous coordinates, this is covered by
the open sets
U
j
= {[z
0
, . . . , z
n
] CP
n
| z
j
6= 0}.
with the chart given by
ϕ
j
: U
j
C
n
[z
0
, . . . , z
n
] 7→
z
0
z
j
, . . . ,
z
j1
z
j
,
z
j+1
z
j
, . . . ,
z
n
z
j
.
One can check that
ϕ
j
ω
F S
=
ϕ
k
ω
F S
. Thus, these glue to give the Fubini–Study
form on CP
n
, making it a ahler manifold.
2.4 Hodge theory
So what we have got so far is that if we have a complex manifold, then we can
decompose
k
(M; C) =
M
p+q=k
p,q
,
and using
¯
: Ω
p,q
p,q+1
, we defined the Dolbeault cohomology groups
H
p,q
Dolb
(M) =
ker
¯
im
¯
.
It would be nice if we also had a decomposition
H
k
dR
=
M
p+q=k
H
p,q
Dolb
(M).
This is not always true. However, it is true for compact ahler manifolds:
Theorem
(Hodge decomposition theorem)
.
Let (
M, ω
) be a compact K
¨
haler
manifold. Then
H
k
dR
=
M
p+q=k
H
p,q
Dolb
(M).
To prove the theorem, we will first need a “real” analogue of the theorem.
This is an analytic theorem that lets us find canonical representatives for each
cohomology class. We can develop the same theory for Dolbeault cohomology,
and see that the canonical representatives for Dolbeault cohomology are the
same as those for de Rham cohomology. We will not prove the analytic theorems,
but just say how these things piece together.
Real Hodge theory
Let
V
be a real oriented vector space
m
with inner product
G
. Then this induces
an inner product on each Λ
k
=
V
k
(V ), denoted h·, ·i, defined by
hv
1
··· v
k
, w
1
··· w
k
i = det(G(v
i
, w
j
))
i,j
Let e
1
, . . . , e
n
be an oriented orthonormal basis for V . Then
{e
j
1
··· e
j
k
: 1 j
1
< ··· < j
k
m}
is an orthonormal basis for Λ
k
.
Definition
(Hodge star)
.
The Hodge
-operator
: Λ
k
Λ
mk
is defined by
the relation
α β = hα, βi e
1
··· e
m
.
It is easy to see that the Hodge star is in fact an isomorphism. It is also not
hard to verify the following properties:
Proposition.
(e
1
··· e
k
) = e
k+1
··· e
m
(e
k+1
··· e
m
) = (1)
k(mk)
e
1
··· e
k
.
∗∗ = α = (1)
k(mk)
α for α Λ
k
.
In general, let (
M, g
) be a compact real oriented Riemannian manifold of
dimension
m
. Then
g
gives an isomorphism
T M
=
T
M
, and induces an inner
product on each
T
p
M
, which we will still denote
g
p
. This induces an inner
product on each
V
k
T
p
M, which we will denote h·, ·i again.
The Riemannian metric and orientation gives us a volume form
Vol
m
(
M
),
defined locally by
Vol
p
(e
1
··· e
m
),
where
e
1
, . . . , e
m
is an oriented basis of
T
p
M
. This induces an
L
2
-inner product
on
k
(M),
hα, βi
L
2
=
Z
M
hα, βi Vol.
Now apply Hodge
-operator to each (
V, G
) = (
T
p
M, g
p
) and
p M
. We then
get
Definition
(Hodge star operator)
.
The Hodge
-operator on forms
: Ω
k
(
M
)
mk
(M) is defined by the equation
α (β) = hα, βi Vol.
We again have some immediate properties.
Proposition.
(i) α = (1)
k(mk)
α for α
k
(M).
(ii) 1 = Vol
We now introduce the codifferential operator
Definition
(Codifferential operator)
.
We define the codifferential operator
δ
:
k
k1
to be the L
2
-formal adjoint of d. In other words, we require
hdα, βi
L
2
= hα, δβi
L
2
for all α
k
and β
k+1
.
We immediately see that
Proposition. δ
2
= 0.
Using the Hodge star, there is a very explicit formula for the codifferential
(which in particular shows that it exists).
Proposition.
δ = (1)
m(k+1)+1
d : Ω
k
k1
.
Proof.
hdα, βi
L
2
=
Z
M
dα β
=
Z
M
d(α β) (1)
k
Z
M
α d(β)
= (1)
k+1
Z
M
α d(β) (Stokes’)
= (1)
k+1
Z
M
(1)
(mk)k
α d(β)
= (1)
k+1+(mk)k
Z
M
hα, d βi.
We can now define the Laplace–Beltrami operator
Definition
(Laplace–Beltrami operator)
.
We define the Laplacian, or the
Laplace–Beltrami operator to be
∆ = dδ + δd : Ω
k
k
.
Example.
If
M
=
R
m
with the Euclidean inner product, and
f
0
(
M
) =
C
(M), then
f =
n
X
i=1
2
f
x
2
i
.
It is an exercise to prove the following
Proposition.
(i) = ∆ : Ω
k
mk
(ii) ∆ = (d + δ)
2
(iii) hα, βi
L
2
= hα, βi
L
2
.
(iv) α = 0 iff dα = δα = 0.
In particular, (iv) follows from the fact that
hα, αi = hdα, dαi + hδα, δαi = kdαk
2
L
2
+ kδαk
2
L
2
.
Similar to IA Vector Calculus, we can define
Definition (Harmonic form). A form α is harmonic if α = 0. We write
H
k
= {α
k
(m) | α = 0}
for the space of harmonic forms.
Observe there is a natural map
H
k
H
k
dR
(
M
), sending
α
to [
α
]. The main
result is that
Theorem
(Hodge decomposition theorem)
.
Let (
M, g
) be a compact oriented
Riemannian manifold. Then every cohomology class in
H
k
dR
(
M
) has a unique
harmonic representation, i.e. the natural map
H
k
H
k
dR
(
M
) is an isomorphism.
We will not prove this.
Complex Hodge theory
We now see what we can do for complex ahler manifolds. First check that
Proposition.
Let
M
be a complex manifold,
dim
C
M
=
n
and (
M, ω
) ahler.
Then
(i) : Ω
p,q
np,nq
.
(ii) ∆ : Ω
p,q
p,q
.
define the
L
2
-adjoints
¯
=
±
¯
and
=
with the appropriate
signs as before, and then
d = +
¯
, δ =
+
¯
.
We can then define
=
+
: Ω
p,q
p,q
¯
=
¯
¯
+
¯
¯
: Ω
p,q
p,q
.
Proposition. If our manifold is ahler, then
∆ = 2∆
= 2∆
¯
.
So if we have a harmonic form, then it is in fact
and
¯
-harmonic, and in
particular it is and
¯
-closed. This give us a Hodge decomposition
H
k
C
=
M
p+q=k
H
p,q
,
where
H
p,q
= {α
p,q
(M) : ∆α = 0}.
Theorem
(Hodge decomposition theorem)
.
Let (
M, ω
) be a compact ahler
manifold. The natural map H
p,q
H
p,q
Dolb
is an isomorphism. Hence
H
k
dR
(M; C)
=
H
k
C
=
M
p+q=k
H
p,q
=
M
p+q=k
H
p,q
Dolb
(M).
What are some topological consequences of this? Recall the Betti numbers
are defined by
b
k
= dim
R
H
k
dR
(M) = dim
C
H
k
dR
(M; C).
We can further define the Hodge numbers
h
p,q
= dim
C
H
p,q
Dolb
(M).
Then the Hodge theorem says
b
k
=
X
p+q=k
h
p,q
.
Moreover, since
H
p,q
Dolb
(M) = H
q,p
Dolb
(M).
So we have Hodge symmetry,
h
p,q
= h
q,p
.
Moreover, the operator induces an isomorphism
H
p,q
Dolb
=
H
np,nq
Dolb
.
So we have
h
p,q
= h
np,nq
.
There is called central symmetry, or Serre duality. Thus, we know that
Corollary. Odd Betti numbers are even.
Proof.
b
2k+1
=
X
p+q=2k+1
h
p,q
= 2
k
X
p=0
h
p,2k+1p
!
.
Corollary. h
1,0
= h
0,1
=
1
2
b
1
is a topological invariant.
We have also previously seen that for a general compact ahler manifold, we
have
Proposition. Even Betti numbers are positive.
Recall that we proved this by arguing that [
ω
k
]
6
= 0
H
2k
(
M
). In fact,
ω
k
H
k,k
Dolb
(M), and so
Proposition. h
k,k
6= 0.
We usually organize these h
p,q
in the form of a Hodge diamond, e.g.
h
0,0
h
1,0
h
0,1
h
2,0
h
1,1
h
0,2
h
2,1
h
1,2
h
2,2
3 Hamiltonian vector fields
Symplectic geometry was first studied by physicists, who modeled their systems
by a symplectic manifold. The Hamiltonian function
H C
(
M
), which returns
the energy of a particular configuration, generates a vector field on
M
which is
the equation of motion of the system. In this chapter, we will discuss how this
process works, and further study what happens when we have families of such
Hamiltonian functions, giving rise to Lie group actions.
3.1 Hamiltonian vector fields
Definition
(Hamiltonian vector field)
.
Let (
M, ω
) be a symplectic manifold. If
H C
(
M
), then since
˜ω
:
T M T
M
is an isomorphism, there is a unique
vector field X
H
on M such that
ι
X
H
ω = dH.
We call X
H
the Hamiltonian vector field with Hamiltonian function H.
Suppose
X
H
is complete (e.g. when
M
is compact). This means we can
integrate X
H
, i.e. solve
ρ
t
t
(p) = X
H
(ρ
t
(p)), ρ
0
(p) = p.
These flow have some nice properties.
Proposition.
If
X
H
is a Hamiltonian vector field with flow
ρ
t
, then
ρ
t
ω
=
ω
.
In other words, each ρ
t
is a symplectomorphism.
Proof. It suffices to show that
t
ρ
t
ω = 0. We have
d
dt
(ρ
t
ω) = ρ
t
(L
X
H
ω) = ρ
t
(dι
X
H
ω + ι
X
H
dω) = ρ
t
(ddH) = 0.
Thus, every function
H
gives us a one-parameter subgroup of symplectomor-
phisms.
Proposition. ρ
t
preserves H, i.e. ρ
t
H = H.
Proof.
d
dt
ρ
t
H = ρ
t
(L
X
H
H) = ρ
t
(ι
X
H
dH) = ρ
t
(ι
X
H
ι
X
H
ω) = 0.
So the flow lines of our vector field are contained in level sets of H.
Example. Take (S
2
, ω = dθ dh). Take
H(h, θ) = h
to be the height function. Then X
H
solves
ι
X
H
(dθ dh) = dh.
So
X
H
=
θ
, ρ
t
(h, θ) = (h, θ + t).
As expected, the flow preserves height and the area form.
We have seen that Hamiltonian vector fields are symplectic:
Definition
(Symplectic vector field)
.
A vector field
X
on (
M, ω
) is a symplectic
vector field if L
X
ω = 0.
Observe that
L
X
ω = ι
X
dω + dι
X
ω = dι
X
ω.
So
X
is symplectic iff
ι
X
ω
is closed, and is Hamiltonian if it is exact. Thus,
locally, every symplectic vector field is Hamiltonian and globally, the obstruction
lies in H
1
dR
(M).
Example.
Take (
T
2
, ω
= d
θ
1
d
θ
2
). Then
X
i
=
θ
i
are symplectic but not
Hamiltonian, since ι
X
i
ω = dθ
2i
is closed but not exact.
Proposition.
Let
X, Y
be symplectic vector fields on (
M, ω
). Then [
X, Y
] is
Hamiltonian.
Recall that if
X, Y
are vector fields on
M
and
f C
(
M
), then their Lie
bracket is given by
[X, Y ]f = (XY Y X)f.
This makes χ(M ), the space of vector fields on M , a Lie algebra.
In order to prove the proposition, we need the following identity:
Exercise. ι
[X,Y ]
α = [L
X
, ι
Y
]α = [ι
X
, L
Y
]α.
Proof of proposition.
We need to check that
ι
[X,Y ]
ω
is exact. By the exercise,
this is
ι
[X,Y ]
ω = L
X
ι
Y
ω ι
Y
L
X
ω = d(ι
X
ι
Y
ω) + ι
X
dι
Y
ω + ι
Y
dι
X
ω ι
Y
ι
Y
dω.
Since
X, Y
are symplectic, we know d
ι
Y
ω
= d
ι
X
ω
= 0, and the last term always
vanishes. So this is exact, and
ω
(
Y, X
) is a Hamiltonian function for [
X, Y
].
Definition
(Poisson bracket)
.
Let
f, g C
(
M
). We then define the Poisson
bracket {f, g} by
{f, g} = ω(X
f
, X
g
).
This is defined so that
X
{f,g}
= [X
f
, X
g
].
Exercise.
The Poisson bracket satisfies the Jacobi identity, and also the Leibniz
rule
{f, gh} = g{f, h}+ {f, g}h.
Thus, if (
M, ω
) is symplectic, then (
C
(
M
)
, {·, ·}
) is a Poisson algebra.
This means it is a commutative, associative algebra with a Lie bracket that
satisfies the Leibniz rule.
Further, the map
C
(
M
)
χ
(
M
) sending
H 7→ X
H
is a Lie algebra
(anti-)homomorphism.
Proposition. {f, g} = 0 iff f is constant along integral curves of X
g
.
Proof.
L
X
g
f = ι
X
g
df = ι
X
g
ι
X
f
ω = ω(X
f
, X
g
) = {f, g} = 0.
Example. If M = R
2n
and ω = ω
0
=
P
dx
j
dy
j
, and f C
(R
2n
), then
X
f
=
X
i
f
y
i
x
i
f
x
i
y
i
.
If ρ
0
(p) = p, then ρ
t
(p) = (x(t), y(t)) is an integral curve for X
f
iff
dx
i
dt
=
f
y
i
,
y
i
t
=
f
x
i
.
In classical mechanics, these are known as Hamilton equations.
3.2 Integrable systems
In classical mechanics, we usually have a fixed H, corresponding to the energy.
Definition
(Hamiltonian system)
.
A Hamiltonian system is a triple (
M, ω, H
)
where (
M, ω
) is a symplectic manifold and
H C
(
M
), called the Hamiltonian
function.
Definition
(Integral of motion)
.
A integral of motion/first integral/constant of
motion/conserved quantity of a Hamiltonian system is a function
f C
(
M
)
such that {f, H} = 0.
For example,
H
is an integral of motion. Are there others? Of course, we
can write down 2
H, H
2
, H
12
, e
H
, etc., but these are more-or-less the same as
H
.
Definition
(Independent integrals of motion)
.
We say
f
1
, . . . , f
n
C
(
M
) are
independent if (d
f
1
)
p
, . . . ,
(d
f
n
)
p
are linearly independent at all points on some
dense subset of M .
Definition
(Commuting integrals of motion)
.
We say
f
1
, . . . , f
n
C
commute
if {f
i
, f
j
} = 0.
If we have
n
independent integrals of motion, then we clearly have
dim M n
.
In fact, the commuting condition implies:
Exercise.
Let
f
1
, . . . , f
n
be independent commuting functions on (
M, ω
). Then
dim M 2n.
The idea is that the (d
f
i
)
p
are not only independent, but span an isotropic
subspace of T M.
If we have the maximum possible number of independent commuting first
integrals, then we say we are integrable.
Definition
(Completely integrable system)
.
A Hamiltonian system (
M, ω, H
)
of dimension
dim M
= 2
n
is (completely) integrable if it has
n
independent
commuting integrals of motion f
1
= H, f
2
, . . . , f
n
.
Example.
If
dim M
= 2, then we only need one integral of motion, which we
can take to be
H
. Then (
M, ω, H
) is integrable as long as the set of non-critical
points of H is dense.
Example.
The physics of a simple pendulum of length 1 and mass 1 can be
modeled by the symplectic manifold
M
=
T
S
1
, where the
S
1
refers to the
angular coordinate
θ
of the pendulum, and the cotangent vector is the momentum.
The Hamiltonian function is
H = K + V = kinetic energy + potential energy =
1
2
ξ
2
+ (1 cos ω).
We can check that the critical points of
H
are (
θ, ξ
) = (0
,
0) and (
π,
0). So
(M, ω, H) is integrable.
Example.
If
dim M
= 4, then (
M, ω, H
) is integrable as long as there exists an
integral motion independent of
H
. For example, on a spherical pendulum, we
have
M
=
T
S
2
, and
H
is the total energy. Then the angular momentum is an
integral of motion.
What can we do with a completely integrable system? Suppose (
M, ω, H
)
is completely integrable system with
dim M
= 2
n
and
f
1
=
H, f
2
, . . . , f
n
are
commuting. Let
c
be a regular value of
f
= (
f
1
, . . . , f
n
). Then
f
1
(
c
) is an
n-dimensional submanifold of M . If p f
1
(c), then
T
p
(f
1
(c)) = ker(df)
p
.
Since
df
p
=
(df
1
)
p
.
.
.
(df
n
)
p
=
ι
X
f
1
ω
.
.
.
ι
X
f
n
ω
,
we know
T
p
(f
1
(c)) = ker(df
p
) = span{(X
f
1
)
p
, . . . (X
f
n
)
p
},
Moreover, since
ω(X
f
i
, X
f
j
) = {f
i
, f
j
} = 0,
we know that T
p
(f
1
(c)) is an isotropic subspace of (T
p
M, ω
p
).
If
X
f
1
, . . . , X
f
n
are complete, then following their flows, we obtain global
coordinates of (the connected components of)
f
1
(
c
), where
q f
1
(
c
) has
coordinates (
ϕ
1
, . . . , ϕ
m
) (angle coordinates) if
q
is achieved from the base point
p
by following the flow of
X
f
i
for
ϕ
i
seconds for each
i
. The fact that the
f
i
are
Poisson commuting implies the vector fields commute, and so the order does not
matter, and this gives a genuine coordinate system.
By the independence of
X
f
i
, the connected components look like
R
nk
×T
k
,
where
T
k
= (
S
1
)
k
is the
k
torus. In the extreme case
k
=
n
, we simply get a
torus, which is a compact connected component.
Definition
(Liouville torus)
.
A Liouville torus is a compact connected compo-
nent of f
1
(c).
It would be nice if the (
ϕ
i
) are part of a Darboux chart of
M
, and this is
true.
Theorem
(Arnold–Liouville thoerem)
.
Let (
M, ω, H
) be an integrable system
with
dim M
= 2
n
and
f
1
=
H, f
2
, . . . , f
n
integrals of motion, and
c R
a regular
value of f = (f
1
, . . . , f
n
).
(i)
If the flows of
X
f
i
are complete, then the connected components of
f
1
(
{c}
)
are homogeneous spaces for
R
n
and admit affine coordinates
ϕ
1
, . . . , ϕ
n
(angle coordinates), in which the flows of X
f
i
are linear.
(ii)
There exists coordinates
ψ
1
, . . . , ψ
n
(action coordinates) such that the
ψ
i
’s
are integrals of motion and ϕ
1
, . . . , ϕ
n
, ψ
1
, . . . , ψ
n
form a Darboux chart.
3.3 Classical mechanics
As mentioned at the beginning, symplectic geometry was first studied by physi-
cists. In this section, we give a brief overview of how symplectic geometry
arises in physics. Historically, there have been three “breakthroughs” in classical
mechanics:
(i)
In
1687, Newton wrote down his three laws of physics, giving rise to
Newtonian mechanics.
(ii) In 1788, this was reformulated into the Lagrangian formalism.
(iii) In 1833, these were further developed into the Hamiltonian formalism.
Newtonian mechanics
In Newtonian mechanics, we consider a particle of mass
m
moving in
R
3
under
the potential
V
(
x
). Newton’s second law then says the trajectory of the particle
obeys
m
d
2
x
dt
2
= −∇V (x).
Hamiltonian mechanics.
To do Hamiltonian mechanics, a key concept to introduce is the momentum:
Definition (Momentum). The momentum of a particle is
y = m
dx
dt
.
We also need the energy function
Definition (Energy). The energy of a particle is
H(x, y) =
1
2m
|y|
2
+ V (x).
We call
R
3
the configuration space and
T
R
3
the phase space, parametrized
by (x, y). This has a canonical symplectic form ω = dx
i
dy
i
.
Newton’s second law can be written as
dy
i
dt
=
V
x
i
.
Combining with the definition of y, we find that (x, y) evolves under.
dx
i
dt
=
H
y
i
dy
i
dt
=
H
x
i
So physical motion is given by Hamiltonian flow under
H
.
H
is called the
Hamiltonian of the system.
Lagrangian mechanics
Lagrangian mechanics is less relevant to our symplectic picture, but is nice to
know about nevertheless. This is formulated via a variational principle.
In general, consider a system with
N
particles of masses
m
1
, . . . , m
N
moving
in
R
3
under a potential
V C
(
R
3N
). The Hamiltonian function can be defined
exactly as before:
H(x, y) =
X
k
1
2m
k
|y
k
|
2
+ V (x),
where
x
(
t
) = (
x
1
, . . . , x
n
) and each
x
i
is a 3-vector; and similarly for
y
with
y
k
=
m
k
dx
t
dt
. Then in Hamiltonian mechanics, we say (
x, y
) evolves under
Hamiltonian flow.
Now fix
a, b R
and
p, q R
3N
. Write
P
for the space of all piecewise
differentiable paths γ = (γ
1
, . . . , γ
n
) : [a, b] R
3N
.
Definition (Action). The action of a path γ P is
A
γ
=
Z
b
a
X
k
m
k
2
dγ
k
dt
(t)
2
V (γ(t))
!
dt.
The integrand is known as the Lagrangian function. We will see that
γ
(
t
) =
x(t) is (locally) a stationary point of A
γ
iff
m
k
d
2
x
t
dt
=
V
x
k
,
i.e. if and only if Newton’s second law is satisfied.
The Lagrangian formulation works more generally if our particles are con-
strained to live in some submanifold
X R
3n
. For example, if we have a
pendulum, then the particle is constrained to live in
S
1
(or
S
2
). Then we set
P
to be the maps
γ
: [
a, b
]
X
that go from
p
to
q
. The Lagrangian formulation
is then exactly the same, except the minimization problem is now performed
within this P.
More generally, suppose we have an
n
-dimensional manifolds
X
, and
F
:
T X R is a Lagrangian function. Given a curve γ : [a, b] X, there is a lift
˜γ : [a, b] T X
t 7→
γ(t),
dγ
dt
(t)
.
The action is then
A
γ
=
Z
b
a
(˜γ
F )(t) =
Z
b
a
F
γ(t),
dγ
dt
(t)
dt.
To find the critical points of the action, we use calculus of variations. Pick a
chart (
x
1
, . . . , x
n
) for
X
and naturally extend to a chart (
x
1
, . . . , x
n
, v
1
, . . . , v
n
)
for T X. Consider a small perturbation of our path
γ
ε
(t) = (γ
1
(t) + εc
1
(t), . . . , γ
n
(t) + εc
n
(t))
for some functions
c
1
, . . . , c
n
C
([
a, b
]) such that
c
i
(
a
) =
c
i
(
b
) = 0. We think
of this as an infinitesimal variation of γ. We then find that
dA
γ
ε
dε
ε=0
= 0
F
x
k
=
d
dt
F
v
F
for k = 1, . . . , n.
These are the Euler–Lagrange equations.
Example. In X = R
3N
and
F (x
1
, . . . , x
n
, v
1
, . . . , v
n
) =
X
k
m
k
2
|v
k
|
2
V (x
1
, . . . , x
n
).
Then the Euler–Lagrange equations are
V
x
k
i
= m
k
d
2
x
k
i
dt
2
.
Example.
On a Riemannian manifold, if we set
F
:
T X R
be (
x, v
)
7→ |v|
2
,
then we obtain the Christoffel equations for a geodesic.
In general, there need not be solutions to the Euler–Lagrange equation.
However, if we satisfy the Legendre condition
det
2
F
v
i
v
j
6= 0,
then the Euler–Lagrange equations become second order ODEs, and there is a
unique solution given
γ
(0) and
˙γ
(0). If furthermore this is positive definite, then
the solution is actually a locally minimum.
3.4 Hamiltonian actions
In the remainder of the course, we are largely interested in how Lie groups can
act on symplectic manifolds via Hamiltonian vector fields. These are known as
Hamiltonian actions. We first begin with the notion of symplectic actions.
Let (M, ω) be a symplectic manifold, and G a Lie group.
Definition
(Symplectic action)
.
A symplectic action is a smooth group action
ψ
:
G Diff
(
M
) such that each
ψ
g
is a symplectomorphism. In other words, it
is a map G Symp(M, ω).
Example.
Let
G
=
R
. Then a map
ψ
:
G Diff
(
M
) is a one-parameter group
of transformations
{ψ
t
:
t R}
. Given such a group action, we obtain a complete
vector field
X
p
=
dψ
t
dt
(p)
t=0
.
Conversely, given a complete vector field X, we can define
ψ
t
= exp tX,
and this gives a group action by R.
Under this correspondence, symplectic actions correspond to complete sym-
plectic vector fields.
Example.
If
G
=
S
1
, then a symplectic action of
S
1
is a symplectic action of
R which is periodic.
In the case where
G
is
R
or
S
1
, it is easy to define what it means for an
action to be Hamiltonian:
Definition
(Hamiltonian action)
.
An action of
R
or
S
1
is Hamiltonian if the
corresponding vector field is Hamiltonian.
Example. Take (S
2
, ω = dθ dh). Then we have a rotation action
ψ
t
(θ, h) = (θ + t, h)
generated by the vector field
θ
. Since
ι
θ
ω
= d
h
is exact, this is in fact a
Hamiltonian S
1
action.
Example. Take (T
2
, dθ
1
dθ
2
). Consider the action
ψ
t
(θ
1
, θ
2
) = (θ
1
+ t, θ
2
).
This is generated by the vector field
θ
1
. But
ι
θ
1
ω
= d
θ
2
, which is closed but
not exact. So this is a symplectic action that is not Hamiltonian.
How should we define Hamiltonian group actions for groups that are not
R
or
S
1
? The simplest possible next case is the torus
G
=
T
n
=
S
1
× ··· × S
1
.
If we have a map
ψ
;
T
n
Symp
(
M, ω
), then for this to be Hamiltonian, it
should definitely be the case that the restriction to each
S
1
is Hamiltonian in
the previous sense. Moreover, for these to be compatible, we would expect each
Hamiltonian function to be preserved by the other factors as well.
For the general case, we need to talk about the Lie group of
G
. Let
G
be a
Lie group. For each g GG, there is a left multiplication map
L
g
: G G
a 7→ ga.
Definition
(Left-invariant vector field)
.
A left-invariant vector field on a Lie
group G is a vector field X such that
(L
g
)
X = X
for all g G.
We write
g
for the space of all left-invariant vector fields on
G
, which comes
with the Lie bracket on vector fields. This is called the Lie algebra of G.
If
X
is left-invariant, then knowing
X
e
tells us what
X
is everywhere, and
specifying
X
e
produces a left-invariant vector field. Thus, we have an isomor-
phism g
=
T
e
G.
The Lie algebra admits a natural action of
G
, called the adjoint action. To
construct this, note that G acts on itself by conjugation,
ϕ
g
(a) = gag
1
.
This fixes the identity, and taking the derivative gives
Ad
g
:
g g
, or equivalently,
Ad
is a map
Ad
:
G GL
(
g
). The dual
g
admits the dual action of
G
, called
the coadjoint action. Explicitly, this is given by
hAd
g
(ξ), xi = hξ, Ad
g
xi.
An important case is when
G
is abelian, i.e. a product of
S
1
’s and
R
’s, in which
case the conjugation action is trivial, hence the (co)adjoint action is trivial.
Returning to group actions, the correspondence between complete vector
fields and
R
/
S
1
actions can be described as follows: Given a smooth action
ψ : G Diff(M) and a point p M, there is a map
G M
g 7→ ψ
g
(p).
Differentiating this at e gives
g
=
T
e
G T
p
M
X 7→ X
#
p
.
We call
X
#
the vector field on
M
generated by
X g
. In the case where
G
=
S
1
or
R
, we have
g
=
R
, and the complete vector field corresponding to the action
is the image of 1 under this map.
We are now ready to define
Definition
(Hamiltonian action)
.
We say
ψ
:
G Symp
(
M, ω
) is a Hamiltonian
action if there exists a map µ : M g
such that
(i)
For all
X g
,
X
#
is the Hamiltonian vector field generated by
µ
X
, where
µ
X
: M R is given by
µ
X
(p) = hµ(p), Xi.
(ii) µ
is
G
-equivariant, where
G
acts on
g
by the coadjoint action. In other
words,
µ ψ
g
= Ad
g
µ for all g G.
µ is then called a moment map for the action ψ.
In the case where G is abelian, condition (ii) just says µ is G-invariant.
Example. Let M = C
n
, and
ω =
1
2
X
j
dz
j
d¯z
j
=
X
i
r
j
dr
j
dθ
j
.
We let
T
n
= {(t
1
, . . . , t
n
) C
n
: |t
k
| = 1 for all k},
acting by
ψ
(t
1
,...,t
n
)
(z
1
, . . . , z
n
) = (t
k
1
1
z
1
, . . . , t
k
n
n
z
n
)
where k
1
, . . . , k
n
Z.
We claim this action has moment map
µ : C
n
(t
n
)
=
R
n
(x
1
, . . . , z
n
) 7→
1
2
(k
1
|z
1
|
2
, . . . , k
n
|z
n
|
2
).
It is clear that this is invariant, and if X = (a
1
, . . . , a
n
) t
n
R
n
, then
X
#
= k
1
a
1
θ
1
+ ··· + k
n
a
n
θ
n
.
Then we have
dµ
X
= d
1
2
X
k
j
a
j
r
2
j
=
X
k
j
a
j
r
j
dr
j
= ι
X
#
ω.
Example.
A nice example of a non-abelian Hamiltonian action is coadjoint
orbits. Let
G
be a Lie group, and
g
the Lie algebra. If
X g
, then we get a
vector field
g
X
#
generated by
X
via the adjoint action, and also a vector field
g
X on g
generated by the co-adjoint action.
If ξ g
, then we can define the coadjoint orbit through ξ
O
ξ
= {Ad
g
(ξ) : g G}.
What is interesting about this is that this coadjoint orbit is actually a symplectic
manifold, with symplectic form given by
ω
ξ
(X
#
ξ
, Y
#
ξ
) = hξ, [X, Y ]i.
Then the coadjoint action of
G
on
O
ξ
has moment map
O
ξ
g
given by the
inclusion.
3.5 Symplectic reduction
Given a Lie group action of
G
on
M
, it is natural to ask what the “quotient” of
M
looks like. What we will study is not quite the quotient, but a symplectic
reduction, which is a natural, well-behaved subspace of the quotient that is in
particular symplectic.
We first introduce some words. Let ψ : G Diff(M) be a smooth action.
Definition (Orbit). If p M, the orbit of p under G is
O
p
= {ψ
g
(p) : g G}.
Definition
(Stabilizer)
.
The stabilizer or isotropy group of
p M
is the closed
subgroup
G
p
= {g G : ψ
g
(p) = p}.
We write g
p
for the Lie algebra of G
p
.
Definition (Transitive action). We say G acts transitively if M is one orbit.
Definition (Free action). We say G acts freely if G
p
= {e} for all p.
Definition
(Locally free action)
.
We say
G
acts locally freely if
g
p
=
{
0
}
, i.e.
G
p
is discrete.
Definition
(Orbit space)
.
The orbit space is
M/G
, and we write
π
:
M M/G
for the orbit projection. We equip M/G with the quotient topology.
The main theorem is the following:
Theorem
(Marsden–Weinstein, Meyer)
.
Let
G
be a compact Lie group, and
(
M, ω
) a symplectic manifold with a Hamiltonian
G
-action with moment map
µ
:
M g
. Write
i
:
µ
1
(0)
M
for the inclusion. Suppose
G
acts freely on
µ
1
(0). Then
(i) M
red
= µ
1
(0)/G is a manifold;
(ii) π : µ
1
(0) M
red
is a principal G-bundle; and
(iii) There exists a symplectic form ω
red
on M
red
such that i
ω = π
ω
red
.
Definition
(Symplectic quotient)
.
We call
M
red
the symplectic quotient/reduced
space/symplectic reduction of (
M, ω
) with respect to the given
G
-action and
moment map.
What happens if we do reduction at other levels? In other words, what can
we say about µ
1
(ξ)/G for other ξ g
?
If we want to make sense of this, we need
ξ
to be preserved under the coadjoint
action of G. This is automatically satisfied if G is abelian, and in this case, we
simply have
µ
1
(
ξ
) =
ϕ
1
(0), where
ϕ
=
µ ξ
is another moment map. So this
is not more general.
If
ξ
is not preserved by
G
, then we can instead consider
µ
1
(
ξ
)
/G
ξ
, or
equivalently take µ
1
(O
ξ
)/G. We check that
µ
1
(ξ)/G
ξ
=
µ
1
(O
ξ
)/G
=
ϕ
1
(0)/G,
where
ϕ : M × O
ξ
g
(ρ, η) 7→ µ(p) η
is a moment map for the product action of G on (M × O
ξ
, ω × ω
ξ
).
So in fact there is no loss in generality for considering just µ
1
(0).
Proof.
We first show that
µ
1
(0) is a manifold. This follows from the following
claim:
Claim. G acts locally freely at p iff p is a regular point of µ.
We compute the dimension of
im
d
µ
p
using the rank-nullity theorem. We
know dµ
p
v = 0 iff hdµ
p
(v), Xi = 0 for all X g. We can compute
hdµ
p
(v), Xi = (dµ
X
)
p
(v) = (ι
X
#
p
ω)(v) = ω
p
(X
#
p
, v).
Moreover, the span of the X
#
p
is exactly T
p
O
p
. So
ker dµ
p
= (T
p
O
p
)
ω
.
Thus,
dim(im dµ
p
) = dim O
p
= dim G dim G
p
.
In particular, dµ
p
is surjective iff G
p
= 0.
Then (i) and (ii) follow from the following theorem:
Theorem.
Let
G
be a compact Lie group and
Z
a manifold, and
G
acts freely
on Z. Then Z/G is a manifold and Z Z/G is a principal G-bundle.
Note that if
G
does not act freely on
µ
1
(0), then by Sard’s theorem,
generically,
ξ
is a regular value of
µ
, and so
µ
1
(
ξ
) is a manifold, and
G
acts
locally freely on
µ
1
(
ξ
). If
µ
1
(
ξ
) is preserved by
G
, then
µ
1
(
ξ
)
/G
is a
symplectic orbifold.
It now remains to construct the symplectic structure. Observe that if
p
µ
1
(0), then
T
p
O
p
T
p
µ
1
(0) = ker dµ
p
= (T
p
O
p
)
ω
.
So
T
p
O
p
is an isotropic subspace of (
T
p
M, ω
). We then observe the following
straightforward linear algebraic result:
Lemma.
Let (
V,
Ω) be a symplectic vector space and
I
an isotropic sub-
space. Then induces a canonical symplectic structure
red
on
I
/I
, given by
red
([u], [v]) = Ω(u, v).
Applying this, we get a canonical symplectic structure on
(T
p
O
p
)
ω
T
p
O
p
=
T
p
µ
1
(0)
T
p
O
p
= T
[p]
M
red
.
This defines
ω
red
on
M
red
, which is well-defined because
ω
is
G
-invariant, and is
smooth by local triviality and canonicity.
It remains to show that dω
red
= 0. By construction, i
ω = π
ω
red
. So
π
(dω
red
) = dπ
ω
red
= di
ω = i
dω = 0
Since π
is injective, we are done.
Example. Take
(M, ω) =
C
n
, ω
0
=
i
2
X
dz
k
d¯z
k
=
X
dx
k
dy
k
=
X
r
k
dr
k
dθ
k
.
We let G = S
1
act by multiplication
e
it
· (z
1
, . . . , z
n
) = (e
it
z
1
, . . . , e
it
z
n
).
This action is Hamiltonian with moment map
µ : C
n
R
z 7→
|z|
2
2
+
1
2
.
The +
1
2
is useful to make the inverse image of 0 non-degenerate. To check this
is a moment map, we compute
d
1
2
|z|
2
= d
1
2
X
r
2
k
=
X
r
k
dr
k
.
On the other hand, if
X = a g
=
R,
then we have
X
#
= a
θ
1
+ ··· +
θ
n
.
So we have
ι
X
#
ω = a
X
r
k
dr
k
= dµ
X
.
It is also clear that µ is S
1
-invariant. We then have
µ
1
(0) = {z C
n
: |z|
2
= 1} = S
2n1
.
Then we have
M
red
= µ
1
(0)/S
1
= CP
n1
.
One can check that this is in fact the Fubini–Study form. So (
CP
n1
, ω
F S
) is
the symplectic quotient of (
C
n
, ω
0
) with respect to the diagonal
S
1
action and
the moment map µ.
Example. Fix k, Z relatively prime. Then S
1
acts on C
2
by
e
it
(z
1
, z
2
) = (e
ikt
z
1
, e
i`t
z
2
).
This action is Hamiltonian with moment map
µ : C
2
R
(z
1
, z
2
) 7→
1
2
(k|z
1
|
2
+ |z
2
|
2
).
There is no level set of µ where the action is free, since
(z, 0) has stabilizer Z/kZ
(0, z) has stabilizer Z/Z
(z
1
, z
2
) has trivial stabilizer if z
1
, z
2
6= 0.
On the other hand, the action is still locally free on
C
2
\ {
(0
,
0)
}
since the
stabilizers are discrete.
The reduced spaces
µ
1
(
ξ
)
/S
1
for
ξ 6
= 0 are orbifolds, called weighted or
twisted projective spaces.
The final example is an infinite dimensional one by Atiyah and Bott. We
will not be able to prove the result in full detail, or any detail at all, but we
will build up to the statement. The summary of the result is that performing
symplectic reduction on the space of all connections gives the moduli space of
flat connections.
Let
G P
π
B
be a principal
G
-bundle, and
ψ
:
G Diff
(
P
) the associated
free action. Let
dψ : g χ(P )
X 7→ X
#
be the associated infinitesimal action. Let
X
1
, . . . , X
k
be a basis of the Lie
algebra
g
. Then since
ψ
is a free action,
X
#
1
, . . . , X
#
k
are all linearly independent
at each p P .
Define the vertical tangent space
V
p
= span{(X
#
1
)
p
, . . . , (X
#
k
)
p
} T
p
P = ker(dπ
p
)
We can then put these together to get V T P , the vertical tangent bundle.
Definition
(Ehresmann Connection)
.
An (Ehresmann) connection on
P
is a
choice of subbundle H T P such that
(i) P = V H
(ii) H is G-invariant.
Such an H is called a horizontal bundle.
There is another way of describing a connection. A connection form on
P
is
a g-valued 1-form A
1
(P ) g such that
(i) ι
X
#
A = X for all X g
(ii) A is G-invariant for the action
g · (A
i
X
i
) = (ψ
g
1
)
A
i
Ad
g
X
i
.
Lemma.
Giving an Ehresmann connection is the same as giving a connection
1-form.
Proof. Given an Ehresmann connection H, we define
A
p
(v) = X,
where v = X
#
p
+ h
p
V H.
Conversely, given an A, we define
H
p
= ker A
p
= {v T
p
P : i
v
A
p
= 0}.
We next want to define the notion of curvature. We will be interested in flat
connections, i.e. connections with zero curvature.
To understand curvature, if we have a connection
T P
=
V H
, then we get
further decompositions
T
P = V
H
, Λ
2
(T
P ) = (
V
2
V
) (V
H
) (
V
2
H
).
So we end up having
1
(P ) = Ω
1
vert
(P )
1
hor
(P )
2
(P ) = Ω
2
vert
mixed
2
hor
If A =
P
k
i=1
A
i
X
i
1
g, then
dA
2
g.
We can then decompose this as
dA = (dA)
vert
+ (dA)
mix
+ (dA)
hor
.
The first part is uninteresting, in the sense that it is always given by
(dA)
vert
(X, Y ) = [X, Y ],
the second part always vanishes, and the last is the curvature form
curv A
2
hor
g.
Definition (Flat connection). A connection A is flat if curv A = 0.
We write
A
for the space of all connections on
P
. Observe that if
A
1
, A
0
A
,
then A
1
A
0
is G-invariant and
ι
X
#
(A
1
A
0
) = X X = 0
for all X g. So A
1
A
0
(Ω
1
hor
g)
G
. So
A = A
0
+ (Ω
1
hor
g)
G
,
and T
A
0
A = (Ω
1
hor
g)
G
.
Suppose
B
is a compact Riemann surface, and
G
a compact or semisimple
Lie group. Then there exists an Ad-invariant inner product on g. We can then
define
ω : (Ω
1
hor
g)
G
× (Ω
1
hor
g)
G
R,
sending
X
a
i
X
i
,
X
b
i
X
i
7→
Z
B
X
i,j
a
i
b
j
(X
i
, X
j
).
This is easily seen to be bilinear, anti-symmetric and non-degenerate. It is also
closed, if suitably interpreted, since it is effectively constant across the affine
space A. Thus, A is an “infinite-dimensional symplectic manifold”.
To perform symplectic reduction, let
G
be the gauge group, i.e. the group of
G
-equivariant diffeomorphisms
f
:
P P
covering the identity.
G
acts on
A
by
V H 7→ V H
f
,
where
H
f
is the image of
H
by d
f
. Atiyah and Bott proved that the action of
G
on (A, ω) is Hamiltonian with moment map
µ : A 7→ Lie(G)
= (Ω
2
hor
g)
G
A 7→ curv A
Performing symplectic reduction, we get
M = µ
1
(0)/G,
the moduli space of flat connections, which has a symplectic structure. It turns
out this is in fact a finite-dimensional symplectic orbifold.
3.6 The convexity theorem
We focus on the case
G
=
T
n
. It turns out the image of the moment map
µ
has
a very rigid structure.
Theorem
(Convexity theorem (Atiyah, Guillemin–Sternberg))
.
Let (
M, ω
) be
a compact connected symplectic manifold, and
µ : M R
n
a moment map for a Hamiltonian torus action. Then
(i) The levels µ
1
(c) are connected for all c
(ii) The image µ(M) is convex.
(iii) The image µ(M) is in fact the convex hull of µ(M
G
).
We call µ(M ) the moment polytope.
Here we identify
G
=
T
n
' R
n
/Z
n
, which gives us an identification
g
=
R
n
and g
=
(R
n
)
=
R
n
.
Example.
Consider (
M
=
CP
n
, ω
F S
).
T
n
acts by letting
t
= (
t
1
, . . . , t
n
)
T
n
=
U(1)
n
send
ψ
t
([z
0
: ···z
n
]) = [z
0
: t
1
z
1
: ··· : t
n
z
n
].
This has moment map
µ([z
0
: ···z
n
]) =
1
2
(|z
1
|
2
, . . . , |z
n
|
2
)
|z
0
|
2
+ ··· + |z
n
|
2
.
The image of this map is
µ(M) =
x R
n
: x
k
0, x
1
+ ··· + x
n
1
2
.
For example, in the CP
2
case, the moment image lives in R
2
, and is just
1
2
1
2
The three vertices are µ([0 : 0 : 1]), µ([0 : 1 : 0]) and µ([1 : 0 : 0]).
We now want to prove the convexity theorem. We first look at (iii). While it
seems like a very strong claim, it actually follows formally from (ii).
Lemma. (ii) implies (iii).
Proof.
Suppose the fixed point set of the action has
k
connected components
Z
=
Z
1
···Z
k
. Then
µ
is constant on each
Z
j
, since
X
#
|
Z
j
= 0 for all
X
. Let
µ
(
Z
j
) =
η
j
R
n
. By (ii), we know the convex hull of
{η
1
, . . . η
k
}
is contained in
µ(M).
To see that
µ
(
M
) is exactly the convex hull, observe that if
X R
n
has
rationally independent components, so that
X
topologically generates
T
, then
p
is fixed by
T
iff
X
#
p
= 0, iff d
µ
X
p
= 0. Thus,
µ
X
attains its maximum on one of
the Z
j
.
Now if
ξ
is not in the convex hull of
{η
j
}
, then we can pick an
X R
n
with
rationally independent components such that
hξ, Xi > hη
j
, Xi
for all
j
, since
the space of such X is open and non-empty. Then
hξ, Xi > sup
p
S
Z
j
hµ(p), Xi = sup
pM
hµ(p), Xi.
So ξ 6∈ µ(M).
With a bit more work, (i) also implies (ii).
Lemma. (i) implies (ii).
Proof.
The case
n
= 1 is immediate, since
µ
(
M
) is compact and connected,
hence a closed interval.
In general, to show that
µ
(
M
) is convex, we want to show that the intersection
of
µ
(
M
) with any line is connected. In other words, if
π
:
R
n+1
R
n
is any
projection and ν = π µ, then
π
1
(c) µ(M) = µ(ν
1
(c))
is connected. This would follow if we knew
ν
1
(
c
) were connected, which would
follow from (i) if
ν
were a moment map of an
T
n
action. Unfortunately, most of
the time, it is just the moment map of an
R
n
action. For it to come from a
T
n
action, we need π to be represented by an integer matrix. Then
T = {π
T
t : t T
n
= R
n
/Z
n
} T
n+1
is a subtorus, and one readily checks that
ν
is the moment map for the
T
action.
Now for any
p
0
, p
1
M
, we can find
p
0
0
, p
0
1
arbitrarily close to
p
0
, p
1
and
a line of the form
π
1
(
c
) with
π
integral. Then the line between
p
0
0
and
p
0
1
is contained in
µ
(
M
) by the above argument, and we are done since
µ
(
M
) is
compact, hence closed.
It thus remains to prove (i), where we have to put in some genuine work.
This requires Morse–Bott theory.
Let M be a manifold, dim M = m, and f : M R a smooth map. Let
Crit(f) = {p M : df
p
= 0}
be the set of critical points. For
p Crit
(
f
) and (
U, x
1
, . . . , x
m
) a coordinate
chart around p, we have a Hessian matrix
H
p
f =
2
f
x
i
x
j
Definition
(Morse(-Bott) function)
. f
is a Morse function if at each
p Crit
(
f
),
H
p
f is non-degenerate.
f
is a Morse–Bott function if the connected components of
Crit
(
f
) are
submanifolds and for all p Crit(p), T
p
Crit(f) = ker(H
p
f).
If
f
is Morse, then the critical points are isolated. If
f
is Morse–Bott, then
the Hessian is non-degenerate in the normal bundle to Crit(f ).
If
M
is compact, then there is a finite number of connected components of
Crit(f). So we have
Crit(f) = Z
1
··· Z
k
,
and the Z
i
are called the critical submanifolds.
For p Z
i
, the Hessian H
p
f is a quadratic form and we can write
T
p
M = E
p
T
p
Z
i
E
+
p
,
where
E
±
p
are the positive and negative eigenspaces of
H
p
f
respectively. Note
that
dim E
±
p
are locally constant, hence constant along
Z
i
. So we can define the
index of Z
i
to be dim E
p
, and the coindex to be dim E
+
p
.
We can then define a vector bundle
E
Z
i
, called the negative normal
bundle.
Morse theory tells us how the topology of
M
c
=
{p M
:
f
(
p
)
c}
changes
with c R.
Theorem (Morse theory).
(i)
If
f
1
([
c
1
, c
2
]) does not contain any critical point. Then
f
1
(
c
1
)
=
f
1
(
c
2
)
and M
c
1
=
M
c
2
(where
=
means diffeomorphic).
(ii)
If
f
1
([
c
1
, c
2
]) contains one critical manifold
Z
, then
M
c
2
' M
c
1
D
(
E
),
where D(E
) is the disk bundle of E
.
In particular, if
Z
is an isolated point,
M
c
2
is, up to homotopy equivalence,
obtained by adding a dim E
p
-cell to M
c
1
.
The key lemma in this proof is the following result:
Lemma.
Let
M
be a compact connected manifold, and
f
:
M R
a Morse–Bott
function with no critical submanifold of index or coindex 1. Then
(i) f has a unique local maximum and local minimum
(ii) All level sets of f
1
(c) are connected.
Proof sketch.
There is always a global minimum since
f
is compact. If there is
another local minimum at c, then the disk bundle is trivial, and so in
M
c+ε
' M
cε
D(E
)
for
ε
small enough, the union is a disjoint union. So
M
c+ε
has two components.
Different connected components can only merge by crossing a level of index 1,
so this cannot happen. To handle the maxima, consider f .
More generally, the same argument shows that a change in connectedness
must happen by passing through a index or coindex 1 critical submanifold.
To apply this to prove the convexity theorem, we will show that for any
X
,
µ
X
is a Morse–Bott function where all the critical submanifolds are symplectic.
In particular, they are of even index and coindex.
Lemma.
For any
X R
n
,
µ
X
is a Morse–Bott function where all critical
submanifolds are symplectic.
Proof sketch.
Note that
p
is a fixed point iff
X
#
p
= 0 iff d
µ
X
p
= 0 iff
p
is a critical
point. So the critical points are exactly the fixed points of X.
(
T
p
M, ω
p
) models (
M, ω
) in a neighbourhood of
p
by Darboux theorem. Near
a fixed point
T
n
, an equivariant version of the Darboux theorem tells us there is
a coordinate chart U where (M, ω, µ) looks like
ω|
U
=
X
dx
i
dy
i
µ|
U
= µ(p)
1
2
X
i
(x
2
i
+ y
2
i
)α
i
,
where α
i
Z are weights.
Then the critical submanifolds of µ are given by
{x
i
= y
i
= 0 : α
i
6= 0},
which is locally a symplectic manifold and has even index and coindex.
Finally, we can prove the theorem.
Lemma. (i) holds.
Proof.
The
n
= 1 case follows from the previous lemmas. We then induct on
n
.
Suppose the theorem holds for
n
, and let
µ
= (
µ
1
, . . . , µ
n
) :
M R
n+1
be
a moment map for a Hamiltonian
T
n+1
-action. We want to show that for all
c = (c
1
, . . . , c
n
) R
n+1
, the set
µ
1
(c) = µ
1
1
(c
1
) ··· µ
1
n+1
(c
n+1
)
is connected.
The idea is to set
N = µ
1
1
(c
1
) ···µ
1
1
(c
n
),
and then show that
µ
n+1
|
N
:
N R
is a Morse–Bott function with no critical
submanifolds of index or coindex 1.
We may assume that d
µ
1
, . . . ,
d
µ
n
are linearly independent, or equivalently,
dµ
X
6= 0 for all X R
n
. Otherwise, this reduces to the case of an n-torus.
To make sense of
N
, we must pick
c
to be a regular value. Density arguments
imply that
C =
[
X6=0
Crit(µ
X
) =
[
XZ
n+1
\{0}
Crit µ
X
.
Since
Crit µ
X
is a union of codimension
2 submanifolds, its complement is
dense. Hence by the Baire category theorem,
C
has dense complement. Then a
continuity argument shows that we only have to consider the case when
c
is a
regular value of µ, hence N is a genuine submanifold of codimension n.
By the induction hypothesis,
N
is connected. We now show that
µ
n+1
|
N
:
N R is Morse–Bott with no critical submanifolds of index or coindex 1.
Let
x
be a critical point. Then the theory of Lagrange multipliers tells us
there are some λ
i
R such that
"
dµ
n+1
+
n
X
n=1
λ
i
dµ
i
#
x
= 0
Thus, µ is critical in M for the function
µ
Y
= µ
n+1
+
n
X
i=1
λ
i
µ
i
,
where
Y
= (
λ
1
, . . . , λ
n
,
1)
R
n+1
. So by the claim,
µ
Y
is Morse–Bott with only
even indices and coindices. Let
W
be a critical submanifold of
µ
Y
containing
x
.
Claim. W intersects N transversely at x.
If this were true, then
µ
Y
|
N
has
W N
as a non-degenerate critical sub-
manifold of even index and coindex, since the coindex doesn’t change and
W
is even-dimensional. Moreover, when restricted to
N
,
P
λ
i
µ
i
is a constant. So
µ
n+1
|
N
satisfies the same properties.
To prove the claim, note that
T
x
N = ker dµ
1
|
x
··· ker dµ
n
|
x
.
With a moments thought, we see that it suffices to show that d
µ
1
, . . . ,
d
µ
n
remain linearly independent when restricted to
T
x
W
. Now observe that the
Hamiltonian vector fields
X
#
1
|
x
, . . . , X
#
n
|
x
are independent since d
µ
1
|
x
, . . .
d
µ
n
|
x
are, and they live in T
x
W since their flows preserve W .
Since
W
is symplectic (by the claim), for all
k
= (
k
1
, . . . k
n
), there exists
v T
x
W such that
ω
X
k
i
X
#
i
|
x
, v
6= 0.
In other words,
X
k
i
dµ
i
(v) 6= 0.
It is natural to seek a non-abelian generalization of this, and it indeed
exists. Let (
M, ω
) be a symplectic manifold, and
G
a compact Lie group with
a Hamiltonian action on
M
with moment map
µ
:
M g
. From Lie group
theory, there is a maximal torus
T G
with Lie algebra
t
, and the Weyl group
W = N(T )/T is finite (where N(T ) is the normalizer of T ).
Then under the coadjoint action, we have
g
/G
=
t
/W,
Pick a Weyl chamber t
+
of t
, i.e. a fundamental domain of t
under W . Then
µ
induces a moment map
µ
+
:
M t
+
, and the non-abelian convexity theorem
says
Theorem (Kirwan, 1984). µ
+
(M) t
+
is a convex polytope.
We shall end with an application of the convexity theorem to linear algebra.
Let
λ
= (
λ
1
, λ
2
)
R
2
and
λ
1
λ
2
, and
H
2
λ
the set of all 2
×
2 Hermitian
matrices with eigenvalues (
λ
1
, λ
2
). What can the diagonal entries of
A H
2
λ
be?
We can definitely solve this by brute force, since any entry in H looks like
A =
a z
¯z b
where a, b R and z C. We know
tr a = a + b = λ
1
+ λ
2
det a = ab |z|
2
= λ
1
λ
2
.
The first implies
b
=
λ
1
+
λ
2
a
, and all the second condition gives is that
ab > λ
1
λ
2
.
a
b
This completely determines the geometry of
H
2
λ
, since for each allowed value
of
a
, there is a unique value of
b
, which in turn determines the norm of
z
.
Topologically, this is a sphere, since there is a
S
1
’s worth of choices of
z
except
at the two end points ab = λ
1
λ
2
.
What has this got to do with Hamiltonian actions? Observe that U(2) acts
transitively on H
2
λ
by conjugation, and
T
2
=

e
1
0
0 e
2

U(2).
This contains a copy of S
1
given by
S
1
=

e
1
0
0 e
1

T
2
.
We can check that
e
0
0 e
a z
¯z b
e
0
0 e
1
=
a e
z
e
z b
Thus, if ϕ : H
2
λ
R
2
is the map that selects the diagonal elements, then
ϕ
1
(a, b) =
a |z|e
|z|e
b
is one
S
1
-orbit. This is reminiscent of the
S
1
action of
S
2
quotienting out to a
line segment.
We can think more generally about H
n
λ
, the n × n Hermitian matrices with
eigenvalues λ
1
··· λ
n
, and ask what the diagonal elements can be.
Take
ϕ
:
H
n
λ
R
n
be the map that selects the diagonal entries. Then the
image lies on the plane
tr A
=
P
λ
i
. This certainly contain the
n
! points whose
coordinates are all possible permutation of the
λ
i
, given by the diagonal matrices.
Theorem
(Schur–Horn theorem)
. ϕ
(
H
n
λ
) is the convex hull of the
n
! points
from the diagonal matrices.
To view this from a symplectic perspective, let
M
=
H
n
λ
, and U(
n
) acts
transitively by conjugation. For A M, let G
A
be the stabilizer of A. Then
H
n
λ
= U(n)/G
A
.
Thus,
T
A
H
n
λ
=
iH
n
g
A
where
H
n
is the Hermitian matrices. The point of this is to define a symplectic
form. We define
ω
A
: iH
n
× iH
n
R
(X, Y ) 7→ i tr([X, Y ]A) = i tr(X(Y A AY ))
So
ker ω
A
= {Y : [A, Y ] = 0} = g
A
.
So
ω
A
induces a non-degenerate form on
T
A
H
n
λ
. In fact, this gives a symplectic
form on H
n
λ
.
Let
T
n
U(
n
) be the maximal torus consisting of diagonal entries. We can
show that the
T
n
action is Hamiltonian with moment map
ϕ
. Since
T
n
fixes
exactly the diagonal matrices, we are done.
3.7 Toric manifolds
What the convexity theorem tells us is that if we have a manifold
M
with a
torus action, then the image of the moment map is a convex polytope. How
much information is retained by a polytope?
Of course, if we take a torus that acts trivially on
M
, then no information is
retained.
Definition
(Effective action)
.
An action
G
on
M
is effective (or faithful) if
every non-identity g G moves at least one point of M.
But we can still take the trivial torus
T
0
that acts trivially, and it will still
be effective. Of course, no information is retained in the polytope as well. Thus,
we want to have as large of a torus action as we can. The following proposition
puts a bound on “how much” torus action we can have:
Proposition.
Let (
M, ω
) be a compact, connected symplectic manifold with
moment map
µ
:
M R
n
for a Hamiltonian
T
n
action. If the
T
n
action is
effective, then
(i) There are at least n + 1 fixed points.
(ii) dim M 2n.
We first state without proof a result that is just about smooth actions.
Fact. An effective action of T
n
has orbits of dimension n.
This doesn’t mean all orbits are of dimension
n
. It just means some orbit
has dimension n.
Proof.
(i)
If
µ
= (
µ
1
, . . . , µ
n
) :
M R
n
and
p
is a point in an
n
-dimensional orbit,
then
{
(d
µ
i
)
p
}
are linearly independent. So
µ
(
p
) is an interior point (if
p
is
not in the interior, then there exists a direction
X
pointing out of
µ
(
M
).
So (d
µ
X
)
p
= 0, and d
µ
X
gives a non-trivial linear combination of the d
µ
i
’s
that vanishes).
So if there is an interior point, we know
µ
(
M
) is a non-degenerate polytope
in
R
n
. This mean it has at least
n
+ 1 vertices. So there are at least
n
+ 1
fixed points.
(ii)
Let
O
be an orbit of
p
in
M
. Then
µ
is constant on
O
by invariance of
µ
.
So
T
p
O ker(dµ
p
) = (T
p
O)
ω
.
So all orbits of a Hamiltonian torus action are isotropic submanifolds. So
dim O
1
2
dim M. So we are done.
In the “optimal” case, we have dim M = 2n.
Definition
((Symplectic) toric manifold)
.
A (symplectic) toric manifold is a
compact connected symplectic manifold (
M
2n
, ω
) equipped with an effective
T
n
action of an n-torus together with a choice of corresponding moment map µ.
Example. Take (CP
n
, ω
F S
), where the moment map is given by
µ([z
0
: z
1
: ··· : z
n
]) =
1
2
(|z
1
|
2
, . . . , |z
n
|
2
)
|z
0
|
2
+ |z
1
|
2
+ ··· + |z
n
|
2
.
Then this is a symplectic toric manifold.
Note that if (
M, ω, T µ
) is a toric manifold and
µ
= (
µ
0
, . . . , µ
n
) :
M R
n
,
then µ
1
, . . . , µ
n
are commuting integrals of motion
{µ
i
, µ
j
} = ω(X
#
i
, X
#
j
) = 0.
So we get an integrable system.
The punch line of the section is that there is a correspondence between toric
manifolds and polytopes of a suitable kind. First, we need a suitable notion of
equivalence of toric manifolds.
Definition
(Equivalent toric manifolds)
.
Fix a torus
T
=
R
2n
/
(2
πZ
)
n
, and
fix an identification
t
=
t
=
R
n
. Given two toric manifolds (
M
i
, ω
i
, T, µ
i
) for
i = 1, 2, We say they are
(i)
equivalent if there exists a symplectomorphism
ϕ
:
M
1
M
2
such that
ϕ(x · p) = x · ϕ(p) and µ
2
ϕ = µ
1
.
(ii)
weakly equivalent if there exists an automorphism
λ
:
T T
and
ϕ
:
M
1
M
2
symplectomorphism such that ϕ(x, p) = λ(x) · ϕ(p).
We also need a notion of equivalence of polytopes. Recall that
Aut
(
T
) =
GL(n, Z), and we can define
Definition.
AGL(n, Z) = {x 7→ Bx + c : B GL(n, Z), c R
n
}.
Finally, not all polytopes can arise from the image of a moment map. It is
not hard to see that the following are some necessary properties:
Definition
(Delzant polytope)
.
A Delzant polytope in
R
n
is a compact convex
polytope satisfying
(i) Simplicity: There exists exactly n edges out meeting at each vertex.
(ii)
Rationality: The edges meeting at each vertex
P
are of the form
P
+
tu
i
for t 0 and u
i
Z
n
.
(iii)
Smoothness: For each vertex, the corresponding
u
i
’s can be chosen to be a
Z-basis of Z
n
.
Observe that all polytopes arising as µ(M) satisfy these properties.
We can equivalently define rationality and smoothness as being the exact
same conditions on the outward-pointing normals to the facets (co-dimension 1
faces) meeting at P .
Example.
In
R
, there is any Delzant polytope is a line segment. This corre-
sponds to the toric manifold
S
2
=
CP
1
as before, and the length of the polytope
corresponds to the volume of CP
1
under ω.
Example. In R
2
, this is Delzant polytope:
On the other hand, this doesn’t work:
since on the bottom right vertex, we have
det
1 1
0 2
= 2 6= ±1.
To fix this, we can do
Of course, we can also do boring things like rectangles.
There is in fact a classification of all Delzant polytopes in
R
2
, but we shall
not discuss this.
Example.
The rectangular pyramid in
R
3
is not Delzant because it is not simple.
The tetrahedron is.
Theorem (Delzant). There are correspondences
symplectic toric manifolds
up to equivalence
Delzant polytopes
symplectic toric manifolds
up to weak equivalence
Delzant polytopes
modulo AGL(n, Z)
Proof sketch.
Given a Delzant polytope in (
R
n
)
with
d
facets, we want to
construct (
M
, ω
, T
, µ
) with
µ
(
M
) = ∆. The idea is to perform the
construction for the “universal” Delzant polytope with
d
facets, and then obtain
the desired
M
as a symplectic reduction of this universal example. As usual,
the universal example will be “too big” to be a genuine symplectic toric manifold.
Instead, it will be non-compact.
If has
d
facets with primitive outward-point normal vectors
v
1
, . . . , v
d
(i.e.
they cannot be written as a
Z
-multiple of some other
Z
-vector), then we can
write as
∆ = {x (R
n
)
: hx, v
i
i λ
i
for i = 1, . . . , d}
for some λ
i
.
There is a natural (surjective) map
π
:
R
d
R
n
that sends the basis vector
e
d
of R
d
to v
d
. If λ = (λ
1
, . . . , λ
d
), and we have a pullback diagram
R
d
λ
(R
n
)
(R
d
)
π
where
R
d
λ
= {X (R
d
)
: hX, e
i
i λ
i
for all i}.
In more down-to-earth language, this says
π
(x) R
d
λ
x ,
which is evident from definition.
Now there is a universal “toric manifold” with
µ
(
M
) =
R
d
λ
, namely (
C
d
, ω
0
)
with the diagonal action
(t
1
, . . . , t
d
) · (z
1
, . . . , z
d
) = (e
it
1
z
1
, . . . , e
it
n
z
n
),
using the moment map
φ(z
1
, . . . , z
d
) =
1
2
(|z
1
|
2
, . . . , |z
d
|
2
) + (λ
1
, . . . , λ
d
).
We now want to pull this back along
π
. To this extent, note that
π
sends
Z
d
to
Z
n
, hence induces a map
T
d
T
n
with kernel
N
. If
n
is the Lie algebra
of N , then we have a short exact sequence
0 (R
n
)
π
(R
d
)
i
n
0.
Since im π
= ker i
, the pullback of C
d
along π
is exactly
Z = (i
φ)
1
(0).
It is easy to see that this is compact.
Observe that
i
φ
is exactly the the moment map of the induced action
by
N
. So
Z/N
is the symplectic reduction of
C
d
by
N
, and in particular has a
natural symplectic structure. It is natural to consider
Z/N
instead of
Z
itself,
since
Z
carries a
T
d
action, but we only want to be left with a
T
n
action. Thus,
after quotienting out by
N
, the
T
d
action becomes a
T
d
/N
=
T
n
action, with
moment map given by the unique factoring of
Z C
d
(R
d
)
through (R
n
)
. The image is exactly ∆.
4 Symplectic embeddings
We end with a tiny chapter on symplectic embeddings, as promised in the course
description.
Definition
(Symplectic embedding)
.
A symplectic embedding is an embedding
ϕ
:
M
1
M
2
such that
ϕ
ω
2
=
ω
1
. The notation we use is (
M
1
, ω
1
)
s
(
M
2
, ω
2
).
A natural question to ask is, if we have two symplectic manifolds, is there a
symplectic embedding between them?
For concreteness, take (
C
n
, ω
0
)
=
(
R
2n
, ω
0
), and consider the subsets
B
2n
(
r
)
and
Z
2n
(
R
) =
B
2
(
R
)
×R
2n2
(where the product is one of symplectic manifolds).
If
r R
, then there is a natural inclusion of
B
2n
(
r
) into
Z
2n
(
R
)? If we only
ask for volume-preserving embeddings, then we can always embed
B
2n
(
r
) into
Z
2n
(
R
), since
Z
2n
(
R
) has infinite volume. It turns out, if we require the
embedding to be symplectic, we have
Theorem
(Non-squeezing theorem, Gromov, 1985)
.
There is an embedding
B
2
(n) Z
2n
(R) iff r < R.
When studying symplectic embeddings, it is natural to consider the following:
Definition
(Symplectic capacity)
.
A symplectic capacity is a function
c
from
the set of 2n-dimensional manifolds to [0, ] such that
(i) Monotonicity: if (M
1
, ω
1
) (M
2
, ω
2
), then c(M
1
, ω
1
) c(M
2
, ω
2
).
(ii) Conformality: c(M, λω) = λc(M, ω).
(iii) Non-triviality: c(B
2n
(1), ω
0
) > 0 and c(Z
2n
(1), ω
0
) < .
If we only have (i) and (ii), this is called a generalized capacity.
Note that the volume is a generalized capacity, but not a symplectic capacity.
Proposition.
The existence of a symplectic capacity is equivalent to Gromov’s
non-squeezing theorem.
Proof.
The
direction is clear by monotonicity and conformality. Conversely,
if we know Gromov’s non-squeezing theorem, we can define the Gromov width
W
G
(M, ω) = sup{πr
2
| (B
2n
(r), ω
0
) (M, ω)}.
This clearly satisfies (i) and (ii), and (iii) follows from Gromov non-squeezing.
Note that Darboux’s theorem says there is always an embedding of
B
2n
(
r
) into
any symplectic manifold as long as r is small enough.