III Symplectic Geometry (Full)

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 K¨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

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 Ω 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

be a real vector space and Ω a skew-

symmetric bilinear form. Then there is a basis

, . . . , u

, e

, . . . , e

, f

, . . . , f

}

of V such that

(i) Ω(u

, v) = 0 for all v ∈ V

(ii) Ω(e

, e

) = Ω(f

, f

) = 0.

(iii) Ω(e

, f

) = δ

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

, . . . , u

of this. Choose any W complementary to U .

First pick

∈ W \{

}

arbitrarily. Since

6∈ U

, we can pick

such that

Ω(e

, f

) = 1. Then define W

= span{e

, f

}, and

Ω

= {w ∈ W : Ω(w, v) = 0 for all v ∈ W

It is clear that

∩ W

Ω

{

}

. Moreover,

⊕ W

Ω

. Indeed, if

v ∈ W

then

v = (Ω(v, f

− Ω(v, e

) + (v − (Ω(v, f

− Ω(v, e

)),

Then we are done by induction on the dimension.

Here

dim U

and 2

dim V − k

are invariants of (

Ω). The number

2n is called the rank of Ω. Non-degeneracy is equivalent to k = 0.

Exercise. Ω is non-degenerate iff Ω

∧n

= Ω ∧ ··· ∧ Ω 6= 0.

By definition, every symplectic vector space is canonically isomorphic to

its dual, given by the map

Ω

. As in the above theorem, a symplectic basis

, . . . , e

, f

, . . . , f

} of V is a basis such that

Ω(e

, e

) = Ω(f

, f

) = 0, Ω(e

, f

) = δ

Every symplectic vector space has a symplectic basis, and in particular has even

dimension. In such a basis, the matrix representing Ω is

Ω



0 I

−I 0



We will need the following definitions:

Definition

(Symplectic subspace)

If (

Ω) is a symplectic vector space, a

symplectic subspace is a subspace W ⊆ V such that Ω|

W ×W

is non-degenerate.

Definition

(Isotropic subspace)

If (

Ω) is a symplectic vector space, an

isotropic subspace is a subspace W ⊆ V such that Ω|

W ×W

= 0.

Definition

(Lagrangian subspace)

If (

Ω) is a symplectic vector space, an

Lagrangian subspace is an isotropic subspace W with dim W =

dim V .

Definition

(Symplectomorphism)

A symplectomorphism between symplectic

vector spaces (

Ω)

(

Ω

) is an isomorphism

V → V

such that

∗

Ω

= Ω.

The standard form theorem says any two 2

-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

of dimension 2

equipped with a 2-form

that is closed (i.e. d

= 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

∧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,

∧n

= 0

∈ H

(

). Since the wedge is

well-defined on cohomology, an immediate consequence of this is that

Proposition.

If a compact manifold

is such that

(

) = 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

= d

for some

α, then

∧ ω

2(n−k)

d(α ∧ ω

2(n−k)

) = 0

by Stokes’ theorem.

Example. S

does not admit a symplectic structure unless

= 2 (or

= 0, if

one insists).

On the other hand, S

is a symplectic manifold.

Example.

Take

. Take the standard embedding in

, and for

p ∈ S

the tangent space consists of the vectors perpendicular to

. We can take the

symplectic form to be

(u, v) = p ·(u × v).

Anti-symmetry and non-degeneracy is IA Vectors and Matrices.

Definition

(Symplectomorphism)

Let (

, ω

) and (

, ω

) be symplectic

manifolds. A symplectomorphism is a diffeomorphism

→ X

such that

∗

= ω

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

are strongly

isotopic if there is an isotopy taking one to the other.

Definition

(Deformation equivalent)

Two symplectic structures

, ω

are deformation equivalent if there is a family of symplectic forms

that start

and end at ω

and ω

respectively.

Definition

(Isotopic)

Two symplectic structures

, ω

are isotopic

if there is a family of symplectic forms

that start and end at

and

respectively, and further the cohomology class [ω

] 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)

is compact with a family

of symplectic forms with

[ω

] constant, then there is an isotopy ρ

: M → M with ρ

∗

= ω

The key idea is to express the condition

∗

as a differential equation

for

, and ODE theory guarantees the existence of a

satisfying the

equation. Then compactness allows us to integrate it up to get a genuine ρ

Proof.

We set

to be the identity. Then the equation

∗

is equivalent

to ρ

∗

being constant. If v

is the associated vector field to ρ

, then we need

0 =

(ρ

∗

) = ρ

∗



dω



So we want to solve

dω

= 0.

To solve for this, since [

dω

] = 0, it follows that there is a family

of 1-forms

such that

dω

= dµ

. Then our equation becomes

+ dµ

= 0.

By assumption, dω

= 0. So by Cartan’s magic formula, we get

dι

+ dµ

= 0.

To solve this equation, it suffices to solve Moser’s equation,

+ µ

= 0,

which can be solved since ω

is non-degenerate.

There is also a relative version of this.

Theorem

(Relative Moser)

Let

X ⊆ M

be a compact manifold of a manifold

and

, ω

symplectic forms on

agreeing on

. Then there are neighbourhoods

, U

of X and a diffeomorphism ϕ : U

→ U

fixing X such that ϕ

∗

= ω

Proof. We set

= (1 − t)ω

+ tω

Then this is certainly closed, and since this is constantly

, it is non-

degenerate on

, hence, by compactness, non-degenerate on a small tubular

neighbourhood U

of X (by compactness). Now

= ω

− ω

and we know this vanishes on

. Since the inclusion of

into a tubular

neighbourhood is a homotopy equivalence, we know [

− ω

] = 0

∈ H

(

Thus, we can find some

such that

− ω

= d

, and by translation by a

constant, we may suppose

vanishes on

. 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

, . . . , x

, y

, . . . , y

) about p on which

ω =

∧ dy

Proof. ω

can certainly be written in this form at

. 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

. In physics,

is called the configuration space, and

∗

called the phase space.

On a coordinate chart (

U, x

, . . . , x

) for

, a generic 1-form can be written

ξ =

i=1

Thus, we have a coordinate chart on

∗

given by (

∗

U, x

, . . . , x

, ξ

, . . . , ξ

On T

∗

U, there is the tautological 1-form

α =

i=1

Observe that this is independent of the chart chosen, since it can be characterized

in the following coordinate-independent way:

Proposition.

Let

∗

X → X

be the projection map, and

∗

X →

∗

M the pullback map. Then for ξ ∈ M, we have α

= π

∗

ξ.

Proof. On a chart, we have

∗



∂

∂x



= ξ



∂

∂x



= ξ

= α



∂

∂x



and similarly both vanish on

∂

∂ξ

This tautological 1-form is constructed so that the following holds:

Proposition.

Let

be a one-form of

, i.e. a section

X → T

∗

. Then

∗

α = µ.

Proof. By definition,

= ξ ◦ dπ.

So for x ∈ X, we have

∗

µ(x)

= µ(x) ◦ dπ ◦ ds

= µ(x) ◦ d(π ◦ s

) = µ(x) ◦ d(id

) = µ(x).

Given this, we can set

ω = −dα =

i=1

∧ dξ

the canonical symplectic form on

∗

. It is clear that it is anti-symmetric,

closed, and non-degenerate, because it looks just like the canonical symplectic

form on R

Example.

Take

, parametrized by

, and

∗

×R

with coordinates

(θ, ξ

). Then

ω = dθ ∧ dξ

We can see explicitly how this is independent of coordinate. Suppose we

parametrized the circle by

= 2

. Then d

= 2d

. However, since we de-

fined ξ

ξ = ξ

(ξ) dθ

for all ξ ∈ T

∗

, we know that ξ

, and hence

dθ ∧ dξ

= dτ ∧ dξ

Of course, if we have a diffeomorphism

X → Y

, then the pullback map

induces a symplectomorphism

∗

Y → T

∗

. However, not all symplecto-

morphisms arise this way.

Example.

and

∗

× R

is given the canonical symplectic

structure, then the vertical translation

(

θ, ξ

) = (

θ, ξ

) is a symplectomorphism

that is not a lift of a diffeomorphism S

→ S

So when does a symplectomorphism come from a diffeomorphism?

Exercise.

A symplectomorphism

∗

X → T

∗

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

is a Lagrangian submanifold if for all

p ∈ L

is a Lagrangian subspace of

. Equivalently, the restriction of

vanishes and dim L =

dim M.

Example.

If (

M, ω

) is a surface with area form

, and

is any 1-dimensional

submanifold, then

is Lagrangian, since any 1-dimensional subspace of

Lagrangian.

What do Lagrangian submanifolds of

∗

look like? Let

be a 1-form on

X, i.e. a section of T

∗

X, and let

= {(x, µ

) : x ∈ X} ⊆ T

∗

The map

X → T

∗

is an embedding, and

dim X

dim T

∗

So this is a good candidate for a Lagrangian submanifold. By definition, X

Lagrangian iff s

∗

ω = 0. But

∗

ω = s

∗

(dα) = d(s

∗

α) = dµ.

is Lagrangian iff

is closed. In particular, if

h ∈ C

∞

(

), then

a Lagrangian submanifold of

∗

. We call

the generating function of the

Lagrangian submanifold.

There are other examples of Lagrangian submanifolds of the cotangent bundle.

Let

be a submanifold of

, and

x ∈ X

. We define the conormal space of

x to be

∗

S = {ξ ∈ T

∗

X : ξ|

= 0}.

The conormal bundle of S is the union of all N

∗

Example. If S = {x}, then N

∗

S = T

∗

Example. If S = X, then N

∗

S = S is the zero section.

Note that both of these are

-dimensional submanifolds of

∗

, 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

, this is certainly a Lagrangian submanifold, and so is

S = {x}. Indeed,

Proposition.

Let

∗

and

∗

. Then

L → M

is a Lagrangian

submanifold.

Proof.

-dimensional, then

S → X

locally looks like

→ R

, and it is

clear in this case.

Our objective is to use Lagrangian submanifolds to understand the following

problem:

Problem.

Let (

, ω

) and (

, ω

) be 2

-dimensional symplectic manifolds.

If f : M

→ M

is a diffeomorphism, when is it a symplectomorphism?

Consider the 4

-dimensional manifold

× M

, with the product form

ω = pr

∗

+ pr

∗

. We have

dω = pr

∗

dω

+ pr

∗

dω

= 0,

and ω is non-degenerate since





∗

∧n

∧ pr

∗

∧n

6= 0.

is a symplectic form on

× M

. In fact, for

, λ

∈ R \ {

}

, the

combination

∗

+ λ

∗

is also symplectic. The case we are particularly interested in is

= 1 and

= −1, where we get the twisted product form.

˜ω = pr

∗

− pr

∗

Let T

be the graph of f, namely

= {(p

, f(p

)) : p

∈ M

} ⊆ M

× M

and let γ

: M

→ M

× M

be the natural embedding with image T

Proposition. f

is a symplectomorphism iff

is a Lagrangian submanifold of

× M

, ˜ω).

Proof.

The first condition is

∗

, and the second condition is

∗

˜ω

= 0,

and

∗

˜ω = γ

∗

= γ

∗

= ω

− f

∗

We are particularly interested in the case where

are cotangent bundles.

, X

are

-dimensional manifolds and

∗

, then we can naturally

identify

∗

× X

)

∼

∗

× T

∗

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

× M

→ M

× M

that fixes M

and sends (x, ξ) 7→ (x, −ξ) on M

. Then σ

∗

ω = ˜ω.

Then, to find a symplectomorphism

→ M

, we can carry out the following

procedure:

(i) Start with L a Lagrangian submanifold of (M

× M

, ω).

(ii) Twist it to get L

= σ(L).

(iii) Check if L

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

, ω

). We pick a generating function

f ∈ C

∞

(

× X

). Then d

is a

closed form on X

× X

, and

L = X

= {(x, y, ∂

(x,y)

, ∂

(x,y)

) : (x, y) ∈ X

× X

}

is a Lagrangian submanifold of (M

× M

, ω).

The twist L

is then

= {(x, y, d

(x,y)

, −d

(x,y)

) : (x, y) ∈ X

× X

}

We then have to check if

is the graph of a diffeomorphism

→ M

. If so,

we win, and we call this the symplectomorphism generated by

. Equivalently,

we say f is the generating function for ϕ.

To check if L

is the graph of a diffeomorphism, for each (x, ξ) ∈ T

∗

, we

need to find (y, η) ∈ T

∗

such that

ξ = ∂

(x,y)

, η = −∂

(x,y)

In local coordinates {x

, ξ

} for T

∗

and {y

, η

} for T

∗

, we want to solve

∂f

∂x

(x, y)

= −

∂f

∂y

(x, y)

The only equation that requires solving is the first equation for

, and then the

second equation tells us what we should pick η

to be.

For the first equation to have a unique smoothly-varying solution for

locally, we must require

det



∂

∂y



∂f

∂x



i,j

6= 0.

This is a necessary condition for

to generate a symplectomorphism

. If this

is satisfied, then we should think a bit harder to solve the global problem of

whether ξ

is uniquely defined.

We can in fact do this quite explicitly:

Example. Let X

= X

= R

, and f : R

× R

→ R given by

(x, y) 7→ −

|x − y|

= −

i=1

− y

)

What, if any, symplectomorphism is generated by f? We have to solve

∂f

∂x

= y

− x

= −

∂f

∂y

= y

− x

So we find that

= x

+ ξ

= ξ

So we have

ϕ(x, ξ) = (x + ξ, ξ).

If we use the Euclidean inner product to identify

∗

∼

, and view

as a

vector, then ϕ is a free translation in R

That seems boring, but in fact this generalizes quite well.

Example.

Let

be a connected manifold, and

a Riemannian

metric on

. We then have the Riemannian distance

X × X → R

, and we

can define

f : X × X → R

(x, y) 7→ −

d(x, y)

Using the Riemannian metric to identify

∗

∼

T X

, we can check that

T X → T X is given by

ϕ(x, v) = (γ(1), ˙γ(1)),

where

is the geodesic with

(0) =

and

˙γ

(0) =

. Thus,

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

M → M

a symplectomorphism. An

-periodic point of

is an

x ∈ M

such that

(x) = x. A periodic point is an n-periodic point for some n.

We are particularly interested in the case where

∗

with the canonical

symplectic form on

, and

is generated by a function

X → R

. We begin

with 1-periodic points, namely fixed points of f.

If ϕ(x

, ξ

) = (x

, ξ

), this means we have

= ∂

)

= −∂

)

In other words, we need

(∂

+ ∂

)

= 0.

Thus, if we define

ψ : X → R

x 7→ f(x, x),

then we see that

Proposition.

The fixed point of

are in one-to-one correspondence with the

critical points of ψ.

It is then natural to consider the same question for

. One way to think

about this is to consider the symplectomorphism ˜ϕ : M

→ M

given by

˜ϕ(m

, m

, . . . , m

) = (ϕ(m

), ϕ(m

), . . . , ϕ(m

n−1

)).

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

, . . . , x

), (y

, . . . , y

)) = f(x

, y

) + f(x

, y

) + ··· + f(x

, y

Thus, we deduce that

Proposition.

The

-periodic points of

are in one-to-one correspondence with

the critical points of

, . . . , x

) = f(x

, x

) + f(x

, x

) + ··· + f(x

, x

Example.

We consider the problem of periodic billiard balls. Suppose we have

a bounded, convex region Y ⊆ R

We put a billiard ball in

and set it in motion. We want to see if it has periodic

orbits.

Two subsequent collisions tend to look like this:

We can parametrize this collision as (

x, v

cos θ

)

∈ ∂Y ×

(

−

1). We can think of

this as the unit disk bundle of cotangent bundle on

∂Y

∼

with canonical

symplectic form d

x ∧

, using a fixed parametrization

∼

R/Z → X

. If

(

x, v

) = (

y, w

), then

is the projection of the unit vector pointing from

onto the tangent line at x. Thus,

v =

χ(y) − χ(x)

kχ(y) − χ(x)k

dχ



s=x

∂

∂x

(−kχ(x) − χ(y)k),

and a similar formula holds for

, 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

-periodic points are given by the critical points

, . . . , x

) 7→ −|χ(x

) − χ(x

)| − |χ(x

) − χ(x

)| − ··· − |χ(x

) − χ(x

)|.

Up to a sign, this is the length of the “generalized polygon” with vertices

, . . . , x

In general, if

is a compact manifold, and

∗

X → T

∗

is a symplecto-

morphism generated by a function

, then the number of fixed points of

are

the number of fixed points of

(

) =

(

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

A →

be an area-preserving diffeomorphism such that

preserves the boundary

components, and twists them in opposite directions. Then

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,

M → M

Then the graph of

is a subset of

M ×M

. Again let

˜ω

be the twisted product form

M ×M

. Then we saw that a morphism

M → M

is a symplectomorphism

iff the graph of

is Lagrangian in (

M ×M, ˜ω

). Moreover, the set of fixed points

is exactly ∆ ∩ graph ϕ, where ∆ = graph id

is the diagonal.

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,

a compact Lagrangian submanifold, and

the canonical symplectic

form on

∗

. Then there exists neighbourhoods

∗

and

in M and a symplectomorphism ϕ : U

→ U sending X to X.

An equivalent theorem is the following:

Theorem

(Weinstein)

Let

be a 2

-dimensional manifold,

X n

-dimensional

compact submanifold, and

X → M

the inclusion, and symplectic forms

, ω

such that

∗

= 0, i.e.

is Lagrangian with respect to both

symplectic structures. Then there exists neighbourhoods

, U

such

that ρ|

= id

and ρ

∗

= ω

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 (

Ω) is a symplectic vector space,

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

∼

∗

. Taking

and L = T

X, we get an isomorphism

NX = T M |

/T X

∼

∗

Thus, by the standard tubular neighbourhood theorem, there is a neighbourhood

and a neighbourhood

, and a diffeomorphism

→ N

. We now have two symplectic forms on

— the one from the

cotangent bundle and the pullback of that from

. 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

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

and

, and

any complement of

. By assumption,

is a Lagrangian subspace of both

(

V, ω

= Ω

) and (

V, ω

= Ω

). We apply the following linear-algebraic

lemma:

Lemma.

Let

be a 2

-dimensional vector space, Ω

Ω

symplectic structures

. Suppose

is a subspace of

Lagrangian with respect to both Ω

and

Ω

, and

is any complement of

. Then we can construct canonically a linear

isomorphism H : V → V such that H|

= id

and H

∗

Ω

= Ω

Note that the statement of the theorem doesn’t mention

, but the con-

struction of

requires a complement of

, so it is canonical only after we pick

a W .

By this lemma, we get canonically an isomorphism

M → T

such

that

and

∗

. The canonicity implies

varies

smoothly with p. We now apply the Whitney extension theorem

Theorem

(Whitney extension theorem)

Let

be a submanifold of

M → T

smooth family of isomorphisms such that

. Then

there exists an neighbourhood

and an embedding

N → M

such that h|

= id

and for all p ∈ X, dh

= H

So at

p ∈ X

, we have

∗

= (d

)

∗

. 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

and ∆ = Γ

are La-

grangian submanifolds of (

M × M, ˜ω

). Then by our theorem, there is a neigh-

bourhood

of ∆ in (

M × M, ˜ω

) that is symplectomorphic to a neighbourhood

of the zero section of (T

∗

M, ω

Suppose

is sufficiently

-close to

. Then Γ

⊆ U

. If

is sufficiently

-close to the identity, then the image of Γ

is a smooth section

for

some 1-form µ.

Now

is a symplectomorphism iff Γ

is Lagrangian iff

is closed. So a small

-neighbourhood of

Sym

(

M, ω

) is “the same as” a

small

-neighbourhood of the zero-section in the space of closed 1-forms on

We can also use this to understand fixed points of symplectomorphisms

(again!).

Theorem.

Let (

M, ω

) be a compact symplectic manifold such that

(

) = 0.

Then any symplectomorphism

M → M

sufficiently close to the identity has

at least two fixed points.

Proof.

The graph of

corresponds to a closed 1-form on

. Since

is closed

and

(

) = 0, we know

= d

for some

h ∈ C

∞

. Since

is compact,

has at least two critical points (the global maximum and minimum). Since the

fixed points corresponding to the points where

vanish (so that Γ

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

, and

M → M

a symplectomorphism. Suppose

is exactly

homotopic to the identity and non-degenerate. Then the number of fixed points

of ϕ is at least

i=0

dim H

(M, R).

We should think of the sum

i=0

dim H

(

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

is exactly homotopic to the identity

if there is isotopy

M → M

such that

and

, and further there

is some 1-periodic family of functions

such that

is generated by the vector

field v

defined by ι

∗

ω = dh

The condition

∗

says

is a Hamiltonian vector field, which we will

discuss soon after this.

Definition

(Non-degenerate function)

A endomorphism

M → M

is non-

degenerate iff all its fixed points are non-degenerate, i.e. if

is a fixed point,

then det(id −dϕ

) 6= 0.

Example.

In the original statement of the Arnold conjecture, which is the case

, dθ

∧ dθ

), any symplectomorphism has at least 4 fixed points.

In the case where

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

K¨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

be a vector space. A complex structure

is a linear J : V → V with J

= −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 (

Ω) be a symplectic vector

space, and

a complex structure on

. We say

is compatible with Ω if

(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

, Ω), we set

) = f

, J

) = −e

We can then check that this is compatible with the symplectic structure, and in

fact gives the standard inner product.

Proposition

(Polar decomposition)

Let (

Ω) be a symplectic vector space,

and

an inner product on

. Then from

, we can canonically construct a

compatible complex structure

on (

Ω). If

for some

, then this

process returns J.

Note that in general, G

(u, v) = Ω(u, Jv) 6= G(u, v).

Proof. Since Ω, G are non-degenerate, we know

Ω(u, v) = G(u, Av)

for some

V → V

. If

−

1, then we are done, and set

. In general,

we also know that

is skew-symmetric, i.e.

(

Au, v

) =

(

u, −Av

), which is

clear since Ω is anti-symmetric. Since

is symmetric and positive definite, it

makes sense to write down

√

(e.g. by diagonalizing), and we take

J =

√

−1

A =

−A

−1

It is clear that

−

1, since

commutes with

√

, so this is a complex

structure. We can write this as

√

, and this is called the (left) polar

decomposition of A.

We now check that

is a compatible, i.e.

(

u, v

) = Ω(

u, Jv

) is symmetric

and positive definite. But

(u, v) = G(u,

√

v),

and we are done since

√

is positive and symmetric.

Notation.

Let (

Ω) be a symplectic vector space. We write

(

Ω) for the

space of all compatible complex structures on (V, Ω).

Proposition. J(V, Ω) is path-connected.

Proof.

Let

, J

∈ J

(

Ω). Then this induces inner products

, G

. Let

= (1

−t

)

be a smooth family of inner products on

. Then apply

polar decomposition to get a family of complex structures that start from

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

a manifold is a smooth field of complex structures on the tangent space

M → T

M, J

= −1.

Example.

Suppose

is a complex manifold with local complex coordinates

, . . . , z

on U ⊆ M. We have real coordinates {x

, y

} given by

= x

+ iy

The tangent space is spanned by

∂

∂x

∂

∂y

. We define J by



∂

∂x



∂

∂y

, J



∂

∂y



= −

∂

∂x

The Cauchy–Riemann equations imply this is globally well-defined. This

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

has an almost complex but no

complex structure.

Definition

(Compatible almost complex structure)

An almost complex struc-

ture

is compatible with a symplectic structure

is compatible with

for all p ∈ M. In this case, (ω, g

, 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

a metric on

. Then

from

we can canonically construct a compatible almost complex structure

As before, in general, g

( ·, ·) 6= g( ·, ·).

Corollary.

Any symplectic manifold has a compatible almost complex structure.

The converse does not hold. For example,

is almost complex but not

symplectic.

Notation.

Let (

M, ω

) be a symplectic manifold. We write

(

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

be an almost complex structure on

that is compatible

with ω

and ω

. Then ω

and ω

are deformation equivalent.

Proof.

Check that

= (1

− t

)

tω

works, which is non-degenerate since

(

·, J ·

) is a positive linear combination of inner products, hence is non-

degenerate.

Proposition.

Let (

M, ω

) be a symplectic manifold,

a compatible almost

complex structure. If

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

open,

i.e.

has no closed connected components. Then there exists a symplectic form

in any even 2-cohomology class and such that

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

-forms in terms of

“how holomorphic” they are.

Let (

M, J

) be an almost complex manifold. By complexifying

T M

T M ⊗C

and then extending J linearly, we can split T M ⊗ C into its ±i eigenspace.

Notation.

We write

1,0

for the +

eigenspace of

and

1,0

for the

−i

eigenspace of

. These are called the

-holomorphic tangent vectors and the

J-anti-holomorphic tangent vectors respectively.

We write

1,0

for the +

eigenspace of

, and

0,1

the

−i

-eigenspace of

Then we have a splitting

T M ⊗ C

∼

→ T

1,0

⊕ T

0,1

We can explicitly write down the projection maps as

1,0

(w) =

(w − iJw)

0,1

(w) =

(w + iJw).

Example.

On a complex manifold with local complex coordinates (

, . . . , z

the holomorphic tangent vectors

1,0

is spanned by the

∂

∂z

, while

0,1

is spanned

by the

∂

∂ ¯z

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,

1,0

is spanned by the d

and

0,1

is spanned

by the d¯z

More generally, we can decompose

∗

M ⊗ C) =

1,0

⊕ T

0,1

) =

p+q=k



1,0



⊗



0,1



≡

p+q=k

p,q

We write

Ω

(M, C) = sections of Λ

∗

M ⊗ C)

Ω

p,q

(M, C) = sections of Λ

p,q

Ω

(M, C) =

p+q=k

Ω

p,q

(M, C).

The sections in Ω

p,q

(M, C) are called forms of type (p, q).

Example. In the complex case, we have, locally,

p,q

= C{dz

∧ dz

: |I| = , |K| = m}

and

Ω

p,q







|I|=p,|K|=q

I,K

∧ d¯z

: b

∈ C

∞

(U, C)







As always, we have projections

p,q

∗

M ⊗ C) → Λ

p,q

Combining the exterior derivative d : Ω

(

M, C

)

→

Ω

k−1

(

M, C

) with projections

yield the ∂ and

∂ operators

∂ : Ω

p,q

→ Ω

p+1,q

∂ : Ω

p,q

→ Ω

p,q+1

Observe that for functions

, we have d

∂f

, but this is not necessarily

true in general.

Definition

(

-holomorphic)

We say a function

-holomorphic if

∂f

= 0,

and J-anti-holomorphic if ∂f = 0.

It would be very nice if the sequence

Ω

p,q

Ω

p,q+1

Ω

p,q+2

···

∂

were a chain complex, i.e.

∂

= 0. This is not always true. However, this is

true in the case where

is integrable. Indeed, if

is a complex manifold and

β ∈ Ω

(M, C), then in local complex coordinates, we can write

β =

p+q=k





|I|=p,|K|=q

I,K

∧ d¯z





dβ =

p+q=k





|I|=p,|K|=q

I,K

∧ dz

∧ d¯z





p+q=k





|I|=p,|K|=q

(∂ +

∂)b

I,K

∧ dz

∧ d¯z





= (∂ +

∂)

p+q=k





|I|=p,|K|=q

I,K

∧ d¯z





Thus, on a complex manifold, d = ∂ +

∂.

Thus, if β ∈ Ω

p,q

, then

0 = d

β = ∂

β + (∂

∂ +

∂∂)β +

∂

β.

Since the three terms are of different types, it follows that

∂

= ∂

∂ +

∂∂ = 0.

In fact, the converse of this computation is also true, which we will not prove:

Theorem (Newlander–Nirenberg). The following are equivalent:

–

∂

= 0

– ∂

= 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

Ω

p,q

Ω

p,q+1

Ω

p,q+2

···

∂

Explicitly,

p,q

Dolb

(M) =

ker(

∂ : Ω

p,q

→ Ω

p,q+1

)

im(

∂ : Ω

p,q−1

→ Ω

p,q

)

2.3 K¨ahler manifolds

In the best possible scenario, we would have a symplectic manifold with a

compatible, integrable almost complex structure. Such manifolds are called

K¨ahler manifolds.

Definition

(K¨ahler manifold)

A K¨ahler manifold is a symplectic manifold

equipped with a compatible integrable almost complex structure. Then

called a K¨ahler form.

Let (

M, ω, J

) be a K¨ahler manifold. What can we say about the K¨ahler form

ω? We can decompose it as

ω ∈ Ω

(M) ⊆ Ω

(M, C) = Ω

2,0

⊕ Ω

1,1

⊕ Ω

2,2

We claim that

Lemma. ω ∈ Ω

1,1

Proof. Since ω( ·, J ·) is symmetric, we have

∗

ω(u, v) = ω(Ju, Jv) = ω(v, JJu) = −ω(v, −u) = ω(u, v).

So J

∗

ω = ω.

On the other hand,

∗

acts on holomorphic forms as multiplication by

and

anti-holomorphic forms by multiplication by

−

1 (by definition). So it acts on

Ω

2,0

and Ω

0,2

by multiplication by

−

1 (locally Ω

2,0

is spanned by d

∧

, etc.),

while it fixes Ω

1,1

. So ω must lie in Ω

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 =

j,k

∧ d¯z

for some

j,k

. The fact that

is real valued, so that

¯ω

gives some constraints

on what the h

can be. We compute

¯ω = −

j,k

d¯z

∧ dz

j,k

∧ ¯z

So we have

= h

The non-degeneracy condition ω

∧n

6= 0 is equivalent to det h

6= 0, since





n! det(h

) dz

∧ d¯z

∧ ··· ∧ dz

∧ d¯z

So h

is a non-singular Hermitian matrix.

Finally, we take into account the compatibility condition

(

v, Jv

)

0. If we

write

v =

∂

∂z

+ b

∂

∂¯z

then we have

Jv = i





∂

∂z

− b

∂

∂¯z





So we have

ω(v, Jv) =

(−ia

− ia

) =

> 0.

So the conclusion is that h

is positive definite.

Thus, the conclusion is

Theorem.

A K¨ahler form

on a complex manifold

is a

∂

- and

∂

-closed

form of type (1, 1) which on a local chart is given by

ω =

j,k

∧ d¯z

where at each point, the matrix (h

) is Hermitian and positive definite.

Often, we start with a complex manifold, and want to show that it has a

K¨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

∗

) is K¨ahler, and this is called a K¨ahler submanifold.

In particular, if we can construct K¨ahler forms on

and

, then we

have K¨ahler forms for a lot of our favorite complex manifolds, and in particular

complex projective varieties.

But we still have to construct some K¨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,



∂

∂z

∂ ¯z



is positive definite.

Proposition.

Let

be a complex manifold, and

ρ ∈ C

∞

(

;

) strictly

plurisubharmonic. Then

ω =

∂

∂ρ

is a K¨ahler form.

We call ρ the K¨ahler potential for ω.

Proof. ω

is indeed a 2-form of type (1, 1). Since

∂

= 0 and

∂

−

∂∂

we know ∂ω =

∂ω = 0. We also have

ω =

j,k

∂

∂z

∂¯z

∧ d¯z

and the matrix is Hermitian positive definite by assumption.

Example. If M = C

∼

, we take

ρ(z) = |z|

¯z

Then we have

∂

∂z

∂¯z

= δ

so this is strictly plurisubharmonic. Then

ω =

∧ d¯z

d(x

+ iy

) ∧ d(x

− iy

)

∧ dy

which is the standard symplectic form. So (

, ω

) is K¨ahler and

|z|

is a

(global) K¨ahler potential for ω

There is a local converse to this result.

Proposition.

Let

be a complex manifold,

a closed real-valued (1

1)-form

and

p ∈ M

, then there exists a neighbourhood

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 K¨ahler, such a ρ is called a local K¨ahler potential for ω.

Note that it is not possible to have a global K¨ahler potential on a closed

K¨ahler manifold, because if ω =

∂

∂ρ, then

ω = d



∂ρ



is exact, and we know symplectic forms cannot be exact.

Example. Let M = C

and

ρ(z) = log(|z|

+ 1).

It is an exercise to check that ρ is strictly plurisubharmonic. Then

F S

∂

∂(log(|z

| + 1))

is another K¨ahler form on C

, called the Fubini–Study form.

The reason this is interesting is that it allows us to put a K¨ahler structure

on CP

Example.

Let

. Using homogeneous coordinates, this is covered by

the open sets

= {[z

, . . . , z

] ∈ CP

| z

6= 0}.

with the chart given by

: U

→ C

, . . . , z

] 7→



, . . . ,

j−1

j+1

, . . . ,



One can check that

∗

F S

∗

F S

. Thus, these glue to give the Fubini–Study

form on CP

, making it a K¨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

Ω

(M; C) =

p+q=k

Ω

p,q

and using

∂ : Ω

p,q

→ Ω

p,q+1

, we defined the Dolbeault cohomology groups

p,q

Dolb

(M) =

ker

∂

It would be nice if we also had a decomposition

∼

p+q=k

p,q

Dolb

(M).

This is not always true. However, it is true for compact K¨ahler manifolds:

Theorem

(Hodge decomposition theorem)

Let (

M, ω

) be a compact K

haler

manifold. Then

∼

p+q=k

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

be a real oriented vector space

with inner product

. Then this induces

an inner product on each Λ

(V ), denoted h·, ·i, defined by

∧ ··· ∧ v

, w

∧ ··· ∧ w

i = det(G(v

, w

))

i,j

Let e

, . . . , e

be an oriented orthonormal basis for V . Then

∧ ··· ∧ e

: 1 ≤ j

< ··· < j

≤ m}

is an orthonormal basis for Λ

Definition

(Hodge star)

The Hodge

∗

-operator

∗

: Λ

→

m−k

is defined by

the relation

α ∧ ∗β = hα, βi e

∧ ··· ∧ e

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

∧ ··· ∧ e

) = e

k+1

∧ ··· ∧ e

– ∗(e

k+1

∧ ··· ∧ e

) = (−1)

k(m−k)

∧ ··· ∧ e

– ∗∗ = α = (−1)

k(m−k)

α for α ∈ Λ

In general, let (

M, g

) be a compact real oriented Riemannian manifold of

dimension

. Then

gives an isomorphism

T M

∼

∗

, and induces an inner

product on each

∗

, which we will still denote

. This induces an inner

product on each

∗

M, which we will denote h·, ·i again.

The Riemannian metric and orientation gives us a volume form

Vol ∈

Ω

(

defined locally by

Vol

∧ ··· ∧ e

where

, . . . , e

is an oriented basis of

∗

. This induces an

-inner product

on Ω

(M),

hα, βi

hα, βi Vol.

Now apply Hodge

∗

-operator to each (

V, G

) = (

∗

M, g

) and

p ∈ M

. We then

get

Definition

(Hodge star operator)

The Hodge

∗

-operator on forms

∗

: Ω

(

)

→

Ω

m−k

(M) is defined by the equation

α ∧ (∗β) = hα, βi Vol.

We again have some immediate properties.

Proposition.

(i) ∗ ∗ α = (−1)

k(m−k)

α for α ∈ Ω

(M).

(ii) ∗1 = Vol

We now introduce the codifferential operator

Definition

(Codifferential operator)

We define the codifferential operator

Ω

→ Ω

k−1

to be the L

-formal adjoint of d. In other words, we require

hdα, βi

= hα, δβi

for all α ∈ Ω

and β ∈ Ω

k+1

We immediately see that

Proposition. δ

= 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

→ Ω

k−1

Proof.

hdα, βi

dα ∧ ∗β

d(α ∧ ∗β) − (−1)

α ∧ d(∗β)

= (−1)

k+1

α ∧ d(∗β) (Stokes’)

= (−1)

k+1

(−1)

(m−k)k

α ∧ ∗ ∗ d(∗β)

= (−1)

k+1+(m−k)k

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

→ Ω

Example.

with the Euclidean inner product, and

f ∈

Ω

(

) =

∞

(M), then

∆f = −

i=1

∂

∂x

It is an exercise to prove the following

Proposition.

(i) ∆∗ = ∗∆ : Ω

→ Ω

m−k

(ii) ∆ = (d + δ)

(iii) h∆α, βi

= hα, ∆βi

(iv) ∆α = 0 iff dα = δα = 0.

In particular, (iv) follows from the fact that

h∆α, αi = hdα, dαi + hδα, δαi = kdαk

+ kδαk

Similar to IA Vector Calculus, we can define

Definition (Harmonic form). A form α is harmonic if ∆α = 0. We write

= {α ∈ Ω

(m) | ∆α = 0}

for the space of harmonic forms.

Observe there is a natural map

→ H

(

), sending

to [

]. The main

result is that

Theorem

(Hodge decomposition theorem)

Let (

M, g

) be a compact oriented

Riemannian manifold. Then every cohomology class in

(

) has a unique

harmonic representation, i.e. the natural map

→ H

(

) is an isomorphism.

We will not prove this.

Complex Hodge theory

We now see what we can do for complex K¨ahler manifolds. First check that

Proposition.

Let

be a complex manifold,

dim

and (

M, ω

) K¨ahler.

Then

(i) ∗ : Ω

p,q

→ Ω

n−p,n−q

(ii) ∆ : Ω

p,q

→ Ω

p,q

define the

-adjoints

∂

∗

± ∗

∂∗

and

∂

∗

− − ∗∂∗

with the appropriate

signs as before, and then

d = ∂ +

∂, δ = ∂

∗

∂

∗

We can then define

∆

∂

= ∂∂

∗

+ ∂

∗

∂ : Ω

p,q

→ Ω

p,q

∆

∂

∗

∂

∗

∂ : Ω

p,q

→ Ω

p,q

Proposition. If our manifold is K¨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

p+q=k

p,q

where

p,q

= {α ∈ Ω

p,q

(M) : ∆α = 0}.

Theorem

(Hodge decomposition theorem)

Let (

M, ω

) be a compact K¨ahler

manifold. The natural map H

p,q

→ H

p,q

Dolb

is an isomorphism. Hence

(M; C)

∼

p+q=k

p,q

∼

p+q=k

p,q

Dolb

(M).

What are some topological consequences of this? Recall the Betti numbers

are defined by

= dim

(M) = dim

(M; C).

We can further define the Hodge numbers

p,q

= dim

p,q

Dolb

(M).

Then the Hodge theorem says

p+q=k

p,q

Moreover, since

p,q

Dolb

(M) = H

q,p

Dolb

(M).

So we have Hodge symmetry,

p,q

= h

q,p

Moreover, the ∗ operator induces an isomorphism

p,q

Dolb

∼

n−p,n−q

Dolb

So we have

p,q

= h

n−p,n−q

There is called central symmetry, or Serre duality. Thus, we know that

Corollary. Odd Betti numbers are even.

Proof.

2k+1

p+q=2k+1

p,q

= 2

p=0

p,2k+1−p

Corollary. h

1,0

= h

0,1

is a topological invariant.

We have also previously seen that for a general compact K¨ahler manifold, we

have

Proposition. Even Betti numbers are positive.

Recall that we proved this by arguing that [

]

= 0

∈ H

(

). In fact,

∈ 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.

0,0

1,0

0,1

2,0

1,1

0,2

2,1

1,2

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

∞

(

), which returns

the energy of a particular configuration, generates a vector field on

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

∞

(

), then since

˜ω

T M → T

∗

is an isomorphism, there is a unique

vector field X

on M such that

ω = dH.

We call X

the Hamiltonian vector field with Hamiltonian function H.

Suppose

is complete (e.g. when

is compact). This means we can

integrate X

, i.e. solve

∂ρ

∂t

(p) = X

(ρ

(p)), ρ

(p) = p.

These flow have some nice properties.

Proposition.

is a Hamiltonian vector field with flow

, then

∗

In other words, each ρ

is a symplectomorphism.

Proof. It suffices to show that

∂

∂t

∗

ω = 0. We have

(ρ

∗

ω) = ρ

∗

ω) = ρ

∗

(dι

ω + ι

dω) = ρ

∗

(ddH) = 0.

Thus, every function

gives us a one-parameter subgroup of symplectomor-

phisms.

Proposition. ρ

preserves H, i.e. ρ

∗

H = H.

Proof.

∗

H = ρ

∗

H) = ρ

∗

(ι

dH) = ρ

∗

(ι

ω) = 0.

So the flow lines of our vector field are contained in level sets of H.

Example. Take (S

, ω = dθ ∧ dh). Take

H(h, θ) = h

to be the height function. Then X

solves

(dθ ∧ dh) = dh.

∂

∂θ

, ρ

(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

on (

M, ω

) is a symplectic

vector field if L

ω = 0.

Observe that

ω = ι

dω + dι

ω = dι

ω.

is symplectic iff

is closed, and is Hamiltonian if it is exact. Thus,

locally, every symplectic vector field is Hamiltonian and globally, the obstruction

lies in H

(M).

Example.

Take (

, ω

= d

∧

). Then

∂

∂θ

are symplectic but not

Hamiltonian, since ι

ω = dθ

2−i

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

and

f ∈ C

∞

(

), 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

, ι

]α = [ι

, L

]α.

Proof of proposition.

We need to check that

[X,Y ]

is exact. By the exercise,

this is

[X,Y ]

ω = L

ω − ι

ω = d(ι

ω) + ι

dι

ω + ι

dι

ω − ι

dω.

Since

X, Y

are symplectic, we know d

= d

= 0, and the last term always

vanishes. So this is exact, and

(

Y, X

) is a Hamiltonian function for [

X, Y

Definition

(Poisson bracket)

Let

f, g ∈ C

∞

(

). We then define the Poisson

bracket {f, g} by

{f, g} = ω(X

, X

This is defined so that

{f,g}

= −[X

, X

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 (

∞

(

)

, {·, ·}

) is a Poisson algebra.

This means it is a commutative, associative algebra with a Lie bracket that

satisfies the Leibniz rule.

Further, the map

∞

(

)

→ χ

(

) sending

H 7→ X

is a Lie algebra

(anti-)homomorphism.

Proposition. {f, g} = 0 iff f is constant along integral curves of X

Proof.

f = ι

df = ι

ω = ω(X

, X

) = {f, g} = 0.

Example. If M = R

and ω = ω

∧ dy

, and f ∈ C

∞

), then



∂f

∂y

∂

∂x

−

∂f

∂x

∂

∂y



If ρ

(p) = p, then ρ

(p) = (x(t), y(t)) is an integral curve for X

iff

∂f

∂y

∂t

= −

∂f

∂x

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

∞

(

), 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

∞

(

)

such that {f, H} = 0.

For example,

is an integral of motion. Are there others? Of course, we

can write down 2

H, H

, H

, e

, etc., but these are more-or-less the same as

Definition

(Independent integrals of motion)

We say

, . . . , f

∈ C

∞

(

) are

independent if (d

)

, . . . ,

)

are linearly independent at all points on some

dense subset of M .

Definition

(Commuting integrals of motion)

We say

, . . . , f

∈ C

∞

commute

if {f

, f

} = 0.

If we have

independent integrals of motion, then we clearly have

dim M ≥ n

In fact, the commuting condition implies:

Exercise.

Let

, . . . , f

be independent commuting functions on (

M, ω

). Then

dim M ≥ 2n.

The idea is that the (d

)

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

is (completely) integrable if it has

independent

commuting integrals of motion f

= H, f

, . . . , f

Example.

dim M

= 2, then we only need one integral of motion, which we

can take to be

. 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

∗

, where the

refers to the

angular coordinate

of the pendulum, and the cotangent vector is the momentum.

The Hamiltonian function is

H = K + V = kinetic energy + potential energy =

+ (1 − cos ω).

We can check that the critical points of

are (

θ, ξ

) = (0

0) and (

π,

0). So

(M, ω, H) is integrable.

Example.

dim M

= 4, then (

M, ω, H

) is integrable as long as there exists an

integral motion independent of

. For example, on a spherical pendulum, we

have

∗

, and

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

and

H, f

, . . . , f

are

commuting. Let

be a regular value of

= (

, . . . , f

). Then

−1

(

) is an

n-dimensional submanifold of M . If p ∈ f

−1

(c), then

−1

(c)) = ker(df)

Since







(df

)

(df

)



















we know

−1

(c)) = ker(df

) = span{(X

)

, . . . (X

)

Moreover, since

ω(X

, X

) = {f

, f

} = 0,

we know that T

−1

(c)) is an isotropic subspace of (T

M, ω

, . . . , X

are complete, then following their flows, we obtain global

coordinates of (the connected components of)

−1

(

), where

q ∈ f

−1

(

) has

coordinates (

, . . . , ϕ

) (angle coordinates) if

is achieved from the base point

by following the flow of

for

seconds for each

. The fact that the

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

, the connected components look like

n−k

×T

where

= (

)

is the

torus. In the extreme case

, 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 (

) are part of a Darboux chart of

, and this is

true.

Theorem

(Arnold–Liouville thoerem)

Let (

M, ω, H

) be an integrable system

with

dim M

= 2

and

H, f

, . . . , f

integrals of motion, and

c ∈ R

a regular

value of f = (f

, . . . , f

(i)

If the flows of

are complete, then the connected components of

−1

(

{c}

)

are homogeneous spaces for

and admit affine coordinates

, . . . , ϕ

(angle coordinates), in which the flows of X

are linear.

(ii)

There exists coordinates

, . . . , ψ

(action coordinates) such that the

’s

are integrals of motion and ϕ

, . . . , ϕ

, ψ

, . . . , ψ

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)

∼

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

moving in

under

the potential

(

). Newton’s second law then says the trajectory of the particle

obeys

= −∇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

We also need the energy function

Definition (Energy). The energy of a particle is

H(x, y) =

|y|

+ V (x).

We call

the configuration space and

∗

the phase space, parametrized

by (x, y). This has a canonical symplectic form ω = dx

∧ dy

Newton’s second law can be written as

= −

∂V

∂x

Combining with the definition of y, we find that (x, y) evolves under.

∂H

∂y

= −

∂H

∂x

So physical motion is given by Hamiltonian flow under

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

particles of masses

, . . . , m

moving

under a potential

V ∈ C

∞

(

). The Hamiltonian function can be defined

exactly as before:

H(x, y) =

+ V (x),

where

(

) = (

, . . . , x

) and each

is a 3-vector; and similarly for

with

. Then in Hamiltonian mechanics, we say (

x, y

) evolves under

Hamiltonian flow.

Now fix

a, b ∈ R

and

p, q ∈ R

. Write

for the space of all piecewise

differentiable paths γ = (γ

, . . . , γ

) : [a, b] → R

Definition (Action). The action of a path γ ∈ P is



dγ

(t)



− V (γ(t))

dt.

The integrand is known as the Lagrangian function. We will see that

(

) =

x(t) is (locally) a stationary point of A

iff

= −

∂V

∂x

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

. For example, if we have a

pendulum, then the particle is constrained to live in

(or

). Then we set

to be the maps

: [

a, b

]

→ X

that go from

. The Lagrangian formulation

is then exactly the same, except the minimization problem is now performed

within this P.

More generally, suppose we have an

-dimensional manifolds

, and

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

(t)



The action is then

(˜γ

∗

F )(t) =



γ(t),

dγ

(t)



dt.

To find the critical points of the action, we use calculus of variations. Pick a

chart (

, . . . , x

) for

and naturally extend to a chart (

, . . . , x

, v

, . . . , v

)

for T X. Consider a small perturbation of our path

(t) = (γ

(t) + εc

(t), . . . , γ

(t) + εc

(t))

for some functions

, . . . , c

∈ C

∞

([

a, b

]) such that

(

) =

(

) = 0. We think

of this as an infinitesimal variation of γ. We then find that

dε



ε=0

= 0 ⇔

∂F

∂x

∂F

∂v

for k = 1, . . . , n.

These are the Euler–Lagrange equations.

Example. In X = R

and

F (x

, . . . , x

, v

, . . . , v

) =

− V (x

, . . . , x

Then the Euler–Lagrange equations are

−

∂V

∂x

= m

Example.

On a Riemannian manifold, if we set

T X → R

be (

x, v

)

7→ |v|

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



∂

∂v



6= 0,

then the Euler–Lagrange equations become second order ODEs, and there is a

unique solution given

(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

G → Diff

(

) such that each

is a symplectomorphism. In other words, it

is a map G → Symp(M, ω).

Example.

Let

. Then a map

G → Diff

(

) is a one-parameter group

of transformations

{ψ

t ∈ R}

. Given such a group action, we obtain a complete

vector field

dψ

(p)



t=0

Conversely, given a complete vector field X, we can define

= exp tX,

and this gives a group action by R.

Under this correspondence, symplectic actions correspond to complete sym-

plectic vector fields.

Example.

, then a symplectic action of

is a symplectic action of

R which is periodic.

In the case where

, it is easy to define what it means for an

action to be Hamiltonian:

Definition

(Hamiltonian action)

An action of

is Hamiltonian if the

corresponding vector field is Hamiltonian.

Example. Take (S

, ω = dθ ∧ dh). Then we have a rotation action

(θ, h) = (θ + t, h)

generated by the vector field

∂

∂θ

. Since

∂

∂θ

= d

is exact, this is in fact a

Hamiltonian S

action.

Example. Take (T

, dθ

∧ dθ

). Consider the action

(θ

, θ

) = (θ

+ t, θ

This is generated by the vector field

∂

∂θ

. But

∂

∂θ

= d

, 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

? The simplest possible next case is the torus

× ··· × S

If we have a map

;

→ Symp

(

M, ω

), then for this to be Hamiltonian, it

should definitely be the case that the restriction to each

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

. Let

be a

Lie group. For each g ∈ GG, there is a left multiplication map

: 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

)

∗

X = X

for all g ∈ G.

We write

for the space of all left-invariant vector fields on

, which comes

with the Lie bracket on vector fields. This is called the Lie algebra of G.

is left-invariant, then knowing

tells us what

is everywhere, and

specifying

produces a left-invariant vector field. Thus, we have an isomor-

phism g

∼

The Lie algebra admits a natural action of

, called the adjoint action. To

construct this, note that G acts on itself by conjugation,

(a) = gag

−1

This fixes the identity, and taking the derivative gives

g → g

, or equivalently,

is a map

G → GL

(

). The dual

∗

admits the dual action of

, called

the coadjoint action. Explicitly, this is given by

hAd

∗

(ξ), xi = hξ, Ad

xi.

An important case is when

is abelian, i.e. a product of

’s and

’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

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

(p).

Differentiating this at e gives

∼

G → T

X 7→ X

We call

the vector field on

generated by

X ∈ g

. In the case where

, we have

∼

, 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

G → Symp

(

M, ω

) is a Hamiltonian

action if there exists a map µ : M → g

∗

such that

(i)

For all

X ∈ g

is the Hamiltonian vector field generated by

, where

: M → R is given by

(p) = hµ(p), Xi.

(ii) µ

-equivariant, where

acts on

∗

by the coadjoint action. In other

words,

µ ◦ ψ

= Ad

∗

◦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

, and

ω =

∧ d¯z

∧ dθ

We let

= {(t

, . . . , t

) ∈ C

: |t

| = 1 for all k},

acting by

,...,t

)

, . . . , z

) = (t

, . . . , t

)

where k

, . . . , k

∈ Z.

We claim this action has moment map

µ : C

→ (t

)

∗

∼

, . . . , z

) 7→ −

, . . . , k

It is clear that this is invariant, and if X = (a

, . . . , a

) ∈ t

∈ R

, then

= k

∂

∂θ

+ ··· + k

∂

∂θ

Then we have

dµ

= d



−



= −

= ι

ω.

Example.

A nice example of a non-abelian Hamiltonian action is coadjoint

orbits. Let

be a Lie group, and

the Lie algebra. If

X ∈ g

, then we get a

vector field

generated by

via the adjoint action, and also a vector field

X on g

∗

generated by the co-adjoint action.

If ξ ∈ g

∗

, then we can define the coadjoint orbit through ξ

= {Ad

∗

(ξ) : g ∈ G}.

What is interesting about this is that this coadjoint orbit is actually a symplectic

manifold, with symplectic form given by

, Y

) = hξ, [X, Y ]i.

Then the coadjoint action of

has moment map

→ g

∗

given by the

inclusion.

3.5 Symplectic reduction

Given a Lie group action of

, it is natural to ask what the “quotient” of

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

= {ψ

(p) : g ∈ G}.

Definition

(Stabilizer)

The stabilizer or isotropy group of

p ∈ M

is the closed

subgroup

= {g ∈ G : ψ

(p) = p}.

We write g

for the Lie algebra of G

Definition (Transitive action). We say G acts transitively if M is one orbit.

Definition (Free action). We say G acts freely if G

= {e} for all p.

Definition

(Locally free action)

We say

acts locally freely if

{

}

, i.e.

is discrete.

Definition

(Orbit space)

The orbit space is

M/G

, and we write

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

be a compact Lie group, and

(

M, ω

) a symplectic manifold with a Hamiltonian

-action with moment map

M → g

∗

. Write

−1

(0)

→ M

for the inclusion. Suppose

acts freely on

−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

∗

red

Definition

(Symplectic quotient)

We call

red

the symplectic quotient/reduced

space/symplectic reduction of (

M, ω

) with respect to the given

-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

to be preserved under the coadjoint

action of G. This is automatically satisfied if G is abelian, and in this case, we

simply have

−1

(

) =

−1

(0), where

µ −ξ

is another moment map. So this

is not more general.

is not preserved by

, then we can instead consider

−1

(

)

, or

equivalently take µ

−1

)/G. We check that

−1

(ξ)/G

∼

−1

)/G

∼

−1

(0)/G,

where

ϕ : M × O

→ g

∗

(ρ, η) 7→ µ(p) − η

is a moment map for the product action of G on (M × O

, ω × ω

So in fact there is no loss in generality for considering just µ

−1

(0).

Proof.

We first show that

−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

using the rank-nullity theorem. We

know dµ

v = 0 iff hdµ

(v), Xi = 0 for all X ∈ g. We can compute

hdµ

(v), Xi = (dµ

)

(v) = (ι

ω)(v) = ω

, v).

Moreover, the span of the X

is exactly T

. So

ker dµ

= (T

)

Thus,

dim(im dµ

) = dim O

= dim G − dim G

In particular, dµ

is surjective iff G

= 0.

Then (i) and (ii) follow from the following theorem:

Theorem.

Let

be a compact Lie group and

a manifold, and

acts freely

on Z. Then Z/G is a manifold and Z → Z/G is a principal G-bundle.

Note that if

does not act freely on

−1

(0), then by Sard’s theorem,

generically,

is a regular value of

, and so

−1

(

) is a manifold, and

acts

locally freely on

−1

(

). If

−1

(

) is preserved by

, then

−1

(

)

is a

symplectic orbifold.

It now remains to construct the symplectic structure. Observe that if

p ∈

−1

(0), then

⊆ T

−1

(0) = ker dµ

= (T

)

is an isotropic subspace of (

M, ω

). We then observe the following

straightforward linear algebraic result:

Lemma.

Let (

Ω) be a symplectic vector space and

an isotropic sub-

space. Then Ω induces a canonical symplectic structure Ω

red

Ω

, given by

Ω

red

([u], [v]) = Ω(u, v).

Applying this, we get a canonical symplectic structure on

)

−1

(0)

= T

[p]

red

This defines

red

, which is well-defined because

-invariant, and is

smooth by local triviality and canonicity.

It remains to show that dω

red

= 0. By construction, i

∗

ω = π

∗

red

. So

∗

(dω

red

) = dπ

∗

red

= di

∗

ω = i

∗

dω = 0

Since π

∗

is injective, we are done.

Example. Take

(M, ω) =



, ω

∧ d¯z

∧ dy

∧ dθ



We let G = S

act by multiplication

· (z

, . . . , z

) = (e

, . . . , e

This action is Hamiltonian with moment map

µ : C

→ R

z 7→ −

|z|

The +

is useful to make the inverse image of 0 non-degenerate. To check this

is a moment map, we compute



|z|



= d



−



= −

On the other hand, if

X = a ∈ g

∼

then we have

= a



∂

∂θ

+ ··· +

∂

∂θ



So we have

ω = −a

= dµ

It is also clear that µ is S

-invariant. We then have

−1

(0) = {z ∈ C

: |z|

= 1} = S

2n−1

Then we have

red

= µ

−1

(0)/S

= CP

n−1

One can check that this is in fact the Fubini–Study form. So (

n−1

, ω

F S

) is

the symplectic quotient of (

, ω

) with respect to the diagonal

action and

the moment map µ.

Example. Fix k,  ∈ Z relatively prime. Then S

acts on C

, z

) = (e

ikt

, e

i`t

This action is Hamiltonian with moment map

µ : C

→ R

, z

) 7→ −

(k|z

+ |z

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

, z

) has trivial stabilizer if z

, z

6= 0.

On the other hand, the action is still locally free on

\ {

}

since the

stabilizers are discrete.

The reduced spaces

−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

→ B

be a principal

-bundle, and

G → Diff

(

) the associated

free action. Let

dψ : g → χ(P )

X 7→ X

be the associated infinitesimal action. Let

, . . . , X

be a basis of the Lie

algebra

. Then since

is a free action,

, . . . , X

are all linearly independent

at each p ∈ P .

Define the vertical tangent space

= span{(X

)

, . . . , (X

)

} ⊆ T

P = ker(dπ

)

We can then put these together to get V ⊆ T P , the vertical tangent bundle.

Definition

(Ehresmann Connection)

An (Ehresmann) connection on

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

a g-valued 1-form A ∈ Ω

(P ) ⊗ g such that

(i) ι

A = X for all X ∈ g

(ii) A is G-invariant for the action

g · (A

⊗ X

) = (ψ

−1

)

∗

⊗ Ad

Lemma.

Giving an Ehresmann connection is the same as giving a connection

1-form.

Proof. Given an Ehresmann connection H, we define

(v) = X,

where v = X

+ h

∈ V ⊕ H.

Conversely, given an A, we define

= ker A

= {v ∈ T

P : i

= 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

∗

P = V

∗

⊕ H

∗

, Λ

∗

P ) = (

∗

) ⊕ (V

∗

∧ H

∗

) ⊕ (

∗

So we end up having

Ω

(P ) = Ω

vert

(P ) ⊕ Ω

hor

(P )

Ω

(P ) = Ω

vert

⊕ Ω

mixed

⊕ Ω

hor

If A =

i=1

⊗ X

∈ Ω

⊗ g, then

dA ∈ Ω

⊗ 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 ∈ Ω

hor

⊗ g.

Definition (Flat connection). A connection A is flat if curv A = 0.

We write

for the space of all connections on

. Observe that if

, A

∈ A

then A

− A

is G-invariant and

− A

) = X − X = 0

for all X ∈ g. So A

− A

∈ (Ω

hor

⊗ g)

. So

A = A

+ (Ω

hor

⊗ g)

and T

A = (Ω

hor

⊗ g)

Suppose

is a compact Riemann surface, and

a compact or semisimple

Lie group. Then there exists an Ad-invariant inner product on g. We can then

define

ω : (Ω

hor

⊗ g)

× (Ω

hor

⊗ g)

→ R,

sending





7→

i,j

∧ b

, X

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

be the gauge group, i.e. the group of

-equivariant diffeomorphisms

P → P

covering the identity.

acts on

V ⊕ H 7→ V ⊕ H

where

is the image of

by d

. Atiyah and Bott proved that the action of

on (A, ω) is Hamiltonian with moment map

µ : A 7→ Lie(G)

∗

= (Ω

hor

⊗ 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

. It turns out the image of the moment map

has

a very rigid structure.

Theorem

(Convexity theorem (Atiyah, Guillemin–Sternberg))

Let (

M, ω

) be

a compact connected symplectic manifold, and

µ : M → R

a moment map for a Hamiltonian torus action. Then

(i) The levels µ

−1

(ii) The image µ(M) is convex.

(iii) The image µ(M) is in fact the convex hull of µ(M

We call µ(M ) the moment polytope.

Here we identify

∼

' R

, which gives us an identification

∼

and g

∗

∼

)

∗

∼

Example.

Consider (

, ω

F S

acts by letting

= (

, . . . , t

)

∈

∼

U(1)

send

([z

: ···z

]) = [z

: t

: ··· : t

This has moment map

µ([z

: ···z

]) = −

(|z

, . . . , |z

)

+ ··· + |z

The image of this map is

µ(M) =



x ∈ R

: x

≤ 0, x

+ ··· + x

≥ −



For example, in the CP

case, the moment image lives in R

, and is just

−

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

connected components

∪···∪Z

. Then

is constant on each

, since

= 0 for all

. Let

(

) =

∈ R

. By (ii), we know the convex hull of

{η

, . . . η

}

is contained in

µ(M).

To see that

(

) is exactly the convex hull, observe that if

X ∈ R

has

rationally independent components, so that

topologically generates

, then

is fixed by

iff

= 0, iff d

= 0. Thus,

attains its maximum on one of

the Z

Now if

is not in the convex hull of

{η

}

, then we can pick an

X ∈ R

with

rationally independent components such that

hξ, Xi > hη

, Xi

for all

, since

the space of such X is open and non-empty. Then

hξ, Xi > sup

p∈

hµ(p), Xi = sup

p∈M

hµ(p), Xi.

So ξ 6∈ µ(M).

With a bit more work, (i) also implies (ii).

Lemma. (i) implies (ii).

Proof.

The case

= 1 is immediate, since

(

) is compact and connected,

hence a closed interval.

In general, to show that

(

) is convex, we want to show that the intersection

(

) with any line is connected. In other words, if

n+1

→ R

is any

projection and ν = π ◦µ, then

−1

(c))

is connected. This would follow if we knew

−1

(

) were connected, which would

follow from (i) if

were a moment map of an

action. Unfortunately, most of

the time, it is just the moment map of an

action. For it to come from a

action, we need π to be represented by an integer matrix. Then

T = {π

t : t ∈ T

= R

} ⊆ T

n+1

is a subtorus, and one readily checks that

is the moment map for the

action.

Now for any

, p

∈ M

, we can find

, p

arbitrarily close to

, p

and

a line of the form

−1

(

) with

integral. Then the line between

and

is contained in

(

) by the above argument, and we are done since

(

) 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

= 0}

be the set of critical points. For

p ∈ Crit

(

) and (

U, x

, . . . , x

) a coordinate

chart around p, we have a Hessian matrix

f =



∂

∂x



Definition

(Morse(-Bott) function)

. f

is a Morse function if at each

p ∈ Crit

(

f is non-degenerate.

is a Morse–Bott function if the connected components of

Crit

(

) are

submanifolds and for all p ∈ Crit(p), T

Crit(f) = ker(H

f).

is Morse, then the critical points are isolated. If

is Morse–Bott, then

the Hessian is non-degenerate in the normal bundle to Crit(f ).

is compact, then there is a finite number of connected components of

Crit(f). So we have

Crit(f) = Z

∪ ··· ∪ Z

and the Z

are called the critical submanifolds.

For p ∈ Z

, the Hessian H

f is a quadratic form and we can write

M = E

−

⊕ T

⊕ E

where

are the positive and negative eigenspaces of

respectively. Note

that

dim E

are locally constant, hence constant along

. So we can define the

index of Z

to be dim E

−

, and the coindex to be dim E

We can then define a vector bundle

−

→ Z

, called the negative normal

bundle.

Morse theory tells us how the topology of

−

{p ∈ M

(

)

≤ c}

changes

with c ∈ R.

Theorem (Morse theory).

(i)

−1

([

, c

]) does not contain any critical point. Then

−1

(

)

∼

−1

(

)

and M

∼

(where

∼

means diffeomorphic).

(ii)

−1

([

, c

]) contains one critical manifold

, then

−

' M

−

∪D

(

−

where D(E

−

) is the disk bundle of E

−

In particular, if

is an isolated point,

−

is, up to homotopy equivalence,

obtained by adding a dim E

−

-cell to M

−

The key lemma in this proof is the following result:

Lemma.

Let

be a compact connected manifold, and

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

Proof sketch.

There is always a global minimum since

is compact. If there is

another local minimum at c, then the disk bundle is trivial, and so in

−

c+ε

' M

−

c−ε

∪ D(E

−

)

for

small enough, the union is a disjoint union. So

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

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

is a Morse–Bott function where all critical

submanifolds are symplectic.

Proof sketch.

Note that

is a fixed point iff

= 0 iff d

= 0 iff

is a critical

point. So the critical points are exactly the fixed points of X.

(

M, ω

) models (

M, ω

) in a neighbourhood of

by Darboux theorem. Near

a fixed point

, an equivariant version of the Darboux theorem tells us there is

a coordinate chart U where (M, ω, µ) looks like

ω|

∧ dy

µ|

= µ(p) −

+ y

)α

where α

∈ Z are weights.

Then the critical submanifolds of µ are given by

= y

= 0 : α

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

= 1 case follows from the previous lemmas. We then induct on

Suppose the theorem holds for

, and let

= (

, . . . , µ

) :

M → R

n+1

a moment map for a Hamiltonian

n+1

-action. We want to show that for all

c = (c

, . . . , c

) ∈ R

n+1

, the set

−1

) ∩ ··· ∩ µ

−1

n+1

)

is connected.

The idea is to set

N = µ

−1

) ∩ ···µ

−1

and then show that

n+1

N → R

is a Morse–Bott function with no critical

submanifolds of index or coindex 1.

We may assume that d

, . . . ,

are linearly independent, or equivalently,

dµ

6= 0 for all X ∈ R

. Otherwise, this reduces to the case of an n-torus.

To make sense of

, we must pick

to be a regular value. Density arguments

imply that

C =

[

X6=0

Crit(µ

) =

[

X∈Z

n+1

\{0}

Crit µ

Since

Crit µ

is a union of codimension

≥

2 submanifolds, its complement is

dense. Hence by the Baire category theorem,

has dense complement. Then a

continuity argument shows that we only have to consider the case when

is a

regular value of µ, hence N is a genuine submanifold of codimension n.

By the induction hypothesis,

is connected. We now show that

n+1

N → R is Morse–Bott with no critical submanifolds of index or coindex 1.

Let

be a critical point. Then the theory of Lagrange multipliers tells us

there are some λ

∈ R such that

dµ

n+1

n=1

dµ

= 0

Thus, µ is critical in M for the function

= µ

n+1

i=1

where

= (

, . . . , λ

∈ R

n+1

. So by the claim,

is Morse–Bott with only

even indices and coindices. Let

be a critical submanifold of

containing

Claim. W intersects N transversely at x.

If this were true, then

has

W ∩ N

as a non-degenerate critical sub-

manifold of even index and coindex, since the coindex doesn’t change and

is even-dimensional. Moreover, when restricted to

is a constant. So

n+1

satisfies the same properties.

To prove the claim, note that

N = ker dµ

∩ ··· ∩ ker dµ

With a moments thought, we see that it suffices to show that d

, . . . ,

remain linearly independent when restricted to

. Now observe that the

Hamiltonian vector fields

, . . . , X

are independent since d

, . . .

are, and they live in T

W since their flows preserve W .

Since

is symplectic (by the claim), for all

= (

, . . . k

), there exists

v ∈ T

W such that



, v



6= 0.

In other words,



dµ



(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

a compact Lie group with

a Hamiltonian action on

with moment map

M → g

∗

. From Lie group

theory, there is a maximal torus

T ⊆ G

with Lie algebra

, 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

∗

∼

∗

/W,

Pick a Weyl chamber t

∗

of t

∗

, i.e. a fundamental domain of t

∗

under W . Then

induces a moment map

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

)

∈ R

and

≥ λ

, and

the set of all 2

2 Hermitian

matrices with eigenvalues (

, λ

). What can the diagonal entries of

A ∈ H

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

+ λ

det a = ab − |z|

= λ

The first implies

− a

, and all the second condition gives is that

ab > λ

This completely determines the geometry of

, since for each allowed value

, there is a unique value of

, which in turn determines the norm of

Topologically, this is a sphere, since there is a

’s worth of choices of

except

at the two end points ab = λ

What has this got to do with Hamiltonian actions? Observe that U(2) acts

transitively on H

by conjugation, and



iθ

0 e

iθ



⊆ U(2).

This contains a copy of S

given by



iθ

0 e

−iθ



⊆ T

We can check that



iθ

0 e

−iθ



a z

¯z b



iθ

0 e

−iθ



−1



a e

iθ

z b



Thus, if ϕ : H

→ R

is the map that selects the diagonal elements, then

−1

(a, b) =



a |z|e

iθ

|z|e

−iθ



is one

-orbit. This is reminiscent of the

action of

quotienting out to a

line segment.

We can think more generally about H

, the n × n Hermitian matrices with

eigenvalues λ

≥ ··· ≥ λ

, and ask what the diagonal elements can be.

Take

→ R

be the map that selects the diagonal entries. Then the

image lies on the plane

tr A

. This certainly contain the

! points whose

coordinates are all possible permutation of the

, given by the diagonal matrices.

Theorem

(Schur–Horn theorem)

. ϕ

(

) is the convex hull of the

! points

from the diagonal matrices.

To view this from a symplectic perspective, let

, and U(

) acts

transitively by conjugation. For A ⊆ M, let G

be the stabilizer of A. Then

= U(n)/G

Thus,

∼

where

is the Hermitian matrices. The point of this is to define a symplectic

form. We define

: iH

× iH

→ R

(X, Y ) 7→ i tr([X, Y ]A) = i tr(X(Y A − AY ))

ker ω

= {Y : [A, Y ] = 0} = g

induces a non-degenerate form on

. In fact, this gives a symplectic

form on H

Let

⊆

) be the maximal torus consisting of diagonal entries. We can

show that the

action is Hamiltonian with moment map

. Since

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

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

, then no information is

retained.

Definition

(Effective action)

An action

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

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

M → R

for a Hamiltonian

action. If the

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

has orbits of dimension n.

This doesn’t mean all orbits are of dimension

. It just means some orbit

has dimension n.

Proof.

(i)

= (

, . . . , µ

) :

M → R

and

is a point in an

-dimensional orbit,

then

{

)

}

are linearly independent. So

(

) is an interior point (if

not in the interior, then there exists a direction

pointing out of

(

So (d

)

= 0, and d

gives a non-trivial linear combination of the d

’s

that vanishes).

So if there is an interior point, we know

(

) is a non-degenerate polytope

. This mean it has at least

+ 1 vertices. So there are at least

+ 1

fixed points.

(ii)

Let

be an orbit of

. Then

is constant on

by invariance of

O ⊆ ker(dµ

) = (T

So all orbits of a Hamiltonian torus action are isotropic submanifolds. So

dim O ≤

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 (

, ω

) equipped with an effective

action of an n-torus together with a choice of corresponding moment map µ.

Example. Take (CP

, ω

F S

), where the moment map is given by

µ([z

: z

: ··· : z

]) = −

(|z

, . . . , |z

)

+ |z

+ ··· + |z

Then this is a symplectic toric manifold.

Note that if (

M, ω, T µ

) is a toric manifold and

= (

, . . . , µ

) :

M → R

then µ

, . . . , µ

are commuting integrals of motion

{µ

, µ

} = ω(X

, X

) = 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

πZ

)

, and

fix an identification

∗

∼

. Given two toric manifolds (

, ω

, T, µ

) for

i = 1, 2, We say they are

(i)

equivalent if there exists a symplectomorphism

→ M

such that

ϕ(x · p) = x · ϕ(p) and µ

◦ ϕ = µ

(ii)

weakly equivalent if there exists an automorphism

T → T

and

→

symplectomorphism such that ϕ(x, p) = λ(x) · ϕ(p).

We also need a notion of equivalence of polytopes. Recall that

Aut

(

) =

GL(n, Z), and we can define

Definition.

AGL(n, Z) = {x 7→ Bx + c : B ∈ GL(n, Z), c ∈ R

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

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

are of the form

for t ≥ 0 and u

∈ Z

(iii)

Smoothness: For each vertex, the corresponding

’s can be chosen to be a

Z-basis of Z

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.

, there is any Delzant polytope is a line segment. This corre-

sponds to the toric manifold

as before, and the length of the polytope

corresponds to the volume of CP

under ω.

Example. In R

, 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

, but we shall

not discuss this.

Example.

The rectangular pyramid in

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 (

)

∗

with

facets, we want to

construct (

∆

, ω

∆

, T

∆

, µ

∆

) with

∆

(

∆

) = ∆. The idea is to perform the

construction for the “universal” Delzant polytope with

facets, and then obtain

the desired

∆

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

facets with primitive outward-point normal vectors

, . . . , v

(i.e.

they cannot be written as a

-multiple of some other

-vector), then we can

write ∆ as

∆ = {x ∈ (R

)

∗

: hx, v

i ≤ λ

for i = 1, . . . , d}

for some λ

There is a natural (surjective) map

→ R

that sends the basis vector

of R

to v

. If λ = (λ

, . . . , λ

), and we have a pullback diagram

∆ R

)

∗

)

∗

where

= {X ∈ (R

)

∗

: hX, e

i ≤ λ

for all i}.

In more down-to-earth language, this says

∗

(x) ∈ R

⇐⇒ x ∈ ∆,

which is evident from definition.

Now there is a universal “toric manifold” with

(

) =

, namely (

, ω

)

with the diagonal action

, . . . , t

) · (z

, . . . , z

) = (e

, . . . , e

using the moment map

φ(z

, . . . , z

) = −

(|z

, . . . , |z

) + (λ

, . . . , λ

We now want to pull this back along

∗

. To this extent, note that

sends

, hence induces a map

→ T

with kernel

. If

is the Lie algebra

of N , then we have a short exact sequence

0 −→ (R

)

∗

−→ (R

)

∗

−→ n

∗

−→ 0.

Since im π

∗

= ker i

∗

, the pullback of C

along π

∗

is exactly

Z = (i

∗

◦ φ)

−1

(0).

It is easy to see that this is compact.

Observe that

∗

◦ φ

is exactly the the moment map of the induced action

. So

Z/N

is the symplectic reduction of

, and in particular has a

natural symplectic structure. It is natural to consider

Z/N

instead of

itself,

since

carries a

action, but we only want to be left with a

action. Thus,

after quotienting out by

, the

action becomes a

∼

action, with

moment map given by the unique factoring of

Z → C

→ (R

)

∗

through (R

)

∗

. 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

→ M

such that

∗

. The notation we use is (

, ω

)

→

(

, ω

A natural question to ask is, if we have two symplectic manifolds, is there a

symplectic embedding between them?

For concreteness, take (

, ω

)

∼

(

, ω

), and consider the subsets

(

)

and

(

) =

(

)

×R

2n−2

(where the product is one of symplectic manifolds).

r ≤ R

, then there is a natural inclusion of

(

) into

(

)? If we only

ask for volume-preserving embeddings, then we can always embed

(

) into

(

), since

(

) 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

(n) → Z

(R) iff r < R.

When studying symplectic embeddings, it is natural to consider the following:

Definition

(Symplectic capacity)

A symplectic capacity is a function

from

the set of 2n-dimensional manifolds to [0, ∞] such that

(i) Monotonicity: if (M

, ω

) → (M

, ω

), then c(M

, ω

) ≤ c(M

, ω

(ii) Conformality: c(M, λω) = λc(M, ω).

(iii) Non-triviality: c(B

(1), ω

) > 0 and c(Z

(1), ω

) < ∞.

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

(M, ω) = sup{πr

| (B

(r), ω

) → (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

(

) into

any symplectic manifold as long as r is small enough.