III Symplectic Geometry - Hamiltonian vector fields

3Hamiltonian vector fields

III Symplectic Geometry

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 ∆.