Contents 1 Recap & Motivation

Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

Contents

1 Recap & Motivation 1

2 Symplectic Toric Manifolds & the Delzant Construction 2

3 Examples 6 1 1 3.1 CP with a T -action ...... 6 2 4 3.2 T -action on C ...... 7

1 Recap & Motivation

Recall from Ana’s talk, the Atiyah-Guillemin-Sternberg convexity theorem [1, 2].

Theorem 1.1 (Atiyah, Guillemin-Sternberg). Let (M, ω) be a compact connected symplectic n n n manifold, and let T be an n-torus. Consider moment map µ : M → R = Lie(T ) for hamiltonian T n-action.

−1 n • the level sets µ (c) are connected ∀c ∈ R ;

• the image µ(M) is convex;

• the image µ(M) is the convex hull of the images of the ﬁxed points of the action.

The image µ(M) of the moment map is called the moment polytope.

2 2 2 Example 1. Consider CP with the Fubini-Study symplectic form (CP , ωFS). The T -action on 2 CP given by

iθ1 iθ2 iθ1 iθ2 (e , e ) · [z0 : z1 : z2] = [z0 : z1e : z2e ] has moment map 1 |z |2 |z |2 µ([z : z : z ]) = − 1 , 2 . 0 1 2 2 kzk2 kzk2 There are three ﬁxed points of the T 2-action, namely

[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1], which get mapped to 1 1 (0, 0), (− 2 , 0), (0, − 2 ),

2 respectively. Hence the moment polytope µ(CP ) is the right-angled triangle:

1 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

(−0.5, 0) (0, 0)

(0, −0.5)

n Question 2. Is this reversible? That is, given a convex polytope ∆ ⊂ R , does there exist a compact connected symplectic manifold (M∆, ω∆) with some Lie group action G M∆, such

that µ∆(M∆) = ∆? If so, is M∆ unique?

A partial converse to the AGS convexity theorem is given by the Delzant construction.

2 Symplectic Toric Manifolds & the Delzant Construction

Deﬁnition 3. A symplectic toric manifold is a compact connected symplectic manifold (M 2n, ω) with an eﬀective Hamiltonian action of a torus T n of dimension equal to half the dimension of the manifold 1 dim T n = dim M 2n, 2

n and with a choice of corresponding moment map µ : M → R .

The previous example is symplectic toric manifold: 2 1 1 1 1 1 2 M = CP , and G = S × S = T , so dimC T = 2 dimC CP .

n Deﬁnition 4. A Delzant polytope ∆ in R is a polytope satisfying:

simplicity: there are n edges meeting at each vertex;

n rationality: each edge that meets a vertex p is of the form p + tui, with ti ≥ 0 and ui ∈ Z ;

n smoothness: for each vertex, the corresponding u1, . . . , un can be chosen to be a Z-basis of Z .

It turns out that the moment polytope of a symplectic toric manifold is Delzant.

Lemma 2.1. For any symplectic toric manifold (M, ω), its moment polytope is Delzant.

Proof. Let x ∈ M be a ﬁxed point of the action, then µ(x) = p is a vertex of ∆ = µ(M). In the proof of the AGS convecity theorem, one proves that there exists a neighbourhood of x that

2 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019 locally looks like the convex cone

n X p + ttui : ti ≥ 0, i = 1, . . . n , i=1

n n where the u1, . . . un are the weights of the linearised T -action on TxM, say π : T → GL(TxM). So ∆ satisﬁes the simplicity and rationality conditions.

Now suppose that ∆ does not satisfy the smoothness condition. Then the matrix U = [u1 u2 . . . un] −1 n n n is not invertible as a Z-matrix. Pick some a ∈ U (Z ) such that a ∈ R − Z , then (using the bi-invariant metric on t) we get that hwi, ai ∈ Z for each i = 1, . . . , n. But as the ui are the ∼ n weights of the action, then (where π = ⊕i=1πi, with each πi of degree one because T is abelian) in the neighbourhood

2πihui,ai πi(exp(a)) · z = e z = z for i = 1, . . . n, for all points z in the convex neighbourhood of x. But exp(a) cannot be the identity element since in a dense open subset of M the action is free.

So this shows that any toric symplectic manifold has, as the image of its moment map, a Delzant polytope associated to it.

Theorem 2.2 (Delzant, [3]). Toric manifolds are classiﬁed by Delzant polytopes. More speciﬁ- cally, the bijective correspondence between these two sets is given by the moment map:

{toric manifolds} {Delzant polytopes} ←→ {T n-equivariant symplectomorphisms} {translations} 2n n (M∆ , ω∆,T , µ) ←→ µ(M∆) = ∆.

Sketch of proof. One can break the proof down into several steps:

1. M is toric =⇒ µ(M) is Delzant (already done).

2. ∆ is Delzant =⇒ there exists a compact connected symplectic manifold (M∆, ω∆).

3. Show that M∆ is toric with µ(M∆) = ∆.

4. Show that if M1 ' M2 =⇒ µ(M1) = µ(M2).

5. Show that if µ(M1) = µ(M2) =⇒ M1 ' M2.

We will prove the surjectivity statement (Step 2.) in the theorem, so that to any Delzant polytope ∆ there exists a corresponding toric symplectic manifold M∆. Then we can mimick the steps with a particular example for some Delzant polytope ∆ or some toric symplectic

3 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

manifold (M, ω) with a hamiltonian torus action T M, and see either what toric symplectic M or Delzant polytope ∆ pops out, respectively.

d n Let π : R → R be the linear map π : ei 7→ vi. The map is surjective as ∆ is Delzant, and maps d n Z → Z , so induces a map between tori (still denoted by π)

d d n n π : R /2πZ −→ R /2πZ , whose kernel N = ker(π) is an (n − d)-dimensional Lie subgroup of T d, with the inclusion i : N,→ T d. The exact sequence

1 N i T d π T n 1 induces the exact sequence of Lie algebras

i∗ d π∗ n 0 n R R 0, and corresponding dual exact sequence

n ∗ π∗ d ∗ i∗ 0 (R ) (R ) n 0.

d d Now let C be acted upon by the diagonal Hamiltonian action of T

iθj iθj e · zj = e z

d d ∗ which has the moment map φ : C → (R )

1 φ(z , . . . , z ) = − (|z |2,... |z |2) + constant, 1 d 2 1 d

d where we choose the constant to be (λ1, . . . , λd). Restricting the action on C to the subgroup N, it remains Hamiltonian [4] with moment map

∗ d ∗ µN := i ◦ φ : C → n .

−1 Set Z := µN (0) to be the zero-level set; it can be shown that Z is compact symplectic and N ∗ ∗ d acts freely on Z (see [4]). Since i is surjective, 0 ∈ n is a regular value of µN , and so Z ⊂ C is a compact submanifold of real dimension

d ∗ dimR Z = dimR C − dimR n = 2d − (d − n) = d + n.

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By the Marsden-Weinstein-Meyer theorem, the orbit space M∆ = Z/N is also a compact manifold, and its dimension is

dim M∆ = dimR Z − dimR N = (n + d) − (d − n) = 2n.

The point-orbit map p : Z → Z/N = M∆ is a principal N-bundle over M∆. By considering the diagram j d Z C p

M∆

d d where j : Z,→ C is the inclusion, and let ω0 be the standard symplectic form on C . Then the Marsden-Weinstein-Meyer theorem guarantees the existence of a symplectic form ω∆ on M∆ satisfying ∗ ∗ p ω∆ = j ω0.

Finally, as Z is connected, the compact symplectic 2n-dimensional manifold (M∆, ω∆) is connected [4]

Remark 5. Step 4 is actually straightforward to prove: if M1 ' M2, then the different moment maps for the same T n-action differ only by a constant in the dual Lie algebra. Thus the corresponding moment polytopes differ by a translation.

d Since the symplectic form ω∆ on M∆ coincides with that of ω0 on C when both are pulled back to Z, we have the following:

Corollary 6. The symplectic manifold (M∆, ω∆) constructed above has a natural K¨ahlerstruc- ture.

n ∗ n Remark 7. Let ∆ be a Delzant polyope in (R ) and with d facets. Let vi ∈ Z , i = 1, . . . d, be the primitive outward-pointing normal vectors to the facets of ∆. Then ∆ can be described as n intersection of half-spaces

n ∗ ∆ = {x ∈ (R ) : hx, vii ≤ λi, i = 1, . . . , d} for some λi ∈ R.

2 2 Example 8. From the T CP example from before:

2 ∗ 1 ∆ = x ∈ (R ) : x1 ≤ 0, x2 ≤ 0, x1 + x2 ≥ − 2 2 ∗ 1 = x ∈ (R ) : hx, (1, 0)i ≤ 0, hx, (0, 1)i ≤ 0, hx, (−1, −1)i ≤ 2

5 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

v1 = (0, 1)

− 1 ( 2 , 0) (0, 0)

v2 = (1, 0)

v3 = (−1, −1)

− 1 (0, 2 )

3 Examples

1 1 3.1 CP with a T -action

∗ Consider ∆ = [−1, 1] ⊂ R , so n = 1 and d = 2. Let v (= 1) be the standard basis vector in R. Then ∗ ∆ = {x ∈ R : hx, 1i ≤ 1, hx, −1i ≤ 1}.

The projection 2 π∗ : R → R : (1, 0) 7→ 1, (0, 1) 7→ −1

it it ∼ 1 2 has kernel ker π∗ = {(t, t): t ∈ R}, hence ker π = {(e , e ): t ∈ R} = S = N. Now i : N,→ T it it it is given by the inclusion e 7→ (e , e ), and so i∗(t) = (t, t). From this we can determine the ∗ 2 ∗ structure of i :(R ) → n using the standard inner-product:

∗ hi (x, y), zi = h(x, y), i∗(z)i = h(x, y), (z, z)i = (x + y)z = h(x + y), zi,

∗ ∗ 2 2 so i (x, y) = x + y ∈ n . The standard diagonal action of T on C has the moment map

|z |2 |z |2 φ(z , z ) = − 1 , − 2 + (λ , λ ), 1 2 2 2 1 2

∗ 2 ∗ ∼ where we will set λ1 = λ2 = 1. The induced moment map µ∆ = i ◦ φ : C → n = R is then

|z |2 |z |2 i∗ ◦ φ(z , z ) = i∗ 1 − 1 , 1 − 2 1 2 2 2 |z |2 |z |2 = 2 − 1 + 2 , 2 2

6 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019 whence ∗ −1 2 2 2 ∼ 3 Z = (i ◦ φ) (0) = {(z1, z2) ∈ C : |z1| + |z2| = 4} = S .

∼ 1 it it Finally, we quotient out by the action of N = S i.e. by the equivalence (z1, z2) ∼ (e z1, e z2) ∼ 3 1 ∼ 1 to get that M∆ = Z/N = S /S = CP .

∼ 1 Remark 9. It should not be hard to see that M∆ = CP for all 1-dimensional Delzant polytopes n n n ∆. In general, the toric action of T on CP has an n-dimensional polytope ∆ ⊂ R formed from n the convex hull of n-ﬁxed points of the action, which are the n + 1 points [..., 0, 1, 0,...] ∈ CP .

2 4 3.2 T -action on C

Consider the polytope

2 ∗ ∆ = (x1, x2) ∈ (R ) : x1 ≥ 0, 0 ≤ x2 ≤ 1, x1 + lx2 ≤ 1 + l 2 ∗ = (x1, x2) ∈ (R ) : hx, (−1, 0)i ≤ 0, hx, (0, −1)i ≤ 0, hx, (0, 1)i ≤ 1, hx, (1, l)i ≤ 1 + l 4 \ 2 ∗ = {(x1, x2) ∈ (R ) : hx, vii ≤ λi} i=1

(0, 1) = v3 (0, 1) (1, 1)

(−1, 0) = v1 4

(0, 0) (1 + l, 0) (0, −1) = v2

We have 4 edges (facets) so d = 4, and the polytope is 2-dimensional so n = 2. From the second equality for ∆ above, we see that (λ1, λ2, λ3, λ4) = (0, 0, 1, 1 + l). Since

4 2 π∗ := R → R : ei 7→ vi,

7 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

π∗ can be written down as a 2 × 4 matrix as π∗ = [v1 . . . v4], namely

  −1 0 0 1 π∗ =   0 −1 1 l hence

π∗(A, B, C, D) = (−A + D, −B + C + lD).

The corresponding group element of T 4 is mapped to an element of T 2 as

π(a, b, c, d) = (a−1d, b−1cdl), hence the kernel of π is

N = ker(π) = {(a, b, c, d): a = d, b = cdl} =∼ T 2.

Now the inclusion map i : N,→ T 4 is

(a, b) 7−→ (a, b, ba−l, a),

4 and its push-forward between Lie algebras i∗ : n → R is

(A, B) 7−→ (A, B, B − lA, A).

∗ 4 ∗ ∗ ∗ We need to ﬁnd the form of the pull-back i :(R ) → n in order to ﬁnd µ∆ = i ◦ φ, 2 which we can determine using the standard inner product on R ; for X = (A, B) ∈ n and ∗ Y = (C1,C2,C3,C4) ∈ n ,

D ∗ E hY,Xi = i (C1,C2,C3,C4), (A, B) D E = (C1,C2,C3,C4), i∗(A, B) D E = (C1,C2,C3,C4), (A, B, B − lA, A)

= (C1 − lC3 + C4)A + (C2 + C3)B D E = (C1 − lC3 + C4,C2 + C3), (A, B) , from which one deduces that

∗ i (C1,C2,C3,C4) = (C1 − lC3 + C4,C2 + C3).

8 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

Next, we consider the moment map

4 4 ∗ φ : C −→ (R ) 1 (z , z , z , z ) 7−→ − |z |2, |z |2, |z |2, |z |2 + (λ , λ , λ , λ ) 1 2 3 4 2 1 2 3 4 1 2 3 4 1 = − |z |2, −|z |2, −|z |2 + 2, −|z |2 + 2l + 2 2 1 2 3 4

d Now we restrict the hamiltonian action on C to the subtorus N with moment map

∗ d ∗ µ∆ = i ◦ φ : C −→ n 1 (z , z , z , z ) 7−→ − |z |2 + l|z |2 − |z |2 + 2, −|z |2 − |z |2 + 2 . 1 2 3 4 2 1 3 4 2 3

−1 The zero-level set Z = µN (0) can now be seen to be determined by

2 2 2 2 2 |z1| − l|z3| + |z4| = 2, and |z2| + |z3| = 2,

so neither of the pairs (z1, z4) and (z2, z3) can equal zero. Finally, dimR Z = 6 and so after taking 2 the quotient of Z by the action of N = T , we get the toric symplectic manifold M∆ = Z/N of dimension dimR M∆ = 4.

th Claim 10. M∆ is the l Hirzebuch surface.

Proof. I will relabel the coordinates as (z1, z2, z3, z4) = (x1, y1, y2, x2) and l as p, so that Z is now determined by the equations

2 2 2 |x1| − p|y2| + |x2| = 2,

2 2 |y1| + |y2| = 2, and where N = S1 × S1 acts on Z as

−p (α, 1) · (x1, x2; y1, y2) = (αx1, αx2; y1, α y2),

(1, β) · (x1, x2; y1, y2) = (x1, x2; βy1, βy2).

The ratio [x1 : x2] is preserved by the action of N, so projecting to the first factors gives us 1 1 a diffeomorphism p : M∆ → CP . The fibre above any given ratio [x1 : x2] ∈ CP has to be invariant under the action of N. The fibre above a point (x0, x1) is invariant under

−p α · (x1, x2), [y1 : y2] = (αx1, αx2), [0 : y1 : α y2] , where [1 : 0 : 0] is the point at infinity in each fibre, disjoint to the affine patches. When

9 Toric Symplectic Manifolds & the Delzant Construction 6th March 2019

U1 = {x1 6= 0} and U2 = {x2 6= 0}, it can be covered aﬃnely by

p x2/x1, x1y2/y1, on U1, p x1/x2, x2y2/y1, on U2,

∗ p so the transition function g12 : U1 ∩ U2 → GL(2, C ) is g12(x1, x2) = diag(1, (x2/x1) ), which is the cocycle that determines the sum OCP ⊕ OCP(−p) bundle. Projectivising (i.e. adding in

[1 : 0 : 0] to each ﬁbre) shows that M∆ is the total space of P(OCP ⊕ OCP(−p)), known as the pth Hirzebuch surface.

Other interpretations of M∆ for each p ≥ 1 is as the blow-up of the weighted projective space P(1, 1, p) at a point (see [5]), or as a rational normal scroll F(p, 0) (see [6], chapter 2). Note 2 that the Delzant polytope here can be obtained from the one for CP by cutting oﬀ one of the vertices (in a prescribed way). This operation is equivalent to a symplectic blow-up (see [4]).

References

[1] V. Guillemin and S. Sternberg. Convexity properties of the moment mapping. Inventiones mathematicae, 67(3):491–513, 1982.

[2] M. F. Atiyah. Convexity and commuting hamiltonians. Bulletin of the London Mathematical Society, 14(1):1–15, 1982.

[3] Thomas Delzant. Hamiltoniens périodiques et images convexes de l’application moment. Bulletin de la SociétéMathématiquede France, 116(3):315–339, 1988.

[4] Ana Cannas Da Silva and A Cannas Da Salva. Lectures on symplectic geometry, volume 3575. Springer, 2001.

[5] Paul Gauduchon. Hirzebruch Surfaces and Weighted Projective Planes, pages 25–48. 07 2010.

[6] Miles Reid. Chapters on algebraic surfaces. arXiv e-prints, pages alg–geom/9602006, Feb 1996.