Geometry of Toric Manifolds

Department of Physics and Astronomy

Advanced Physics - Project Course, 10c

Nikolaos Iakovidis

Supervisor: Jian Qiu

8 January 2016 Academic Year 2015/2016 Contents

1 Background 2 1.1 Complex Manifolds ...... 2 1.2 Almost Complex Structure ...... 2 1.3 Hermitian Manifolds ...... 3 1.4 K¨ahlerManifolds ...... 4 1.5 K¨ahlerManifold Examples ...... 5 1.5.1 Complex Space Cn ...... 5 1.5.2 Complex Projective Space CPn ...... 6

2 Toric Manifolds 8 2.1 Group Actions ...... 8 2.1.1 Orbits and Stabilizers ...... 9 2.1.2 Types of Group Action ...... 9 2.2 Hamiltonian Actions and the Moment Map ...... 9 2.3 K¨ahlerReduction ...... 10 2.4 Delzant Construction ...... 13

3 Examples 16 3.1 Circle action T1 on the complex plane C ...... 16 3.2 Torus action Td on Cd ...... 18 3.3 Circle action T1 on Cd ...... 19 3.4 Torus action Tn on CPn ...... 20 3.5 Delzant Polytope and Symplectic Reduction ...... 21 3.6 K¨ahlerReduction on C3 by circle action T1 ...... 24

1 Abstract This project is an overview of Hamiltonian geometry of Kählermanifolds and of Kähler reduction. In the first section we define complex manifolds, give their basic properties and build some structures on them. We are mainly interested in Kählermanifolds which are a subset of symplectic manifolds. In the second section we discuss group actions on manifolds. We are only concerned with Hamiltonian actions for which we can compute their moment maps. From these we prove how to construct new manifolds using a process called Kählerreduction. Finally we define toric manifolds and review Delzant polytopes and their corresponding manifolds. In the third section we present detailed examples in order to clarify all previously given definitions.

1 Background

1.1 Complex Manifolds A complex manifold M is deﬁned through the following axioms

1. M is a topological space 2. M is provided with a family of pairs {(Ui, ϕi)} 3. {Ui} is a collection of open sets which covers M. The map ϕi is a homeomorphism from m Ui to an open subset U of C (meaning that M is even dimensional) −1 4. Given Ui and Uj such that Ui ∩Uj 6= 0, the “transition map” ψji = ϕj ◦ϕi from ϕi(Ui ∩Uj) to ϕj(Ui ∩ Uj) is holomorphic.

We denote the complex dimension m of the manifold as dimC M = m and its real dimension as m dimR M = 2m. The simplest example of a complex manifold is C itself, where there is only one chart U1 (which covers the whole space) and ϕ is the identity map. Another important example is the complex projective space CPn. We take the n-tuple z = (z0, z1, . . . , zn) ∈ Cn+1 and define the equivalence relation w ∼ z if w can be written as w = λz, where λ 6= 0. Then the complex projective space is CPn ≡ (Cn+1)∗/ ∼. The homogeneous coordinates are denoted 0 1 n 0 1 n by [z : z : ··· : z ] and are defined by the identification of (z0, z1, . . . , zn) with λ(z , z , . . . , z ) (λ 6= 0). Hence, a point in CPn is a line through the origin in Cn+1. In CPn there are n charts Ui defined as Ui = {[z0 : z1 : ··· : zn] | zi 6= 0} (1.1) ν ν µ In a chart Uµ we define the inhomogeneous coordinates as ξ(µ) ≡ z /z . In the overlap of two n n charts Uµ ∩ Uν the transition function ψµν : C → C is zν ξλ 7→ ξλ = ξλ (1.2) (ν) (µ) zµ (ν) 1.2 Almost Complex Structure

Suppose we have a complex manifold M with dimC M = m. A point p belonging to a chart µ µ µ (U, ϕ) has coordinates z = x + iy . The tangent space TpM at p is then spanned by the 2m basis vectors ∂ ∂ ∂ ∂ ,..., ; ,..., (1.3) ∂x1 ∂xm ∂y1 ∂ym ∗ The dual tangent space Tp M has as basis

dx1,..., dxm; dy1,..., dym (1.4)

2 Using the above we can deﬁne 2m complex vectors ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ ≡ − i and ≡ + i (1.5) ∂zµ 2 ∂xµ ∂yµ ∂z¯µ 2 ∂xµ ∂yµ

C C These form a basis of the complexified tangent space TpM where we note that dimC TpM = m. ∗ C Similarly we obtain a basis for the complexified dual tangent space Tp M by using dzµ ≡ dxµ + i dyµ and d¯zµ ≡ dxµ − i dyµ (1.6) An almost complex structure J of a complex manifold M is a tensor field which at a point p acts as ∂ ∂ ∂ ∂ J = and J = − (1.7) p ∂xµ ∂yµ p ∂yµ ∂xµ

Thus Jp : TpM → TpM is a real tensor of type (1, 1). An important property of an almost complex structure is that 2 Jp = −idTpM (1.8)

It can be easily proven that the action of Jp is independent of the chart, and whichever coordinates we choose its form is always 0 −Im Jp = (1.9) Im 0 with respect to the aforementioned basis. It can be proven that any 2m-dimensional manifold admits locally (on each tangent space) a tensor field J (linear complex structure) which squares to I2m. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined (J is integrable). Now we extend Jp so that it can act on the complexified tangent C space TpM Jp (X + i Y ) ≡ JpX + iJpY (1.10) From our previous definitions it follows that ∂ ∂ ∂ ∂ J = i and J = −i (1.11) p ∂zµ ∂zµ p ∂z¯µ ∂z¯µ

This means that we can express Jp as ∂ ∂ J = i dzµ ⊗ − i d¯zµ ⊗ (1.12) p ∂zµ ∂z¯µ or in matrix form iIm 0 Jp = (1.13) 0 −iIm

1.3 Hermitian Manifolds

Let M be a complex manifold with dimC M = m and g a Riemannian metric of M. For two C elements of TpM (lets call them Z = X + i Y and W = U + iV ) we have

gp(Z,W ) = gp(X,Y ) − gp(Y,V ) + i [gp(X,V ) + gp(Y,V )] (1.14) The components of the metric with respect to the basis vectors ∂/∂zµ and ∂/∂z¯µ are ∂ ∂ ∂ ∂ g (p) = g , g (p) = g , (1.15) µν p ∂zµ ∂zν µν¯ p ∂zµ ∂z¯ν

3 ∂ ∂ ∂ ∂ g (p) = g , g (p) = g , (1.16) µν¯ p ∂z¯µ ∂zν µν p ∂z¯µ ∂z¯ν and its easy to verify that g is symmetric (for all index combinations) and also

gµν¯ = gµν¯ and gµν = gµν (1.17)

A Hermitian metric is a Riemannian metric of a complex manifold M that satisﬁes

gp(JpX,JpY ) = gp(X,Y ) (1.18)

∀p ∈ M and ∀X,Y ∈ TpM. The pair (M, g) is then called a Hermitian manifold. We note that the vector JpX is orthogonal to X with respect to a Hermitian metric

2 gp(JpX,X) = gp(Jp X,JpX) = gp(−X,JpX) = −gp(JpX,X) = 0 (1.19) We also note that a complex manifold always admits a Hermitian metric. The components of C the Hermitian metric with respect to the basis of TpM that we previously defined also have some nice properties. Specifically, we find ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ g (p) = g , = g J ,J = g i , i = −g , = 0 µν p ∂zµ ∂zµ p p ∂zµ p ∂zµ p ∂zµ ∂zµ p ∂zµ ∂zµ (1.20) and similarly gµ¯ν¯(p) = 0. So, finally the Hermitian metric can be written as

µ ν µ ν g = gµν¯ dz ⊗ d¯z + gµν¯ d¯z ⊗ dz (1.21)

1.4 K¨ahlerManifolds

Let (M, g) be a Hermitian manifold. We deﬁne a tensor ﬁeld Ω whose action on X,Y ∈ TpM is

Ωp(X,Y ) = gp(JpX,Y ) (1.22)

Ω is called K¨ahlerform. We can easily see that Ω has the following properties (we imply the point p from here on) • Ω is antisymmetric

Ω(X,Y ) = g(JX,Y ) = g(J 2X,JY ) = g(−X,JY ) = −g(JY,X) = −Ω(Y,X) (1.23)

• Ω is invariant under J

Ω(JX,JY ) = g(J 2X,JY ) = g(JX,Y ) = Ω(X,Y ) (1.24)

C If we extend the domain to TpM we ﬁnd that Ω is a 2-form of bidegree (1, 1) ∂ ∂ ∂ ∂ Ω , = g J , = i g = 0 (1.25) ∂zµ ∂zν ∂zµ ∂zν µν and similarly Ωµ¯ν¯ = 0. We also have ∂ ∂ ∂ ∂ Ω , = i g = −Ω , (1.26) ∂zµ ∂z¯ν µν¯ ∂z¯ν ∂zµ So we can write Ω as µ ν ν µ Ω = i gµν¯ dz ⊗ d¯z − i gνµ¯ d¯z ⊗ dz (1.27)

4 or µ ν Ω = i gµν¯ dz ∧ d¯z (1.28) A K¨ahlermanifold is a Hermitian manifold (M, g) whose K¨ahlerform is closed. Since dΩ = 0, we can obtain some more relations. We have ¯ µ ν dΩ = ∂ + ∂ i gµν¯ dz ∧ d¯z λ µ ν λ µ ν = i ∂λgµν¯ dz ∧ dz ∧ d¯z + i ∂λ¯gµν¯ d¯z ∧ dz ∧ d¯z 1 λ µ ν 1 λ µ ν = 2 i (∂λgµν¯ − ∂µgλν¯) dz ∧ dz ∧ d¯z + 2 i ∂λ¯gµν¯ − ∂ν¯gµλ¯ d¯z ∧ dz ∧ d¯z = 0

λ µ 1 λ µ µ λ where we used dz ∧ dz = 2 dz ∧ dz − dz ∧ dz and renaming of indices (also, we didn’t write any terms with only barred or unbarred indices for the metric because they are already zero). From the above we get

∂g ∂g ∂g ∂g ¯ µν¯ = λν¯ and µν¯ = µλ (1.29) ∂zλ ∂zµ ∂z¯λ ∂z¯ν

A metric which satisﬁes these conditions on a chart Ui is

gµν¯ = ∂µ∂ν¯Ki , Ki ∈ Fi (1.30) It can be proven that any Kähler metric can be written locally as the above expression. The function K is called Kählerpotential. Now we can write the Kählerform as ¯ Ω = i ∂∂Ki (1.31) on the chart Ui.

1.5 KählerManifold Examples 1.5.1 Complex Space Cn We take M = Cn. There is only one chart U and the coordinates of a point p are given by ϕ(p) = (z1, . . . , zn). We identify Cn ∼= R2n by zµ → xµ + iyµ so we have the Euclidean metric g with ∂ ∂ ∂ ∂ g , = g , = δ (1.32) ∂xµ ∂xν ∂yµ ∂yν µν and ∂ ∂ g , = 0 (1.33) ∂xµ ∂yν Its easy to verify that g(JX,JY ) = g(X,Y ) for all X,Y and thus, g is Hermitian. Using the above we find for the complex coordinates ∂ ∂ ∂ ∂ g , = g , = 0 (1.34) ∂zµ ∂zν ∂z¯µ ∂z¯ν and ∂ ∂ ∂ ∂ 1 g , = g , = δ (1.35) ∂zµ ∂z¯ν ∂z¯µ ∂zν 2 µν We now have the components of the Kählerform Ω, so we can write

n i i X Ω = δ dzµ ∧ d¯zν = dzµ ∧ d¯zµ (1.36) 2 µν 2 µ=1

5 Its obvious that Ω is closed (its coeﬃcients are constant) and the K¨ahlerpotential is

n 1 X K = zµz¯µ (1.37) 2 µ=1

1.5.2 Complex Projective Space CPn CPn µ Let M = and (Ui, ϕi) be a chart with inhomogeneous coordinates ϕi(p) = ξ(i) for µ 6= i. We introduce the coordinate notation  µ µ ξ(i) = ζ(i) µ ≤ i − 1 n µ o  ζ(i) | 1 ≤ µ ≤ m where (1.38)  µ+1 µ ξ(i) = ζ(i) µ ≥ i so that we have continuous counting for µ. We now choose the positive deﬁnite function zµz¯µ K (p) ≡ ζµ ζ¯µ + 1 = (1.39) i (i) (i) ziz¯i where summation over µ is implied. We deﬁne (locally) a closed 2-form Ω by

∂2 log K Ω = i∂∂¯log K = i i dζµ ∧ dζ¯ν (1.40) i ∂ζµ∂ζ¯ν

By substituting Ki we obtain δ ζλζ¯λ + 1 − ζ¯µζν Ω = i µν dζµ ∧ dζ¯ν (1.41) ζλζ¯λ + 12 where summation over µ, ν and λ is implied. There is also a Hermitian metric g that corresponds n n n to this K¨ahlerform Ω. We take X,Y ∈ TpCP and deﬁne g : TpCP ⊗ TpCP → R by g(X,Y ) = Ω(X,JY ). The components of the metric are

λ ¯λ ¯µ ν δµν ζ ζ + 1 − ζ ζ gµν¯ = −i Ωµν¯ = (1.42) ζλζ¯λ + 12 and thus µ ¯ν ¯ν µ g = gµν¯ dζ ⊗ dζ + dζ ⊗ dζ (1.43) It can be proven that g(JX,JY ) = g(X,Y ) and that g is positive deﬁnite, meaning that g is indeed a Hermitian metric.

Now we prove a relation that will be useful later on. We know that Ω is a real and deﬁ- nite positive 2-form of bidegree (1, 1). We also know that Ω is closed (dΩ = 0). These two properties mean that Ω is locally exact Ω = dα, where α is a real 1-form. Any real 1-form can be written as the sum α = β + β¯ where β is a 1-form of bidegree (1, 0). Also, since Ω is (1, 1) we need to have ∂β = 0 and ∂¯β¯ = 0. So Ω is written as

Ω = ∂β¯ + ∂β¯ (1.44)

From ∂β = 0 we have that locally β = ∂ϕ where ϕ is a smooth complex function. Thus, we can ﬁnally write Ω as Ω = ∂∂ϕ¯ + ∂∂¯ϕ¯ = ∂∂¯(ϕ ¯ − ϕ) = i∂∂f¯ (1.45)

6 where now f is a real function. We already know that i Ω = ∂∂¯log ζλζ¯λ + 1 (1.46) 2 where again, summation over λ is implied. Hence, our real function f is i f = log ζλζ¯λ + 1 (1.47) 2

Our ﬁnal goal is to ﬁnd α. So we trace our steps from before. We can write ϕ = ϕ1 + iϕ2 and have i 1 ϕ¯ − ϕ = log ζλζ¯λ + 1 ⇒ ϕ = − log ζλζ¯λ + 1 (1.48) 2 2 4 Now β and β¯ become

∂ϕ ∂ 1 β = ∂ϕ = 1 dζµ + i − log ζλζ¯λ + 1 dζµ ∂ζµ ∂ζµ 4 ∂ϕ i ζ¯µ = 1 dζµ − dζµ ∂ζµ 4 ζλζ¯λ + 1 and ∂ϕ ∂ 1 β¯ = ∂¯ϕ¯ = 1 dζ¯µ − i − log ζλζ¯λ + 1 dζ¯µ ∂ζ¯µ ∂ζ¯µ 4 ∂ϕ i ζµ = 1 dζ¯µ + dζ¯µ ∂ζ¯µ 4 ζλζ¯λ + 1

So for α we have i ζµ dζ¯µ − ζ¯µ dζµ α = dϕ + (1.49) 1 4 ζλζ¯λ + 1 and for Ω i ζµ dζ¯µ − ζ¯µ dζµ Ω = dα = d (1.50) 4 ζλζ¯λ + 1 because d2 = 0. So we can deﬁne α0 as i ζµ dζ¯µ − ζ¯µ dζµ α0 = (1.51) 4 ζλζ¯λ + 1 and Ω = dα0. In homogeneous coordinates, this expression is written as

i zµd¯zµ − z¯µdzµ α0 = (1.52) 4 zλz¯λ where λ is summed over all values but µ cannot be i (this is easily seen if we perform all the previous calculations for the homogeneous coordinates).

7 2 Toric Manifolds

2.1 Group Actions If G is a group and X is a set, then a left group action ϕ of G on X is a function ϕ : G × X → X (g, X) → ϕ (g, X) which satisﬁes the following axioms (we denote ϕ(g, X) ≡ g.x) 1. e.x = x ∀x ∈ X (where e is the identity element of G) 2.( gh).x = g.(h.x) ∀g, h ∈ G, ∀x ∈ X From these axioms it follows that ∀g ∈ G the map ϕ is a bijection from X to X (with the inverse function mapping x to g−1x). If X is a topological space and ϕ is continuous, then ϕ is called continuous group action and the pair (X, ϕ) is called G-space.

Let G be a Lie group and M be a manifold. A group action σ : G × M → M is a Lie group action if it is differentiable. The orbit map at a specific point of the manifold, σp : G → M such that σp(g) = σ(p, g), is differentiable and one can compute its differential at the identity element of G deσp : g → TpM (2.1)

If X ∈ g, then its image under the map above is a vector belonging to TpM, and by varying p we obtain a vector field on M. The minus of this vector field is called the fundamental vector field associated with X and is denoted by X#. The fundamental vector field is then given by the following equivalent definitions

# d tX X (p) = deσp(X) = d(e,p)σ X, 0TpM = σ e , p (2.2) dt t=0 where d is the diﬀerential of a smooth map and 0TpM is the zero vector in TpM.

Let g be the Lie algebra of G. We deﬁne the map

Ψ: G → AutG , g → Ψg (2.3) where AutG is the automorphism group of G and Ψg is deﬁned by −1 Ψg(h) = ghg , ∀h ∈ G (2.4)

The diﬀerential of Ψg at the identity is an automorphism of the Lie algebra g and its called the adjoint action of the group G on its algebra g

deΨg ≡ Adg : g → g (2.5) If we use the exponential map we can write this as

d tX −1 Adg(X) = g e g (2.6) dt t=0 To deﬁne the group action on the dual of the Lie algebra g∗, consider the dual of the adjoint action ∗ ˜ ˜ ˜ ∗ hAdgX,Xi = hX, Adg−1 Xi ,X ∈ g, X ∈ g (2.7) This induces a group action on g∗ ˜ ˜ ˜ ∗ g.X = Adg−1 X, X ∈ g (2.8) which is called the coadjoint action.

8 2.1.1 Orbits and Stabilizers Consider a group G acting on a set X. The orbit of a point x ∈ X is the set of elements of X to which x can be moved by the elements of G. We denote the orbit of x by

G.x = {g.x | g ∈ G} (2.9)

The set of orbits of points x ∈ X under the action of G forms a partition of X. The associated equivalence relation is deﬁned by x ∼ y if there exists a g ∈ G such that g.x = y (meaning that x and y are in the same orbit). The orbits are then the equivalence classes under this relation. Two elements x, y are are equivalent only if their orbits are the same G.x = G.y. The set of orbits of X under the action of G is written as X/G and it is called the quotient of the action (or orbit space).

Given g ∈ G and x ∈ X with g.x = x, we say that x is a fixed point of g and g fixes x. ∀x ∈ X we define the stabilizer subgroup of G with respect to x as the set of all elements in G that fix x Gx = {g ∈ G | g.x = x} (2.10)

Gx is a subgroup of G, typically not a normal one.

2.1.2 Types of Group Action • Proper: A group action of a topological group G on a topological space X is said to be a proper action if the mapping G × X → X × X ,(g, x) → (g.x, x), is a proper map, i.e. the preimage of a compact set is compact. Actions of compact groups are always proper.

• Transitive: The action of a group G on X is transitive if X is non-empty and ∀x, y ∈ X there exists g ∈ G such that g.x = y. This means that the group action possesses only one orbit G.x = X. Also, in this case X is isomorphic to the left cosets of the isotropy (stabilizer) ∼ group X = G/Gx. • Eﬀective (or Faithful): The group action of G is eﬀective if for any distinct g, h ∈ G there exists an x ∈ X such that g.x 6= h.x, or equivalently, if for any g 6= e in G there exists an x ∈ X such that g.x 6= x. This means that there are no trivial actions. In particular, there is no element of the group (besides the identity element) which does nothing, leaving every point x as it is. This can also be expressed as \ Gx = {e} (2.11) x∈X

where Gx is the stabilizer subgroup and e is the identity. • Free: A group action G × X → X is called free if ∀x ∈ X, g.x = x implies g = e. In other words, the action is free if the map G × X → X × X,(g, x) → (α(g, x), x) is injective, so that α(g, x) = x implies that g = e, ∀g ∈ G and ∀x ∈ X. This means that all stabilizers are trivial.

2.2 Hamiltonian Actions and the Moment Map A Kählermanifold (K, ω) is a symplectic manifold equipped with an integrable almost-complex structure which is compatible with the symplectic form ω. All the following definitions will be given for a Kählermanifold, but they also apply to symplectic manifolds (in fact, their original

9 form is written for symplectic manifolds). However, we will only consider Kählermanifolds, so we just confine our definitions there.

Hamiltonian vector field: Let (M, J, ω) be a Kählermanifold. A vector field X on M is Kählerif the contraction ιX ω is closed. A vector field X on M is Hamiltonian if ιX ω is exact. Also, the flow of X preserves ω, since the Lie derivative is zero

LX ω = d ◦ ιX ω + ιX ◦ dω = 0 (2.12) where we used Cartan’s magic formula. If a vector ﬁeld is Hamiltonian we can write ιX ω = dH for some smooth function H : M → R. Then, the ﬂow of X also preserves H

LX H = ιX dH = ιX ιX ω = 0 (2.13)

This means that each integral curve (ﬂow) of X, {ρt(x) | t ∈ R} must be contained in a level set of H ∗ R H(x) = (ρt H)(x) = H (ρt(x)) , ∀t ∈ (2.14) The function H is called the Hamiltonian function for the Hamiltonian vector ﬁeld X.

Moment Map: Let (M, J, ω) be a Kählermanifold and G a Lie group that acts on it. Also, suppose that the action of each g ∈ G preserves the Kählerform ω. Let g be the lie algebra of G, g∗ its dual, and h·, ·i : g∗ × g → R the pairing between them. Any X ∈ g induces a vector # field X on M (as we defined before). Let ιX# ω denote the contraction of this vector field with ω. Since the action of G preserves the Kählerform, it follows that ιX# ω is closed ∀X ∈ g

∗ σ (ω) = ω ⇒ LX# ω = 0 ⇒ d (ιX# ω) = 0 (2.15)

A moment map for the G-action on (M, J, ω) is a map µ : M → g∗ such that

dhµ, Xi = ιX# ω , ∀X ∈ g (2.16) where hµ, Xi is a function from M → R deﬁned by hµ, Xi(x) = hµ(x),Xi. We also use the notation µX (x) ≡ hµ(x),Xi (2.17) denoting the component of µ along X. Then we have

X dµ = ιX# ω (2.18) meaning that the function µX is a Hamiltonian function for the vector ﬁeld X#. The map µ should also be equivariant with respect to the coadjoint action on g∗

µ(g.x) = g.µ(x) = Adg−1 µ(x) (2.19)

If such a map µ exists, then the action is called Hamiltonian action and the vector (M, ω, G, µ) is called a Hamiltonian G-space.

2.3 K¨ahlerReduction We start by making two assumptions that apply to everything from here on

• All group actions are proper

10 • If the stabilizer group of a point is zero-dimensional, then it is trivial The Kählerreduction is the process of creating a principal G-bundle from a Hamiltonian G- space (M, ω, G, µ), where M is Kähler,equipped with a Kählerform on the base of the bundle. This is stated by the Marsden, Weinstein, Meyer theorem for a symplectic manifold.

Marsden - Weinstein - Meyer Theorem: Consider a Hamiltonian action of a Lie group G on a symplectic manifold (M, ω) with a corresponding moment map µ : M → g∗. Suppose 0 is a regular value of µ. Then µ−1(0) is a submanifold of M. Moreover, the action of G on µ−1(0) has zero-dimensional stabilizer groups (locally free action). Let i : µ−1(0) ,→ M be the inclusion map. Then,

−1 • The orbit space M0 = µ (0)/G is a manifold • π : µ−1(0) → µ−1(0)/G is a principal G-bundle

∗ ∗ • There is a symplectic form ω0 on M0 satisfying i ω = π ω0

Then (M0, ω0) is called the reduction of (M, ω) or symplectic quotient and we write

M//G := µ−1(0)/G (2.20)

All of the above apply to K¨ahlermanifolds since they are a subset of symplectic manifolds. In order to prove the theorem we have to show that • the action of G on µ−1(0) is free

−1 • µ (0) is a principal bundle over M0

• ω0 exists and it is closed and non-degenerate The necessary steps for the proof are the following 1. Let x ∈ M and G acting on M. The tangent space of the orbit G.x is

# Tx(G.x) = {X (x) | X ∈ g} (2.21)

2. The Lie algebra of the stabilizer subgroup Gx is

# gx = {X ∈ g | X (x) = 0} (2.22)

3. If V is a vector space and U ⊂ V is a subspace, then the annihilator U 0 of U in V ∗ is the subspace U 0 = {` ∈ V ∗ | `(u) = 0, ∀u ∈ U} (2.23)

4. If (V, ω) is a symplectic vector space and U ⊂ V is a subspace, then the symplectic perpendicular of U in V is the subspace

U ω = {v ∈ V | ω(v, u) = 0, ∀u ∈ U} (2.24)

∗ 5. Lemma: If x ∈ M, v ∈ TxM, X ∈ g and µ : M → g we have

# hdµx(v),Xi = ωx(X (x), v) (2.25)

which is easily proven using the deﬁnition of the moment map.

11 6. The diﬀerential of the moment map at a point x ∈ M is

∗ dµx : TxM → g (2.26)

The annihilator of the image of the moment map is

0 (Im(dµx)) = {X ∈ g | X(`) = 0, ∀` ∈ Im(dµx)} (2.27)

where ` = dµx(v), ∀v ∈ TxM. This can also be written as

hdµx(v),Xi = 0 (2.28)

and using the lemma we easily ﬁnd that

0 (Im(dµx)) = gx (2.29)

The symplectic perpendicular to Tx(G.x) is

ω # # (Tx(G.x)) = {v ∈ TxM | ω(v, X ) = 0, ∀X ∈ Tx(G.x)} (2.30)

By using the lemma again we have (∀X ∈ g)

hdµx(v),Xi = 0 ⇒ dµx(v) = 0 ⇒ v ∈ ker(dµx) (2.31)

So we ﬁnally get ω ker(dµx) = (Tx(G.x)) (2.32)

7. Now if we count dimensions we can get

ω dim(TxM) = dim(Tx(G.x)) + dim (Tx(G.x)) (2.33)

or, using the previous step

dim(ker(dµx)) = dim(TxM) − dim(Tx(G.x)) (2.34)

From the rank-nullity theorem we obtain for the map dµx

dim(Im(dµx)) + dim(ker(dµx)) = dim(TxM) (2.35)

Combining the above we ﬁnd

dim(Im(dµx)) = dim(Tx(G.x)) (2.36)

A point x ∈ M is a regular point of µ if dµx is surjective. From the previous discussion we conclude that this is equivalent to saying that the stabilizer of x in G is ﬁnite (Gx is zero-dimensional). So if 0 is a regular value of µ (meaning that all preimage points µ−1(0) −1 are regular) then the action of G on µ (0) is locally free. Using our assumption (if Gx is zero-dimensional then it is trivial) we have that the action of G on µ−1(0) is free.

8. Theorem: Suppose a Lie group G acts freely and properly on a manifold P . Then the quotient B = P/G is a Hausdorﬀ manifold and the orbit map π : P → B makes P into a principal bundle over B.

Using this theorem for P = µ−1(0) and B = µ−1(0)/G proves that µ−1(0) is a principal bundle over M0.

12 9. Proposition: Let G → P → B be a principal G-bundle. If a form ω ∈ Ω•(P ) satisﬁes

∗ σ ω = ω and ιX# ω = 0 , ∀X ∈ g (2.37)

∗ • where σ is the pullback of the group action, then there exists a form ω0 ∈ Ω (B) such that ∗ ω = π ω0.

∗ −1 We have ω on M and i ω = ω|µ−1(0) on µ (0). We also know that ω is invariant with respect to the G-action since the action is Hamiltonian. So, in order to show that there exists ∗ ∗ ∗ a 2-form ω0 on M0 with π ω0 = i ω it is enough to check that ιX# (i ω) = 0 ∀X ∈ g. For any −1 −1 x ∈ µ (0) the tangent space Txµ (0) is the kernel of dµx

ker(dµx) = {v ∈ TxM : dµx(v) = 0} (2.38)

−1 or v ∈ Txµ (0). For all v ∈ ker dµx we have

# ωx(X (x), v) = hdµx(v),Xi = h0,Xi = 0 (2.39)

∗ ∗ ∗ Therefore i ω has the necessary properties and so we have i ω = π ω0.

∗ ∗ 10. Now we have to show that ω0 is closed. Since π is injective, it is enough to prove that π ω0 is closed. We have ∗ ∗ ∗ d(π ω0) = d(i ω) = i (dω) = 0 (2.40)

11. The last step is to show that ω0 is non-degenerate. If (V, ω) is a symplectic vector space and U is a subspace with the property U ⊂ U ω, then the quotient U ω/U is naturally a −1 symplectic vector space. Therefore, the form ω0 is non-degenerate iﬀ ∀x ∈ µ (0) we have −1 ωx −1 Txµ (0) = Tx(G.x) so that for M0 = µ (0)/G we have

−1 ωx T[x]M0 = Txµ (0)/Tx(G.x) = Tx(G.x) /Tx(G.x) (2.41) being a symplectic manifold. We have

ωx −1 ker(dµx) = Tx(G.x) and ker(dµx) = Txµ (0) (2.42) So we can conclude that −1 ωx Txµ (0) = Tx(G.x) (2.43)

2.4 Delzant Construction Delzant showed that symplectic toric manifolds can be classiﬁed (as Hamiltonian spaces) by a set of polytopes. Before we prove this statement we need to give the deﬁnitions of a toric manifold and a polytope, and also describe their properties.

A (symplectic) toric manifold is a 2n-dimensional compact connected symplectic manifold (M 2n, ω) equipped with an eﬀective Hamiltonian action of a n-torus Tn and a corresponding moment map µ : M → Rn. note: The Lie group Tn is G = U(1)n and its Lie algebra is g = Rn.

A Delzant polytope ∆ in (Rn)∗ is a convex polytope satisfying • simplicity: there are n edges meting at each vertex

• rationality: the edges meeting at the vertex p are rational in the sense that each edge is n ∗ of the form p + tui, t ≥ 0 where ui ∈ (Z )

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

Let ∆ be a Delzant polytope in (Rn)∗ (space of the image of the moment map of a toric manifold, n n dual to R ) with d facets (a facet is a (d − 1)-dimensional face of the polytope). Let vi ∈ Z , n i = 1, . . . , d be the primitive (vi cannot be written as vi = kvj where k ∈ Z and |k| > 1), outward-pointing normal vectors to the facets of ∆. The Delzant polytope ∆ can then be discribed as an intersection of halfspaces

n ∗ ∆ = {x ∈ (R ) | hx, vii ≤ λi , i = 1, . . . , d , λi ∈ R} (2.44)

Delzant Theorem: Toric manifolds are classiﬁed by Delzant polytopes. Speciﬁcally, there is a 1 − 1 correspondence between a toric manifold and a polytope

{toric manifold} −→ {Delzant polytope} (M 2n, ω, Tn, µ) 7−→ µ(M)

Next we will show that the theorem is surjective, meaning that given a Delzant polytope we can recover the corresponding toric manifold

∆n −→ (M 2n, ω, Tn, µ) (2.45)

d Let ei with i = 1, . . . , d be the standard Cartesian basis of R . We deﬁne the map π that maps Rd to Rn (n is the dimension of the polytope)

d n π∗ : R −→ R (2.46)

ei −→ vi (2.47)

d n Its easily proven that the map π is surjective and maps Z onto Z . Thus, π∗ induces a surjective map (π) between tori

Rd/2πZd π∗ Rn/2πZn

Td Tn 1

We give the following names to our groups and their algebras

G = Td , g = Rd ,H = Tn , h = Rn (2.48)

We deﬁne a subgroup of G named N by

N ≡ ker(π) ⊂ G (2.49) and n is its Lie algebra. This enables us to create an exact sequence of tori, which in turn induces an exact sequence on their Lie algebras

1 N i G = Td π H = Tn 1

0 n i∗ g = Rd π∗ h = Rn 0

The dual exact sequence of these algebras is

14 ∗ ∗ 0 h∗ = (Rn)∗ π g∗ = Rd∗ i n∗ 0

Now we specify our manifold M to be Cd. This is a K¨ahlermanifold and admits the sym- i P µ µ plectic form ω = 2 dz ∧ d¯z . We make this into a toric manifold by equipping it with the standard Hamiltonian action of G = Td

iθ1 iθd iθ1 iθd (e ,..., e ) · (z1, . . . , zd) = (e z1,..., e zd) (2.50) The moment map for this action is µ : Cd → Rd∗ (we prove this in the next section) 1 µ(z , . . . , z ) = − |z |2,..., |z |2 + const. (2.51) 1 d 2 1 d d and we choose the constant to be equal to (λ1, . . . , λd). If we restrict the action on C to the subgroup N we ﬁnd that it is Hamiltonian with moment map

i∗ ◦ µ : Cd → n∗ (2.52)

∗ We will use the notation i ◦ µ ≡ µN for convenience.

µ Cd g∗ = Rd∗

µN i∗ n∗

−1 Let Z = µN (0) be the zero-level set. It can be proven that the set Z is compact and N acts ∗ freely on Z. Then 0 ∈ n is a regular value of µN . The real dimension of Z can be found from

d dim(ker dµN (z)) = dim(TzC ) − dim(Tz(N.z)) = 2d − (d − n) = d + n (2.53)

−1 but TzµN (0) = ker(dµN (z)), so we have

dimR Z = d + n (2.54)

Using the Marsden-Weinstein-Meyer theorem we know that the orbit space M∆ = Z/N is also a compact manifold and its dimension is

dimR M∆ = d + n − (d − n) = 2n (2.55) where d−n is the dimension of the group (and its algebra) N. The point-orbit map p : Z → M∆ d is a principal N-bundle over M∆ and the inclusion of Z is j : Z,→ C . There also exists a symplectic form ω∆ on M∆ obeying ∗ ∗ p ω∆ = j ω (2.56) In diagrammatic form the principal bundle is

j N Z Cd p

M∆

It can be proven that the symplectic manifold (M∆, ω∆) is a toric manifold equipped with the action of Tn and its moment map image is

µ∆(M∆) = ∆ (2.57)

15 3 Examples

3.1 Circle action T1 on the complex plane C The circle group T1 = S1 = U(1) is deﬁned as T1 = {ξ ∈ C : |ξ| = 1} (3.1) so a group element can be represented by eit. The tangent space at the identity element of the group can be identiﬁed with the imaginary line in the complex plane {it : t ∈ R}. The exponential map is it → eit. The toric action on C, which is represented by the coordinates φ(p) = z, is σ : T1 × C → C (eit, z) → eitz Every component for the moment map of an action is given by

X dµ = ιX# ω (3.2) Our goal will be to compute the fundamental vector ﬁelds of the manifold that correspond to each Lie algebra vector X and also to write ω = dα so that we will have

X dµ = ιX# dα = −d(ιX# α) (3.3) where in the last step we used Cartan’s formula and assumed that the action also preserves the form α (Lie derivative equal to zero)

ιX# ω = ιX# dα = LX# α − d ◦ ιX# α = −d ◦ ιX# α (3.4) We will prove this assumption at the end of this subsection. First we calculate the 1-form α. The standard form ω on C is i ω = dz ∧ d¯z (3.5) 2 The most general form for α is α = Adz + Bd¯z (3.6) where A and B are functions of z andz ¯. Its exterior derivative gives us ∂A ∂B dα = d¯z ∧ dz + dz ∧ z¯ ∂z¯ ∂z ∂B ∂A = − dz ∧ d¯z ∂z¯ ∂z so we need ∂B ∂A i − = ∂z¯ ∂z 2 i i We can of course choose A = − 4 z¯ and B = 4 z so that we finally get i α = (zd¯z − z¯dz) (3.7) 4 Now we have to find the fundamental vector field that corresponds to the generator of the group i (the group and its algebra are 1-dimensional, so we only have one vector field to calculate). By definition we have

# d it d it X (z) = σ(e , z) = e z = iz (3.8) dt t=0 dt t=0

16 In order to ﬁnd the form of X# we observe that if we write

z = x + i y = r cos θ + i r sin θ (3.9) we have ∂z ∂z¯ = iz and = −i¯z (3.10) ∂θ ∂θ and thus, the fundamental vector ﬁeld is written as ∂ X# = (3.11) ∂θ We also want to write ∂/∂θ as a function of z andz ¯ so that we can contract it with α. We have

∂ ∂z ∂ ∂z¯ ∂ ∂ ∂ = + = i z − z¯ (3.12) ∂θ ∂θ ∂z ∂θ ∂z¯ ∂z ∂z¯

In our case, the moment map is µ : C → R∗ because g∗ ∼= R∗ and for its only component we have X # dµ = −d(ιX# α) = −dhα, X i (3.13) by the deﬁnition of the interior product for an 1-form. The inner product gives us

∂ ∂ hα, X#i = h i (zd¯z − z¯dz) , i z − z¯ i 4 ∂z ∂z¯ ∂ ∂ = 1 hzd¯z, z¯ i + 1 hz¯dz, z i 4 ∂z¯ 4 ∂z ∂ ∂ = 1 |z|2hd¯z, i + 1 |z|2hdz, i 4 ∂z¯ 4 ∂z 1 2 = 2 |z| So we ﬁnd that the moment map (up to constant) is 1 µ(z) = − |z|2 + λ (3.14) 2 where λ is constant and will be chosen depending on the situation. For example, if we chose λ = 1/2 we can see that the zero-level set

µ(z) = 0 =⇒ |z|2 = 1 (3.15) is the unit sphere µ−1(0) = {z ∈ C : |z|2 = 1} = S1.

# Here we prove that for our ﬁeld X , the Lie derivative LX# α is equal to zero and thus the form α is preserved by the toric action. This proof can easily be generalized to d dimensions. We adopt the notation i i α = − 4 z¯dz + 4 zd¯z = Aµdzµ (3.16) i i where A1 = − 4 z¯, A1¯ = 4 z and dz1 = dz, dz1¯ = d¯z (In d dimensions we would have d barred and d unbarred dimensions). We also drop the # from our ﬁeld and have

∂ ∂ ∂ X = iz − i¯z = Xµ (3.17) ∂z ∂z¯ ∂zµ

17 where again we have X1 = iz and X1¯ = −i¯z. The Lie derivative of this 1-form is written as

ν ν LX α = (X ∂νAµ + ∂µX Aν) dzµ (3.18) where we have summation over µ and ν. Now we compute each component of LX α. We have ν ν (LX α)1 = X ∂νA1 + ∂1X Aν 1 1¯ 1 1¯ = X ∂1A1 + X ∂1¯A1 + ∂1X A1 + ∂1X A1¯ 1 i 1¯ i 1 i = X ∂z(− 4 z¯) + X ∂z¯(− 4 z¯) + ∂zX (− 4 z¯) i i = −i¯z(− 4 ) + i(− 4 z¯) = 0

Similarly, we ﬁnd that (LX α)1¯ is also zero, hence the Lie derivative is zero.

3.2 Torus action Td on Cd This example is just the generalization of the previous one. We now have the toric manifold (Cd, Td) equipped with the toric action

it1 itd it1 itd σ : (e ,..., e ), (z1, . . . , zd) → e z1,..., e zd (3.19) The standard form on Cd is d i X ω = dz ∧ d¯z (3.20) 2 i i i=1 and the 1-form α is d i X α = (z d¯z − z¯ dz ) (3.21) 4 i i i i i=1 # Now we have n fundamental vector ﬁelds Xi , since the group and its Lie algebra are d- dimensional Td = G = U(1) × · · · × U(1) (3.22) and td = g = u(1) ⊕ · · · ⊕ u(1) ∼= R ⊕ · · · ⊕ R = Rd (3.23) # Thus, the vector ﬁelds Xi are given by d d # iti iti Xi (z) = σ((0,..., e ,... 0), z) = e zi = izi (3.24) dt t=0 dt t=0 where z = (z1, . . . , zd). Using the results of the previous example, we have # ∂ ∂ ∂ Xi = = i zi − z¯i (3.25) ∂θi ∂zi ∂z¯i The moment map is µ : Cd → (Rd)∗ and each of its components is given by

Xi # dµ = −d(ι # α) = −dhα, Xi i (3.26) Xi where again we used the fact that the Lie derivative LX# α is zero (the proof is the same as it # was in one complex dimension). In the inner product between α and Xi only the ith component of α will be picked up, so we get

d X ∂ ∂ 1 hα, X#i = h i (z d¯z − z¯ dz ) , i z − z¯ i = |z |2 (3.27) i 4 j j j j i ∂z i ∂z¯ 2 i j=1 i i

18 and ﬁnally, the moment map is written as 1 µ(z) = − |z |2,..., |z |2 + λ (3.28) 2 1 d where λ = (λ1, . . . , λd) is constant.

3.3 Circle action T1 on Cd This is a continuation from our previous example, and its the restriction of the Td action to the subgroup T1. We name our groups G = Td and N = T1 and create the map

i : N → G (3.29) and its pullback for their dual Lie algebras

i∗ : n∗ = R∗ → g∗ = (Rn)∗ (3.30)

We deﬁne the group action as

σ : i(N) × Cd → Cd it it it i(e ), (z1, . . . , zd) → e z1,..., e zd meaning that the group acts diagonally. Note that this is not a toric action since the dimension of the group is not half the dimension of the manifold. Then the fundamental vector ﬁeld is

# d it d it it X (z) = σ(e , z) = e z1,..., e zd = i(z1, . . . , zd) (3.31) dt t=0 dt t=0 This means that X# is

d ∂ ∂ X ∂ ∂ X# = + ··· + = i z − z¯ (3.32) ∂θ ∂θ i ∂z i ∂z¯ 1 d i=1 i i The moment map will be i∗ ◦ µ : Cd → n∗ = R∗ (3.33) d ∗ where µ is the moment map for the full group G = T . We will again use the notation i ◦µ ≡ µN to denote the moment map of the restricted action. To ﬁnd the moment map we ﬁrst calculate the inner product between α and X#

d d d i X X ∂ ∂ 1 X 1 hα, X#i = h (z d¯z − z¯ dz ) , i z − z¯ i = z z¯ = |z|2 4 i i i i i ∂z i ∂z¯ 2 i i 2 i=1 i=1 i i i=1 Thus, the moment map is 1 µ (z) = − |z|2 + λ (3.34) N 2 For λ = 1/2, the zero level set

2 2 µN (z) = 0 =⇒ |z1| + ··· + |zd| = 1 (3.35) is the unit sphere −1 2d−1 Cd 2 2 µN (0) = S = {z ∈ : |z1| + ··· + |zd| = 1} (3.36)

19 −1 We can check that the real dimension of this manifold Z = µN (0) is d + n where n = d − 1 (because N is 1-dimensional). This is in accordance with the result we got during the proof of the Marsden-Weinstein-Meyer theorem.

−1 1 Now using Z = µN (0) as total space and N = S as ﬁber, we can create a principal N-bundle over M∆ 1 2d−1 S → S → M∆ (3.37) 2d−1 1 ∼ d−1 where M∆ = S /S = CP is the orbit space. This is the K¨ahlerquotient

S2d−1/S1 = Cd//U(1) = Cd/C∗ (3.38)

Thus, by the MWM theorem, CPd−1 is a toric manifold equipped with the action of the torus Td−1.

3.4 Torus action Tn on CPn We will use homogeneous coordinates and work, without loss of generality, on the coordinate neighborhood U0 where U0 = {[z0 : z1 : ··· : zn] | z0 6= 0} (3.39) The standard action of the torus Tn on CPn is given by

σ : Tn × CPn → CPn

it1 itn it1 itn (e ,..., e ), [z0 : z1 : ··· : zn] → z0 : e z1 : ··· : e zn

n The K¨ahlerform on CP is deﬁned (locally on U0) as i ¯ zµz¯µ Ω0 = ∂∂ log (3.40) 2 z0z¯0 where we sum over µ, and as we proven before, we can write Ω = dα with α being i z d¯z − z¯ dz α = µ µ µ µ (3.41) 4 zλz¯λ

# where µ 6= 0 but λ summed over all values. The fundamental vector ﬁelds Xi can be found using the same method as before and they are # ∂ ∂ ∂ Xi = = i zi − z¯i (3.42) ∂θi ∂zi ∂z¯i since the action has the same form as on Cn. Each component of the moment map is given by

Xi # dµ = −d(ι # α) = −dhα, Xi i (3.43) Xi For the inner product we have

2 # i zµd¯zµ − z¯µdzµ ∂ ∂ 1 |zi| hα, Xi i = h 4 , i zi − z¯i i = 2 (3.44) zλz¯λ ∂zi ∂z¯i 2 |z|

2 Pn where |z| = λ=1 zλz¯λ. The moment map is then given by 1 µ(z) = − |z |2,..., |z |2 + λ (3.45) 2|z|2 1 n

20 2 2 where λ = (λ1, . . . , λn) is constant. As a concrete example we take CP with the action of T

σ : T2 × CP2 → CP2

it1 it2 it1 it2 (e , e ), [z0 : z1 : z2] → z0 : e z1 : e z2

The moment map of this action is

2 2 1 |z1| |z2| µ([z]) = − 2 2 2 , 2 2 2 (3.46) 2 |z0| + |z1| + |z2| |z0| + |z1| + |z2| where we chose λ = 0. The image of this moment map is a triangle with vertices at (0, 0), (−1/2, 0) and (0, −1/2).

Using the same logic, the action of T3 on CP3 has the moment map

1 1 2 2 2 µ([z]) = − |z1| , |z2| , |z3| (3.47) 2 P3 2 i=0 |zi| The image of this moment map is a tetrahedron in R3. The vertices and the corresponding CP3 points are

[1 : 0 : 0 : 0] −→ (0, 0, 0) 1 [0 : 1 : 0 : 0] −→ (− 2 , 0, 0) 1 [0 : 0 : 1 : 0] −→ (0, − 2 , 0) 1 [0 : 0 : 0 : 1] −→ (0, 0, − 2 )

3.5 Delzant Polytope and Symplectic Reduction In this example we will start from a polytope ∆ and construct the corresponding toric manifold M∆ with momentum map image µ∆ = ∆. This example demonstrates in practice the theory we described in the Delzant construction section. The polytope we chose is

2 ∗ ∆ = {(x1, x2) ∈ (R ) | x1 ≥ 0, 0 ≤ x2 ≤ 1, x1 + ` x2 ≤ ` + 1} (3.48) and is shown in the ﬁgure below.

Figure 1: Polytope 1

21 We have 4 faces so we get d = 4 and we are in 2 dimensions, so n = 2. We denote x = (x1, x2) and we use the 4 equations hx, vii ≤ λi to ﬁnd the outward pointing vectors vi. The 4 λs are given by our deﬁnition of the polytope. We have

−x1 ≤ 0 ⇒ h(x1, x2), (−1, 0)i ≤ 0 ⇒ v1 = (−1, 0)

−x2 ≤ 0 ⇒ h(x1, x2), (0, −1)i ≤ 0 ⇒ v2 = (0, −1)

x2 ≤ 1 ⇒ h(x1, x2), (0, 1)i ≤ 1 ⇒ v3 = (0, 1)

x1 + ` x2 ≤ ` + 1 ⇒ h(x1, x2), (1, `)i ≤ ` + 1 ⇒ v4 = (1, `) where we had λ1 = λ2 = 0, λ3 = 1 and λ4 = ` + 1. Now its easy to compute the map π∗ where

4 2 π∗ : R −→ R (3.49)

ei −→ vi (3.50) and then the map π between the groups G = T4 and H = T2 which are part of the exact sequence

1 N i G = T4 π H = T2 1

0 n i∗ g = R4 π∗ h = R2 0 with N = ker π (3.51)

The map π∗ is of course a 2 × 4 matrix, and its entries can be found by solving the equations

π∗(1, 0, 0, 0) = (−1, 0)

π∗(0, 1, 0, 0) = (0, −1)

π∗(0, 0, 1, 0) = (0, 1)

π∗(0, 0, 0, 1) = (1, `)

We write down a general form for the map π∗ a11 a12 a13 a14 π∗ = (3.52) a21 a22 a23 a24 and we ﬁnd −1 0 0 1 π = (3.53) ∗ 0 −1 1 ` A general element of g = R4 will get mapped to h = R2 as follows

π∗(A, B, C, D) = (−A + D, −B + C + `D) (3.54)

The corresponding group element of G = T4 will be mapped to H = T2

π(a, b, c, d) = a−1d, b−1cd` (3.55)

The kernel of this map is our group N

N = ker π = {(a, b, c, d) | a = d, b = cd`} ∼= T2 (3.56)

22 The inclusion map i : N 7→ G is

i :(a, b) 7→ (a, b, ba−`, a) (3.57) and its push-forward between their Lie algebras i∗ : n 7→ g is

i∗ :(A, B) 7→ (A, B, B − `A, A) (3.58) We also have the exact sequence between the dual algebras

∗ ∗ 0 h∗ = (R2)∗ π g∗ = (R4)∗ i n∗ 0

In order to compute the pullback of i we take the inner product between an element of the Lie algebra n and an element of its dual algebra n∗ with

∗ ∗ X = (A, B) ∈ n and Y = i (C1,C2,C3,C4) ∈ n (3.59)

∗ where (C1,C2,C3,C4) ∈ g and use the inner product property as follows

∗ hY,Xi = hi (C1,C2,C3,C4), (A, B)i

= h(C1,C2,C3,C4), i∗(A, B)i

= h(C1,C2,C3,C4), (A, B, B − `A, A)i

= C1A + C2B + C3(B − `A) + C4A

= (C1 − `C3 + C4)A + (C2 + C3)B from which it is immediately obvious that the pullback i∗ is

∗ i :(C1,C2,C3,C4) 7→ (C1 − `C3 + C4,C2 + C3) (3.60) The moment map of the large group G is

µ : C4 → (R4)∗ (3.61) where 1 µ(z , z , z , z ) = − |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 |z |2 |z |2 |z |2 |z |2 = − 1 , − 2 , − 3 + 1, − 4 + ` + 1 2 2 2 2 If we constrain the action to the small group N we have the moment map

∗ 4 ∗ µN = i ◦ µ : C → n (3.62)

∗ Having calculated the map i it is now easy to ﬁnd µN

∗ µN (z1, z2, z3, z4) = i ◦ µ(z1, z2, z3, z4) |z |2 |z |2 |z |2 |z |2 = i∗ − 1 , − 2 , − 3 + 1, − 4 + ` + 1 2 2 2 2 |z |2 |z |2 |z |2 |z |2 |z |2 = − 1 − ` − 3 + 1 − 4 + ` + 1, − 2 − 3 + 1 2 2 2 2 2 |z |2 |z |2 |z |2 |z |2 |z |2 = − 1 + ` 3 − 4 + 1, − 2 − 3 + 1 2 2 2 2 2

23 −1 The zero level set Z = µN (0) is then |z |2 |z |2 |z |2 |z |2 |z |2 1 − ` 3 + 4 = 1 and 2 + 3 = 1 (3.63) 2 2 2 2 2

From these equations we see that the pair (z2, z3) and the pair (z1, z4) cannot be zero. The real dimension of Z is d + n, so we have dimR Z = 6 (3.64) 2 Now we mod out Z by our group N = T to get M∆ = Z/N which gives us the principal bundle N → Z → M∆. A point of M∆ is z = (z1, z2, z3, z4) ∈ Z obeying the equivalence relation (because of the ﬁber N) −` (z1, z2, z3, z4) ∼ (az1, bz2, ba z3, az4) (3.65)

The real dimension of M∆ is 2n which in our case means that

dimR M∆ = 4 (3.66)

This is easily seen from the fact that we start with the 6 dimensional manifold Z and mod out 2 by the 2 dimensional group N. Now we have a toric manifold (M∆, T ) equipped with a moment map having the property µ∆(M∆) = ∆ (3.67) where ∆ is the trapezoid we started from. From the above discussion we can conclude that the 1 ∼ 1 manifold M∆ is the total space of a CP ﬁbration with ﬁber = CP

[z1 : z4] × [z2 : z3] (3.68)

This is called a Hirzebruch surface.

3.6 K¨ahlerReduction on C3 by circle action T1 C3 is a toric manifold equipped with the action of T3 = U(1) × U(1) × U(1). We are interested in restricting this action to the group N which is 1 dimensional with weights (1, 1, −p)

N = {eit, eit, e−ipt ∈ U(1) × U(1) × U(1)} ∼= T1 (3.69)

We will use the notation eit = a like we did in the previous example. Our full group G = T3 has dimension 3 and the group N has dimension 1, so we have d = 3 and n = 2. Our exact sequence is

1 N i G = T3 π H = T2 1

0 n i∗ g = R3 π∗ h = R2 0

Since we already know N we also know ker π

N = ker π = {(a, b, c) | a = b, c = a−p} (3.70)

From this we can construct the map π, which in our case will not be unique since we only know its kernel. A possible action of π on an element (a, b, c) ∈ G is

π(a, b, c) = (a−1b, b−pc−1) (3.71)

24 from which we can easily check that N = ker π (we chose this action because it produces a polytope in the ﬁrst quadrant of R2). The corresponding push forward between the lie algebras g ∼= R3 and h ∼= R2 is π∗(A, B, C) = (−A + B, −pB − C) (3.72)

This means that the matrix form of π∗ : ei → ui is −1 1 0 π = (3.73) ∗ 0 −p −1

3 where the vectors ei are the Cartesian basis of R and ui are the normal vectors to the facets of the Delzant polytope. By using π∗ei = ui we ﬁnd

u1 = (−1, 0) , u2 = (1, −p) , u3 = (0, −1) (3.74)

The corresponding polytope is shown in ﬁgure 2. From the polytope we obtain the relations 1 x ≥ 0 , y ≥ 0 , y ≥ (x − λ) ⇒ x − py ≤ λ (3.75) p so we have

(x, y) · (−1, 0) ≤ λ1 ⇒ −x ≤ λ1 ⇒ λ1 = 0

(x, y) · (1, −p) ≤ λ2 ⇒ x − py ≤ λ2 ⇒ λ2 = λ

(x, y) · (0, −1) ≤ λ3 ⇒ −x2 ≤ λ3 ⇒ λ3 = 0

The inclusion map i : N → G and its push-forward are

i : a 7→ (a, a, a−p) (3.76)

i∗ : A 7→ (A, A, −pA) (3.77) ∗ Again we need the pullback i in order to compute the moment map µN that corresponds to to the action of N on C3 ∗ 3 ∗ µN = i ◦ µ : C → n (3.78)

Figure 2: Polytope 2

25 The moment map for G is 1 µ(z , z , z ) = − |z |2, |z |2, |z |2 + (0, λ, 0) (3.79) 1 2 3 2 1 2 3 ∗ ∗ ∗ In order to ﬁnd i we use the same method as before. We take X ∈ n, Y = i (C1,C2,C3) ∈ n and compute their inner product

∗ hY,Xi = hi (C1,C2,C3),Xi

= h(C1,C2,C3), i∗Xi

= h(C1,C2,C3), (X,X, −pX)i

= XC1 + XC2 − pXC3

= (C1 + C2 − p C3) X so we obtain ∗ i :(C1,C2,C2) 7→ C1 + C2 − p C3 (3.80)

Therefore, the moment map µN is 1 µ (z , z , z ) = i∗ − |z |2, |z |2, |z |2 + (0, λ, 0) N 1 2 3 2 1 2 3 |z |2 |z |2 |z |2 = − 1 − 2 + λ + p 3 2 2 2 −1 The zero level set Z = µN (0) then consists of points obeying the relation 2 2 2 |z1| + |z2| − p |z3| = C (3.81) where C = 2λ. Since |z3| ≥ 0 we have the restriction

2 2 |z1| + |z2| ≥ C (3.82) meaning that (z1, z2) 6= (0, 0) and for each value of z3 we will obtain a 3-sphere with size 2 2 C + p|z3| . The projection of this set on R (where (x, y, z) are the moduli of (z1, z2, z3)) is shown in the ﬁgure below

Figure 3: Projection of the set Z

The manifold Z is of course a subset of C3 with dimension

dimR Z = d + n = 5 (3.83)

26 Now ﬁnally, the reduction of Z by our group N is a manifold with (z1, z2, z3) ∈ Z equipped with the equivalence relation it it −ipt (z1, z2, z3) ∼ (e z1, e z2, e z3) (3.84)

So M∆ = Z/N is deﬁned by

3 2 2 2 it it −ipt M∆ = {(z1, z2, z3) ∈ C : |z1| + |z2| − p|z3| = C and (z1, z2, z3) ∼ (e z1, e z2, e z3)}

The dimension of M∆ = Z/N is

dimR M∆ = dimR Z − dimR N = 4 (3.85) which is equal to dimR M∆ = 2n (3.86) as expected by the MWM theorem (note that in this example the Delzant polytope was un- bounded and the set Z and the manifold Z/N are not compact. However, the action of N is free on Z so the theory is the same). We will now prove that the manifold M∆ is diﬀeomorphic to the total space of the line bundle O(−p) over CP1 which is deﬁned as

3 −p O(−p) = {(z1, z2, z3) ∈ C | (z1, z2, z3) ∼ (αz1, αz2, α z3) , α ∈ C} (3.87)

1 where (z1, z2) ∈ CP is the base and z3 ∈ C is the fiber. From the definition of the set Z we know that z1 and z2 cannot be zero at the same time. So we define 2 patches U1 and U2 where

2 2 U1 = {(z1, z2) ∈ C | z1 6= 0} ,U2 = {(z1, z2) ∈ C | z2 6= 0} (3.88)

iϕi iϕi We also deﬁne zi = rie andz ˆi = zi/|zi| = e . On patch U1 × C we take t = −φ1 and the equivalence relation becomes z1 z2 p (z1, z2, z3) ∼ , , z3zˆ1 = (u1, u2, u3) (3.89) zˆ1 zˆ1

On patch U2 × C we take t = −φ2 and we obtain z1 z2 p (z1, z2, z3) ∼ , , z3zˆ2 = (w1, w2, w3) (3.90) zˆ2 zˆ2 The relation of the coordinates between the two patches is p |z1||z2| zˆ2 w1 = and w3 = u3 (3.91) u2 zˆ1

Since z1 6= 0 in U1 and z2 6= 0 in U2 we can define new coordinates, diffeomorphic to the old ones, by u2 w1 p p ue2 = , we1 = , ue3 = u3|z1| , we3 = w3|z2| (3.92) |z1| |z2| Now the relation between the coordinates are p 1 z2 we1 = and we3 = ue3 (3.93) ue2 z1 These are the transition functions (the first one for CP1 and the second for the fibers) between the 2 patches of the OCP1 (−p) bundle. Thus, we can see that the manifold M∆ = Z/N is diffeomorphic to the total space of the line bundle O(−p).

27 References

[1] G. Szkelyhidi, An Introduction to Extremal Kaehler Metrics. American Mathematical Soc., Jun. 2014.

[2] M. Nakahara, Geometry, Topology and Physics, Second Edition. CRC Press, Jun. 2003.

[3] E. Meinrenken, Group actions on manifolds, 2003. [Online]. Available: http: //www.math.toronto.edu/mein/teaching/action.pdf

[4] J. Lee, Introduction to Topological Manifolds. Springer Science & Business Media, Dec. 2010.

[5] A. C. da Silva, Lectures on Symplectic Geometry, ser. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001, vol. 1764.

[6] E. Lerman, Symplectic Geometry and Hamiltonian Systems. [Online]. Available: http://www.math.uiuc.edu/∼lerman/467/v3.pdf