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Geometry of Toric Manifolds

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Geometry of Toric Manifolds Department of Physics and Astronomy Advanced Physics - Project Course, 10c Geometry of Toric Manifolds 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 K¨ahlermanifolds and of K¨ahler reduction. In the first section we define complex manifolds, give their basic properties and build some structures on them. We are mainly interested in K¨ahlermanifolds 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 K¨ahlerreduction. 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 f(Ui;'i)g 3. fUig is a collection of open sets which covers M. The map 'i 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" ji = 'j ◦'i from 'i(Ui \Uj) to 'j(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 ' is the identity map. Another important example is the complex projective space CPn. We take the n-tuple z = (z0; z1; : : : ; zn) 2 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 = f[z0 : z1 : ··· : zn] j zi 6= 0g (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 µν : 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; ') 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 coordi- nates 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) 8p 2 M and 8X; Y 2 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 2 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 2 Fi (1.30) It can be proven that any K¨ahler metric can be written locally as the above expression.
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