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Complex Projective Space the Complex Projective Space Cpn Is the Most Important Compact Complex Manifold

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Complex Projective Space the Complex Projective Space Cpn Is the Most Important Compact Complex Manifold Complex projective space The complex projective space CPn is the most important compact complex manifold. By definition, CPn is the set of lines in Cn+1 or, equivalently, CPn := (Cn+1 0 )/C∗, \{ } ∗ n+1 n where C acts by multiplication on C . The points of CP are written as (z0, z1, ..., zn). ∗ Here, the notation intends to indicate that for λ C the two points (λz0, λz1, ..., λzn) and n ∈ (z0, z1, ..., zn) define the same point in CP . We denote the equivalent class by [z0 : z1 : ... : n zn]. Only the origin (0, 0, ..., 0) does not define a point in CP . n We take the standard open covering of CP . Let Ui be the open set n Ui := [z : ... : zn] zi =0 CP . { 0 | } ⊂ Consider the bijective maps n τi : Ui C → z0 zi−1 zi+1 zn [z0 : ... : zn] , ..., , , ..., → zi zi zi zi For the transition maps −1 τij = τi τ : τj(Ui Uj) τi(Ui Uj) ◦ j ∩ → ∩ w1 wi−1 wi+1 wj−1 1 wi+1 wn (w1, ..., wn) , ..., , , ..., , , , ..., → wi wi wi wi wi wi wi is biholomorphic. In fact, −1 τij(w , ..., wn)= τi τ (w , ..., wn) 1 ◦ j 1 = τi([w1 : ... : wj−1 : 1 : wj+1 : ... : wn]) w1 wi−1 wi+1 wj−1 1 wi+1 wn = τi : ... : : 1 : : ... : : : : ... : wi wi wi wi wi wi wi w w − w w − 1 w w = 1 , ..., i 1 , i+1 , ..., j 1 , , i+1 , ..., n wi wi wi wi wi wi wi In particular, when n = 1, CP1 = U U where 0 ∪ 1 z1 1 U0 = [z0 : z1] z0 =0 = [1 : z0 =0 = [1 : w] w C S , { | } { z0 | } { | ∈ } ≃ − {∞} and z0 1 U1 = [z0 : z1] z1 =0 = [ :1 z1 =0 = [w : 1] w C S 0 . { | } { z1 | } { | ∈ } ≃ −{ } 8 Then 1 τ = τ τ −1(w)= τ ([w : 1]) = , and τ = τ −1. (2) 01 0 ◦ 1 0 w 10 01 Complex tori We’ll study “genus” g of a compact Riemann surface M, the number of “holes” of M. When g = 0, M is biholomorphic to CP1. When g = 2, it is torus. Geometrically, a torus can be “glued” as follows. Gluing to construct a torus Analytically, we let M = C as a topological space and Γ= g(z)= z + m + m √ 1, m , m Z { 1 2 − 1 2 ∈ } as a subgroup of Aut(C). We define an equivalence relation: z z if and only if there is ∼ some g Γ such that g(z)= z. In other words, z z if and only if z z = m1 + m2√ 1 for ∈ ∼ e− − some integers m1 and m2. We denote by [z] the equivalence class represented by z. Then from the natural projection e e e π : M = C M/Γ= C/ , z [z], → ∼ → we get a quotient space M/ or M/Γ, and we can define a quotient topology on M/Γ. Namely, Uˆ M/Γ is open if and∼ only if π−1(Uˆ) is open in M. Let ⊂ ν = [U]= U/ ∼: U is open in M such that g(U) U = for g = Id, g Γ . { ∩ ∅ ∈ } Then ν forms a basis of the topology of M/Γ. We notice that the map πM M/Γ is a covering map. → Now, for any p M/Γ, p has a neighborhood [Up] ν. Then we have disjoint union ∈ ⊂ −1 π ([Up]) = g(Up) and , g[∈Γ 9 ′ ′ g(Up) g (Up) = g = g . ∅⇔ \ −1 Moreover, π g(Up) : g(Up) [Up] is a homeomorphism. By regaring (π g(Up)) as coordinate map, it can| be verified that→ the torus T := M/Γ is a complex manifold.| [Example] In general, such “gluing process” may not produce a smooth manifold. For example, let g : C2 C2, (z , z ) ( z , z ) (3) → 1 2 → − 1 − 2 be an element in Aut(C2). Then Γ = g,Id defines a subgroup. C2/Γ is not a smooth manifold. { } In order to make quotient space a smooth manifold, we introduce some notions as follows. Let M be a complex manifold of dimension n. Write Aut(M)= f : M M, f biholomorphic . { → } Then Aut(M) is a group under the composition law, called the automorphism group of M. Let Γ Aut(M) be a subgroup. ⊂ (i) Γ is called discrete if p M, Γ(p )= r(p ) : r Γ is a discrete subset. ∀ 0 ∈ 0 { 0 ∈ } (ii) Γ is said to be fixed point free if for any g Γ, g = id, g has no fixed point. ∈ (iii) Γ is called properly discontinuous if for any K ,K M, r Γ : r(K ) K = 1 2 ⊂⊂ { ∈ 1 2 ∅} is a finite set of Γ. T Theorem 1.2 8 Let M be a complex manifold and Γ Aut(M) be a subgroup. If Γ is fixed ⊂ point free and properly discontinuous. M/Γ has a canonical structure of a complex maniofld induced from that of M. Going back to (3), when M = C2 and Γ = g,Id where g(z)= z, we see that Γ is not fixed point free because g(0, 0) = (0, 0) so that{g has} a fixed point (0−, 0). In fact, consider a Γ-invariant map (i.e., each component function is Γ invariant) L : C2 C3, (z , z ) (z2, z2, z z ). → 1 2 → 1 2 1 2 Notice L(z1, z2)= L(z1, z2) if and only if either (z1, z2)=(z1, z2)or(z1, z2)=( z1, z2). It induces a quotient map − − e e e e e e L : C2/Γ A = (z , z , z ) C3, z z = z2 . → { 1 2 3 ∈ 1 2 3 } Here C2/Γ can be identified with A which is a variety on C2 with singularity 0. 8cf. K.Kodaira, Complex manifolds and deformation of complex structures, Spring-Verlag, 1985, theorem 2.2, p.44 10 2 De Rham Theorem and Dolbeault Theorem Homology For a topological space X, it can associates some invariant groups called “homology groups” Hp(X) in the sense that if f : X Y is a homeomorphism, it induces → a group isomorphism f∗ : Hp(X) Hp(Y ), p. → ∀ Let X be a topological space. A chain complex C(X) is a sequence of abelian groups or modules with homomorphisms ∂n : Cn Cn− which we call boundary operators. That is, → 1 ∂n+1 ∂n ∂n−1 ∂2 ∂1 ∂0 ... Cn Cn− ... C C 0 −−−→ −→ 1 −−−→ −→ 1 −→ 0 −→ where 0 denotes the trivial group and Cj = 0 for j < 0. We also require the composition of any two consecutive boundary operators to be zero. That is, for all n, ∂n ∂n =0. ◦ +1 This means im(∂n ) ker(∂n). +1 ⊆ Now since each Cn is abelian, im(Cn) is a normal subgroup of ker(Cn). We define the n-th homology group of X with respec to the chain complex C(X) to be the factor group (or quotient module) Hn(X)= ker(∂n)/im(∂n+1) We also use the notation Zn(X) := ker(∂n) and Bn(X) := im(∂n+1), so Hn(X)= Zn(X)/Bn(X). Simplicial homology and singular homology 9 The simplicial homology groups Hn(X) are defined by using the simplicial chain complex C(X), with C(X)n the free abelian group generated by the n-simplices of X. Here an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. 1-simplex, 2-simplex, 3-simplex, 4-simplex, and 5-simplex 9cf., en.wikipedia.org: simplex, simplicial homology. 11 If σn = [p0, ..., pn], then n k ∂nσn := ( 1) [p , ..., pk− ,pk , ...pn]. − 0 1 +1 Xk=0 We can verify that ∂n+1 ∂n = 0. For example, if σ = [p0,p1,p2] is a 2-simplex. ∂2(σ) = [p ,p ] [p ,p ] + [p ,p ]◦ and ∂ ∂ (σ) = [p ] [p ] [p ] + [p ] + [p ] [p ]=0. 1 2 − 0 2 0 1 1 ◦ 2 2 − 1 − 2 0 1 − 0 12.
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