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.