Complex projective space The complex projective space CP n is the most important compact complex manifold. By definition, CP n is the set of lines in C n+1 or, equivalently, CP n := (C n+1 \{0})/C, where C acts by multiplication on C n+1. The points of CP n are written as (z 0, z 1,..., z n ). Here, the notation intends to indicate that for λ C the two points (λz 0, λz 1,..., λz n ) and (z 0, z 1,..., z n ) define the same point in CP n. We denote the equivalent class by [z 0 : z 1 :... : z n ]. Only the origin (0, 0,..., 0) does not define a point in CP n. We take the standard open covering of CP n. Let U i be the open set Consider the bijective maps For the transition maps U i := {[z 0 :... : z n ] 0} CP n. τ i : U i ( C n z [z 0 :... : z n ] 0 zi,..., 1, +1,..., zn ) τ ij = τ i τj 1 : τ j (U i U j ) ( τ i (U i U j ) ) w (w 1,..., w n ) 1,..., 1, +1,..., w j 1, 1, +1,..., wn is biholomorphic. In fact, τ ij (w 1,..., w n ) = τ i τ 1 j (w 1,..., w n ) = τ i ([w 1 :... : w j 1 : 1 : w j+1 :... : w n ]) ([ w1 = τ i :... : 1 : 1 : +1 :... : w j 1 1 : : +1 w ( i w1 =,..., 1, +1,..., w j 1, 1, +1,..., w ) n In particular, when n = 1, CP 1 = U 0 U 1 where :... : w ]) n and U 0 = {[z 0 : z 1 ] z 0 0} = {[1 : z 1 z 0 z 0 0} = {[1 : w] w C} S 1 { }, U 1 = {[z 0 : z 1 ] z 1 0} = {[ z 0 z 1 : 1 z 1 0} = {[w : 1] w C} S 1 {0}. 8
Then τ 01 = τ 0 τ 1 1 (w) = τ 0 ([w : 1]) = 1 w, and τ 10 = τ 1 01. (2) 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 CP 1. 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 1 + m 2 1, m1, m 2 Z} as a subgroup of Aut(C). We define an equivalence relation: z f and only if there is some g Γ such that g(z) = z. In other words, z f and only if z z = m 1 +m 2 1 for some integers m 1 and m 2. We denote by [z] the equivalence class represented by z. Then from the natural projection 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, Û M/Γ is open if and only if π 1 (Û) 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 [U p ] ν. Then we have disjoint union π 1 ([U p ]) = g Γ g(u p ) and, 9
g(u p ) g (U p ) g = g. Moreover, π g(up) : g(u p ) [U p ] is a homeomorphism. By regaring (π g(up)) 1 as coordinate map, it can be verified that the torus T := M/Γ is a complex manifold. [Example] example, let In general, such gluing process may not produce a smooth manifold. For g : C 2 C 2, (z 1, z 2 ) ( z 1, z 2 ) (3) be an element in Aut(C 2 ). Then Γ = {g, Id} defines a subgroup. C 2 /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 0 M, Γ(p 0 ) = {r(p 0 ) : r Γ} is a discrete subset. (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 1, K 2 M, {r Γ : r(k 1 ) K 2 } is a finite set of Γ. 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 = C 2 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 : C 2 C 3, (z 1, z 2 ) (z 2 1, z 2 2, z 1 z 2 ). Notice L(z 1, z 2 ) = L( z 1, z 2 ) if and only if either (z 1, z 2 ) = ( z 1, z 2 ) or (z 1, z 2 ) = ( z 1, z 2 ). It induces a quotient map L : C 2 /Γ A = {(z 1, z 2, z 3 ) C 3, z 1 z 2 = z 2 3}. Here C 2 /Γ can be identified with A which is a variety on C 2 with singularity 0. 8 cf. 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 H p (X) in the sense that if f : X Y is a homeomorphism, it induces a group isomorphism f : H p (X) H p (Y ), p. Let X be a topological space. A chain complex C(X) is a sequence of abelian groups or modules with homomorphisms n : C n C n 1 which we call boundary operators. That is,... n+1 C n n 1 n Cn 1... 2 C 1 1 0 C0 0 where 0 denotes the trivial group and C j = 0 for j < 0. We also require the composition of any two consecutive boundary operators to be zero. That is, for all n, This means im( n+1 ) ker( n ). n n+1 = 0. Now since each C n is abelian, im(c n ) is a normal subgroup of ker(c n ). 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) H n (X) = ker( n )/im( n+1 ) We also use the notation Z n (X) := ker( n ) and B n (X) := im( n+1 ), so H n (X) = Z n (X)/B n (X). Simplicial homology and singular homology 9 The simplicial homology groups H n (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 9 cf., en.wikipedia.org: simplex, simplicial homology. 11
If σ n = [p 0,..., p n ], then n σ n := n ( 1) k [p 0,..., p k 1, p k+1,...p n ]. k=0 We can verify that n+1 n = 0. For example, if σ = [p 0, p 1, p 2 ] is a 2-simplex. 2 (σ) = [p 1, p 2 ] [p 0, p 2 ] + [p 0, p 1 ] and 1 2 (σ) = [p 2 ] [p 1 ] [p 2 ] + [p 0 ] + [p 1 ] [p 0 ] = 0. 12