Download Complex projective space The complex projective space CPn is the

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∗,
where C∗ acts by multiplication on Cn+1 . The points of CPn are written as (z0 , z1 , ..., zn ).
Here, the notation intends to indicate that for λ ∈ C∗ the two points (λz0 , λz1 , ..., λzn ) and
(z0 , z1 , ..., zn ) define the same point in CPn . We denote the equivalent class by [z0 : z1 : ... :
zn ]. Only the origin (0, 0, ..., 0) does not define a point in CPn .
We take the standard open covering of CPn . Let Ui be the open set
Ui := {[z0 : ... : zn ] | zi 6= 0} ⊂ CPn .
Consider the bijective maps
τi :
→ Cn
zi−1 zi+1
z0
zn
[z0 : ... : zn ] 7→
, ..., zi , zi , ..., zi
zi
Ui
For the transition maps
τij = τi ◦ τj−1 : τj (Ui ∩ Uj ) → τi (Ui ∩ Uj )
wj−1 1 wi+1
wi−1 wi+1
w1
wn
(w1 , ..., wn ) 7→
, ..., wi , wi , ..., wi , wi , wi , ..., wi
wi
is biholomorphic. In fact,
τij (w1 , ..., wn ) = τi ◦ τj−1 (w1 , ..., wn )
= 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
wi−1 wi+1
wj−1 1 wi+1
wn
w1
, ...,
,
, ...,
, ,
, ...,
=
wi
wi
wi
wi wi wi
wi
In particular, when n = 1, CP1 = U0 ∪ U1 where
z1
U0 = {[z0 : z1 ] | z0 6= 0} = {[1 : | z0 6= 0} = {[1 : w] | w ∈ C} ≃ S 1 − {∞},
z0
and
U1 = {[z0 : z1 ] | z1 6= 0} = {[
z0
: 1| z1 6= 0} = {[w : 1] | w ∈ C} ≃ S 1 − {0}.
z1
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Then
τ01 = τ0 ◦ τ1−1 (w) = τ0 ([w : 1]) =
1
−1
, and τ10 = τ01
.
w
(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 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 + m1 + m2 −1, m1 , m2 ∈ Z}
as a subgroup of Aut(C). We define an equivalence relation: z ∼ ze if and only if √there is
some g ∈ Γ such that g(z) = ze. In other words, z ∼ ze if and only if z − ze = m1 + m2 −1 for
some integers m1 and m2 . We denote by [z] the equivalence class represented by z. Then
from the natural projection
π : M = C → M/Γ = C/ ∼, z 7→ [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 6= 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
[
π −1 ([Up ]) =
g(Up ) and ,
g∈Γ
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g(Up )
\
g ′ (Up ) 6= ∅ ⇔ g = g ′ .
Moreover, π|g(Up ) : g(Up ) → [Up ] 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
2
g : C2 → C2 , (z1 , z2 ) 7→ (−z1 , −z2 )
(3)
2
be an element in Aut(C ). Then Γ = {g, Id} defines a subgroup. C /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 ∀p0 ∈ M, Γ(p0 ) = {r(p0 ) : r ∈ Γ} is a discrete subset.
(ii) Γ is said to be fixed point free if for any g ∈ Γ, g 6= id, g has no fixed point.
T
(iii) Γ is called properly discontinuous if for any K1 , K2 ⊂⊂ M, {r ∈ Γ : r(K1 ) K2 6= ∅}
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 = 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 , (z1 , z2 ) 7→ (z12 , z22 , z1 z2 ).
Notice L(z1 , z2 ) = L(e
z1 , ze2 ) if and only if either (z1 , z2 ) = (e
z1 , ze2 ) or (z1 , z2 ) = (−e
z1 , −e
z2 ). It
induces a quotient map
L : C2 /Γ → A = {(z1 , z2 , z3 ) ∈ C3 , z1 z2 = z32 }.
Here C2 /Γ can be identified with A which is a variety on C2 with singularity 0.
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cf. K.Kodaira, Complex manifolds and deformation of complex structures, Spring-Verlag, 1985, theorem
2.2, p.44
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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−1 which we call boundary operators. That is,
∂n+1
∂
∂n−1
∂
∂
∂
n
2
1
0
... −−−→ Cn −→
Cn−1 −−−→ ... −
→
C1 −
→
C0 −
→
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,
This means im(∂n+1 ) ⊆ ker(∂n ).
∂n ◦ ∂n+1 = 0.
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).
The simplicial homology groups
Simplicial homology and singular homology 9
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
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cf., en.wikipedia.org: simplex, simplicial homology.
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If σn = [p0 , ..., pn ], then
∂n σn :=
n
X
(−1)k [p0 , ..., pk−1 , pk+1, ...pn ].
k=0
We can verify that ∂n+1 ◦ ∂n = 0. For example, if σ = [p0 , p1 , p2 ] is a 2-simplex. ∂2 (σ) =
[p1 , p2 ] − [p0 , p2 ] + [p0 , p1 ] and ∂1 ◦ ∂2 (σ) = [p2 ] − [p1 ] − [p2 ] + [p0 ] + [p1 ] − [p0 ] = 0.
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