Espace projectif complexe Pn(C)

Pn(C)
Pn(C) = {Cn+1},
=©[Z0:... :Zn] : (Z0, ..., Zn)∈Cn+1\{0}ª,
Cn+1 [Z0:... :Zn]
Pn(C)
Cn+1\{0} −→ Pn(C),(Z0, ..., Zn)7−→ [Z0:... :Zn].
Pn(C)
n+ 1 U0, ..., Un
Ui={[Z0:... :Zn]∈Pn(C) : Zi6= 0},
Zi6= 0 i= 0, ..., n
Ui
ϕi:Ui−→ Cn,[Z0:... :Zn]7−→ µZ0
Zi
, ..., Zi−1
Zi
,Zi+1
Zi
, ..., Zn
Zi¶≡(z1, ..., zn),
zk=
Zk−1
Zj
k≤j
Zk
Zi
k > j
1≤k≤n ϕi(Ui) = Cnϕi
(z1, ..., zn)7−→ [z1, ...zi,1, zi+1, ..., zn].

Pn(C)n(Ui, ϕi)
0≤i≤nPn(C)
Ui∩ Uj={[Z0:... :Zn]∈Pn(C) : Zi6= 0 Zj6= 0}, i 6=j
j > i
ϕi(Ui∩ Uj) = {(z1, ..., zn)∈Cn:zj6= 0},
ϕj(Ui∩ Uj) = {(z1, ..., zn)∈Cn:zi+1 6= 0}.
ϕji ≡ϕjϕ−1
i:ϕi(Ui∩ Uj)−→ ϕj(Ui∩ Uj),
(z1, ..., zn)7−→ µz1
zj
, ..., czj
zj
, ..., 1
zj
, ..., zn
zj¶, zj6= 0
bϕij ≡ϕiϕ−1
j
ϕji ϕij
(Ui, ϕi) (Uj, ϕj)Sn
i=0 Ui=Pn(C)
Pn(C)
n= 1
P1(C) = C∪ {∞}
Ui=P1(C)\{∞} =C,
Uj=P1(C)\{0}=C∗∪ {∞},
ϕi:Ui−→ C, ,
ϕi:Uj−→ C, z 7−→ ϕj(z) = (1
zz∈C∗
0z=∞
ϕiϕjUi
UjUi∩ Uj6=∅P1(C)
ϕi(Ui∩ Uj) = ϕj(Ui∩ Uj) = C∗,
ϕjϕ−1
i:C∗−→ C∗, z 7−→ 1
z,
S2−→ C∪ {∞},(z0, z1, z3)7−→ 
z0+√−1z2
1−z3
z6= 1
∞z3= 1

S2={(z0, z1, z3)∈C3:|z0|2+|z1|2+|z2|2},
C3
P1(C)S2
Pn(C)
Cn+1\{0}
[Z0:... :Zn]∼[λZ0:... :λZn], λ ∈C∗.
Pn(C) = {[Z]6= 0 ∈Cn+1}
[Z]∼[λZ].
Hi={[Z0:... :Zn]∈Pn(C) : Zi= 0},
Pn(C)Zi= 0 0 ≤i≤n
n−1
ϕi:Pn(C)\Hi−→ Cn,[Z0:... :Zn]7−→ µZ0
Zi
, ..., Zi−1
Zi
,Zi+1
Zi
, ..., Zn
Zi¶,
Pn(C)\HiCn
Ui=Pn(C)\Hi={[Z0:... :Zn]∈Pn(C) : Zi6= 0}.
HiUi
Pn(C)Cn+1\{0}
Pn(C)
Cn+1\{0} −→ Pn(C)ϕi
Pn(C)
(Ui, ϕi)zk
Ui=Pn(C)\HiHiPn−1(C)
Pn(C) = {[Z0:... :Zn]∈Pn(C) : Zi6= 0}∪{[Z0:... :Zn]∈Pn(C) : Zi= 0},
=Ui∪Hi,
= (Pn(C)\Hi)∪Hi,
'Cn∪Hi.
Tn
i=0 Hi=∅[Z0:... :Zn]
Zi
n
[
i=0 Ui=
n
[
i=0
(Pn(C)\Hi) = Pn(C).
ξ∈Pn(C)Z0, ..., Zn
ξ i Zi6= 0 ξ∈ Ui
Pn(C) (n+ 1) Cn