11th for KARCEVSKIJ Sergej
1
Open set of Cn Ω
Holomorphic function over Ω f
Set of all the holomorphic function over Ω A ( Ω )
Open set U ⊂ Ω
f・A ( U ) is called f ’s divisor class at U.
Divisor class is notated by D ( f, U ).
2
n-dimensional polydisk is defined by the next.
Open set {z | | zj-aj | < r, 1≤j≤n }
n-dimensional polydisk is notated by ∆(a, r) (r = (r1, …, rn))
∆(0, 1) is notated by ∆.
∆n = ∆×…×∆ (Number of ∆ is n.)
∆(a, r) and ∆n are biholomorphic equivalent.
Hartogs figure Tε = {(z1, z2) ∈∆ 2 | |z1| <ε}
When holomorphic map from Hartogs figure to Ω is always expanded to holomorphic map from ∆ 2 to Ω, Ω is called Hartogs pseudo-convex.
3
C is Hartogs pseudo-convex.
Cn is Hartogs pseudo-convex.
Holomorphic open set is Hartogs pseudoconvex.
4
Subharmonic function is defined by the next.
Open set at complex plane Ω
Semicontinuous function that is valued at [-∞, ∞) ψ : Ω → [-∞, ∞)
⊂ Ω
ψ(z ) ≤ (z + reiθ)dθ
5
Plurisubharmonic function is defined by the next.
Open set at complex plane Ω
Semicontinuous function that is valued at [-∞, ∞) ψ : Ω → [-∞, ∞)
(z, ω) ∈Ω×Cn
Function ψ( z+ζω )
When ψ( z+ζω ) is subharmornic as ζ ‘s function, ψ( z+ζω ) is called plurisubharmonic function.
Set of all the plurisubharmonic functions PSH (Ω)
6
What Ω is pseudoconvex is defined by the next.
Continuous plurisubharmonic function ψ : Ω → R
Arbitrary c∈R
Ωψc : = {z∈Ω |ψ(z) < c }
Ωψc is relatively compact in Ω .
7
Pseudoconvex open set Ω
H2 (Ω, Z) = {0}
Open subset of Ω U
g ∈A(U)
Element of A(U) f
When V(g) is closed set of Ω, there exists D ( f, U ).∋g.
8
Locally finite open ball Bj = B (pj, Rj)
Family of Bj { Bj }∞j = 1
Ω = ∪∞j = 1 Bj
Bj ∩V(g) ≠Ø ⇒ Bj ⊂ U
gj ∈A(Bj) is defined by the next.
gj = g | Bj (Bj ∩V(g) ≠ Ø)
gj = 1 (Bj ∩V(g) = Ø)
gjk ∈A(Bj∩Bk) : = gj / gk (Bj∩Bk ≠ Ø)
Bj∩Bk is convex and simply connected.
gjk has not zero point.
(j, k) has one to one correspond with branch ujk of loggjk
uijk over Bi∩ Bj∩Bk is defined by the next.
uijk : = uij + ujk+ uki
9
Language is defined by the next.
Meaning minimum : = Bj
Word : = gjk
Sentence : = uijk
[References]
Tokyo June 19, 2008
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