1
Open set of Cn Ω
Closed subset of Ω X
Arbitrary point of X x0
Neighborhood of x0 at Cn U
Set of all the holomorphic functions over Ω A (Ω)
System of functions {fα}α∈Λ⊂A (U)
X ∩U = {z∈U | fα (z) = 0, α∈Λ}
X is called analytic subset
{fα}α∈Λ is called local defining functions over U.
Element of A (U) F
When F satisfies F | U ∩X = f | U ∩X |, function f over X is called holomorphic function†.
2
r graded differential form over Ω
Element of C∞0 (Ω)’ u
∑I, J uIJdzI∧d
J, uIJ = sgn( 
)sgn( 
) uI’J’
J, uIJ = sgn( )sgn( ) uI’J’
I, J Multiple index from natural number 1 to n
When longitude of I, J is constant p, q, u is called (p, q) type differential form.
Set of (p, q) type differential form is notated C p, q (Ω).
(1, 0) type complex exterior differentiation operator ∂ : C p, q (Ω) → C p+1, q (Ω)
(0, 1) type complex exterior differentiation operator 
: C p, q (Ω) → C p, q+1 (Ω)
: C p, q (Ω) → C p, q+1 (Ω)
∂ ( ∑’I, J uIJdzI∧dJ) = ∑’I, J∑k 
dzk∧dzI∧d
J
J) = ∑’I, J∑k dzk∧dzI∧dJ
( ∑’I, J uIJdzI∧dJ) = ∑’I, J∑k dk∧dzI∧dJ
3
L : = {z∈Cn | z | = … = zn-m = 0}
Holomorphic function over Ω∩L f = {zn-m+1, …, zn}
W : = {z∈Cn | ( 0, …, 0, zn-m+1, …, zn ) ∈Ω∩L}
Holomorphic function over W 
(z) : = f ( 0, …, 0, zn-m+1, …, zn )
(z) : = f ( 0, …, 0, zn-m+1, …, zn )
C∞ class function ρW : W → [0, 1]
supp (ρW – 1 ) ∩L = Ø
supp ρW ∩∂Ω= Ø
ρW ‘s trivial expansion to Ω 
∈C∞ (Ω) | Ω∩L = f
Therefore
u = 
u | Ω∩L = 0 . (1)
H p, q (Ω) : = Ker ∩C p, q (Ω) / Im ∩C p, q (Ω)
∩C p, q (Ω) / Im ∩C p, q (Ω)
H p, q (Ω) is called 
cohomology of type (p, q).
cohomology of type (p, q).
4
(i)
Serre’s condition
H 0, q (Ω) ={ 0 } ( 1 ≤ q ≤ n-1 )
(ii)
Arbitrary z0∈∂Ω
(iii)
Sequence pμ in Ω that is convergent to z0, there exists f ∈A (Ω).
From (i) (ii) (iii)
μ→∞ | f (pμ) | = ∞ (2)
5
From (1) and (2), solution on the domain and the equation is expanded to mathematical formality of word, i.e. language.
Space in which word and sentence is generated : = Ω
The space is called language space. Notation is LS.
Base meaning that becomes root of word : = x0 and sequence pμ that is convergent to x0 in Ω
Additional meaning† : = sequence pμ
Word and sentence, i.e. language : = f ∈A ( Ω )
Language in LS is considered at 
μ→∞ | f (pμ) | = ∞.
μ→∞ | f (pμ) | = ∞.
Tokyo June 12, 2008
[Postscript June 19]
On holomorphic, refer to the next.
