f O(U) (f n ) O(Ω) f f n ; L (K) 0(n )

30 11 http://www.ozawa.phys.waseda.ac.jp/index2.html Ω C OΩ M Ω f M Ω Polf C PC RC 1 Ω C K C K Ω 1 K U Ω U f OU f n OΩ f f n ; L K 0n 2 K U Ω U f OU f n OΩ f f n ; L K 0n 3 z Ω \ K f OΩ f; L K < fz 4 K Ω K Ω : {z Ω; fz f; L K f OΩ} : K = K Ω 5 Ω \ K V V Ω 6 Ω \ K V V Ω 7 Ω \ K V V Ω 8 Ω \ K V V Ω 1

Ω K 1-2 3-4 5-8 Ω K 0 r<1/4 r Ω={z C; r< z < 1}, K= {z C; 2r z 1/2} Ω \ K Ω \ K V 1 = {z C; 1/2 < z < 1} V 2 = {z C; r< z < 2r} Ω \ K = V 1 V 2 V 1 V 2 Ω \ K V 1 = {z C; 1/2 z 1} V 2 = {z C; r z 2r} {z C; z =1} V 1 \ Ω {z C; z = r} V 2 \ Ω V 1 Ω V 2 Ω V 1 Ω ={z C; z =1} V 2 Ω ={z C; z = r} 5-8 Ω={z C; z < 1} 0 <r<1 r K = {z C; r/2 z <r} Ω\K Ω\K V 1 = {z C; r< z < 1} V 2 = {z C; z <r/2} Ω \ K = V 1 V 2 V 1 V 2 Ω \ K V 1 = {z C; r z 1} V 2 = {z C; z r/2} V 1 Ω V 2 Ω V 2 Ω = 5-8 Ω C K C K Ω f OΩ f n ; n Z 1 RC ξ j ;1 j N Ω n Z 1 Polf n ={ξ j ;1 j n} Ω f n f; L K 0n 1/2 n Q n j, k; j, k Z Ω Ω n n Z 1 { } z C;, Q n j, k := Ω n := Int Ω n ; n Z 1 j j +1 Re z 2n 2, k n 2 2 n n Im z k +1 j,k I n Q n j, k, I n := {j, k Z Z; Q n j, k Ω} Ω= n 1 Int Ω n Ω N 0 Z 1 n N 0 n Z 1 K Int Ω n δ 0 := dk, Ω N0 δ 0 > 0 n N 0 δ 0 dk, Ω n dk, Ω n+1 dk, C \ Ω 2

N N 0 n N 2 2 n 1 δ 0 n N K Int Ω n, 2 2 δ n 1 0 dk, Ω n dk, Ω n+1 dk, C \ Ω, 2 d Ω n, C \ Ω 2 n Ω N C 1 γ 1/2 N γ 1,,γ N j {1,,N} γ j Q N k, l Ω γ j Q N k,l Ω Q N k,l Q N k, l =γ j Q N k,l Ω Q N k,l ξ j Q N k,l Ω z K ζ γ j 2 ζ ξ j 2 δ 0 N 2 1 2 dk, Ω N 1 dk, Ω 2 1 2 z ξ j z K ζ γ j 1/z ζ K γ j 1 z ζ = 1 1 ζ ξ j k = z ξ j 1 ζ ξ j z ξ j k+1 z ξ j n Z 1 f n := N R jn, R jn z := 1 2πi f n RC n γ j ζ ξ j k fζdζz ξ j k 1 Polf n ={ξ 1,,ξ N } Ω 3

f f n ; L K =sup fz f n z z K N 1 fζ =sup z K 2πi γ ζ z dζ R jn z 1 N 1 n =sup z K 2πi γ j ζ z + ζ ξ j k fζdζ z ξ j k+1 1 N ζ ξ j k =sup fζdζ z K 2πi γ j z ξ j k+1 k=n+1 1 2π f; L Ω N 1 k 1 Lγ δ 0 2 k=n+1 1 = f; L Ω 2 n+1 N Lγ πδ 0 0 n Lγ γ K C V C \ K V a, b Polf ={a} f n RC f n RC n Polf n ={b} f n f; L K 0n f n RC p j ;0 j m m p j z fz =,z C\{a} z a j j=0 a, b V I =[0, 1] γ : I C γi V, γ0 = a, γ0 = b γi K δ := dγi, K δ>0 γ I t ν ;0 ν l 0=t 0 <t 1 < <t l =1, max 1 ν l γt ν γt ν 1 δ 2 a ν = γt ν a 0 = a, a l = b max sup a ν 1 a ν 1 ν l z K z a ν 1 2 K δ/2 1 = 1 1 z a ν 1 z a ν 1 a ν 1 a ν = z a ν j 1 1 z a ν 1 = j j + k! j 1! k! 4 a ν 1 a ν k z a ν k+1 a ν 1 a ν k z a ν k+j+1

K R n ν,j RC R ν,j n z := j + k! j 1! k! a ν 1 a ν k z a ν k+j+1,z C \{a ν} j 1 max sup 1 1 ν l z K z a ν 1 j Rν,j n z =max sup j + k! a ν 1 a ν k 1 ν l z K j 1! k! z a ν k+j+1 k=n+1 k max sup j + k! a ν 1 a ν 1 1 ν l z K j 1! k! z a ν z a ν j+1 k=n+1 1 j + k! 1 δ j+1 j 1! k! 2 k 0 k=n+1 n fz =p 0 z+ f 1 n := p 0 + f n 1 RC, Polf n 1 ={a 1 } m m p j z z a 0 j p j R 1,j n f f 1 n ; L K 0 n ε>0 N 1 1 f f n 1 ; L K ε/l f 1 N 1 = p 0 z+ m N 1 k 1 =0 k 1 + j! j 1! k 1! a 0 a 1 k 1 p j z z a 1 k 1+j+1 f 2 n = p 0 + m N 1 k 1 =0 f n 2 RC, Polf n 2 ={a 2 } f 1 k 1 + j! j 1! k 1! a 0 a 1 k 1 p j R 2, k 1+j+1 n N 1 f n 2 ; L K n 5

N 2 1 f 1 N 1 f 2 N 2 ; L K ε/l 2 ν l ν N ν 1 f ν n := p 0 + m N 1 k 1 =0 N ν 1 k ν 1 =0 k 1 + + k ν 1 + j + ν 2! j 1! p j R ν, k 1+ +k ν 1 +j+ν 1 n ν 1 μ=1 a μ 1 a μ kμ k μ! f ν n RC, Polf n ν ={a ν } f n := f l n f f n ; L K f f 1 N 1 ; L K + ν 1 max f N 2 ν l 1 ν 1 f ν N ν ; L K ε/l f n RC, Polf n ={b} l 1 ν=2 f ν 1 N ν 1 l 1 ε + f l 1 N l l 1 f n l ; L K lim sup n f ν N ν ; L K + f l 1 N l 1 f n l ; L K f f n ; L K l 1 ε + lim f l 1 l n N l 1 f n l ; L K = l 1 ε l N l 1 n N l n f f n ; L K <ε f n RC : 1 2: 1 2 2 1 2 U U 0 K U Ω U f OU U 0 U f U 0 OU 0 2 1 U U 0 f U 0 f OU 0 2 1 3 4: K K Ω 3 4 K K Ω z 0 K Ω \ K z 0 K Ω \ K : f OΩ, fz 0 f; L K 6

5 6: 6 7 8: V Ω V Ω=Ω Ω V Ω V Ω = V Ω = V Ω V Ω V Ω 1 5: Ω\ K V V Ω V K V K a V \ K V V Ω a Ω \ K Ω \ K a ε - Ba; ε Ω \ K a V Ba; ε V Ba; ε V Ω \ K V Ba; ε V V V K V V K f OΩ f; L V = f; L V f; L K z 0 V z 0 K δ := dz 0,K > 0 U := {z Ω; dz, K <δ/2} U K U Ω z 0 U z U fz = 1 z z 0 f OU 1 f n OΩ sup z K f n f; L K 0 n f n zz z 0 1 =sup z z 0 f n z fz f m f n OΩ z K sup z z 0 f n f; L K 0 n z K f m f n ; L V = f m f n ; L V f m f n ; L K 0 m, n g CV g V OV f n g; L V 0 n 7

sup gzz z 0 1 z V =sup z z 0 gz f n z + z z 0 f n z 1 z V sup z z 0 gz f n z +sup z z 0 f n z 1 z V sup z V z V z z 0 g f n ; L V +sup z V f n zz z 0 1 sup z z 0 g f n ; L V +sup f n zz z 0 1 z V z K 0n z V gzz z 0 =1 z = z 0 7 2: U C K U U Ω U f OU ξ j U R jn ; n Z 1 RC j =1,,N : R n = N R jn RC j f n = PolR jn ={ξ j } n Z 1 R n f; L K 0n f j n ; n Z 1 OΩ Rjn f n j ; L K 0n N f j n OΩ f n ξ j U Ω \ K Ω \ K V ξ j V V 7 V Ω η j V Ω V ξ j, η j ξ j, η j K ξ j, η j C \ K n Z 1 R jnl ; l Z 1 RC PolR jnl ={η j } n, l Z 1 R jnl R jn ; L K 0l 8

l n ; n Z 1 Z 1 l 1 <l 2 < <l n n R jnln R jn ; L K < 1/n n f j n := R jnln Ω f j n ; n Z 1 OΩ V K B0; r r>0 V C\B0; r V η j C \ B0; r V V ξ j, η j ξ j, η j K ξ j, η j C \ K n Z 1 R jnl ; l Z 1 RC PolR jnl ={η j } n, l Z 1 R jnl R jn ; L K 0l l n ; n Z 1 Z 1 l 1 <l 2 < <l n n R jnln R jn ; L K < 1/n n l k k, l Z 1 1 l sup z K z η j + m! k m k +1!η m+1 z m k+1 m=k 1 j 1 1 =sup z K ηj k 1 z/η j 1 l m k+1 m! z k η j m k +1! η j m=k 1 = 1 η j sup m k+1 m! z k z K m k +1! η j m=l+1 1 m k+1 m! r 0l η j k m k +1! η j m=l+1 p jn ; n Z 1 PC f n j p jn R jnln ; L K < 1/n n := p jn Ω f n j ; n Z 1 OΩ 3 5: Ω\K V V Ω 1 5 V K f OΩ 3 f; L V = f; L V f; L K 9

7 3: 1 2 5 6 7 8 3 z 0 Ω \ K δ := dz 0 ; K > 0 Bz 0 ; δ/4 Ω \ K K := K Bz 0 ; δ/4 Ω \ K V V Bz 0 ; δ/4 = Ω \ K 7 V Ω U := K δ/2 Bz 0 ; δ/2, K δ/2 := {z C; dz; K <δ/2} U K U Ω K δ/2 Bz 0 ; δ/2 = K δ/2 g 0 Bz 0 ; δ/2 g 1 g OU K g K f 1 g f OΩ f g; L K < 1/2 f; L K = f g; L K < 1/2 f 1; L Bz 0 ; δ/4 = f g; L Bz 0 ; δ/4 < 1/2 fz 0 = 1 1 fz 0 1 fz 0 1 1 f 1; L Bz 0 ; δ/4 > 1/2 > f; L K K C 1 K U U f OU p n PC p n f; L K 0n 2 K U : f OU p n OC p n f; L K n 3 C \ K Ω =C 10

Ω C K Ω \ K Ω C K C K Ω V λ ; λ Λ Ω \ K V λ Ω Λ K V λ ; λ Λ K 0 := K λ Λ V λ Λ λ Λ V λ v λ Q 2 V λ λ Λ v λ Q 2 V λ ϕ :Λ λ ϕλ :=v λ Q 2 λ, μ Λ λ μ V λ V μ = ϕ Q 2 ϕλ ϕ :Λ λ ϕλ ϕλ Λ K 0 K 0 K R>0 K B0; R K 0 K 0 B0; 2R K 0 B0; 2R v K 0 v > 2R v v K 0 \ K λ Λ v V λ V := C \ B0; R V V C \ K v V V V λ V λ K 0 K 0 \ K 0 ζ K 0 K 0 = K λ Λ V λ K Ω \ K =Ω K 0 Ω=Ω Ω ζ Ω ζ Ω ζ Ω ζ K 0 ζ Ω \ K Ω \ K V ζ V ζ K 0 V K 0 V z V K 0 V Ω \ K z λ Λ V λ z V λ λ Λ V λ V λ = V V λ ζ Ω δ := dk, C \ Ω > 0 ζ K 0 z n K 0 z n ζn ζ Ω dz n, C \ Ω = inf{ z n ξ ; ξ C \ Ω} inf{ z n ζ + ζ ξ ; ξ C \ Ω} = z n ζ 0n 11

dz n, C\Ω <δ/2 z n K 0 Bz n ; dz n, C\Ω Ω \ K z z n <dz n, C \ Ω z C z Ω dz n, C \ Ω = inf{ z ξ ; ξ Ω} z n z dz n, C \ Ω = inf{ z n ξ ; ξ C \ Ω} inf{ z z n + z n ξ ; ξ C \ Ω} = z z n + inf{ z n ξ ; ξ C \ Ω} < 2dz n, C \ Ω <δ z K z n K 0 Ω \ K = λ Λ V λ z n V λ λ Λ B z n ; dz n, C\Ω Ω \ K B z n ; dz n, C\Ω V λ = Ω B z n ; dz n, C\Ω Ω V λ V λ Ω K 0 \ K 0 Ω C K C K Ω K Ω K Ω K Ω := {z Ω; fz f; L K f OΩ} V λ ; λ Λ Ω \ K V λ Ω Λ K Ω K Ω = K λ Λ V λ K 0 K Ω = K 0 K 0 ˆK Ω V λ K Ω λ Λ V λ Ω \ K λ Λ V λ Ω 1 5 V λ K f OΩ f; L V λ = f; L V λ f; L K V λ K Ω K 0 K Ω 12

ˆK Ω K 0 K 0 K 0 Ω Ω \ K 0 V V Ω K 0 =K 0 Ωˆ K K 0 K Ω K 0 Ωˆ K Ω K 0 Ωˆ = K 0 Ω C Ω U n ; n Z 1 iii i n Z 1 U n U n U n+1 ii U n Ω U n n Z 1 V n := {z Ω; dz, Ω > 1/n, z <n} V n V n+1 n Z 1, Ω= n Z 1 V n V n V n+1 V n 0 V n+1 \ V n V n+1 Ṽ n V n 0 = V n Ṽn V n 0 W n := IntV n 0 V n V n V n 0 V n =IntV n IntV n 0 = W n W n =V n 0 W n =IntV n 0 V n 0 W n V n 0 =V n 0 V n Ṽn V n 0 V n Ṽn =IntV n Ṽn IntV n 0 = W n V n 0 = V n Ṽn V n Ṽn = V n Ṽn W n U 1 := W 1, n 1 := 1 U 1 = W 1 =V 1 0 Ω n 2 n 1 U 1 V n2 U 2 := W n2 U 1 U 1 V n2 W n2 = U 2 n 1 n 2 n j, U j = W nj, U j V nj+1 j Z 1 U j U j V nj+1 W nj+1 = U j+1 Ω= U j U j U j = W nj =V nj 0 j Z 1 U j Ωˆ = V nj 0 ˆ =V Ω nj 0 = U j U j Ω U j 13

Ω C u z = f f :Ω C u :Ω C z := 1 2 x + i y f C0 C z C uz := 1 2πi = 1 π C R 2 fζ dζ d ζ ζ z fζ ζ z dξdη ζ = ξ +iη C R 2 dζ d ζ =dξ +idη dξ idη = 2idξ dη C z uz C u u C C C u z = f Ω C f C Ω u C Ω Ω U n ; n Z 1 ϕ j ; j Z 1 C 0 Ω U j+1 ϕ j =1 ψ 1 = ϕ 1,ψ j = ϕ j ϕ j 1 j 2 j 2 U j ϕ j 1 =1 U j U j+1 ϕ j =1 U j ψ j =0 ψ j ; j Z 1 Ω ψ j =1 f C Ω j 1 fψ j C 0 Ω C\Ω fψ j C 0 C u j C C C u j z = fψ j 14

j 2 U j fψ j =0 u j U j ii u j U j OU j Ω U j 1 j 2 v j OΩ u j v j ; L U j 1 2 j v 1 OΩ u := u j v j k 2 u j v j U j OU j u j v j Ω C Ω u j v j j=k L U k 1 u j v j U k 1 OU k 1 u = j=k u j v j Ω u C Ω u z = z u j v j = z u j = fψ j = f u C Ω Ω C U λ ; λ Λ f λμ O ; λ, μ Λ 2 I, Λ 2 I := {λ, μ Λ2 ; U λ U μ } iii i alternating condition λ, μ Λ 2 I U λ U μ f λμ = f μλ ii cocycle condition λ, μ, ν Λ 3 I := {λ, μ, ν Λ3 ; U λ U μ U ν } U λ U μ U ν f λμ + f μν + f νλ =0 f λ OU λ ; λ Λ λ, μ Λ 2 I U λ U μ f λμ = f μ f λ 15

ϕ λ ; λ Λ U λ ; λ Λ λ Λ ϕ λ C 0 Ω; R ϕ λ 0 U λ supp ϕ λ U λ Ω K {λ Λ; supp ϕ λ K} 1 λ Λ ϕ λ =1 Ω λ Λ g λ := μ Λ λ ϕ μ f μλ Λ λ := {μ Λ; λ, μ Λ I } λ, μ Λ 2 I U λ U μ g λ g μ = ϕ ν f νλ f νμ ν Λ λ Λ μ = ϕ ν f νλ + f μν ν Λ λ Λ μ = ϕ ν f λμ ν Λ λ Λ μ = ϕ ν f μλ = f μλ ϕ ν = f μλ ϕ ν = f μλ ν Λ λ Λ μ ν Λ λ Λ μ ν Λ f μλ OU λ U μ U λ U μ g λ z = g μ z λ Λ U λ f U λ := g λ z f :Ω C g λ C Ω f C Ω Ω u z = f u C Ω f λ := g λ + u λ, μ Λ 2 I U λ U μ f λμ = g λ g μ =f λ u f μ u =f λ f μ f λ U λ f λ OU λ f λ z = g λ z + u z = g λ z + f =0 16

Ω C U λ ; λ Λ V =f λ M U λ ; λ Λ λ, μ Λ 2 I := {λ, μ Λ2 ; U λ U μ } f μ f λ OU λ U μ f M Ω λ Λ f f λ OU λ λ, μ Λ I f λμ := f μ f λ OU λ U μ f λμ = f μ f λ = f λ f μ = f μλ, f λμ + f μν + f νλ =f μ f λ +f ν f μ +f λ f ν =0 f λμ ;λ, μ Λ I g λ OU λ ; λ Λ λ, μ Λ 2 I U λ U μ f λμ = g μ g λ λ, μ Λ 2 I U λ U μ f μ f λ = g μ g λ OU λ U μ U λ U μ f μ g μ = f λ g λ M U λ U μ λ Λ U λ f U λ := f λ g λ f M Ω f f λ = g λ OU λ X V F V U V U, V ι V U : F V F U 17

S0 U ι U U : F U F U U V W U, V, W ι W V : F W F V, ι V U : F V F U, ι W U : F W F U compatibility condition ι V U ι W V F = F V, ι V U X presheaf of vector spaces on X X presheaf on X X F X Ω U =U λ ; λ Λ { F U λ := f :Λ } F U λ ; λ Λ, fλ F U λ λ Λ λ Λ = ι W U f =f λ ; λ Λ λ Λ F U λ f λ ; λ Λ + g λ ; λ Λ := f λ + g λ ; λ Λ af λ ; λ Λ := af λ ; λ Λ λ Λ F U λ Λ 2 I := {λ, μ Λ2 ; U λ U μ } F U λμ := f :Λ2 I F U λμ ; λ, μ Λ 2 I,fλ, μ F U λμ λ,μ Λ 2 I λ,μ Λ 2 I f =f λμ ;λ, μ Λ 2 I U λμ := U λ U μ F U λμ λ,μ Λ 2 I fλμ ;λ, μ Λ 2 I + gλμ ;λ, μ Λ 2 I := fλμ + g λμ ;λ, μ ΛI 2 a f λμ ;λ, μ ΛI 2 := afλμ ;λ, μ Λ 2 I F U λμ λ,μ Λ 2 I f F Ω λ Λ ι Ω U λ f F U λ εf := ι Ω U λ f; λ Λ εf λ Λ F U λ f,g F Ω εf + g = ι Ω U λ f + g; λ Λ = ι Ω U λ f+ι Ω U λ g; λ Λ = ι Ω U λ f; λ Λ + ι Ω U λ g; λ Λ = εf+εg, εaf = ι Ω U λ af; λ Λ = aι Ω U λ f; λ Λ = a ι Ω U λ f; λ Λ = aεf 18

ε : F Ω f εf λ Λ F U λ f λ ; λ Λ F U λ λ, μ Λ 2 I λ Λ ι U λ f λ F δ f λ ; λ Λ := δ f λ ; λ Λ F U λμ λ,μ Λ 2 I f λ ; λ Λ, g λ ; λ Λ λ Λ F U λ ιuμ f μ ι Uμ f μ ι U λ f λ ; λ, μ Λ 2 I δ f λ ; λ Λ + g λ ; λ Λ = δ f λ + g λ ; λ Λ = ι Uμ f μ + g μ ι U λ f λ + g λ ; λ, μ Λ 2 I = ι Uμ f μ ι U λ f λ + ι Uμ g μ ι U λ g λ ;λ, μ Λ 2 I = ι Uμ f μ ι U λ f λ ; λ, μ Λ 2 I + ι Uμ g μ ι U λ g λ ; λ, μ Λ 2 I = δ f λ ; λ Λ + δ g λ ; λ Λ, δ af λ ; λ Λ = δ af λ ; λ Λ = ι Uμ af μ ι U λ af λ ; λ, μ Λ 2 I = a ι Uμ f μ ι U λ f λ ;λ, μ Λ 2 I = a ι Uμ f μ ι U λ f λ ; λ, μ Λ 2 I = aδ f λ ; λ Λ δ : λ Λ F U λ f λ ; λ Λ δ f λ ; λ Λ F U λμ λ,μ Λ 2 I 0 F Ω ε F U λ δ F U λμ 4.1 λ Λ λ,μ Λ 2 I 0 {0} 0 F Ω f F Ω S0 δ εf =δ ι Ω U λ f; λ Λ = ι Uμ ι Ω U μ f ι U λ ι Ω Uλ f ;λ, μ Λ 2 I = ι Ω f ι Ω f; λ, μ Λ 2 I = 0; λ, μ Λ 2 I =0 19

δ ε =0 Im ε Ker δ ε : F Ω λ Λ F U λ Ker ε =0 S1 S1 f F Ω λ Λ ι Ω U λ f =0 f =0 Im ε Ker δ Im ε = Ker δ S2 S2 f λ ; λ Λ λ Λ F U λ λ, μ Λ 2 I ι U λ f λ =ι Uμ f μ f F Ω λ λ ι Ω U λ f =f λ X F Ω U S1S2 Ω U 4.1 exact sequence X sheaf on X F X p Z 0 U F p alternating p-cochain module { C p U, F := f = f λ0 λ p ;λ 0,,λ p Λ p+1 I F U λ0 λ p ; λ 0,,λ p Λ p+1 I f λσ0 λ σp =sgnσf λ0 λ p σ S p+1 } U λ0 λ p := p j=0 U λj Λ p+1 I := { λ 0,,λ p Λ p+1 ; U λ0 λ p } S p+1 {0,,p} p +1sgnσ σ p =0 C 0 U, F = λ Λ F U λ f C p U, F λ 0,,λ p Λ p+1 I p+1 δf λ0 λ p+1 := 1 k ι Uλ 0 λ k λ p+1 U λ0 λ p+1 f λ0 λ k λ p+1 λ k λ k 20

δf λ0 λ p+1 F U λ0 λ p+1 σ S p+2 p+1 δf λσ0 λ σp+1 = 1 k ι Uλ σ0 λ k λ σp+1 p+1 = p+1 = U λσ0 λ σp+1 f λσ0 λ k λ σp+1 1 k ι Uλ σ0 λ k λ σp+1 U λσ0 λ σp+1 sgnσf λ0 λ k λ p+1 1 k sgnσι Uλ σ0 λ k λ σp+1 U λσ0 λ σp+1 f λ0 λ k λ p+1 p+1 =sgnσ 1 k ι Uλ σ0 λ k λ σp+1 U λσ0 λ f σp+1 λ0 λ k λ p+1 =sgnσδf λ0 λ σp+1 δf C p+1 U, F U V ι V U δ : Cp U, F f δf C p+1 U, F δ p f C p U, F δ p+1 δ p f =δ p+1 δ p f p+2 = 1 k ι Uλ 0 λ k λ p+2 δ p f λ0 λ k λ p+2 p+2 = U λ0 λ p+2 1 k ι Uλ 0 λ k λ p+2 U λ0 λ p+2 j<k 1 j ι Uλ 0 λ j λ k λ p+2 U f λ0 λ λ0 λ k λ p+2 j λ k λ p+2 + j>k 1 j 1 ι Uλ 0 λ k λ j λ p+2 U f λ0 λ λ0 λ k λ p+2 k λ j λ p+2 p+2 = 1 k 1 j ι Uλ 0 λ j λ k λ p+2 U λ0 λ f p+2 λ0 λ j λ k λ p+2 j<k p+2 1 k 1 j ι Uλ 0 λ k λ j λ p+2 U λ0 λ f p+2 λ0 λ k λ j λ p+2 j>k p+2 = 1 j+k ι Uλ 0 λ j λ k λ p+2 f λ0 λ j λ k λ p+2 j<k U λ0 λ p+2 p+2 1 j+k ι Uλ 0 λ k λ j λ p+2 j=0 k<j U λ0 λ p+2 p+2 = 1 j+k ι Uλ 0 λ j λ k λ p+2 =0 j<k U λ0 λ p+2 p+2 1 k+j ι Uλ 0 λ j λ k λ p+2 j<k U λ0 λ p+2 f λ0 λ k λ j λ p+2 f λ0 λ j λ k λ p+2 f λ0 λ j λ k λ p+2 21

δ p+1 δ p =0 C p U, F δ p δ p 1 Z p U, F := Ker δ p B p U, F :=Imδ p 1 p p-cocycle module p p-coboundary module B 0 U, F =0 δ p+1 δ p =0 B p U, F Z p U, F H p U, F :=Z p U, F /B p U, F H p U, F U F p cohomology module of order k p =0 H 0 U, F =FΩ iii Ω U =U λ ; λ Λ OΩ f λμ ; λ, μ Λ 2 I OΩ U U OΩ U V ι V U : F V f f U OU f λμ Z 1 U, OΩ f λμ B 1 U, OΩ Ω C U OΩ p 1 H p U, OΩ = 0,,,,,, F. Haslinger, Complex Analysis, De Gruyter 22