15, pp.61-80 Abel-Jacob I 1 Introducton Remann Abel-Jacob X g Remann X ω 1,..., ω g Λ = {( γ ω 1,..., γ ω g) C g γ H 1 (X, Z)} Λ C g lattce Jac(X) = C g /Λ Le Abel-Jacob (Theorem 2.2, 4.2) Jac(X) Pcard 0 Pc 0 (X) D. Mumford ([Mum, Chapter II]) [Sh, ] [Lan] [Iwa] Mumford theta (Remann ) Remann ( 6.2) Jacob 2 Jacob 3 4 Jacob 0 Abel-Jacob ( 5) theta ( 6) 7 Abel-Jacob 0 Z O X O X 0 H 1 (X, Z) H 1 (X, O X ) H 1 (X, O X ) ϕ H 2 (X, Z) Poncaré H 1 (X, Z) H 1 (X, Z) Serre H 1 (X, O X ) H 0 (X, Ω X ) Jac(X) = H 0 (X, Ω X ) /H 1 (X, Z) Ker ϕ = Pc 0 (X) COE E-mal gunj@ms.u-tokyo.ac.jp 61
62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P X D Remann-Roch L(D) = {X f (f) + D 0} P n P ( deg D > 2g 2 ) L(D) X X C X = U α U α U β α, β U α U β g αβ : (1) g αα = 1. (2) g βα = g 1 αβ. (3) g αβ g βγ = g αγ (U α U β U γ ). (U α C) : U α C (x α, u α ) (x β, u β ) U β C x α = x β u α = g αβ (x α )u β. L = (U α C)/ ( ) π : L X, (x α, u α ) x α well-defned
Abel-Jacob I 63 L X {g αβ } 2 L L {g αβ } {g αβ } ( ) U α h α g αβ = h 1 α g αβ h β X Pcard Pc(X) L g αβ M h αβ g αβ h αβ L M g j 1 X C O X Γ(X, L) = {s: X hol L π s = d X } L Γ(X, O X ) X C L Γ(X, L) 0 hol Γ(X, L) = {f α : U α C f α /f β = g αβ } 2.2 D = r =1 n P X X X = k U k P U P j (j ) k U k f k : P U f P n ( n ) P / U k (1 r) f k = 1 U U j g j = f /f j O X (D) O X (D) f f f u u U g j u g j u 1 j X Dv(X) Pc(X) ϕ ϕ X f (f) Dv l (X) ker ϕ = Dv l (X) D = n P = (f) U P f U g j = f/f = 1 ϕ(d) = O X ϕ(d) = O X D = n P U P n f U h 1h j f h U U j X f (f) = n P ker ϕ = Dv l (X) ϕ Dv(X)/ Dv l (X) Pc(X) Dv(X)/ Dv l (X) Pc(X) X ( ) X f /f j = h 1 0 O X K X K X /O X 0
64 H 0 (X, K X ) H0 (X, K X /O X ) H1 (X, O X ) H1 (X, K X ) = 0 K X X 0 D = n P Dv(X) P U f P n ( n ) hol Γ(X, O X (D)) = {h α : U α C h α /h β = f α /f β } h α fα 1 X ψ (ψ) + D 0 Γ(X, O X (D)) L(D), (h ) ψ = (h f 1 ) L(D) D 2.1 (1) Dv(X)/ Dv l (X) Pc(X), D O X (D) (2) D Dv(X) Γ(X, L) L(D) Γ(X, L) C- 2.3 Pcard Jacob X Pc(X) Pc(X) ( ) Pc(X) Pc(X) Pc(X) Pc(X) L = O X (D) D deg L Pc n (X) n Pc(X) Pc 0 (X) Pc(X) = n Pcn (X) X P Pc 0 (X) Pc n (X), L L O(nP ) Pc 0 (X) Dv 0 (X)/ Dv l (X) Dv 0 (X) 0 2.2 X g C g Λ Pc 0 (X) g- C g /Λ Pc 0 (X) C g /Λ X 0 C g /Λ
Abel-Jacob I 65 C g /Λ C g /Λ (C ) 2.1 g = 0 Pc 0 (X) = {1 } P 1 D = r =1 (P Q ) P 1 0 P l = P l+1 = = P r = P (1 < l), Q (1 r) C l 1 =1 ϕ(x) = (x P k) r =1 (x Q ) C {P 1,... P l 1 } {Q 1,..., Q r } r l (ϕ) = D Abel g = 1 Remann-Roch 2.2 g = 1 1 D P D P L(K X D) = 0 Remann- Roch dm L(D) = deg D 1 + g = 1 D P X X Pc 0 (X), P P Pc 0 (X) X = C/Λ X C g /Λ Jacob Jac(X) Pc 0 (X) C Sch/C Set Pc 0 X T {X T 0 } {pr T (L) L T } pr T X T T X T 0 T 0 Pc 0 X (representable) C Pc 0 (X) Pc 0 X (T ) = Hom C-Sch(T, Pc 0 (X)) ( ) Pc 0 (X) (fne modul) 2.2 : Pc 0 X Pc0 (X) C g /Λ T = Spec C X Pc 0 X(C) C g /Λ 2.2
66 0 Pc X Pc 0 (X) X ( 0 ) X Pc 0 (X) X T = X Pc 0 X( X) = Hom( X, X) X X dx b Poncaré (cf [Ko]) Jacob Albanese X X Alb(X) ι: X Alb(X) : A X A f g : Alb(X) A f = g ι Alb(X) X Albanese X Alb(X) = Jac(X) X Pcard Albanese ( Kähler X Pc 0 (X) Alb(X) ) 3 Jacob X g X 2g path α 1,..., α g, β 1,..., β g α 1 α 2 α 3 α g β 1 β 2 β 3 β g H 1 (X, Z) 2g H 1 (X, Z), X 1-cycle γ γ γ γ 1 γ, γ = 1, γ, γ = 1 α, α j = β, β j = 0, α, β j = β, α j = δ j : Poncaré H 1 (X, Z) H 1 (X, Z) 1 [Mum, P.137] α, β
Abel-Jacob I 67 H 1 (X, Z) H 1 (X, Z) Z, de Rham C- de Rham H dr (X, C) sngular H sng 1 (X, C) 3.1 (de Rham ) H dr (X, C) H sng (X, C) = H (X, C), [ω] ( ) γ ω γ dual -form -cycle H 0 (X, Ω X ) ω closed 1-form dω = 0 d = δ + δ d ω δω = 0 δω 2-form X 1 δω = 0 H 1 (X, C) 2g de Rham ω 1,..., ω g, ω 1,..., ω g ( ) : 3.2 H 1 (X, Z) H 0 (X, Ω X ), γ ( ) ω ω γ H 1 (X, Z) g- C- H 0 (X, Ω X ) 2g H 0 (X, Ω X ) C g H 1 (X, Z) Λ Λ 2g Λ 3.3 (Remann ) (1) ω, η X 1-form ( ) ( ) = 0. =1 α ω β η =1 =1 α ω α η (2) ω 0 1-form ( ) Im > 0. β ω β ω
68 X α β 4g 1-cycle α, β α +, β+ α, β β α β + α + 4g X 0 X 0 = g =1 (α+ α ) + g =1 (β+ β ) 2 X 0 X 0 f 1-form η η = df fω 1- d(fω) = 0 Green 0 = d(fω) X 0 = fω X 0 ( ) = fω fω + fω fω =1 = α + (f α + α α β + f α ) ω + (f β + β β f β ) ω. df = η X η α + = η α f α + f α β α + α 3 β η = α η f β + f β 0 = β η α ω + (1) (2) f = dω f d(fω) = ω ω + ( ω ω = 2 Im X 0 β α α β α η β ω β ω α ω ) 2 [Mum]. 3 β α
Abel-Jacob I 69 d(fω) = d f df = df dz 2 dz dz z = x + y dz dz = 2 dx dy ω 0 (1/2) fω > 0 3.4 ω 1,..., ω n H 0 (X, Ω X ) A j = α ω j B j = β ω j det(a j ),j 0 τ = A 1 B H g H g = {Z M g (C) t Z = Z, Im(Z) > 0 ( )} ω H 0 (X, Ω X ) α α ω = 0 3.3 (2) ω = 0 det(a j ) 0 H 0 (X, Ω X ) α ω j = δ j 3.3 (1) ω = ω, η = ω j τ j = τ j (2) ω = a ω a = t (a 1,..., a g ) t a Im(τ)a > 0 (a 0) H 0 (X, Ω X ) ω 1,..., ω g α ω j = δ j H 0 (X, Ω) C g H 0 (X, Ω X ) /H 1 (X, Z) = C g /Λ τ, Λ τ = Z g + τz g 3.5 Λ τ C g (lattce) ( ) 1 g τ det 0 1 g τ Λ τ R = C g 3.1 Jac(X) = C g /Λ τ X Jacob 4 Abel-Jacob X X path γ ( ) ω := ω 1,..., ω g C g γ γ γ
70 Λ τ = { ω C g γ H 1 (X, Z)} γ 4.1 D = (P Q ) Dv 0 (X) 0 (P Q ) I : Dv 0 (X) Jac(X) I(D) = P Q ω mod Λ P Q ω mod Λ τ Q P path path 1-cycle α, β Λ τ - I D = (P Q ) well-defned 4.1 D Dv l (X) I(D) = 0 I Pc 0 (X) = Dv 0 (X)/ Dv l (X) Jac(X) I X f f X P 1 t P 1 f 1 (t) = D(t) X deg D(t) t n X φ: P 1 Jac(X), t D(t) n ω mod Λ τ P 1 φ φ: P 1 C g P 1 φ φ ϕ(0) = ϕ( ) I(D) = φ(0) φ( ) 4.2 (Abel-Jacob ) I : Pc 0 (X) Jac(X) Abel Jacob 5
Abel-Jacob I 71 5.1 X g X D Dv 0 (X) P 1,..., P g X D P 1 + + P g g D + g g Remann-Roch l(d + g ) = l(k x D g ) + deg(d + g ) g + 1 g g + 1 = 1 D + g P 1 + + P g Pc 0 (X) g =1 P g J : X g := X } {{ X } Jac(X), g (P 1,..., P g ) P ω mod Λ τ Im J = Im I J P X (g) = X g /S g (S g g ) J X (g) Jac(X) 5.2 X (g) g- X (g) X X (g) j j P X t P 1 = = P g = P t σ (t) (P,..., P ) g X (g) (P,..., P ) Im I = Im J I J J J ( ) X (g) Jac(X) 5.1 ( 2 ) C 2 C y 2 = x 5 + (x, y) (x, y) (hyperellptc nvoluton) ι (x, y) x f : C P 1 x P 1 P C f 1 (x) = {P, ι(p )} C/ ι P 1 ι K C C P C Remann-Roch l(p ) l(k C P ) = deg P g + 1 = 0
72 l(p ) = 1 l(k C P ) = 1 K C P +Q Q C Q ι(p ) C (2) Pc 0 (C) C Ψ: C (2) Pc 0 (C), (P 1, P 2 ) P 1 + P 2 2 5.1 Ψ Remann-Roch l(p 1 + P 2 ) = l(k C P 1 P 2 ) + deg(p 1 + P 2 ) + 1 g = 1 + l(k C P 1 P 2 ) P 2 ι(p 1 ) l(k C P 1 P 2 ) = 0 ( 0 ) l(p 1 + P 2 ) = 1 P 1 + P 2 D = {(P, ι(p )) C (2) } C (2) D Ψ P C P +ι(p ) 2 K C Ψ(D) = {0} D C/ ι P 1 (D 2 ) = 1 Castelnuovo ([Har, Theorem 5.7, Chapter V]) D Z π : C (2) Z z 0 Z π 1 (z 0 ) = D π : C (2) D Z {z 0 } (z 0 blow-up) Ψ Ψ: Z Pc 0 (C) Z Jacob Jac(C) (D 2 ) = 1 pr: C C C (2) : C C C, x (x, x) ϕ: C C 1 ι C C pr C (2) D = ϕ( (C)) deg ϕ = 2 C C ( (C) 2 ) = 2 C C 0 I O C C O C 0 I I O C C ( (C)) ([Har, Proposton 6.18, Chapter II]) ( (C) 2 ) = deg O C C ( (C)) = deg I I 0 I 2 I O C I 0 I/I 2 O C I Ω 1 C = (I/I 2 ) I ( C O C F F = F ) ( (C) 2 ) = deg Ω 1 C = 2 2g = 2
Abel-Jacob I 73 6 Remann theta Remann theta Remann theta C g - z H g - τ z C g /Λ τ (z = 0 ) τ I Dv l (X) X X Jac(X) = C g /Λ τ X Jac(X), P P ω mod Λ τ X 4 6.1 z C g τ H g ϑ(z, τ) = l Z g exp(π t lτl + 2π t lz) C g H g C g 6.1 ϑ(z, τ) : (1) ϑ(z + m, τ) = ϑ(z, τ), m Z g ; (2) ϑ(z + τm, τ) = exp( π t mτm 2π t mz)ϑ(z, τ). (1), (2) C g ϑ(z, τ) f (1) f(z) = l Z g c l e 2πt lz Fourer (2) Fourer c n C g u 1,..., u g c l+uk = exp(2π t lτu k + π t u k τu k ) c l (1), (2) 1 ϑ(z, τ) 4 Remann theta
74 Jac(X) O X (Θ) Γ(X, O X (Θ)) 1 O X (Θ) ample Jac(X) (cf. [Ko]) Remann 6.2 X z C g X ( ) (P ) = ϑ(z + P ω, τ) Remann C g : 0 g Q 1,..., Q g Q ω z + mod Λ τ =1 (P ) P path path ϑ(z, τ) 0 (P ) 3.3 Q ( ) D d / X 0 D 1-form closed form ( ) dfz 0 = = = X 0 S D d (X 0 S D ) ( ) dfz D d + k=1 α + k α k d + k=1 β + k β k α k path α k β k β+ k path β + = k β 3 0 k path β k exp( π t u k τu k 2π P ω k + t u k z) (u k C g ), β k α + k α k path 2 { } d(log ) d(log ) = 2πω k = 2πg. α k k=1 α k α + k k=1 d.
Abel-Jacob I 75 0 = d D + 2πg g X 0 ω k = dg k g k ( ) = 0 1-form g k d / 0 = =1 D g k d + l=1 α + l α l g k d + 1 d g k = 2πg k (Q ) = 2π D 3 g k β + l d g k = δ kl β + l β l g k β l d β l l=1 Qk ω k β + l β l g k d = α l ω k = δ kl = δ kl = δ kl ( π t u l τu l 2π β l d(log ) P1 ω l 2π t u l z + 2πm l ) P 1 path α, β m l 2 g k α + g k l α l β l ω k = τ kl ) α + l α l g k d = α l [ ( dfz (g k τ kl ) = 2πτ kl n l + 2π α + l 2πω l g k ω l 2πτ kl ] d g k [ Q τ P1 kk ω k = z k + =1 2 ω k + l τ kl l α + k g k ω l ] + m k + l τ kl n l 7 Abel-Jacob Remann Abel-Jacob f z Remann f z 0 Q 1,..., Q g g D = Q J(D) = Q =1 ω mod Λ τ = z
76 { ( E = z Jac(X) f z(p ) = ϑ z + P ) } ω = 0, P X E Jac(X) U = Jac(X) E Im J U X (g) Im J J 7.1 (Jacob ) (1) J : X (g) Jac(X) = C g /Λ τ J : J 1 (U) U (2) P 1,..., P g X z C g g P =1 ω z mod Λ τ f z (P ) = ϑ( z + P ω) = 0 z U f z P P ω z mod Λ τ =1 X (g) Jac(X) W { ((P1 W =,..., P g ), z ) X (g) Jac(X) P ω z mod Λ τ, f z (P ) = 0 (1 g) } W 2 pr 1 X (g) W pr 2 Jac(X) 6.2 pr 2 : pr 1 2 (U) U z U f z {P 1,..., P g } z ( (P 1,..., P g ), z) ) pr 2 Im pr 2 U (W ) pr 2 dm W g W pr 1 W J {(x, J(x)) x X (g) } (2) (2) (1) Abel 7.2 (Abel ) P 1,..., P r, Q 1,..., Q r X I( r =1 (P Q )) = 0 {P 1,..., P r } {Q 1,..., Q r } X
Abel-Jacob I 77 (P ) = ϑ(z + P ω, τ) ϑ(z 0, τ) = 0 z 0 C g z 0 X ( ) ϕ P (x) ϕ P (x) = ϑ(z 0 + x P ω, τ) x = P f(x) = r =1 ϕ P (x) ϕ Q (x) ϕ P 0 ϕ P 2 7.3 P X { y D P = z Jac(X) ϑ(z + P } ω) = 0, y X 2 R (1 r) ϕ R ϑ z 0 D D P X P = { y ω y X} X P P + D Jac(X) ϑ dm(x P + D) g 1 dm D = g 1 D = D+X P D = D+X P = D+X P + X P + + X p = }{{} g Jac(X) (I ) dm D g 2. 7.4 z 0 C g ϑ(z 0 ) = 0 X X Φ z0 (P, Q) = ϑ(z 0 + Q ω) 0 P 2g 2 R 1,..., R g 1, S 1,..., S g 1 : {(P, Q) X X Φ z0 (P, Q) = 0} g 1 g 1 ={(P, P ) X X} ({R } X) (X {S }). =1 =1 y Φ z0 (R, y) 0 R 6.2 Φ z0 (R, y) y 1,..., y g g 7.1 (y 1,..., y g ) y ω z 0 P0 =1 R ω mod Λ τ
78 X (g) Φ z0 (R, R) = ϑ(z 0 + R R ω) = 0 y 1 = R (y 2,..., y g ) y =2 z 0 mod Λ τ X (g 1) R S = y +1 Φ z0 (, S ) 0, (1 g 1) y Φ z0 (R, y) 0 {R, S 1,..., S g } y Φ z0 (R, y) 0 R g 1 S 1,..., S g S 0 Φ(, S 0 ) 0 Φ z0 (x, S 0 ) = 0 x x = S 0 y Φ z0 (x, y) 0 x Φ z0 (x, S 0 ) = Φ z0 (S 0, x) x g S 0, R 1,..., R g 1 Φ z0 (R, ) 0 7.2 7.3 P, Q ϕ P 0 ϕ Q 0 z 0 z 0 1 X ( ) r ϕ P (x) f(x) = ϕ Q (x) =1 f(x) X x P Q x path I( (P Q )) = 0 r P r Q ( ) ω ω mod Λ τ =0 =0 P Q path γ, δ ( ) (modλ τ ) γ, δ f(x) path : x path γ δ P, Q x path x path α ω β w Λ τ theta f(x) X well-defned ϕ P (ϕ Q ) 7.4 P, S 1,..., S g 1 (Q, S 1,..., S g 1 ) g f(x) P 1,..., P r Q 1,..., Q r (f) = D [BLR] S. Bosch; W. Lütkebohmert; M. Raynaud, Néron models. Ergebnsse der Mathematk und hrer Grenzgebete (3) [Results n Mathematcs and Related Areas (3)], 21. Sprnger-Verlag, Berln, 1990.
Abel-Jacob I 79 [Har] R. Hartshorne, Algebrac geometry, Graduate Texts n Mathematcs, No. 52. Sprnger-Verlag, New York-Hedelberg, 1977. [Iwa], 1973. [Ko] [Lan] [Ml] Algebrac theory va schemes S. Lang, Introducton to algebrac and abelan functons, Second edton. Graduate Texts n Mathematcs, 89. Sprnger-Verlag, New York-Berln, 1982. J. S. Mlne, Jacoban varetes Chapter VII of Arthmetc geometry, Papers from the conference held at the Unversty of Connectcut, Storrs, Connectcut, July 30 August 10, 1984. Edted by Gary Cornell and Joseph H. Slverman. Sprnger-Verlag, New York, 1986. [Mum] D. Mumford, Tata lectures on theta I, Progress n Mathematcs, 28. Brkhäuser Boston, Inc., Boston, MA, 1983. [Ser] [Sh] J. P. Serre, Algebrac groups and class felds, Graduate Texts n Mathematcs, 117. Sprnger-Verlag, New York, 1988 1992. [Wel] A. Wel, Varétés abélennes et courbes algébrques Actualtés Sc. Ind., no. 1064 = Publ. Inst. Math. Unv. Strasbourg 8 (1946). Hermann & Ce., Pars, 1948.
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