9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb / 33

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Transcription:

9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb 2008 1 / 33

1 NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) ( GCA95 ) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) ( GCA95 ) X NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) ( GCA95 ) X D = i D i X NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) ( GCA95 ) X D = i D i X f : C X NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

1 Green-Griffiths (1972) X f : C X f (C) X (1970) P n (C) 2n 1 (n 3) ( GCA95 ) X D = i D i X f : C X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + ϵt f (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33

Griffihts NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

Griffihts 1.1 (Griffiths et al., 1972 ) NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

Griffihts 1.1 (Griffiths et al., 1972 ) X n D = i D i X NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

Griffihts 1.1 (Griffiths et al., 1972 ) X n D = i D i X f : C n X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + S f (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

Griffihts 1.1 (Griffiths et al., 1972 ) X n D = i D i X f : C n X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + S f (r) ϵ, ϵ > 0. S f (r) = O(δ log r + log + T f (r)) δ, δ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

Griffihts 1.1 (Griffiths et al., 1972 ) X n D = i D i X f : C n X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + S f (r) ϵ, ϵ > 0. S f (r) = O(δ log r + log + T f (r)) δ, δ > 0. Green-Griffiths NOGUCHI (UT) HDVT 9 Feb 2008 3 / 33

NOGUCHI (UT) HDVT 9 Feb 2008 4 / 33

1.2 (Ein 88 91, Xu 94, Voisin 96) X P n (C) 2n 1(n 3) X NOGUCHI (UT) HDVT 9 Feb 2008 4 / 33

1.2 (Ein 88 91, Xu 94, Voisin 96) X P n (C) 2n 1(n 3) X 2 NOGUCHI (UT) HDVT 9 Feb 2008 4 / 33

1.2 (Ein 88 91, Xu 94, Voisin 96) X P n (C) 2n 1(n 3) X 2 E = µ=1 ν µz µ C z µ C. { ν µ, z = z µ, ord z E = 0, z {z µ }. E l( ) NOGUCHI (UT) HDVT 9 Feb 2008 4 / 33

n l (t; E) = min{ν µ, l}, N l (r; E) = { z µ <t} r 1 n l (t; E) dt. t l = n(t; E) = n (t; E), N(r; E) = N (r; E). NOGUCHI (UT) HDVT 9 Feb 2008 5 / 33

n l (t; E) = min{ν µ, l}, N l (r; E) = { z µ <t} r 1 n l (t; E) dt. t l = n(t; E) = n (t; E), N(r; E) = N (r; E). X O X I O X f : C X, f (C) Supp O X /I NOGUCHI (UT) HDVT 9 Feb 2008 5 / 33

X {U j } (i) U j σ jk Γ(U j, I), k = 1, 2,..., U j I (ii) {U j } {c j } NOGUCHI (UT) HDVT 9 Feb 2008 6 / 33

X {U j } (i) U j σ jk Γ(U j, I), k = 1, 2,..., U j I (ii) {U j } {c j } ρ I (x) = ( j c j(x) k σ jk(x) 2 ) 1/2 C Cρ I (x) 1, x M. X log ρ I ( Weil {U j }, {c j } NOGUCHI (UT) HDVT 9 Feb 2008 6 / 33

X {U j } (i) U j σ jk Γ(U j, I), k = 1, 2,..., U j I (ii) {U j } {c j } ρ I (x) = ( j c j(x) k σ jk(x) 2 ) 1/2 C Cρ I (x) 1, x M. X log ρ I ( Weil {U j }, {c j } f I Y = (Supp O X /I, O/I) m f (r; I) = m f (r; Y ) 1 dθ = log Cρ I (f (z)) 2π z =r ( 0) NOGUCHI (UT) HDVT 9 Feb 2008 6 / 33

ρ I f (z) C \ f 1 (Supp Y ) NOGUCHI (UT) HDVT 9 Feb 2008 7 / 33

ρ I f (z) C \ f 1 (Supp Y ) z 0 f 1 (Supp Y ) U f I = ((z z 0 ) ν ), ν N U ψ(z) log ρ I f (z) = ν log z z 0 + ψ(z), z U. NOGUCHI (UT) HDVT 9 Feb 2008 7 / 33

ρ I f (z) C \ f 1 (Supp Y ) z 0 f 1 (Supp Y ) U f I = ((z z 0 ) ν ), ν N U ψ(z) log ρ I f (z) = ν log z z 0 + ψ(z), z U. N(r; f I) N l (r; f I) ν NOGUCHI (UT) HDVT 9 Feb 2008 7 / 33

ρ I f (z) C \ f 1 (Supp Y ) z 0 f 1 (Supp Y ) U f I = ((z z 0 ) ν ), ν N U ψ(z) log ρ I f (z) = ν log z z 0 + ψ(z), z U. N(r; f I) N l (r; f I) ν ω I,f = ω Y,f = dd c ψ(z) = i 2π ψ(z) = dd c 1 log (z U), ρ I f (z) C (1,1) NOGUCHI (UT) HDVT 9 Feb 2008 7 / 33

ρ I f (z) C \ f 1 (Supp Y ) z 0 f 1 (Supp Y ) U f I = ((z z 0 ) ν ), ν N U ψ(z) log ρ I f (z) = ν log z z 0 + ψ(z), z U. N(r; f I) N l (r; f I) ν ω I,f = ω Y,f = dd c ψ(z) = i 2π ψ(z) = dd c 1 log (z U), ρ I f (z) C (1,1) f I Y r dt T (r; ω I,f ) = T (r; ω Y,f ) = t 1 ω I,f z <t NOGUCHI (UT) HDVT 9 Feb 2008 7 / 33

I X Cartier D m f (r; I) = m f (r; D) + O(1), T (r; ω I,f ) = T f (r; [D]) + O(1). NOGUCHI (UT) HDVT 9 Feb 2008 8 / 33

I X Cartier D m f (r; I) = m f (r; D) + O(1), T (r; ω I,f ) = T f (r; [D]) + O(1). X ω X r dt (1) T f (r) = T (r; ω X ) = f ω X. t 1 z <t NOGUCHI (UT) HDVT 9 Feb 2008 8 / 33

2 f : C X I NOGUCHI (UT) HDVT 9 Feb 2008 9 / 33

2 f : C X I (i) ( ) T (r; ω I,f ) = N(r; f I) + m f (r; I) m f (1; I) NOGUCHI (UT) HDVT 9 Feb 2008 9 / 33

2 f : C X I (i) ( ) T (r; ω I,f ) = N(r; f I) + m f (r; I) m f (1; I) (ii) X T (r; ω I,f ) = O(T f (r)) NOGUCHI (UT) HDVT 9 Feb 2008 9 / 33

2 f : C X I (i) ( ) T (r; ω I,f ) = N(r; f I) + m f (r; I) m f (1; I) (ii) X T (r; ω I,f ) = O(T f (r)) (iii) ϕ : X 1 X 2 Y i X i (i = 1, 2) ϕ(y 1 ) Y 2 m f (r; Y 1 ) m ϕ f (r; Y 2 ) + O(1) NOGUCHI (UT) HDVT 9 Feb 2008 9 / 33

3 Cartan f : C P n (C) f = (f 0 : : f n ) T f (r) = T (r; f (F.-S.)) Fubini-Study NOGUCHI (UT) HDVT 9 Feb 2008 10 / 33

3 Cartan f : C P n (C) f = (f 0 : : f n ) T f (r) = T (r; f (F.-S.)) Fubini-Study 3 (Cartan 1933) f NOGUCHI (UT) HDVT 9 Feb 2008 10 / 33

3 Cartan f : C P n (C) f = (f 0 : : f n ) T f (r) = T (r; f (F.-S.)) Fubini-Study 3 (Cartan 1933) f H j, 1 j q (q n 1)T f (r) q N n (r; f H i ) + S f (r) j=1 NOGUCHI (UT) HDVT 9 Feb 2008 10 / 33

3 Cartan f : C P n (C) f = (f 0 : : f n ) T f (r) = T (r; f (F.-S.)) Fubini-Study 3 (Cartan 1933) f H j, 1 j q (q n 1)T f (r) q N n (r; f H i ) + S f (r) j=1 S f (r) = O(δ log + r + log + T f (r)) δ, δ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 10 / 33

Cartan n + 1 W (f 0,, f n ) NOGUCHI (UT) HDVT 9 Feb 2008 11 / 33

Cartan n + 1 W (f 0,, f n ) Riesz j Q log Ĥj f W ( f ) j R = log Ĥj f + log Ĥj f W ( f ) j Q\R Q = {1,..., q}, R Q, R = n + 1. NOGUCHI (UT) HDVT 9 Feb 2008 11 / 33

Cartan n + 1 W (f 0,, f n ) Riesz j Q log Ĥj f W ( f ) j R = log Ĥj f + log Ĥj f W ( f ) j Q\R Q = {1,..., q}, R Q, R = n + 1. 4 (Cartan Nochka 1982) f (C) P n (C) l NOGUCHI (UT) HDVT 9 Feb 2008 11 / 33

Cartan n + 1 W (f 0,, f n ) Riesz j Q log Ĥj f W ( f ) j R = log Ĥj f + log Ĥj f W ( f ) j Q\R Q = {1,..., q}, R Q, R = n + 1. 4 (Cartan Nochka 1982) f (C) P n (C) l H j, 1 j q (q 2n + l 1)T f (r) q N n (r; f H i ) + S f (r). j=1 NOGUCHI (UT) HDVT 9 Feb 2008 11 / 33

4 P n (C) NOGUCHI (UT) HDVT 9 Feb 2008 12 / 33

4 P n (C) Eremenko-Sodin(1992) P n (C) D j, 1 j q, NOGUCHI (UT) HDVT 9 Feb 2008 12 / 33

4 P n (C) Eremenko-Sodin(1992) P n (C) D j, 1 j q, D j, 1 j q R {1,..., q} R n codim j R D j = R R > n j R D j = 5 (Eremenko-Sodin 1992) D j 1 j q P n (C) f : C P n (C) (q 2n)T f (r) q j=1 1 deg D j N(r; f D j ) + ϵt f (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 12 / 33

Cartan Weyls-Ahlfors NOGUCHI (UT) HDVT 9 Feb 2008 13 / 33

Cartan Weyls-Ahlfors N(r; f D j ) NOGUCHI (UT) HDVT 9 Feb 2008 13 / 33

Cartan Weyls-Ahlfors N(r; f D j ) Corvaja Zannier Schmidt Min Ru B. Shiffman 1979 NOGUCHI (UT) HDVT 9 Feb 2008 13 / 33

Cartan Weyls-Ahlfors N(r; f D j ) Corvaja Zannier Schmidt Min Ru B. Shiffman 1979 6 (M. Ru, 2004) D j P n (C), 1 j q, f : C P n (C) (q n 1)T f (r) q j=1 1 deg D j N(r; f D j ) + ϵt f (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 13 / 33

Cartan Weyls-Ahlfors N(r; f D j ) Corvaja Zannier Schmidt Min Ru B. Shiffman 1979 6 (M. Ru, 2004) D j P n (C), 1 j q, f : C P n (C) (q n 1)T f (r) q j=1 1 deg D j N(r; f D j ) + ϵt f (r) ϵ. N(r; f D j ) NOGUCHI (UT) HDVT 9 Feb 2008 13 / 33

5 Siu X n L > 0 D j L, 1 j q, s j H 0 (X, L) NOGUCHI (UT) HDVT 9 Feb 2008 14 / 33

5 Siu X n L > 0 D j L, 1 j q, s j H 0 (X, L) Γ γ αβ Γ γ αβ T(X ) F t H 0 (X, F ) tγ γ αβ NOGUCHI (UT) HDVT 9 Feb 2008 14 / 33

5 Siu X n L > 0 D j L, 1 j q, s j H 0 (X, L) Γ γ αβ Γ γ αβ T(X ) F t H 0 (X, F ) tγ γ αβ f : C X 1 (i) α, β td α β s j α s j, β s j and s j C α (1, 0) D α Γ γ αβ (1, 0) 2 (ii) f z td z f z (td z ) (n 1) f z K 1 X F n(n 1)/2, 0. NOGUCHI (UT) HDVT 9 Feb 2008 14 / 33

7 (Y. Siu, 1987) n(n 1) qt f (r; L) + T f (r; K X ) T f (r; F ) 2 q N(r; f D j ) + ϵt f (r) ϵ. j=1 NOGUCHI (UT) HDVT 9 Feb 2008 15 / 33

7 (Y. Siu, 1987) n(n 1) qt f (r; L) + T f (r; K X ) T f (r; F ) 2 q N(r; f D j ) + ϵt f (r) ϵ. j=1 P n (C) H j Cartan ( q n 1 ) n(n 1) T f (r; L) 2 q N(r; f H j ) + ϵt f (r) ϵ j=1 NOGUCHI (UT) HDVT 9 Feb 2008 15 / 33

6 Deligne X n D X σ 1 σ l = 0 σ i NOGUCHI (UT) HDVT 9 Feb 2008 16 / 33

6 Deligne X n D X σ 1 σ l = 0 σ i ( d j ) ν σ i, 1 j k, σ i k Ω 1 X (log D) D Deligne 1 NOGUCHI (UT) HDVT 9 Feb 2008 16 / 33

6 Deligne X n D X σ 1 σ l = 0 σ i ( d j ) ν σ i, 1 j k, σ i k Ω 1 X (log D) D Deligne 1 α : X \ D A X Albanese Zariski Y NOGUCHI (UT) HDVT 9 Feb 2008 16 / 33

8 ( 1977/81) (i) dim Y = n (ii) Y NOGUCHI (UT) HDVT 9 Feb 2008 17 / 33

8 ( 1977/81) (i) dim Y = n (ii) Y κ > 0 f : C X κt f (r) N 1 (r, f D) + S f (r). NOGUCHI (UT) HDVT 9 Feb 2008 17 / 33

8 ( 1977/81) (i) dim Y = n (ii) Y κ > 0 f : C X κt f (r) N 1 (r, f D) + S f (r). X \ D q(x \ D) = h 0 (X, Ω 1 X (log D)); D = q(x ) = q(x ) NOGUCHI (UT) HDVT 9 Feb 2008 17 / 33

8 ( 1977/81) (i) dim Y = n (ii) Y κ > 0 f : C X κt f (r) N 1 (r, f D) + S f (r). X \ D q(x \ D) = h 0 (X, Ω 1 X (log D)); D = q(x ) = q(x ) 9 Bloch- q(x \ D) > n f : C X \ D NOGUCHI (UT) HDVT 9 Feb 2008 17 / 33

10 Bloch- 1926/77;... q(x ) > n f : C X NOGUCHI (UT) HDVT 9 Feb 2008 18 / 33

10 Bloch- 1926/77;... q(x ) > n f : C X A. Bloch (1926) Nevanlinna (1977) NOGUCHI (UT) HDVT 9 Feb 2008 18 / 33

10 Bloch- 1926/77;... q(x ) > n f : C X A. Bloch (1926) Nevanlinna (1977) 11 Borel 1897 H j P n (C), 1 j n + 2, f : C P n (C) \ n+2 j=1 H j NOGUCHI (UT) HDVT 9 Feb 2008 18 / 33

10 Bloch- 1926/77;... q(x ) > n f : C X A. Bloch (1926) Nevanlinna (1977) 11 Borel 1897 H j P n (C), 1 j n + 2, f : C P n (C) \ n+2 j=1 H j ( q P n (C) \ ) n+2 j=1 H j = n + 1 > n Bloch- Borel Bloch- NOGUCHI (UT) HDVT 9 Feb 2008 18 / 33

7 Bloch- M f : (C, 0) (M, x) (f (0)) = x) f (1) (0) T (M) x x M NOGUCHI (UT) HDVT 9 Feb 2008 19 / 33

7 Bloch- M f : (C, 0) (M, x) (f (0)) = x) f (1) (0) T (M) x x M k f (ζ) f (0) + 1 1! f (1) (0)ζ + + 1 k! f (k) (0)ζ k, NOGUCHI (UT) HDVT 9 Feb 2008 19 / 33

7 Bloch- M f : (C, 0) (M, x) (f (0)) = x) f (1) (0) T (M) x x M k f (ζ) f (0) + 1 1! f (1) (0)ζ + + 1 k! f (k) (0)ζ k, (f (1) (0),..., f (k) (0)) J k (M) x x J k (M) M k NOGUCHI (UT) HDVT 9 Feb 2008 19 / 33

f : C M J k (f ) : C J k (M) NOGUCHI (UT) HDVT 9 Feb 2008 20 / 33

f : C M J k (f ) : C J k (M) J k (M) Ψ : J k (M) C { } k NOGUCHI (UT) HDVT 9 Feb 2008 20 / 33

f : C M J k (f ) : C J k (M) J k (M) Ψ : J k (M) C { } k D M σ 1 σ l = 0 σ i NOGUCHI (UT) HDVT 9 Feb 2008 20 / 33

f : C M J k (f ) : C J k (M) J k (M) Ψ : J k (M) C { } k D M σ 1 σ l = 0 σ i ( d j ) ν σ i, 1 j k, σ i k NOGUCHI (UT) HDVT 9 Feb 2008 20 / 33

12 M Ψ : J k (M) C { } k m Ψ Jk (f )(r; ) = S f (r). NOGUCHI (UT) HDVT 9 Feb 2008 21 / 33

12 M Ψ : J k (M) C { } k m Ψ Jk (f )(r; ) = S f (r). 8 NOGUCHI (UT) HDVT 9 Feb 2008 21 / 33

12 M Ψ : J k (M) C { } k m Ψ Jk (f )(r; ) = S f (r). 8 A NOGUCHI (UT) HDVT 9 Feb 2008 21 / 33

12 M Ψ : J k (M) C { } k m Ψ Jk (f )(r; ) = S f (r). 8 A 0 (C ) t A A 0 0, A 0 NOGUCHI (UT) HDVT 9 Feb 2008 21 / 33

12 M Ψ : J k (M) C { } k m Ψ Jk (f )(r; ) = S f (r). 8 A 0 (C ) t A A 0 0, A 0 J k (A) A k f : C A J k (f ) : C J k (A) k X k (f ) J k (f )(C) J k (A) Zariski NOGUCHI (UT) HDVT 9 Feb 2008 21 / 33

13 ( -Winkelmann- 2007 to appear in Forum Math.) f : C A NOGUCHI (UT) HDVT 9 Feb 2008 22 / 33

13 ( -Winkelmann- 2007 to appear in Forum Math.) f : C A (i) Z X k (f ) (k 0) X k (f ) T (r; ω Z,Jk (f ) ) N 1(r; J k (f ) Z) + ϵt f (r) ϵ, ϵ > 0, Z X k (f ) Z NOGUCHI (UT) HDVT 9 Feb 2008 22 / 33

13 ( -Winkelmann- 2007 to appear in Forum Math.) f : C A (i) Z X k (f ) (k 0) X k (f ) T (r; ω Z,Jk (f ) ) N 1(r; J k (f ) Z) + ϵt f (r) ϵ, ϵ > 0, Z X k (f ) Z (ii) codim Xk (f )Z 2 T (r; ω Z,J k (f ) ) ϵt f (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 22 / 33

13 ( -Winkelmann- 2007 to appear in Forum Math.) f : C A (i) Z X k (f ) (k 0) X k (f ) T (r; ω Z,Jk (f ) ) N 1(r; J k (f ) Z) + ϵt f (r) ϵ, ϵ > 0, Z X k (f ) Z (ii) codim Xk (f )Z 2 T (r; ω Z,J k (f ) ) ϵt f (r) ϵ, ϵ > 0. (iii) k = 0 Z A D A Ā A f T f (r; L( D)) N 1 (r; f D) + ϵt f (r; L( D)) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 22 / 33

(iii) Lang NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D 9 NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D 9 Bloch- NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D 9 Bloch- X κ(x ) A NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D 9 Bloch- X κ(x ) A 15 ( -Winkelmann- 2007) (i) π : X A (ii) κ(x ) > 0. NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

(iii) Lang 14 (Y.T. Siu-Z.K. Yeung 1996 1998 M. McQuillan 2001) A D f : C A \ D 9 Bloch- X κ(x ) A 15 ( -Winkelmann- 2007) (i) π : X A (ii) κ(x ) > 0. f : C X NOGUCHI (UT) HDVT 9 Feb 2008 23 / 33

κ(x ) > 0 X Bloch- 15 X Green-Griffiths NOGUCHI (UT) HDVT 9 Feb 2008 24 / 33

κ(x ) > 0 X Bloch- 15 X Green-Griffiths 16 X Albanese κ(x ) > 0 q(x ) dim X f : C X NOGUCHI (UT) HDVT 9 Feb 2008 24 / 33

κ(x ) > 0 X Bloch- 15 X Green-Griffiths 16 X Albanese κ(x ) > 0 q(x ) dim X f : C X 17 D = q i=1 D i P n (C) q n + 1 deg D n + 2 f : C P n (C) \ D NOGUCHI (UT) HDVT 9 Feb 2008 24 / 33

1987 Borel NOGUCHI (UT) HDVT 9 Feb 2008 25 / 33

1987 Borel P 2 (C), f : C P 2 (C) \ {z 0 = 0} {z 1 = 0} {z 2 0 + z 2 1 + z 2 2 = 0} M. Green (1974) 10 NOGUCHI (UT) HDVT 9 Feb 2008 25 / 33

1987 Borel P 2 (C), f : C P 2 (C) \ {z 0 = 0} {z 1 = 0} {z 2 0 + z 2 1 + z 2 2 = 0} M. Green (1974) 10 Acta(2004) Nevannlinna Ahlfors P 1 D. Drasin Math. Review This outstanding paper... NOGUCHI (UT) HDVT 9 Feb 2008 25 / 33

1987 Borel P 2 (C), f : C P 2 (C) \ {z 0 = 0} {z 1 = 0} {z 2 0 + z 2 1 + z 2 2 = 0} M. Green (1974) 10 Acta(2004) Nevannlinna Ahlfors P 1 D. Drasin Math. Review This outstanding paper... p : X S K X /S NOGUCHI (UT) HDVT 9 Feb 2008 25 / 33

18 2004 dim X /S = 1 D X f : C X g = p f : C S ϵ > 0 C(ϵ) > 0 T f (r; [D]) + T f (r; K X /S ) N 1 (r; f D) + ϵt f (r) + C(ϵ)T g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 26 / 33

18 2004 dim X /S = 1 D X f : C X g = p f : C S ϵ > 0 C(ϵ) > 0 T f (r; [D]) + T f (r; K X /S ) N 1 (r; f D) + ϵt f (r) + C(ϵ)T g (r) ϵ. f : C X s-dim(f ) Y X f (C) C(Y ) Y transc-deg C C(Y ) = dim Y dim X. NOGUCHI (UT) HDVT 9 Feb 2008 26 / 33

18 2004 dim X /S = 1 D X f : C X g = p f : C S ϵ > 0 C(ϵ) > 0 T f (r; [D]) + T f (r; K X /S ) N 1 (r; f D) + ϵt f (r) + C(ϵ)T g (r) ϵ. f : C X s-dim(f ) Y X f (C) C(Y ) Y transc-deg C C(Y ) = dim Y dim X. S(f ) = {ϕ C(Y ); T (r; f ϕ) ϵt f (r) ϵ, ϵ > 0}. NOGUCHI (UT) HDVT 9 Feb 2008 26 / 33

S(f ) C(Y ) NOGUCHI (UT) HDVT 9 Feb 2008 27 / 33

S(f ) C(Y ) s-dim(f ) = transc-deg C S(f ) NOGUCHI (UT) HDVT 9 Feb 2008 27 / 33

S(f ) C(Y ) 19 s-dim(f ) = transc-deg C S(f ) f : C X s-dim(f ) < dim X 9 ( Bloch- ) 14 (Lang ) f J k (f )(k >> 1) s-dim(j k (f )) = dim J k (f )(C) Zar NOGUCHI (UT) HDVT 9 Feb 2008 27 / 33

20 f : C X κ(x ) = dim X = 2, s-dim(f ) = 1. f f NOGUCHI (UT) HDVT 9 Feb 2008 28 / 33

20 f : C X f κ(x ) = dim X = 2, s-dim(f ) = 1. f S p : X S g = p f : C S T g (r) ϵt f (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 28 / 33

20 f : C X f κ(x ) = dim X = 2, s-dim(f ) = 1. f S p : X S g = p f : C S T g (r) ϵt f (r) ϵ. S K X /S NOGUCHI (UT) HDVT 9 Feb 2008 28 / 33

20 f : C X f κ(x ) = dim X = 2, s-dim(f ) = 1. f S p : X S g = p f : C S T g (r) ϵt f (r) ϵ. S K X /S T f (r; K X /S ) T f (r). NOGUCHI (UT) HDVT 9 Feb 2008 28 / 33

20 f : C X f κ(x ) = dim X = 2, s-dim(f ) = 1. f S p : X S g = p f : C S T g (r) ϵt f (r) ϵ. S K X /S T f (r; K X /S ) T f (r). T f (r) ϵt f (r) ϵ. (q.e.d) NOGUCHI (UT) HDVT 9 Feb 2008 28 / 33

11 NOGUCHI (UT) HDVT 9 Feb 2008 29 / 33

11 X D = i D i X f : C X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + ϵt f (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 29 / 33

11 X D = i D i X f : C X T f (r; [D]) + T f (r; K X ) i N 1 (r; f D i ) + ϵt f (r) ϵ, ϵ > 0. D NOGUCHI (UT) HDVT 9 Feb 2008 29 / 33

X n X P N (C) π : X P n (C) D X E = π D P n (C) NOGUCHI (UT) HDVT 9 Feb 2008 30 / 33

X n X P N (C) π : X P n (C) D X E = π D P n (C) π π NOGUCHI (UT) HDVT 9 Feb 2008 30 / 33

X n X P N (C) π : X P n (C) D X E = π D P n (C) π π f : C P n (C) {deg E n 1}T f (r) N 1 (r; f E) + ϵt f (r) ϵ, ϵ > 0, NOGUCHI (UT) HDVT 9 Feb 2008 30 / 33

X n X P N (C) π : X P n (C) D X E = π D P n (C) π π f : C P n (C) {deg E n 1}T f (r) N 1 (r; f E) + ϵt f (r) ϵ, ϵ > 0, NOGUCHI (UT) HDVT 9 Feb 2008 30 / 33

X n X P N (C) π : X P n (C) D X E = π D P n (C) π π f : C P n (C) {deg E n 1}T f (r) N 1 (r; f E) + ϵt f (r) ϵ, ϵ > 0, f : C X T f (r; [D]) N 1 (r; f D) + m f (r, D) + ϵt f (r) ϵ, ϵ > 0. X NOGUCHI (UT) HDVT 9 Feb 2008 30 / 33

Green-Griffits NOGUCHI (UT) HDVT 9 Feb 2008 31 / 33

Green-Griffits f : C X π : X P n (C), D X E P n (C) NOGUCHI (UT) HDVT 9 Feb 2008 31 / 33

Green-Griffits f : C X π : X P n (C), D X E P n (C) (deg E n 1)T g (r) N 1 (r; g E) + ϵt g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 31 / 33

Green-Griffits f : C X π : X P n (C), D X E P n (C) (deg E n 1)T g (r) N 1 (r; g E) + ϵt g (r) ϵ. N(r; g E) N 1 (r; g E) (n + 1)T g (r) m g (r; E) + ϵt g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 31 / 33

N 1 (r; f D) N(r; g E) N 1 (r; g E) (n + 1)T g (r) m g (r; E) + ϵt g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 32 / 33

N 1 (r; f D) N(r; g E) N 1 (r; g E) (n + 1)T g (r) m g (r; E) + ϵt g (r) ϵ. T f (r; [D]) N 1 (r; f D) + m f (r, D) + ϵt f (r) ϵ NOGUCHI (UT) HDVT 9 Feb 2008 32 / 33

N 1 (r; f D) N(r; g E) N 1 (r; g E) (n + 1)T g (r) m g (r; E) + ϵt g (r) ϵ. T f (r; [D]) N 1 (r; f D) + m f (r, D) + ϵt f (r) ϵ m f (r; D) m g (r; E) T f (r; [D]) (n + 1)T g (r) m g (r; D) + ϵt g (r) + m f (r; D) (n + 1)T g (r) + ϵt g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 32 / 33

K X = π K P n (C) + D T f (r; K X ) = (n + 1)T g (r) + T f (r; [D]). NOGUCHI (UT) HDVT 9 Feb 2008 33 / 33

K X = π K P n (C) + D T f (r; K X ) = (n + 1)T g (r) + T f (r; [D]). T f (r; K X ) ϵt g (r) ϵ. NOGUCHI (UT) HDVT 9 Feb 2008 33 / 33

K X = π K P n (C) + D T f (r; K X ) = (n + 1)T g (r) + T f (r; [D]). T f (r; K X ) ϵt g (r) ϵ. K X T f (r; K X ) T g (r) NOGUCHI (UT) HDVT 9 Feb 2008 33 / 33

K X = π K P n (C) + D T f (r; K X ) = (n + 1)T g (r) + T f (r; [D]). T f (r; K X ) ϵt g (r) ϵ. K X T f (r; K X ) T g (r) T g (r) ϵt g (r) ϵ, ϵ > 0. NOGUCHI (UT) HDVT 9 Feb 2008 33 / 33

K X = π K P n (C) + D T f (r; K X ) = (n + 1)T g (r) + T f (r; [D]). T f (r; K X ) ϵt g (r) ϵ. K X T f (r; K X ) T g (r) T g (r) ϵt g (r) ϵ, ϵ > 0. P n (C) ( 13) NOGUCHI (UT) HDVT 9 Feb 2008 33 / 33

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