Геометрическая теория функций на компактной римановой поверхности

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Author: Чуешев В.

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Δ

f θ˜ +

df ∧ θ˜ = b+ j

f θ˜ +

Δ

a− j

˜ = d(f θ)

f θ˜ +

b− j

∂Δ

f θ˜ =

˜ f θ).

z0 Δ, z ∈ Δ " f (z) = z + − z θ. 8 z z # aj aj ∂Δ,

0

a+ j

f θ˜ +

a− j

f θ˜ =

a+ j

˜ + f (z)θ(z)

a− j

˜ ) = f (z )θ(z

a+ j

˜ − f (z)θ(z)

a+ j

[

z

z0

a+ j

θ−

˜ = f (z )θ(z)

z

θ]θ˜ =

z0

a+ j

aj

(−

bj

˜ = [f (z) − f (z )]θ(z)

θ)θ˜ = −

aj

θ˜ ·

θ. bj

m a− j z + I bj

m b− j6

R

6

z 6 + aj

z0

G m m

b+ j

f θ˜ +

b− j

A), # b+j b−j ,

f θ˜ = [

m

z0

bj

θ−

m z0

θ]θ˜ =

˜ θ.

θ aj

bj

1

/ θ )

F g > 0,

g 0 ≤ ||θ|| = (θ, θ) = [ θ ∗θ − θ ∗θ]. 2

aj

j=1

bj

bj

aj

8 !0 J

Γ = (γjk ) 9:C;

3)γkj =

F

ωk ∧ ∗ωj =

∗ω = , k = 1, ..., g, bk j Ng+k ∗ω j . − ak−g ∗ωj = − Nk−g ∗ωj , k = g + 1, ..., 2g;

5' F Γ " (Γ > 0). / Γ=

A B C D

Γ(tΓ = Γ) # + t A = A,t D = D, B =t C, Γ > 0 0 A > 0, D > 0. 8 2 " ∗ R %C ' !

! % '

8 ω1, ..., ω2g ∗ ! Λ 2g, ∗ωk = 2g j=1 λkj ωj , k = 1, ..., 2g, t ∗A = ΛA, A = (ω1, ..., ω2g). K ∗∗ = −id Λ2 = −I2g , −A = ∗ ∗ A = ∗(ΛA) = Λ · Λ · A. K " Λ, Γ J Γ = Λt J γlk = (ωl , ωk ) = (∗ωl , ∗ωk ) = (

2g

λlj ωj , ∗ωk ) =

j=1

2g

λlj

j=1

F

ωk ∧ ωj ,

l, k = 1, ..., 2g. 8

# 9:C; K ! ω1, ..., ω2g

" φj = ωj +i∗ωj , j = 1, ..., 2g. [70] φ1 , ..., φ2g J

" I I 1 1 (φk , φj ) = (φj , φk ) = (ωj , φk ) = 2 2

−i bj φk , j = 1, ..., g, i aj−g φk , j = g + 1, ..., 2g.

$ % 9443:C; * F

g > 0 Ω1(F ), !

F, φ1, ..., φg ,

H(F ) = Ω1(F ) ⊕ Ω1(F )

$87G(G1/6H.1 8 Ω1(F ) ∩ Ω1(F ) = {0}, ω = udz + vdz ∈ Ω1(F ), v = 0, ω = udz ∈ Ω1(F ) u = 0. " ! " ω ∈ H(F ). $ ω + i ∗ ω ∈ Ω1(F ) ω − i ∗ ω ∈ Ω1 (F ), ω + i ∗ ω ∈ Ω1(F ). 8 ω = 12 (ω + i ∗ ω) + 12 (ω − i ∗ ω). / ω = ω1 + ω2 , ωj ∈ Ω1(F ), j = 1, 2, ∗ω1 = −iω1 , ∗ω2 = −iω2 ∗ω2 = iω2 . # (ω1, ω 2) = (∗ω1, ∗ω 2) = (−iω1, iω 2) = −(ω1, ω 2 )

(ω1, ω 2 ) = 0. 8 " ω → ω R Ω (F ) Ω1 (F ), dimC Ω1(F ) = dimC Ω1(F ) = 12 dimH(F ) = g. * φ1, ..., φg C, Ω1(F ). ( φg+1, ..., φ2g " Ω1(F ). " $ % ! 9:C; * F g ≥ 1 ! ζ1, ..., ζg Ω1(F ) 1

aj ζk

= δjk , j, k = 1, ..., g.

8 #

(Ig , Ω), Ω = (πjk ), πkj =

bj

ζk ,

tΩ = Ω, JmΩ > 0. 1 Ω1(F ) a1, ..., ag , b1, ..., bg F. $ % " 9:,; F g ≥ 1, ω, ϕ Ω1(F ). / ω + ϕ = df, f ∈ C ∞(F ), ω = 0 = ϕ F. $87G(G1/6H.1 8 " ω = h(z)dz, ϕ = g(z)dz, h(z), g(z) z F. 1 ω ∧ ϕ = 0.

" ϕ = 0, i 2

F

ϕ∧ϕ=

F

|g(z)|2 dx ∧ dy > 0,

z

= x + iy. * ϕ ∧ ϕ = ϕ ∧ ω + ϕ ∧ ϕ = ϕ ∧ df. L " F ϕ∧df = 0, ϕ = 0 F. ( d(f ϕ) = df ∧ ϕ + f dϕ = df ∧ ϕ, F ϕ∧ϕ = − F d(f ϕ) = − ∂F f ϕ = 0. ! " ω = 0 " . ω = 0 = ϕ F. " a− b− ω1, ..., ωg Ω1(F ), (π1, ..., π2g), πj = t ( Nj ω1, ..., Nj ωg ), j = 1, ..., 2g. $ % # a− b− ω1 , ..., ωg Ω1(F ), R $87G(G1/6H.1 8 8 " π1, ..., π2g

R ! ! x1, ..., x2g % ' x1π1 + ... + x2g π2g = 0. 1"

x1π1 + ... + x2g π2g = 0. 8 π1 π2 ... π2g , Ω = π 1 π 2 ... π 2g

2g. ! ! " 2g Ω # x1, ..., x2g, . Ω @ 2g. 8

! ξ 1, ..., ξ g, η1, ..., ηg % ' ! 2g Ω # #

(

g

Ni j=1

j

ξ ωj +

g

η j ωj ) = 0, i = 1, ..., 2g.

j=1

g g j 1 j ω = j=1 ξ ωj , ϕ = j=1 η ωj ∈ Ω (F ), ∞ Ni (ω + ϕ) = 0, i = 1, ..., 2g. # ω + ϕ = df, f ∈ C (F ). " )=)> ω = 0 = ϕ, # C− ω1, ..., ωg. "

M @ % ' A + )' / θ, θ˜ F, 0=

F

g ˜ θ ∧ θ˜ = [ θ θ θ˜ − θ]; aj

j=1

bj

bj

aj

= ζj , θ˜ = ζk bj ζk − bk ζj = 0, πkj = πjk , k, j = 1, ..., g, Ω = (πjk )

,' / θ = θ F, g 2 0 ≤ ||θ|| = (θ, θ) = i [ θ θ− θ θ].

θ

j=1

aj

bj

,' θ = k = 1, ..., g,

aj

bj

g

k=1 ck ζk ,

ck ∈ C,

ck

g 0 < ||θ||2 = i[ (cj (c1 π1j + ... + cg πgj ) − cj (c1 π1j + ... + cg πgj )] = j=1

2

ImΩ > 0,

ak

θ = ck ,

g

cj ck Imπjk

j,k=1 bk

θ = c1 π1k + ... + cg πgk , k = 1, ..., g.

& θ

F g > 0. / a θ b θ θ ! θ = 0 F.

$87G(G1/6H.1 8 K

2

0 ≤ ||θ|| = (θ, θ) = i

F

g θ∧θ =i [ θ θ− θ θ]. j=1

aj

bj

bj

aj

0 ||θ||2 = 0 θ = 0 F. . 6 % ' F g ≥ 0 $ ! f, ω = df # ω = 0 F f F. " " 1 ! C Ω1 (F ). A

g " ω → ( a ω, ..., a ω) Ω (F ) C . ! & a ω ω = 0 F. & g g E " C Ω1(F ) Cg . e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), ...., eg = (0, 0, ..., 0, 1), ζ1, ..., ζg Ω1(F ). G 0 a b ! F g > 0. G G ! G " F. P, Q ∈ F, P = Q, τ = τP Q = τ (z)dz F \{P, Q}, ordP τ = −1 = ordQτ, resP τ = 1 = −resQ τ. 1 c τ (z)dz, c F, c F. 1 c ∼ c F c, c P, Q, ! n ∈ N 1

g

c τ (z)dz

− c τ (z)dz = 2πin. θ F P1, ..., Pk, k ≥ 2. " P1, ..., Pk. 0 " cj Pj ,

j = 1, ..., k,

F.* F = F \{P1, ..., Pk} ck

g k−1 + + − − j=1 aj bj aj bj + l=1 cl , a1 , ..., ag , b1 , ..., bg , c1 , ..., ck−1 F . / c F, F k−1 j=1 nj cj , nj ∈ Z. $ θ " F . 8

c

θ=

k−1

nj

j=1

cj

θ = 2πi

k−1

nj resPj θ.

j=1

/ c % ' F , c a1, ..., ag, b1, ..., bg c1 , ..., ck−1. . c θ c θ, ..., c θ. τP Q !

" a τP Q = 0, j = 1, ..., g;

N τP Q, k = 1, ..., 2g, @ τ τP Q ωP Q aj Nk 0 K @ A " @

θ = ζj , θ = τP Q, 9443:C; df = ζj Δ, Δ ζj ∧ τP Q = ∂Δ fτP Q = b τP Q. . ∂Δ f τP Q = 2πi(f (P ) − f (Q)). 1

1

k−1

j

k

j

2πi

P Q

ζj =

bj

τP Q ,

" 2πi

P Q

ζj =

bj

ωP Q −

g l=1 πjl al

Δ. G ωP Q , j = 1, ..., g.

$ θ = τP Q, θ = τRS , P, Q, R, S Δ. 0 Δ O ∈ ∂Δ P Q γ1 γ2 z Δ . * Δ θ = df, f (z) = z θ.

0

∂Δ

= 2πi[f (R) − f (S)] = 2πi f θ = 2πi[resR f θ + resS f θ]

R S

θ.

. ∂Δ

g + f θ, θ] f θ = [ θ θ − θ al

l=1

bl

bl

al

c

c O Q % +), O % −), " P. (

c

f θ = 2πi[

P

O

θ −

Q O

= 2πi θ]

P

θ.

Q

8 R

S

P

τP Q =

Q

τRS ,

" Δ = Δ \ROS, ROS O R O S Δ .

−

a O b-− 2 -2 zγ2+ a+ 1 −Q γ R1 R + + −P b1

b+ I2

a+ 2

− a− 1 I b1

A)4 G Re SR ωP Q = Re QP ωRS . * θ 0 ! n P, z(P ) = 0, B θ = zdz , n ≥ 2. * a

τP(n). 8 # ∞ (j) l z P, z(P ) = 0. θ = ζj = l=0 dl z dz P. 1 @ n

bj

j = 1, ..., g.

(n)

τP =

2πi (j) d , n − 1 n−2

$ "

θ = ζj = df, f (z) =

z

z0 ζj

Δ.

.

∂Δ

g = f θ = [ θ θ − θ θ] θ, l=1

al

bl

bl

al

bj

(j)

d f θ = 2πiresP f θ = n−2 2πi, n−1 ∂Δ

f (z) = (... +

(j) dn−2

(j)

d z n−1 1 + ...), f θ = (... + n−2 + ...)dz n−1 n − 1z

P. 0 !

6 ω1 ω2 F. 1 ϕ = ω1 − ω2 F. / d1, ..., dg a ϕ − d1ζ1 − ... − dg ζg a " " 0 F. ω1 = ω2 + d1ζ1 + ... + dg ζg . G

a

F g ≥ 1. 944; 6 " "

F g ≥ 1. $87G(G1/6H.1 8 ω c1, ..., cn P1 , ..., Pn, n ≥ 2. P0 = Pj , j = 1, ..., n. τP P , ..., τP P . 1 n n j=1 cj = 0, j=1 cj τP P

P0 cj Pj , j = 1, ..., n. / n ω − j=1 cj τP P a d1, ..., dg , ω2 = ω − g n j=1 cj τP P − k=1 dk ζk 1 944; / F g ≥ 1, 1 0

j

j

j

0

0

0

n 0

% ' F. 8 df f 944; * F !

P1, ..., Pl −2 (k) ωP = j=−n dj z j dz, k = 1, ..., l, nk ≥ 2, # 9443:C; @ A b− " F g ≥ 1. ! P1 , ..., Pn, n ≥ 2, F g ≥ 1. 1 F, F \{P1 , ..., Pn} P1 , ..., Pn, n + g − 1. n $87G(G1/6H.1 8 1 ω = j=1 cj τP P + g k=1 dk ζk , P0 = Pj , j = 1, ..., n, c1 + ... + cn = 0, dk , k = 1, ..., g, # a ω. # c1 , ..., cn−1, d1, ..., dg , # n − 1 + g. 1 k

k

j

0

§

F D = P1n ...Pkn , Pj ∈ F, nj ∈ Z, j = 1, ..., k. Div(F ) F Div(F ) ! "

# D k degD = j=1 nj . $ deg % (Div(F ), ·) (Z, +). f ∈ M ∗(F ), f F, &' %( % % (f ) = P ∈F P ord f ∈ Div(F ). & () M ∗ (F ) Div0(F ) )( & " % # DivH (F ) & () M ∗(F ). DivH (F ) & ( F. * Div(F )/DivH (F ) D D1 & " # + (D ∼ D1), D/D1 f ∈ M(F )\C, & " max{−ord f,0} (f )∞ = , (f )0 =

# P ∈F P max{ord f,0} . ,( (f ) = (f(f)) . f, g P ∈F P F (f ) = (g), (f(g)) = 1 f = cg, c ∈ C∗ = C\0, & F - f : F1 → F2 ( !# f ∗ : Div(F2) → Div(F1) . /# f ∗ ( f ∗((g)) = (f ∗(g)), g F2. 0# D1 ∼ D2 F2, f ∗(D1) ∼ f ∗(D2) F1. ω = 0 q ( % (ω) = P ∈F P ord ω . q q - q = 1 Z. ω1 ω2 q ( ( ωω ∈ M ∗(F ). $ ( (ω1) (ω2) "+ q # 1

k

P

P

0

P

∞

P

1 2

D = P ∈F P n(P ) ( n(P ) ≥ 0 P ∈ F, D ≥ 1. -+ ( D ≥ D1, DD1−1 ≥ 1. *& f ∈ M ∗ (F ) " q ω = 0) D, (f )D−1 ≥ 1((ω)D−1 ≥ 1). 1 (0)D−1 ≥ 1 & D ∈ Div(F ). 2 ( f D, f = 0, ordP f ≥ n(P ) P ∈ F. $ ( f P ∈ F, n(P ) ≥ 0. f ≥ n(P ) P, n(P ) > 0. f & ≤ −n(P ) P, n(P ) < 0. & D F L(D) = {f ∈ M(F ) : (f ) ≥ D}. r(D) & D. !# D ≤ D1 , L(D1) ⊂ L(D); /# L(1) = C r(1) = 1; 0# deg D > 0, r(D) = 0. & D ∈ Div(F ) Ωq (D), ' ω ( ω q− F (ω) ≥ D. iq (D) = dimCΩq (D) D q = 1. Ω1(D) = Ω(D) i1 (D) = i(D). & D ∈ Div(F ) degD, r(D), i(D) D, & 3 ( ω = 0 ( i(D) = r(D(ω)−1).

34542 67$28 - D1 ∼ D2, D1D2−1 = (f ), f 2

∈ M ∗ (F ).

L(D2) h → hf ∈ L(D1)

C ( r(D1) = r(D2). 5 Ω(D) ω0 →

ω0 ∈ L(D(ω)−1) ω + i(D) = r(D(ω)−1).

C ( - F g ≥ 1 i(1) = g.

! "9 9 # :00;( : 0 & P ∈ F. -

:00; = F g ≥ 1 g P1 , ..., Pg ( ' & P1, ..., Pg ( 1 &' ! 34542 67$28 & D r(D−1) ≥ 1, L(D−1) & degD + i(D) − g ≥ 0. - D = P1...Pn n P1 , ..., Pn ∈ F. 2 i(D) ≥ g − n, + n ≤ g. @ ( i(P1 ) ≤ g−1, % ' i(P1) = g−1 & P1 F. 5 Ω(P1P2 ) ⊂ Ω(P1) i(P1P2) = g − 1, g − 1 Ω(P1 ) ' & P2. = + & Ω(P1), &

Ω(P1) ' P2. - g ≥ 2 '

ϕ ∈ Ω(P1) ϕ = 0. 8% P2 ( ϕ & 2 i(P1P2) ≤ g − 2, % ' i(P1P2 ) = g − 2 P2 ∈ F. - + n ( ( g ≥ n ' n P1 , ..., Pn F ( i(P1 ...Pn) = g − n. 8 ( g 1 1 F i(P1 ...Pg ) = 0. & r( P ...P ) = 1 L( P ...P ) 2 - Ω2 F, Ωe Ω2, '

! :00;( : 2g − 2). - i(P n ) !( r(P −n ) % ( ! " 1 & 6 P ). 2 2 degP n = n, r(P −n) ≥ 2 n > g "n ≥ g + 1). -+ ' ( & & P ' n. ? 9 9 r(P −g ) = g − g + 1 + i(P g ) ( r(P −g ) ≥ 2, i(P g ) > 0.

% "8 1 # :00; $' P F g ≥ 1, i(P g ) > 0. 34542 67$28 - Pn F, i(Png ) > 0, n = 1, 2, 3, ... A & P0 ∈ F. 8 z z(P0 ) = 0 U (P0). 2 ζj = fj (z)dz, j = 1, ..., g, fj , j = 1, ..., g, U (P0).

$' N ( Pn ∈ U (P0), n > N, z(Pn) = zn. ? i(Png ) > 0 ( & zn '

ϕ = c1 ζ1 + ... + cg ζg ∈ Ω(Png ),

n

2 i=1 |ci |

>0

c1 f1 (zn ) + c2 f2 (zn ) + ... + cg fg (zn) = 0

c1 f1 (zn ) + c2 f2 (zn ) + ... + cg fg (zn) = 0 ... (g−1)

c1 f1

(g−1)

(zn ) + c2 f2

(zn ) + ... + cg fg(g−1)(zn ) = 0.

2 ( 8 Wg (z) + ' zn, % zn → 0 n → ∞. = Wg (z) z U (P0). ? ( f1, ..., fg U (P0). - 2 -+ ' P F ( + & ≤ g, & F. A & 8 1 F. - g = 0, 1 ' 8 1 P0 ∈ F &' 8 1 ( ' f F & & g + 1 P0 F. ( i(P0g ) = 0 r(P0−g ) = 1. = −(g+1)

r(P0

(g+1)

) = g + 1 − g + 1 + i(P0

)≥2

' & & ≤ (g + 1) P0 F. -% + n & g + 1, n ≤ g r(P0−g ) ≥ 2. - 2 (

3 F g + (g + 1)− 5 ( 1( g + 1 ( P0 8 1 & & & P0 & ≤ g. = ( & " # F g ≥ 2 8 1 8 1 =

1 8 1 -+ F ' & 8 1 - +(

& 8 1 ( F : 2g. 34542 67$28 g = 2, 1 %( α1 = 3 α2 = 4, ( ' j = 1, 0 < 1 < 2 α1 + α1 = 6 > 4. g = 3, {3, 4, 6}, {3, 5, 6} {4, 5, 6}, ( 2( ' 1 2g3 (< g) ' '% ≤ 2g - α ( &' "!# r, 1 ≤ r ≤ [ α2g ] < g − 1, 2g3 < g − 1 g ≥ 4, 1

&

rα1 < α < (r + 1)α1.

α1 , α2 = 2α1 , ..., αr = rα1 , αr+1 = α,

( (

αg−1 = 2g − α1 , ..., αg−r = 2g − rα1 , αg−(r+1) = 2g − α.

F ( ≥ αg−(r+1) < 2g. α1 + αg−(r+1) = α1 + 2g − α = 2g − (α − α1 ) > 2g − rα1 = αg−r .

-+ ' j, αg−r < j < 2g, ' αg−1, ..., αg−(r+1). - - # 8 g−1

αj ≥ g(g − 1),

j=1

% ( α1 = 2. 34542 67$28 ? ' 2

g−1

αj ≥ 2g(g − 1).

j=1

α1 = 2, α1 > 2( $ @ ( j ≥ 1 P F ⇔ r(P −j ) − r(P −(j−1)) = 0 ⇔ i(P j−1) − i(P j ) = 1 ⇔ ' F % j − 1 P. & P 0 = n1 − 1 < n2 − 1 < ... < ng − 1 ≤ 2g − 2, nj P F. & P

F g ≥ 2 ' ω ( & P, ordP ω = 0. ( 0 = n1 −1 P - E D ⊂ C, dimC E = n ≥ 1, z D. J ϕ1, ..., ϕn E z, ordz ϕ1 < ordz ϕ2 < ... < ordz ϕn.

(

0 ≤ μ1 = minϕ∈E {ordz ϕ}

ϕ1 ∈ E C ordz ϕ1 = μ1. 5 (n−1)− E1 = {ϕ ∈ E : ordz ϕ > μ1}. - μ2 = minϕ∈E {ordz ϕ} = ordz ϕ2 , ϕ2 ∈ E1 -

z ( μj = ordz ϕj , j = 1, ..., n. ( z, ∞ 2 ϕj (t) = k=0 akj (t−z)k U (z) ⊂ D. - ( aμ j = 1 k = j, aμ j = 0 k = j, j, k = 1, ..., n. A ordz ϕj = μj ϕj (t) = aμ j (t − z)μ + ..., aμ j = 0, ( aμ j , ϕj C aμ j = 1, j = 1, ..., n. 1

k

k

j

j

j

j

j

+ aμ 1 = 0, ϕ1 ϕ1 − aμ 1ϕ2,

aμ 1 = 0. 4 ( aμ 1 = ... = aμ 1 = 0. - ( & ) * = K K ( z, U (z), 2 & + ( + - & ( μj ≥ j − 1, 0 < 1 < ... < j − 1 μ1 < μ2 < ... < μj . 8 z E 2

2

2

3

n

τ (z) = τE (z) =

n

(μj − (j − 1)).

j=1

: 0} D; 0# = D0 ⊂ D > & z ∈ D0 ϕ1, ..., ϕn E ordz ϕj = j − 1, j = 1, .., n. - ' Ωq (F ) F g > 0 q > 0. 2 P ∈ F q− 8 1 ( % Ωq (F ) . ( ' q− F,

P 1 ( dimΩq (F ). 8 ( 1− 8 1 " # 8 1 F. g > 1, q > 0, τ (P ) P, q Ω (F ), Wq Ωq (F ) d = dq = dimΩq (F ).

$ 8 ' Wq m− ( m = (d/2)(2q − 1 + d), τ (P ) = (g − 1)d(2q − 1 + d).

P ∈F

34542 67$28 = ( Wq m− - ζ1 , ..., ζd Ωq (F ), z z = f (z) &' - ζj = ϕj (z)dz q = ϕ j ( z )d zq ,

ϕj (f (z))f (z)q = ϕj (z). ?

det[ϕ1, ..., ϕd] = det[(ϕ 1f )(f )q , ..., (ϕ df )(f )q )] =

(

(f )m det[(ϕ 1, ..., ϕ d)f ]

m = q + (q + 1) + ... + (q + d − 1) = (d/2)(2q − 1 + d), (det[ϕ1, ..., ϕd])dz m = (det[ϕ 1, ..., ϕ d])d zm.

2 ? + ( & g > 1 ' & q− 8 1 q > 0, &

m−

% : 1 & ( Ω1 (F ), 1 g(g − 1)/2. A P, / 34542 67$28 ? ' τ (P ) = (g − 1)g(g + 1).

P ∈F

- 2 ≤ α1 < α2 < ... < αg = 2g g P. 2 ( nj , j = 1, ..., g, αj , j = 1, ..., g, {1, 2, ..., 2g}. & (

g 2g g g τ (P ) = (nj − j) = j− αj − j= j=1

P ∈F 2g−1

j=g+1

j−

g−1 j=1

αj ≤

j=1

j=1

j=1

3g (g − 1) − g(g − 1) = g(g − 1)/2. 2

3 ( & ! 2g − 2 ( k ( 1 &' k, k − g, degDk > 2g − 2 i(Dk ) = 0. $ ( ' g =% 1 & 2g − 1. 2 F " &# 1 & Hhol (F ) 9 ( F, " F d). - ! F g & ( 1 Ω1(1) HDR (F ) B 1 1 Ω (1) J ( dimC Hhol (F ) = g " # 1 Ω1(1) ∼ (F ). = Hhol

' : 0, P1 , ..., Pk F (k ≥ 1). 1 F = F \ {P1 , ..., Pk }. 2 dimC Hhol (F ) = 2g + k − 1. 3 (

1 + Hhol(F ) F, F , % & ≤ 2g P1 1 & P2 , P3, ..., Pk. 34542 67$28 9 H1(F , Z) " F ) 2g + k − 1. D ! F , H1(F , Z), $ 1 ( dimC Hhol (F ) ≤ 2g + k − 1, H1 (F , Z) × 1 Hhol (F ) → C,

([γ], [ω]) →

ω. γ

1 A ( Hhol (F ) 1 C2g+k−1, Hhol (F ) 2g +

( ( + + B d ( ' ' -+ ( ( ! F, F , d F, F , 2g + k − 1. + k. - k = 1. n, n ≥ 2g, ' f F, F \ {P1}, &' & n P1, n(≥ 2g) P1 F. -+ & ! F, F , + & & 1 2g P1. 8 Ω(P1−2g )/d(L(P1−2g+1)). i(P1−2g ) − (r(P1−2g+1) − 1) = 3g − 1 − (g − 1) = 2g, d r( P 1 ) = 2g − 1 − g + 1 + i(P12g−1) = g " degP12g−1 = 2g − 1 > 2g − 2), 0 = r(P12g ) = −2g − g + 1 + i( P1 ) " degP12g > 0). $ ( k = 1 1 - k > 1 F = F \{P1, ..., Pk−1}, ( ( F . - =% ! 0, D0 = P1 ...Pg & (

U (D0) Fg . $ 9::; 1. ! "% # 4;

# 4;

1, 3, ..., 2g − 1. 5 & # ω " < P 2g−2. . ωq " q− P q(2g−2). q(2g − 2) > (2q − 1)(g − 1) − 1, P # q− 4;

F.

5 z 6 " F g ≥ 1 P1 , ..., P2g+2 F. > " #& % z(Pj ) = ∞, j = 1, ..., 2g + 2. 5 2g+2 & w = j=1 (z − z(Pj )) F. 789: ? w g+1 F P1 ...P2g+2/Qg+1 1 Q2 , " Q1 Q2 z. 3 & z w

" F % 2

2

2g+2

SF = {(z, w) ∈ C : w =

(z − z(Pj ))}.

j=1

5 # " SF , # g SF # F. 5 "

g = 2 : +( > C (e1, e2 ), (e3, e4 ) (e5, e6 ), -

b1

a1

R

e1

e2

b2

a2

-

R

e3

e4

e5

e6

P uc.14

0 # % & @ @ % ; @ @ % ej = z(Pj ), j = 1, 2, ..., 6.

6( 3 @ @A

b-1 a-2

a-1 P1

P2

b-2

P3

P4

P5

P6

P uc.15

-

z wdz , j = 0, ..., g − 1, # # # " " F g ≥ 1. -.$/0/1,234. B % z(Pj ) = 0, j = 1, ..., 2g + 2, 3 Q4 (z) = Q Q1 Q2 . C % C = % # " F. ) (dz) = P1Q...P21Q2g+2

% 2 2 j

z j dz ) = Q1g−j−1Q2g−j−1Qj3 Qj4 ≥ 1 ( w

0 ≤ j ≤ g − 1. 3 j 0 % z wdz j ≥ g # g F

Q1 , Q2. 5 j = g z wdz # # " F

Q1, Q2, j Q1 = Q2. 5 j ≥ g + 1 z wdz # # " F

Q1 Q2 j − g + 1(≥ 2). 5 % 7DDE+DE+9DEF8:% " " " F g ≥ 2, w2 = (z − e1 )(z − e2 )...(z − e2g+1),

" ej = ek , j = k, ej = ∞, j, k = 1, ..., 2g + 1. 1 ? f1 = z−e ' i, i = 1, ..., 2g + 1) # i " (z = ei , w = 0), # - % z = a = ∞, a = ei , 1 i = 1, ..., 2g + 1, t = z − a f1 = t+a−e i √

t = 0; z = ei % t = z − ei f1 = t12 2 " t = 0; z = ∞, t = √1z f1 = 1−tt 2ei

t = 0. . % (z = ei , w = 0), i = 1, ..., 2g + 1, 4;

F, "

f1 2 ≤ g. f2 = z z = ∞ " % t = √1z f2 = z = t12 t = 0. 5 (z = ∞, w = ∞) 4;

4 " # 2g+2 4;

F. 0 % 1 ei f1 = z−e i (z = ei , w = 0) # " " F. 5 " F, # 6 P0 P0 # " 5 z(P0 ) = a = ej #" j = 1, ..., 2g + 1, a = ∞, " t = z − a w z = a : w = f (z) = b0 + b1(z − a) + b2(z − a)2 + ..., b0 = f (a) = 0.

(a)+f (a)(z−a) " f (z)+f(z−a) dz " 2 f (z) P0 - %

1 1 2 )dz = [2b + 2b (z − a) + b (z − a) + ...]( 0 1 2 (z − a)2 b0 + b1(z − a) + ... 1 1 b1 2 [2b + 2b (z − a) + b (z − a) + ...][ − (z − a) + ...]dz = 0 1 2 (z − a)2 b0 b20 2 1 2b1 2b0b1 ( [ + − 2 ] + ...)dz = 2 (z − a) z − a b0 b0 2 + c0 + c1 (z − a) + ...)d(z − a). ( (z − a)2 5 z = b = a

A G( b = ej , j = 1, ..., 2g + 1, t = z − b, % # dz # t = 0; √ #( b = e1 , t = z − e1, t2 = (z − e1 ), dz = 2tdt dz √ 2tdt2 , % t = 0; " f (z) t (t +e −e ) j=1

1

j

b = ej , j = 2, 3, ..., 2g + 1; ( b = ∞, t = √1z , z = t12 , dz = − t23 dt

1 b0 + b1 (z − a) ]dz = [1 + (z − a)2 f

t4 t2 b0 + b1 − at2 b1 2g−1 −2dt (1 + ) 3 t (1 − t2 a)2 t 2 j (1 − ej t )

t = 0. √ dz 5 z(P0 ) = e1 , " t = z − e1 (z−e 1 )f P0 6 - % t2 = z − e1 , dz = 2tdt dz 2tdt = = (z − e1 )f 3 t j=1 (z − ej ) 2 ( t2

1

2 j=1 (t

+ e1 − ej )

)dt =

2 ( t2

1 j=1 (e1

− ej )

+ c2 t2 + ...)dt.

5 z = b = e1

A

( b = e2, e3 , ..., e2g+1, t = z − b, f (b) = 0 # dz #

t = 0; √ #( b = eν , ν = 1, t = z − eν , t2 = z − eν , 2tdt = dz dz 2tdt = = (z − e1 )f 2 (t + eν − e1 )( j=ν (z − ej ))t (t2 + eν − e1 )(

t = 0; ( b = ∞, t =

√1 z

2dt

2 j=ν (t

+ eν − ej ))

% dz = − t23 dt

− t23 dt dz = = (z − e1 )f 1 1 ( t2 − e1 ) j ( t2 − ej ) −

1 2dt 2g t (1 − t2 e1 ) 2 j (1 − t ej )

t = 0. g = % z(P0 ) = ∞, t = √1z z fdz

" P0 P0 . - % 1 (− t23 dt) z g dz 2 t−2 dt t2g = − 2 (1 + c2 t2 + ...)dt. = = −2 f t 1 2 j (1 − t ej ) j ( t2 − ej )

4 z = b = ∞ " % & 5 " % &

+ P0 + P1 . 0 P0 % " z(P0 ) = ∞. 5 P1 # % z(P1 ) = a = eν , ν = 1, 2, ..., 2g + 1, a = ∞. 4 P1 t = z − a f (z) = 0 dz b0 + b1(z − a) + b2 (z − a)2 + ... " 12 f (z)+b z−a f (z) " - % 1 1 f (a) 1 (1 + )dz = (1 + c1 (z − a) + ...)d(z − a) 2z − a f (z) z−a

P1 . 0 P0 t =

√1 z

;

1 − t23 dt f (a) )= (1 + 1 2 t12 − a j ( t2 − ej ) −

dt f (a)t2g+1 dt 1 ) = − (1 + (1 + c1 t + ...). t 1 − t2 a t 2 j (1 − t ej )

4 % P0 P1 # 1 P1 % z(P1 ) = eν , " dz F. - % z = eν , # 12 z−e ν √ t = z − eν 1 dz 2tdt dt = 2 = , 2 z − eν 2t t z = ∞, t = √1z , z = t12

1 dz 1 2dt dt(1 + c1 t + ...) ( . =− ) = − 2 z − eν 2t 1 − t2 eν t

5 eν " # " n ≥ 3. = % d dz dz ( )= deν z − eν (z − eν )2

P1 (z(P1 ) = eν ) " P0 # - % z = eν , t2 = z − eν , ;

dt 2tdt t4 = 2 t3 , z = ∞, t =

√1 , z z

=

1 t2 ,

;

− t23 t4 dt − t23 dt = (1 − eν t2)2 ( t12 − eν )2

t = 0. - #" g ≥ 3 & " 4 "

% 7DD:% " F % "# {(z, f ) : f 4 = z 4 − 1}. 5 *

F % 1, i, −1, −i, % g = 3, 2g −2 = 4(0−2)+4(4−1). = "

dz zdz f dz , , . f3 f3 f3

- "

√ z = ±1, ±i z = ∞. 4 z = 1% t = 4 z − 1, % t4 = z − 1, 4t3dt = dz f 4 = t4 (t4 + 2)(t4 + 1 + i)(t4 + 1 − i). . dz 4dt = 3, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 4 zdz 4(t4 + 1)dt = 3, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 4 4tdt f dz = 1, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 2 # t = 0. / "

z = ±i, −1. .

z = ∞. 4 t = 1z , 4 " dz = − dtt2 , f 4 = 1−t t4 . ) dz tdt zdz dt f dz dt = − , = − , = − 3 3 1 f3 (1 − t4 ) 4 f 3 (1 − t4 ) 4 f 3 (1 − t4 ) 2

t = 0. ! C, & " ; 1, z, f. 5 # # Ω1(1). 0 % ; " " f, ; " " z. 5

; " ' ( M(F ) F. 4 " ; dz zdz z g−1 dz " # ; # % f , f , ..., f z, M(F ) g ≥ 2. 789: = "

F g ≥ 2 # " # (2g − 1)− (3g − 3)− "

! "% " z k+j dz 2 , 0 ≤ k + j ≤ 2(g − 1), w2

(1)

(2g − 1)− +( 2g − 1 = 3g − 3 ⇐⇒ g = 2; 6( #

Ω2 (1) " F g ≥ 3, '+( # & j 2

z wdz , j = 0, ..., g − 3, g − 2 " F. - % z j dz z j dz 2 )=( )(dz) = ( w w P1 ...P2g+2 = Q1g−j−1Q2g−j−1Qj3Qj4 Q21Q22

Q1g−j−3Q2g−j−3Qj3Qj4 P1 ...P2g+2 ≥ 1

g − j − 3 ≥ 0, 0 ≤ j ≤ g − 3. = " F g ≥ 2 q ≥ 3 q− # "

# (q(g − 1) + 1)−

(2q − 1)(g − 1)− Ωq (1). - " # # " q− z j1 ...z jq q dz , 0 ≤ jk ≤ g − 1, k = 1, ..., q. (2) wq " 0 ≤ j1 +...+jq ≤ q(g−1) # # (q(g−1)+1)−

* (2q − 1)(g − 1) = q(g − 1) + 1 %

q = 2, g = 2. # % q ≥ 3

" # # g ≥ 2. .# C# ϕ : F −→ J(F ) H " # > "% e ∈ J(F ) n(n ∈ N), ne = 0 ' J(F )), me = 0 0 < m < n. !789: 5 F " g ≥ 2 4# # P0 ϕ 4;

F " ϕ(P ) 6% P 4 ;

F. -.$/0/1,234. 5 P (= P0 ) & 4;

F. 3& f F, (f )∞ = P 2 . 2 P0 # f, (f − f (P0)) = PP02 . . P02 ∼ P 2 /# 2ϕ(P ) = ϕ(P 2) = ϕ(P02 ) = 0 J(F ), ϕ(P ) = 0 H ϕ. 5 "789: 5 F g ≥ 2. " F " % & J F, J 2 = id, 2g + 2 1 g ≥ 2, " 4;

1 F " g ≥ 2, 2

w =

2g+2

(z − ej ),

j=1

" e1 , ..., e2g+2 z− % " J, # (z, w) → (z, −w), z −1 (ej ), j = 1, ..., 2g + 2, 0 z −1 (∞) F. 5 " F g ≥ 2,

w2 = z(z − 1)

2g−1

(z − λk ),

k=1

" λ1 , ..., λ2g−1 C\{0, 1}, P1 = z −1 (0), P2 = z −1 (1), Pj+2 = z −1 (λj ), j = 1, ..., 2g − 1, P2g+2 = z −1 (∞). 5

# $789% DI9: "

" " $ "% "

§

D = P ∈F P α(P ) , α(P ) ∈ Z, α(P ) = 0 F

g ≥ 1. α(P ) P D. |D| D D1 D. !"# $%!&' ( ) * D + F g ≥ 1 |D| ) + ) P L(D−1) * L(D−1)). (,-./.0123405,% D1 ∈ |D|, * D1 ≥ 1 D1 = D(f ) f = 0 6 6

F, D1 ≥ 1 # f ∈ L(D−1). ,) # ) f ∈ L(D−1) D1 = D(f ) ≥ 1

D1 D. ( 6

f g L(D−1) # f = cg, c ∈ C∗. % 2 DV |D| # + 7 P V P L(D−1), * V + + L(D−1). 8 (degD = d, dimCV = r + 1).

9

DV + # )7 DV . 1 P + ) DV , (f )D ≥ P P f ∈ V, # V ⊂ L( D ). 5 # P D # P ) |D|, ) D1 ∈ |D| P, ) f ∈ L(D−1) ) 7 P, L( D1 ) = L( DP ). % &&' + F g ≥ 1 : ; 1 degD < 0, |D| = ∅; |D| ) # ) * P ∈ F dimCL( DP ) = dimCL( D1 ) − 1; ? ,) A ) @ # |D|,

# |D| ) % A+ A |D|. 0 * L( DA ) = L( D1 ). ( +

% 0 A ≥ 1, DA ≤ D

A L( D ) ⊂ L( D1 ). ,) # f ∈ L( D1 ), D(f ) ≥ 1 D(f ) ∈ |D|. /+ (f )D = AD, D ≥ 1, # (fA)D = D ≥ 1 f ∈ L( DA ). | DA | |D|. B+ # ) |D| ) 6 = C ( ) * D, degD ≥ 2g, |D| ) % ( # dimL( DP ) = dimL( D1 ) − 1 ) P ∈ F. 0 degD deg DP @ 2g − 1, i(D) = 1 P D i( D P ) = 0, # dimL( D ) = degD + 1 − g dimL( D ) = deg( P ) + 1 − g = D - |Z| ) % (+ # 6

P ∈ F. A # L( PZ ) = L( Z1 ), # r( PZ ) = r( Z1 ) − 1 = g − 1. E+ r( P1 ) = 1 6 F +F 1 = r( P1 ) = r( PZ ) + degP + 1 − g

dimL( PZ ) = g − 1; ! D = P1...Pn, n ≥ 1. 0 * 7 * 6 66 ω F # (ω) # 7 D. ( * D L( D1 ) + @ s(≥ 1), D s − 1 G ) G % 9 # 6

!"'% D +

F g ≥ 1. 0 * r( D1 ) ≥ s, ) * * * D, degD ≤ s − 1, 7 D # DD ∼ D. - * # # D ) ) * * Fs−1 (s − 1)+ F ). 5 ) B ) ϕ : Fn → J(F ) + J(F ). 1 F +

g ≥ 1, ) ) * * * * ) {a, b} H1(F, Z), # * * * ) ζ1, ..., ζg Ω1(1), Cg /L(F ). 5 # + 7 ) ) {a, b}, 6 * *% 4 # ) # 6

# ) {a, b} F, # ) @ L(F ), ) * * + ) B ) # %% * *

6 % +

(F, {a, b}) (J(F ), L(F )) *# * ) ) P0 )+

B ) % ,) Wn = ϕ(Fn). B # W0 = {0}, Wn ⊂ Wn+1 Wg = J(F ). Wnr = ϕ({D ∈ Fn : r(D−1) ≥ r + 1}) = ϕ(Fnr ),

%% J(F ), 7 ) ϕ D # degD = n, r(D−1) ≥ r + 1. E .) # W1 = ϕ(F ) 6 F ϕ

J(F ). , K = (KF ) = ϕ(Z) + J(F ). g−1 ! K = W2g−2 .

(,-./.0123405,% D + # degD = 2g−2, D ≥ 1. 0 * r(D−1) ≥ g ⇔ i(D) ≥ 1 ⇔ D + % + % # @ J(F ) % !"# $% ;C"' : ; D ∈ Fn. 0 * ) ) ϕ : Fn → Wn ⊂ J(F ) D * n + 1 − r(D−1)(= n − ν); , ) ϕ : Fn → Wn 6 Fn \ Fn1 Wn \ Wn1; ? n ≤ g G )7* G D ∈ Fn i(D) = g − n; C 1 1 ≤ n ≤ g − 1, u ∈ Wn ) Wn,

u ∈ Wn1; 1 D F r Wn+1 , 1 ≤ n ≤ g − 1, + 2n − g ≤ r ≤ n − 1, r = n − 1 *# F + * + % " ??# % ;D 0, $$ D $ 567 7.89: .25$ - s 3 Θ(Ws+1 − Ws − e) = 0. . 0 ≤ s ≤ g − 1. 2 P1 , ..., Ps+1, Q1, ..., Qs , Θ(ϕ(P1...Ps+1) − ϕ(Q1...Qs) − e) = 0. 0 # P → Θ(ϕ(P ) + ϕ(P2...Ps+1) − ϕ(Q1...Qs) − e). 5 #, P1. - D s) g Q1, ..., Qs, T1, ..., Tg−s Θ(Wk+1 − Wk − e) ≡ 0

k < s. 5 # ϕ(Q1...QsT1...Tg−s) + K = ϕ(Q1...Qs) − ϕ(P2...Ps+1) + e

e = ϕ(T1...Tg−sP2 ...Ps+1) + K ∈ Wg + K.

D = T1...Tg−sP2...Ps+1 s $ . , r(D−1) ≥ s + 1 i(D) ≥ s. % 0 ψ = 0. 2 ## Q ∈ F ψ(Q) = 0. - " D ∈ Fg−1 , ϕ(D) = ϕ(QD ). .! ψ(Q) = Θ(ϕ(Q)−e) = Θ(ϕ(D)−ϕ(D )−e) = Θ(−ϕ(D )−K) = 0. %

#, , ψ ≡ 0. ! 3

, e = ϕ(D) + K, D ∈ Fg , ψ = 0 i(D) = 0, D ψ. 8 D ψ, ϕ(D) = ϕ(D ). 1 i(D) = 0 7 D = D . . , D = D , ϕ( DD ) = 0 deg( DD ) = 0, , " f ∈ L( DD ). . , r( DD ) > 1 r( D1 ) ≥ r( DD ) > 1,

r(

1 ) = degD − g + 1 + i(D) = g − g + 1 + 0 = 1. D

- $ , D = D D ψ F. . $ %

" 6 ! " C @ # a ∈ J(F ) " Da ∈ Fg , ϕ(Da) = a J(F ). , a ∈ Cg # a ∈ J(F ) Cg . -! e = a + K # ψ : P → Θ(ϕ(P ) − e). 8 ψ = 0, Da ∈ Fg ϕ(Da) + K = e = a + K. 8 ψ ≡ 0, e = ϕP (Q1...Qg−1) + K = ϕP (P0 Q1...Qg−1) + K ! Da = P0 Q1...Qg−1. * D0< &'(, $ ?;')$ - s 3 , Θ(Ws−1 − Ws−1 − e) ≡ 0, Θ(Ws − Ws − e) = 0. . e = ϕ(D) + K D ∈ Fg−1 i(D) = s. 6 ,

Θ 3 s ( e, s−

Θ # e. 5 , e ,

Θ 3 s ( e, " s−

, ( e, e = ϕ(D) + K, D ∈ Fg−1 i(D) = s.

0

0

# # K 0 (F, {a, b}) g ≥ 1. + &'()$ - D 2g − 2 F. . D F, ϕ(D) = −2K J(F ). 567 7.89: .25$ -!, −2K F. 2 D0

g − 1. . e = ϕ(D0) + K Θ , (−e) ! / , $ . , −e = ϕ(D1) + K, D1 ∈ Fg−1. , ϕ(D0D1 ) = −2K. 5

, D0D1 F. , D0 D1 g − 1 , , r( D 1D ) ≥ g i(D0D1 ) ≥ 1. . , " ω, (ω) ≥ D0D1 ≥ 1, , (ω) = D0D1 / $ # ω1 (ω1) ∼ (ω) ϕ((ω1)) = ϕ((ω)) = −2K. 5 ,

!, D 2g − 2 F ϕ(D) = −2K. . # ω D D ϕ( (ω) ) = ϕ(D) − ϕ((ω)) = 0 deg (ω) = 0. - 7 " D . , (f ω) = D, D f F, (f ) = (ω) f ω F. . $ ! 2 K 0 = J(F ), P02g−2 , P0 ϕ : F → J(F ). , K =, −2K = 0

1

0 = ϕ(P02g−2).

*! / Θ J(F ) Wg−1 + K, P0. 1 , ! Θ P0. &'(, A$ ?;J)$ 5 !,

Fg−1, D ϕP (D) + KP , P0 F. 567 7.89: .25$ , Θ(e) = 0, ψ(P ) = Θ(ϕP (P ) − e) = 0, , e = ϕP (P1 ...Pg ) + KP , D = P1 ...Pg ψ F. 2 0

0

0

0

ϕP1 (P ) = ϕP1 (P0 ) + ϕP0 (P )

0

# P0, P1, P

∈ F.

5 #

Θ(ϕP0 (P ) − e) = Θ(ϕP1 (P ) − ϕP1 (P0 ) − e),

/

ϕP1 (P0 ) + e = ϕP1 (P1...Pg ) + KP1 .

, e

ϕP1 (P0 ) + ϕP0 (P1 ...Pg ) + KP0 = ϕP1 (P1 ...Pg ) + KP1 .

K ϕP (Pj ) = ϕP (Pj ) − ϕP (P0), j = 1, ..., g, , 0

1

1

ϕP1 (P0) + ϕP1 (P1 ...Pg ) − gϕP1 (P0 ) + KP0 = ϕP1 (P1...Pg ) + KP1

KP

0

= ϕP1 (P0g−1) + KP1 .

1, # D ∈ Fg−1

ϕP1 (D) + KP1 = ϕP1 (P0g−1) + ϕP0 (D) + KP1 = ϕP0 (D) + KP0 ,

ϕP1 (Pj ) = ϕP1 (P0 ) + ϕP0 (Pj ), j = 1, ..., g − 1.

. $

! "

ϕQ(P

g−1

).

# P Q F KP

= KQ +

5

, &'(), ! ! J(M) : ;< ! Θzero = Wg−1 + K / Θ > 1 =< ! Θsing = Wg−1 + K, " e, Θ

# e > ?< ! Θsuperzero = Wg1 + K, " e, ψ(P ) = Θ(ϕ(P ) − e) ! # F. *! 3 Θzero ⊃ Θsuperzero ⊃ Θsing . 1 2# Θzero ⊃ Θsing Wg−1 ⊃ Wg−1 . 1 1 2# Θsuperzero ⊃ Θsing , Wg + K ⊃ Wg−1 + K. 2# Θzero ⊃ Θsupersero , e ∈ Θsupersero, ψ(P ) = Θ(ϕP (P ) − e) ≡ 0 P ∈ F, , 0 = Θ(ϕP (P0) − e) = Θ(e), e ∈ Θzero . - / ! @ ;< Θsuperzero g ≥ 2; 0

0

=< Θsing g ≥ 4 / g = 3; ?< dim Θzero = g − 1; E< dim Θsuperzero = g − 2; L< g − 4 ≤ dim Θsing ≤ g − 3; F< *! Θzero Θsing Π (F, {a, b}) g ≥ 1, P0 ! ϕ : F → J(F ); # Ee (P, Q) = Θ(e + ϕ(Q) − ϕ(P )),

e ∈ Cg − , Θ(e) = 0, P, Q ∈ F. % , ,

a− ! / !

b−$ 5 #" , $&==, A$ ;==)$ - e ∈ Cg , Θ(e) = 0, Ee(P, Q) = 0. . " # 2g − 2 R1 , ..., Rg−1, S1, ..., Sg−1 ∈ F , Ee(P, Q) = 0, @ < P = Q, < P = Ri , < Q = Si , i = 1, ..., g − 1.

.

# Ee F × F 7$ D7< &==, A$;;E)$ 1 F

g ≥ 2 D = QP ...Q ...P ( " f (f ) = D, ϕ(D) = 0 J(F ). 567 7.89: .25$ 1 $ # t ∈ C D(t) ! D 0, 1 r( P Q ...Q ) ≥ α+g−1−g+1 = α, L( P QP...Q ) ⊂ L( P Q 1...Q ) 0

0

α 0

1

g−1

α

α 0

1

α 0

g−1

α + g − 1 − g + 1 + i(P0α Q1...Qg−1) = r(

1

1

g−1

) P0α Q1...Qg−1

α 0

1

g−1

≥ α + 1,

, i(P0αQ1...Qg−1) ≥ 1. 5 , i(P0αQ1...Qg−1) > 0, 00 1 r( P Q ...Q ) ≥ α + 1. -/ " /

, 3 α # P ∈ F. , α 0

1

g−1

Θ(ϕP0 (P α )−ϕP0 (P0α Q1 ...Qg−1)−KP0 ) = Θ(ϕP0 (P α )−ϕP0 (P α R1...Rg−1)−KP0 )

. $

= Θ(−ϕP0 (R1 ...Rg−1) − KP0 ) = 0.

! * 8 α ≥ g, P0 ,

F. 567 7.89: .25$ 8 ψ ≡ 0,

" i(P0α Q1...Qg−1) > 0,

α ≥ g ! i(P0g Q1...Qg−1) = 0 deg(P0g Q1 ...Qg−1) > 2g − 2. - $ $ %

( &'(, A$ ?;?)$ +

" , e = ϕP (Q1...Qg−1) + KP i(Q1...Qg−1) = 1, " ! P0, ψ ≡ 0. , " ω (ω) ≥ Q1...Qg−1 2g − 2 , (ω) = Q1...Qg−1R1...Rg−1. . R1 ...Rg−1 ! P0 , i(P0α Q1...Qg−1) = 1 > 0. 6 , i(Q1...Qg−1) > 1 α = 1, # P0 i(P0Q1...Qg−1) = i(Q1...Qg−1), i(P0Q1...Qg−1) + 1 = i(Q1...Qg−1) ψ(P ) = Θ(ϕ(P ) − e) ≡ 0 F.

+ &'(, A$ ???)$ - e ∈ J(F ). . ψ : P −→ Θ(ϕ(P ) ± e) ! F g ≥ 3

# P0 ! ϕ, e ∈ Θsing . ψ≡0

0

0

567 7.89: .25$ K e ∈ Θsing / D 1 Θsing = Wg−1 + KP ) " # P1 , ..., Pg−1 F , e = ϕP (P1...Pg−1) + KP i(P1...Pg−1) ≥ 2. - 0 / / Θ(W1 − W1 ± e) ≡ 0. - ! $ &'(, A$ ??;)$ 9# Q = P0 ψ : P → Θ(gϕP (P ) + KP )(e = −KP ) 2 3 F g ≥ 1. 5 , # 2 3 F / $ 567 7.89: .25$ - !, Q = P0 / $ . 0

0

0

0

0

0

gϕP0 (Q) + KP0 = ϕP0 (R1...Rg−1) + KP0 ,

,

ϕP0 (Qg ) = ϕP0 (R1 ...Rg−1) = ϕP0 (P0R1 ...Rg−1).

- 7 " L( Q1 ) F. -/ Q 2 3 , g

g

i(Q ) > 0.

5 , Q(= P0) 2 3 F. . ≥ 2 " R1 ...Rg , !

! ! P0 / Qg . , (f ) ≥ Q1 , (f − f (P0)) ≥ QP , , " # R1, ..., Rg−1 , (f − f (P0)) = R ...RQ P . 5 # , gϕP (Q) + KP = ϕP (R1...Rg−1) + KP Q ψ $ 5 , P0 / , Θ(KP ) = 0. . $ 4

" ψ g3, / P1 , ..., Pg # # ϕP (P1...Pg ) = g(g+1) 2 (−2KP ), / g(−KP ) = ϕ(P1...Pg ) + g2KP . -!, P0 ψ 3 g. , # P F D , , P = P0) r( Q1g )

0 g

g

1

0

0

g−1 0 g

0

0

0

3

0

3

0

3

0

0

Θ(gϕP0 (P )+KP0 ) = Θ(ϕP0 (P )+KP ) = Θ(−ϕP (P0 )+KP ) = Θ(ϕP (P0 )−KP ),

0 = ϕP (P ) = ϕP (P0 ) + ϕP0 (P ), KP = KP0 + ϕP0 (P g−1) = KP0 + gϕP0 (P ) − ϕP0 (P ).

2 / ! P P0 F. 2 P #" 2 3 $ . P0 → Θ(ϕP (P0) − KP ) g P0 = P, Θ(gϕP (P ) + KP ) = 0. 8 P 2 3 , 3 ! # P0 F, 0

0

0 = Θ(gϕP0 (P ) + KP0 ) = Θ(ϕP (P0) − KP )

# P0 F. 5 # , P = P0 ψ 3 g. 3 ψ ! g(g + 1) (−2KP0 ), 2

ϕP0 (P0g P1 ...Pg3−g ) =

, , ϕP0 (P1...Pg3−g ) =

g(g + 1) (−2KP0 ). 2

&'(, A$ ??=)$ . P1, ..., Pg −g # g(g+1) 2 − F 2 3 F. 8

2 3 , g(g+1) − Φ 2 F. H , ! (Φ) = (ψ) ψ(P ) = Θ(gϕP (P ) + KP ). K! , / $ 1 " , / ! # $ " 2 3 #

, , / ! # # F. - / " # !, # D # # l, 2 ≤ l ≤ k1 + 1 P1;

# l, 2 ≤ l ≤ k2 + 1 P2

# P1 P2. . P ∈ F 2 3 /

, P g+k1 +k2 +1 i( k1 +1 k2 +1 ) > 0. P1 P2

+ " / / , f (g + k1 + k2 + 1)2g Q1 , ..., Q(g+k +k +1) g , #" 3# 1

2

2

(g + k1 + k2 + 1)((k2 + 1)ϕP1 (P2) − KP1 ) = ϕP1 (Q1...Q(g+k1+k2 +1)2g ) + (g + k1 + k2 + 1)2KP1 ,

ϕP1 (

Q1...Q(g+k1+k2 +1)2 g (g+k1 +k2 +1)(k2 +1)

P2

,

(g + k1 + k2 + 1) + (g + k1 + k2 + 1)2 (−2KP1 ). )= 2 Q1...Q(g+k1+k2 +1)2 g

(g+k1 +k2 +1)(k1 +1)

P1

(g+k1 +k2 +1)(k2 +1)

P2

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, # # P1 P2 k1 + 1 k2 + 1 $ 2

V, (P0, k)

P0 k ∈ Z+ ∪ {0}. P ∈ F 1

D0 =

P0k+1 , D1

2

1

2

P0k+1 P0k+1 P0k+1 , ..., Dj = = , ..., D2g+k = 2g+k . P Pj P

! Dj

L(Dj )

r(Dj ). 1 , r ≥ 1

, r(Dr ) − r(Dr−1 ) = 0;

/

$ + 00

0 = r(D0) = r(D1) = ... = r(Dk )

r(D2g+k) = g. 5 # # g = r(D2g+k ) − r(D1) = (r(D2g+k ) − r(D2g+k−1)) + ...

+(r(D2) − r(D1)),

( )

" g

$ , g + k

1, ..., 2g +k, 1, ..., k

$ . 0 0 !

, P0k+1 Pr r( r ) = r − (k + 1) − g + 1 + i( k+1 ), P P0

i(P0−(k+1)) = dimC V

= g + k. i(

6 , r

,

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8 P = P0 2 3 ,

≥ g +1. 6 , P0 2 3 V, 2 3 $ !, P = P0 2 3 V,

≥ g + k + 1. -

! 2 3 ,

" / , , ! # P0

V

3 $ &FJ, A$M;)$ - F / g ≥ 2, P0 ∈ F k = 1. 0 2 3

V, " # P0 3 $ " , , P0 2 3 $ - P0 2 3 F P = P0 # 2 3 F. 2 /

@ r(P02 )

P02 P02 P02 = 0 = r( ), r( 2 ) = 1 = r( 3 ), ... P P P

P02 P02 P02 P02 r( 2i ) = i = r( 2i+1 ), ..., r( 2g ) = g = r( 2g+1 ). P P P P . , !

P 1, 3, ..., 2g + 1 2 3 P /

0 + 1 + 2 + ... + g =

g(g + 1) . 2

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V. 2 / ,

!, P = P0, / 2 3 , # 3 2 P0. 2 / (−2) + (−1) + 0 + ... + (g − 2) =

(g − 4)(g + 1) . 2

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$ , ! 2 3 V ! 2 3 , / $ 8 P0 2 3 , #" P = P0, 2 3 , r(P02 )

P02 P02 P02 = 0 = r( ) = r( 2 ) = r( 3 ), P P P

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D 2 3 < P

V ! 1, 2, 3, 5, 7, ..., 2g − 1, (g−2)(g−1) . - / 2 3 V, 2 g ≥ 3. " 2g + 2 , / "

V (g − 2)(g − 1)(g + 1).

- ! 2 3 , !

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r(P0

−(g+2)

) = 2, r(P0

) = 3, ..., r(P0−2g) = g + 1,

P0

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2 3

$ 8

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! 2 3

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V. 8 P0 2 3 ,

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7

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! "# !' ' ## #

# ! - 3 "

# 1## ( ) # α = ψm = km(z, ζ )dz %5 & (ψm, ψ0) = 2πkm (ζ, ζ ). % " !. 5% α = ψ0, &

(ψ0, ψm ) =

2π dm m k0 (ζ , ζ). (m + 1)! dζ

# -# "## !"&# & & 1 dm km (ζ, z) = k0(z, ζ), (m + 1)! dz m

km & # " k0

& ) km(z, ζ) & ζ. & " &$ # # ϕm = hm(z, ζ)dz. & " & (α, ϕm) = (α, ϕm + ψm) = (α, τm) = (α, θm) = 0. 3 Ω - # 3#

§

! G(w, z) = 0,

(1)

G(w, z) ! " z w. # $ % &' ' $ " F, $ !( ( )& F $ * "& g ≥ 1, ** ! {(z, w) ∈ C2 : G(w, z) = 0}, z w $ + F. )& ω $ !

+$ F, dzω * ** + F, '

+* +& %* z w, dzω = R(w, z). )$ ! !, !

+ F & R(w, z)dz, R(w, z) $ +&* +* z w. -! *

R(w, z)dz =

(w,z) (b,a)

R(w, z)dz + C,

(2)

R(w, z) $ +&* +* z w, . (w, z) $

*( ( /0' C $ ' *, &$ ! ! (b, a). 1 % 2g−1 " F, ! 2 w = z(z − 1) k=1 (z − λk ), * * ! "

+ ." " ! & !,( ( "& F, ( ! /0 2 ' W $ * * +* "3 F, W = R(w, z) $ +&* +* z w. 4$ %' ' N * " !" " Φ(w, z) = 0 F (w, z) = 0 .* !+& " ! *"' ! *" z = g(w, z) w = h(w, z) ∂(g,h) +& +* g, h, ∂(w,z) = 0 ' 3 5 !$ *' (!( ( !(

( % ! & ( ( !$ ( ( G(w, z) = wm + A1(z)wm−1 + ... + Am(z),

(3)

G(w, z) $ Ak (z) $ 6 k, k = 1, ..., m. 7, ( ! * ! $

' 385 9 ! /0 ! % &

Q(w, z) dz, Gw (w, z)

(4)

Q $ 6 m − 3 / % ' 5: 0 9 " F g > 0, * /0 /0' . {ak , bk }gk=1 ! ( *( "& F1. ;. + ∂F1 a+1b+1a−1 b−1 ...a+g b+g a−g b−g . )& w = u + iv $ *' (,( !+ a1 . . . ag b1 . . . bg u α1 . . . αg β1 . . . βg v α1 . . . αg β1 . . . βg

. u|a

(5)

= u|a−k + αk , u|b+k = u|b−k + βk , k = 1, ..., g. ! " ; w = u + iv $ ' $ *' * u v, !+ /0' + k

0
0 *$ * & ! G(w, z) = 0 m n & w z 4 ! $ *(*

Q(w, z)dz , Gw (w, z)

(16)

Q $ 6 m − 2 n − 2 & w z ' 3 B: 9' + * " F g ≥ 2 , Γ = L1, ..., Lg , Lg+1, ..., L2g :

g

[Lj , Lg+j ] = 1,

j=1

* F U, F U/Γ. = ' &( !& F1 * Γ % !& D. g k=1[ak , bk ] z0 ∈ U, %, ! O * * ) + − + Lj : a− j → aj , Lg+j : bj → bj , j = 1, ..., g.

E * * +* ψ F * π $ + U, *(, * ψ(Lt) = ψ(t), t ∈ U, L ∈ Γ.

9 ! F ' $

uj |a+j = uj |a−j + πi, uj |a+k = uj |a−k , k = j, k, j = 1, ..., g,

(* + U * uj (Lj (t)) = uj (t) + πi, uj (Lk (t)) = uj (t), k = j, k, j = 1, ..., g, uj (Lg+k (t)) = uj (t) + ajk , k, j = 1, ..., g.

)

uj (t) =

t c

φj (t)dt =

a

z

ϕj (z)dz, j = 1, ..., g, t ∈ U,

φj (Lk (t))Lk (t) = φj (t), φj (Lg+k (t))Lg+k (t) = φj (t), k, j = 1, ..., g, U. F * ustj = uj (s) − uj (t) = ts duj , j = 1, ..., g, ' ustj, +* s, (,( !+ a1 a2 . . . ag b1 ust πi 0 . . . 0 a11 1 st u2 0 πi . . . 0 a21 ... ... ... ... ... ... ust 0 0 . . . πi ag1 g

b2 a12 a22 ... ag2

... ... ... ... ...

bg a1g a2g .

(17)

agg

- ' ! Yξ (z) (m) Yξ (z) (* " " + Yτ (t) (m) Yτ (t) U ( τ & ! F1 , Yξ (z) = Yτ (t) dzdt (ξ) U. ( ' ( (,( !+ Yτ (t) (m) Yτ (t)

a1 . . . ag 0 ... 0

b1 −2φ1 (τ )

0 ... 0

(τ ) 1 −2 φ(m−1)!

(m−1)

... ...

bg −2φg (τ ) . (m−1) (τ ) g . . . −2 φ(m−1)!

(18)

9 ! & Πxy ξη = Πξη (x) − Πξη (y)

(* Πstστ U, Πst στ = log(s − σ) + r(s) = −log(t − σ) + r(t) = log(t − τ ) + r1(t) = −log(s − τ ) + r1(s)

(," *" ) ** $ στ Πst στ = Πst .

(19)

! " G+* Πst στ − log

(s − σ)(t − τ ) = G(s, t, σ, τ ), (s − τ )(t − σ)

& % & ! F1, ! $ F14. = ' Πstστ $ +*

& " "' (,( + s t σ τ

a1 0 0 0 0

... ... ... ... ...

ag b1 0 2uστ 1 0 −2uστ 1 0 2ust 1 0 −2ust 1

... bg . . . 2uστ g . . . −2uστ g . st . . . 2ug . . . −2ust g

+( t+ t,τ + τ X(t, τ, t, τ ) = t τ exp(−Πt,τ )=

(t − τ + t)(τ − t + τ )exp(−G(t + t, τ + τ, t, τ )).

(21)

4 . t → 0, τ → 0 ! +* X(t, τ ) = −(t − τ )2exp(−G(t, τ, t, τ )) = (t − τ )2 Ψ(t, τ ),

(22)

* * ** F12. ( X(t, τ ) Ψ(t, τ ) $ + )& c, c + c ∈ b−j , cj , cj + cj " ! Lg+j (t) b+j . 2&* !+ ' 6* c+ c,τ + τ j ,τ + τ Πccjj + c = Πcτ + 2ujc+ c,τ + τ + 2ucτ ,τ j + 2ajj .

(23)

7 ' /50 ( j ,τ + τ X(cj , τ, cj , τ ) = cj τ exp(−Πccjj + c ), ,τ

c+ c,τ + τ ). X(c, τ, c, τ ) = c τ exp(−Πcτ

)

X(cj , τ, cj , τ ) =

cj X(c, τ, c, τ )exp(−2ujc+ c,τ + τ − 2ucτ j − 2ajj ).

c

)& " c → 0, τ → 0, 6*

X(cj , τ ) = Lg+j (c)exp(−4ucτ j − 2ajj )X(c, τ ),

6(

X(t , τ ) = Lg+j (t)exp(−4utτ j − 2ajj )X(t, τ ), t = Lg+j (t), j = 1, ..., g.

(24)

- * 6

X(t , τ ) = Lj (t)X(t, τ ), t = Lj (t), j = 1, ..., g.

(25)

) ( $ +( /=0 Ω(t, τ ) = C

X(t, τ ) = (t − τ )Q(t, τ ), C = 0,

(26)

Q(t, τ ) $ * +* F12, !,(,** & t τ, Q(t, τ ) = Q(τ, t). 4& $ % Γ & ! F1 U . + Ω(t, τ ) Q(t, τ ) F12. >' /580' /50'/5:0 * *(* " ' &(' $ &( %* ( ' *( $ +( 7 ,&( + *(* ! ."

& ! F1 : Πst στ = log

Ω(s, σ)Ω(t, τ ) tt , Πστ = 0, Πst τ τ = 0; Ω(s, τ )Ω(t, σ)

Yτst =

Ω(t, τ ) Ωτ (t, τ ) Ωτ (s, τ ) ∂ st ∂ Πστ = log = − , ∂τ ∂τ Ω(s, τ ) Ω(t, τ ) Ω(s, τ )

Yτ(m)st

∂ m st ∂m 1 1 Ω(t, τ ) . = Π = log στ (m − 1)! ∂τ m (m − 1)! ∂τ m Ω(s, τ )

2

(

(27)

st st Πst σg+j ,τ = Πστ + 2uj

1 Ω(s, σg+j )Ω(t, σ) ; ust j = log 2 Ω(s, σ)Ω(t, σg+j ) φj (t) =

Ω(t, σg+j ) 1∂ log ; 2 ∂t Ω(t, σ)

Yτ (tg+j ) = Yτ (t) − 2φj (τ ), j = 1, ..., g.

(28)

7 -!* * ! " ." )& H(t) $ ** * & Γ $ * +* U, * α1 , ..., αm ( β1, ..., βm F1 / . 0 & ! w(t) $

U !+ a1 . . . ag b1 . . . bg w ω1 . . . ωg ωg+1 . . . ω2g . 1 m1 . . . mg 2πi logH(t) −n1 . . . −ng

1 ** 2πi ∂F w(t)dlogH(t), ' +' $ "' 1

wα1 β1 + ... + wαm βm = m1 ω1 + ... + mg ωg + n1ωg+1 + ... + ng ω2g ,

(29)

mj , nj ∈ Z, j = 1, ..., g. ' w1, ..., wg $ ! " ! " ' m

wjαk βk ≡ 0(modωj1, ..., ωj,2g ), j = 1, ..., g,

(29 )

k=1

( + 6 2g, %. !+ t

(ω11, ..., ω1g ), ...,t (ω2g,1, ..., ω2g,g ).

% ! % , $ + * α1 , ..., αm ( β1, ..., βm F1, αj = βk , j, k = 1, ..., m, * m

uαj k βk = mj πi + n1 aj1 + ... + ng ajg , j = 1, ..., g,

k=1

* ** (29 ). 4 + % &

Ω(t, α1)...Ω(t, αm) tτ exp(−2(n1utτ 1 + ... + ng ug )), C = 0. Ω(t, β1)...Ω(t, βm) 2πi1 Πtcστ dlogH(t), .* + ∂F1 ' *(, σ τ !"*, σ τ F1. ) *' H(t) = C

m

Παστk βk

k=1

g H(σ) +2 = log nj uστ j . H(τ ) j=1

(30)

>& nj , j = 1, , , ., g, + ' * ! ! & ! σ τ. - * (* 6* m k=1

Yταk βk

g H (τ ) −2 =− nj φj (τ ); H(τ ) j=1

m k=1

Yτ(l)αk βk

g

1 2 dl (l−1) =− logH(τ ) − nj φj (τ ); l (l − 1)! dτ (l − 1)! j=1

(31)

>' +& !* ' *(, αk βk , k = 1, ..., m, % &' " /510'/0 + $

( 76* /510 $ /0 * *(* !" * -!* * , * + $ * ( " g > 0. ) (29 ) * ** % / ! " 0 " ! " " -!* * (, ! & /-!*0 5' 5' 58 7 (! ! $ R(x, y)dx C, F (x, y) = 0, * " !, & N * C $ C , F (x, y) = 0 m, +& +

+ F , % $ +& " + " %

+$ ' . % % * %& & /-!*0 )& ** * C * $

* C m. 4 , ! R(x, y)dx C & " ' " $ * C C , ! (' Nj=1 (x(x ,y,y )) R(x, y)dx

+ F * C . & /-!*0 B' 3 11 )& D $ & $ " F g > 0. 4 D * ** $ + F, & F , 1−+& γ *' . + ∂γ = D

γ

j

j

0

0

− → φ =0

(∗)

! H! J(F − ) * (! ! ! "

+ → t φ1 , ..., φg ' φ = (φ1 , ..., φg ). -4;?@74 A+ ∂γ = D + γ * ** $+&( 7$ , r ≥ 1, Pk = Qj , k, j = 1, ..., r. r = 0 )& D = QP ...P ...Q → φ , 7 Qj Pj . γj F. ϕ(D) = rj=1 γ − 1

r

1

r

j

− t → φ = (φ1 , ..., φg ) φ1 , ..., φg $ (! ! " ! "

$ + F, ϕ $ !% H! 2 * (∗) ϕ(D) = 0 J(F ). → − → φ = γ0 φ , γ0 $ !' ϕ(D) = 0 J(F ), ϕ(D) = rj=1 γj − + F. )% γ = γ1 + ... + γr − γ0, ∂γ = D − ∂γ0 = D. 4

-!* ). & ! " 6 $ 6 7 ,&( " ! " Yτts

$ . ( +(

Yτts Yτts1

φ1 (τ ) φ1 (τ1)

... ...

φ (τ ) φ (τ ) g g 1

. . . Yτtsg . . . φ1 (τg ) ... ... . . . φg (τg )

= F (t, s; τ, τ1, ..., τg ).

(32)

! τ1, ..., τg ∈ F1 ' !

φ1 (τ1) . . . φ1 (τg )

Δ =

... ... ...

φg (τ1) . . . φg (τg )

= 0.

(33)

% & /50 !+ Yτts = φ1 (τ )Y1ts + ... + φg (τ )Ygts +

' ' Y1ts

1

=

Δ

Yτts1 φ2 (τ1) φ3 (τ1) ... φg (τ1)

. . . Yτtsg . . . φ2 (τg ) . . . φ3 (τg ) ... ... . . . φg (τg )

F (t, s; τ, τ1, ..., τg ) , Δ

Yτts1

ts −1 φ1 (τ1)

, Y2 =

φ3 (τ1)

Δ

...

φ (τ ) g 1

. . . Yτtsg . . . φ1 (τg ) . . . φ3 (τg ) ... ... . . . φg (τg )

, ...

) ' + t, ( (,( !+ a1 0 0 ... ... Ygts 0

Y1ts Y2ts

... ... ... ... ...

ag b1 0 −2 0 0 ... ... 0 0

. . . bg ... 0 ... 0 . . . . ... . . . −2

H' Y1ts, ..., Ygts C. = ' 2g + ts ts ts uts 1 , ..., ug , Y1 , ..., Yg % C F1 . ; w1, ..., wg $ & ' g

wj =

)& " ( a1 w1 ω11 ... ... wg ωg1

λjk uk , j = 1, ..., g.

(33 )

k=1

. . . bg . . . ω1,2g . . . . ... . . . ωg,2g ( λkj = ωπikj πiwj = gk=1 ωjk uk , j, k = 1, ..., g, . +(ωkj )gk,j=1 (ωk,g+j )gk,j=1 ( = &6' ! & wj = ψj (t)dt, ' ψj (t) = gk=1 λjk φk (t), j = 1, ..., g. . ,. Pτtsσ

... ... ... ...

=

ag b1 ω1g ω1,g+1 ... ... ωgg ωg,g+1

Πts τσ

−

g

τσ ckj uts k uj .

j,k=1

1 O0 = ak ∩ bk . ( 3 Fg

[F, f ], F 4 g > 1 f ) ) F0 F. ak,f = f (ak ), bk,f = f (bk ) Of = f (O0). 5 9 0 9 Fg 3 g.

(g, s, m) " ! * !

! = * G

- G, ! * G. +! - !

@D# E #'AC < G =< T1 , ..., Tg, U1, V1, ..., Us, Vs , W1, ..., Wm >

EST − (g, s, m), - G,

* & ! * - . #$ Tj → Ti, Ti → Tj , i = j; '$ Ti → Ti−1; ($ Ti → Tk Ti, i = k; B$ Wj → Wi, Wi → Wj , i = j; A$ Wi → Wi−1; D$ Wi → Wk Wi, i = k; F$ Uj → Vj , Vj → Uj ; G$ Uj → Uj−1; H$ Ui → UiVi; #I$ Uj → TiUj Ti−1, Vj → TiVj Ti−1; ##$ Uj → WiUj Wi−1, Vj → WiVj Wi−1; #'$ Uj → ViUj Vi−1, Vj → ViVj Vi−1, i = j; #($ Ti → Vj Ti; #B$ Ti → Wj Ti; #A$ Uj → Uk , Uk → Uj , Vj → Vk , Vk → Vj , j = k. ; - 4 $ ! 0 @D#C - G

!

0 & - Q∗ = Q∗(g, s, m), Q∗(g, s, m) Q∗A (g, s, m), Q∗B (g, s, m). +

AutQ∗(g, s, m). AutQ∗(g, s, m) Q∗(g, s, m).

/+9:;:060)+ AutQ∗(g, s, m)(= AutQ∗)

Q∗, τ ∈ Q∗ !

∗ ∗ 3(g+s)+2m−3

- {fn∗}∞ ), n=1 AutQ fn (τ ) → τ 4 C ∗ ∞ ∗ " ! {fn }l=1, fn (τ ) = τ l, fn∗ (τ ) = τ l. + G =< T1, ..., Wm > τ : - fn∗ &

- fn G, < T1, ..., Wm > < fn(T1), ..., fn(Wm) > 4 $

* + " < fn(T1), ..., fn(Wm) > ξ3n, ξ2n, ξ1n Bn ! ξ3n, ξ2n, ξ1n 1, 0, ∞ 0 fn∗(τ ) < Bnfn(T1)Bn−1, ..., Bnfn(Wm)Bn−1 > . 0 4 Q∗) * AutQ∗ 2 # ! τ ∈ Q∗ ε > 0 U (τ, ε) ⊂ Q∗ * 4 $ AutQ∗. / * τ < Bn fn(T1)Bn−1, ..., Bnfn (Wm)Bn−1 >, !

< T1, ..., Wm >, " l

l

l

fn∗(τ ) = τ, n ∈ N.

(2)

∞ {fn (T1)}∞ n=1, ..., {fn(Wm )}n=1 ∞ {Ln,1}∞ n=1 , ..., {Ln,g+2s+m}n=1, T1 , ..., Wm L1 , ..., Lg+2s+m.

#$ J! k = k0(k = 1, .., g + 2s + m) ∞ {Ln,k }∞ n=1 " {Ln ,k }l=1, Ln ,k = Ln ,k l ? Bn Ln ,k Bn−1 → Lk l → ∞ Bn = Bn l, {Bn }∞ n=1 4 $ * Bn → B ∈ MC l → ∞. 9

! k = k1(k1 = k0) {Ln,k }∞ n=1

4 $ * 4'$ + Ln ,k → Bn−1Lk Bn , Ln ,k → B −1Lk B, ! * G, ! * MC . 6 G

τ ∈/ Q∗. '$ 9 g + 2s + m 4 $ * $ < ! {fn∗ }∞ l=1 (ξ3,n , ξ2,n , ξ1,n ) (ξ3,n , ξ2,n , ξ1,n ), Bn = Bn 0

l

0

0

l

0

l

l

0

0

0

0

0

l

l

l

1

l

1

0

1

0

l

1

1

l

l

l

l

0

0

0

l

0

l. 0 Ln ,1 → Bn−1L1Bn τ ∈/ Q∗. $ {(ξ3n, ξ2n, ξ1n)}∞ n=1 4 $ ! 3 {(ξ3,n , ξ2,n , ξ1,n )}∞ ! C (ξ30, ξ20, ξ10). 0 {Bn }∞ l=1 l=1 B ∈ MC , B ξ30, ξ20, ξ10 1, 0, ∞ Bn 7 {Ln ,1}∞ l=1 G Ln ,1 + Bn Ln ,1 Bn−1 Bn Ln ,1Bn−1 → L1 ∈ MC . 0 Bn → B ∈ MC Ln ,1 → B −1 L1B ∈ MC . 6 τ ∈ / Q∗. 0 ; EST − G MC " " EST −

! T1, ..., Tg , ..., U1, V1 , ..., Us, Vs , ..., W1, ..., Wm, ..., " T1, ..., Tg, U1, V1, ..., Us, Vs, W1, ..., Wm EST − (g, s, m), G 9 @(DC " " EST − (g, s, m). ) * l

l

l

0

0

l

l

l

l

l

l

l

l

l

l

l

l

l

G =< T1, ..., U1, V1, ..., W1, ... > .

" " EST − Gn =< T1n, ..., U1n, V1n , ..., W1n, ... > 4 !$ G, " T1, ..., U1, V1, ..., W1, ..., #$ T1, ..., U1, V1, ..., W1, ... MC , '$ Tjn(z) → Tj (z), Uin(z) → Ui(z), Vin(z) → Vi(z), Wtn(z) → Wt(z) i, j, t n → ∞ C.

/ (1, 0, 0), (0, 1, 0) (0, 0, 1) & * " EST − (g, s, m) 2 3(g + s) + 2m − 3 > 0. ) @DIC ! (g, 0, 0)(g > 1). Gn =< T1n, ..., U1n, V1n , ..., W1n, ... >

" " EST − G, " T1, ..., U1, V1, ..., W1, ..., n → ∞. 0 G

K ! Q(g, s, m), @#'IC 0 L 2 / 2 ! &- EST − = EST − (0, 0, m) 1

m. /+9:;:060)+ W ! Σ Σ , (p, q, r) ξ

W, ξ = Σ∩Σ . M ! U (ξ, ε) Σ Σ IW IW h IW ∩ {z ∈ C : |z − ξ| = ε} W, Tn (p, q, r−(1/n)) n > n(h)

Σ\{Σ ∩ IW } W Tn 2 h/2, IT 4 Tn$ Σ U (ξ, ε) 0 Σ Tn(Σ) ! Tn Tn(z) → W (z) n → ∞ C. / (0, 0, m) 2 ! W1, ..., Wm n ! Σ1, T1n(Σ1), ..., Σm, Tmn(Σm) 3 Tin Wi, i = 1, ..., m. = /+9:;:060)+ 4 ''F$ 6 {Gn }∞ n=1 EST − (g, s, m). G !

* Sl = Sl (T1, ..., Wm) = 1 G, ! 1 l → ∞. D @DIC Ψn : Gn → G

n

Ψn (Tjn) = Tj , Ψn (Uin) = Ui , Ψn(Vin) = Vi , Ψn (Wtn) = Wt, j = 1, ..., g; i = 1, ..., s; t = 1, ..., m, - Gn G. ? ! Gn ! G

!

{Sln = Sln(T1n, ..., Wmn)}∞ n=1,l=1

$ Sln ∈ Gn n, $ Sln ! Gn , ! Sl ! G, $ Sln = 1 l, n 4 D @DIC$ $Sln(z) → Sl (z) C n → ∞. ) N N

! 1 l → ∞. / l " Pn

! Gn G˜ n = {Sl,n , Pn } ! Sl,n Pn . ) {Pn }∞ l=1 " ! Gn , ! ! G. * {T1,n }∞ l=1 4 $ ! ! ! T1 G. ' @D#C l G˜ n EST − G˜ n

! An , Bn . 0 G˜ n =< An , Bn > " . (2, 0, 0), (1, 0, 1), (0, 0, 2). ''G ˜ n =< A , B > ˜ n 1

G G n n iθ (p, q, re ) An An , Bn Bn Sl,n Sl,n , T1,n T1,n 2 1/nl , Sl,n , T1,n ! An , Bn , ! Sl,n , T1,n ! An , Bn . 0 l Sl,n , T1,n ! G˜ n ,

' @DIC G˜ n =< Sl,n , T1,n > . 6

1

G˜ n =< Sl,n , T1,n > T1,n (z) → T1(z) Sl,n (z) → 1 l → ∞, * B @DIC / " " EST − #$ {Pn }∞ l=1 !

! '$ l G˜ n = {Sl,n , Pn } EST − / '$ " " Gn , " Pn ! Gn , Sl,n . 0 G˜ n "

" EST − ' @D#C " EST − / #$ {Sl,n }∞ l=1 . $ $ * ! Tj,n , j = 1, 2, ..., Wk,n , k = 1, 2, ..., Ui,n , Vi,n , i = 1, 2, ... ) $ * {Pn }∞ l=1 ! {Sl,nl }∞ l=1,

l

l

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l

l

l

l

l

l

l

l

l

) $ ! Tj,n , Wk,n , Ui,n , Vi,n {Sl,n }∞ l=1. * " ! ! ! * a (z) → ! T1,n a b b c d c d T1 (z) = 1, ( W1,n (z) → W1 (z) = 1, U1,n V1,n (z) → U1 V1 (z) = 1) & a, b, c, d(a · b · c · d = 0), Sl,n (z) → 1 l → ∞ 6 $ 0 l

l

l

l

l

l

l

l

l

l

§

! ! " #$%&' ( " R3 )

! *!+ #,$' -" ( ( ) ! ! . #$$,' / / ( / /

0! *! 1 #$' ( ! 2! #34' 5

! 6!7! 0 #48' 5

)

!

6 !" ! . ( ( )

! 9 F ) ( R3 : C ∞) " ( C, ( )

! ( F. ; F, ! 5( ( ) F R3 F ! 9 ( ( ( ( ! '

( R3. ? - @! 2 " $>>A ! 2 A F ( ! ( ! ) = Φ(A).

(4)

B;21J159*K.56;! ) = ∪s=22 Φ(Ans ) ⊆ ∪s=22 Φ(Am(s)O(A)−n(s)ns ) = Φ(A).

; Φ(A) ⊆ Φ (< A >) ! 5 Φ (< A >) = Φ(A). 5 ! J ( )

: C

! B"

2" ( R3 ) ! 6

2"! . A!A 4!$ #$A,' : C

2" (

2" ! 7

G

2" (g, s, i1, ..., ip) 2 EST −

Gg,s (g, s) p

Gi , ..., Gi i1, ..., ip, ( Γi , ..., Γi #$A$G $A,'! 6 A!A Gg,s. 6

Gi " (

Gi 3ik − 3

Fi = IntΓi /Gi

( 5 Ti C3i −3, k = 1, ..., p, (IntΓi ( Γi ) #$G A'! h/N // γ W1(γ), ( W1(z) = ze2πi/N , z ∈ C, 2g + s + p ( T1, ..., Tg, U1, V1, ..., U(h/N )−g, V(h/N )−g ∈ MC , (

2" (g, s, i1, ..., ip); AC

G " T1, ..., V(h/N )−g, T12, ..., V(h/N )−g,N , −(i−1)

−(i−1)

Tki = W1i−1Tk W1 , Uji = W1i−1Uj W1 1, ..., (h/N ) − g, i = 2, ..., N, k = 1, ..., g,

−(i−1)

, Vji = W1i−1Vj W1 ,j =

G F

G 8C W F 1 H/G, ) W1 ∈ MC . W B (F, W ) C! 5 $ #$%&' (F, W ) ( R3 .

F W C C C ! ) M(ξl ) = ξl , M(W j (ξl )) = W j (M(ξl )) = W j (ξl ), l = 1, ..., 2k, j = 1, ..., N − 1,

W j (ξl) ( M. 6 " ( M, ξ1, [ξ1] = {ξ1, W (ξ1), ..., W N −1(ξ1)} < W > . 5 2k > N, " ( ξi, [ξ1]. 6 ( ( M

i = 2. 5 [ξ2] = {ξ2, W (ξ2), ..., W N −1(ξ2)}. . B l = 1, ..., m, j = 1, ..., N − 1,

W j (γl ∪ M(γl )) = W j (γl ) ∪ W j (M(γl )) = W j (γl ) ∪ M(W j (γl )).

F/ < W > . H ) M ◦ π. π ◦ M = M " M 2 = id M

M 2 = id. ◦ π ◦ M M (! π ◦ M 2 = M

B p1 p2 F ( π(p1) = π(p2 ), ! ! < W > p1

< W > p2. 5 M M ( ! ( M . 5 P s γj P s Mγj ( I" l−1 n γj π 1 ≤ s ≤ l, π −1 ( γj ) = ∪n=0 P (γj ∪ Mγj ) l j = 1, ..., k. . E( 2k ( M l + 1

l

( 2m ( (

) < P >, l + 1

( ( p q; 2k = 2ml + 2. ; γ γk , p q. J l

ξ1, ..., ξ2m ( F0 = Int(γ ∪ P (γ)) ∪ Int(M(γ) ∪ M(P (γ))), P i ({ξ1, ..., ξ2m}) ⊂ P i (F0), i = 1, ..., l−1. 1 : C

( γj ∪M(γj ), ( ξ2j−1 ξ2j , j = 1, ..., m, ( F0, ( γj ∪ M(γj ), j = 1, ..., k,

< P > . F F0 F 2(g + s) + m ) γ ∪ P (γ) ∪ M(γ) ∪ MP (γ). ; / / M. ,

- :,C :3C :>C F D = ∪T ∈GT (H).

8C B P M F 1 H/G, ) P1, M P1 , M1 ∈ MC ! % & 5 A!8!$& A!8!A% F ( ) ! ' QRSTUTVWX M! B! J #,>' !

§

!" ! # $% &! '( ! !" " !" ! ) * " " " ! ! $ # ! ! ) ! " !" ! # ) " " ! " + ! ! ! ,# ) ! ! * " #!"

$ " $% &! '( - F ! ! g > 1, Ω2 ! " ! " * ! F, W "# ! F N. . ! W ! # ! ! Ω2, + W [φ] = φW −1, φ ∈ Ω2. / ) ! " W ! " * 3g − 3. 0 * ! W ! ! " ! N −" *" ! ! ! W N = 1. / nk ! ! εk # *" k = 0, 1, ..., N − 1, ε !"# N −"#

*" $ 12 ! 34 " 56 7 ν !! !# *

F → F/W ! F/W ! g1 > 0, n0 ! ΩW (F ), W − ! " " ! " * ! F, ! 3g1 − 3 + ν; 6 nk = 0 k ≥ 1, !" ! ! 3g1 − 3 + ν

2(N − 1) N −1 ≥ nk ≥ 3g1 − 3 + ν ; N N

!6 ! k∗, 1 ≤ k∗ ≤ N − 1, nk = 0; 6 g1 > 0, n0 ≤ 3g − 5, ! # ! y2 = x6 + Ax4 + Bx2 + 1, g = 2, g1 = 1, n0 = 2 = 3g − 4 = 3g1 − 3 + ν. 1 ! # " 8 8 8 %! * . + !2) #9 ∗

:12 ! 34 6 - g # !# ! F, t ! 2" ! W # ! N 8 ! ΩW (F ) W − ! " " ! " * ! F ! 3g − 3 + t, g ! F/W . 1 ! # " 8 8 !# ! " π : F → F/W . ; + "! 2 F/W W − ! ! * φ(z)dz2 2 9 " ) " ! ! 2" ! W. / Mg∗,t 2 ! " !" ! # ! g > 1, ) "# ! W N t ! 2" Mg,t 2 ! ! !) ! ! ,# ) Tg . " < '( 1! " " !" ! [F, α] [F ∗, α∗] g !+ W −! !" F, F ∗ ∈ Mg∗,t f : [F, α] → [F ∗ , α∗] *

W ∗f W −1, f ∈ α∗ α−1 , α α∗ " ! ) * ) !# # ! !

F0 .

:$% &! '( 6 02 ! Mg,t ! ! ,# ) Tg # 3g −3+t ≤ 2g−1, !" ! " " !) ! # " ModTg ! ,# ) Tg . 0 ) Mg,t ! !) ! # 2 # ! ! 8 Tg /ModTg ! ! , Tg /τg , τg , !) # # ! ModTg . - !+ ) ! # " ! '( $ ! + " # ! !" ! # . "! W −! ! ! # ! ! - 2 ! Mg,t " W −! ! ! ! ,# ) Tg . = "! 2"# # W −! ! ! !

! ,# ) Tg . > ! ! " !) ! 2 ! Mg,t, 2 ! Mg,t " !" - #

# " ! 2 ! Mg,t # ! " ! ! Tg , ! !) # [F, α] [F, β], ! F ! " " ? " 8 %! * 2g − 2 = N (2 g − 2) + t(N − 1)

π : F → F/W ! ! 3g − 3 + t ≤ 2g − 1, N ≥ 2. ; @ ! ,# ) Tg,t, ! ! g t " (3g − 3 + t > 0)

!) ) -"! ! !2 Tg,t !) 2 ! Mg,t, ! ! # 3g − 3 + t. $! !" Mg,t " ! ! Tg + ! 2 ! Mg,t : " ! " !) " ! Tg ). A* "! Mg,t ! 2 ! !)# " ! Tg , Mg,t ! Tg . . ) "# " )

! ,# ) '( - ! ,# ) Tg 2 ! ! !! ! ! " !) ! ) Mg,t. .# # [F, α] " 2 ! " " !" ! # " [F, α] " ! " 2 B!" ! " * φ = φ(z)dz2, " 2 ! ΩW (F ), " k 2 [0; 1). " ; ! ! F g " ! # "# ! J, J 2 = id. - F

! J n ! 2" ! !" # / g ! F/ < J > . , g = 2g + n − 1 ) ! F 2 ! F/ < J > . %"# * φ = φ(z)dz2 # ! F "! * & φ(Jz)dJz2 = φ(z)dz2. / )

! 2# ! !" ; " " !" ! # g, "# ! J " n ! 2" !" # Ug∗, Ug ! !) 2 ! ! ! ,# ) . # ! ! " C'( 2 )) $ '( 02 ! Ug ! ! ! 6g − 6 + 3n ! ! ,# ) 6 g −6+3n > 0, !" !" " !) " ! Tg . % & > ! 6 g − 6 + 3n > 0 ) # g = 1. / ! ! U0 ! : ! "!# " ! ! F ). A* 2 ! Tg , !" Ug. ' '( - ! ,# ) Tg 2 ! ! !! ! ! " !) ! ) Ug. .# # [F, α] " 2 ! " " !" ! # " [F, α] " ! " 2 B!" " ! " * !) * & " k 2 [0; 1). % & % 8 DE 2 ! ! ,# ) Tg , " !" " ! [F, α], ) $#9 !" !" n, 2 ≤ n ≤ g. F ) ) n + 2g − 3 ≥ 2g − 1. 0 !" ! # ) φ(z)

2g − 1.

( )*

$ #9 σ !"# "# *" *" g, s, i1, ..., ip, !!) ! ) ik =

|σ| = g + s + i1 + ... + ip. 1"! !! # " Qth 2 ! ! ! ;+ Qσ , ! !) " ;+ " σ = (0, 0, h, 0, ..., 0), ) " !" ! F h, h ≥ 2, h = |σ|, ) "# ! W N t ! 2" = h ! F/ < W > . ; 2 ! Qth. - 1, k = 1, ..., p,

G0 =< T1 , ..., Tg, U1, V1, ..., Uh−g, Vh−g >

! ;+ " σ. ;! "# ! f 9 # # C "! ! # * # " G0, μf f !! ! G μf |Λ(G0 ) = 0, μf (T (z))T (z)/T (z) = μf (z), T ∈ G0 , z ∈ Ω(G0), μf 2 " 9 M0 (C) L∞ (C) 3 1! ! " *

f f1 " G0 "!) ! !" ! B ∈ MC # f1T f1−1 = Bf T f −1B −1 ! T ∈ G0. F ! ! ) f1|Λ(G0 ) = Bf |Λ(G0). / Vσ 2 ! ! ! ! [f ] ! ! " * # f " G0 < f T1f −1, ..., f Uh−gf −1, f Vh−g f −1 > # # ;+ " σ. = dT Vσ , 2 !

dT ([f1], [f2]) = inf lnK(f1f2−1),

K(f1f2−1) f1f2−1 < 3 + ! ! f1 ∈ [f1], f2 ∈ [f2]. , (Vσ , dT ) ! 1! " " ;+

< T1, ..., Tg , U1, V1, ..., Uh−g, Vh−g >

< T1 , ..., Tg, U1, V1 , ..., Uh−g, Vh−g >

" σ "!) # ! !" ! B ∈ MC

Tj = BTj B −1, j = 1, ..., g, Uk = BUk B −1, Vk = BVk B −1, k = 1, ..., h − g.

- ! Vσ ! ! : ##6 ! ! [G], [G] 2 # ) ;+ G "

σ. $! ! ) τ, n → ∞, !) !

[Gn] → [G]

< T1n , ..., Tgn, U1n, V1n, ..., Uh−g,n, Vh−g,n >∈ [Gn ]

< T1, ..., Tg , U1, V1, ..., Uh−g, Vh−g >∈ [G]

Tkn(z) → Tk (z), k = 1, ..., g, Ujn(z) → Uj (z), Vjn(z) → Vj (z), j = 1, ..., h − g, ! ! C n → ∞. ,2 + dT Vσ , 2 ! dT ([G], [G ]) = inf lnK(f ), + ! ! " * f

Tk = f Tk f −1, Uj = f Uj f −1, Vj = f Vj f −1, k = 1, ..., g, j = 1, ..., h − g,

< T1, ..., Vh−g >∈ [G], < T1, ..., Vh−g >∈ [G ]. , (Vσ , dT ) ! ! + - ! (Vσ , dT ), (Vσ , dT ) (Vσ , τ ) " 1/;=,7 H.,$/ = 2 ϕ1 : V σ → Vσ , 2 ! ϕ1([f ]) = [< f T1f −1, ..., f Vh−gf −1 >]. I ϕ1 : 3< C CC 6 -2 ! ) 2 ϕ1. - [f1] [f2] !" [< f1T1f1−1, ..., f1Vh−g f1−1 >] = [< f2T1f2−1, ..., f2Vh−g f2−1 >]. , ! B ∈ MC f1T f1−1 = Bf2T f2−1B −1, T = f1−1Bf2T (f1−1Bf2)−1, ! T ∈ G0 . , f ≡ f1−1Bf2 # ! Ω(G0), Ω(G0) ! C, 2 @0 J' ! !"# ! "# ! f C # f|Ω(G0) = f f(z) = z, z ∈ Λ(G0 ). / ) f1−1Bf2(z) = z, z ∈ Λ(G0 ), f1|Λ(G0 ) = Bf2|Λ(G0 ), [f1] = [f2]. 1 2 "! ϕ1, dn = dT ([fn], [f ]) → 0 n → ∞, [Gn ] → [G] !

τ n → ∞. 1# ! 2 n !) !" ! fn ∈ [fn], f ∈ [f ] ! ! ! ln K(fnf −1) ≤ dn + (1/n). , K(fnf −1) → 1, μf f → 0 ! ) ! C n → ∞. / ) !" 2 fμ id : 2 ! 2 C). .! fn f −1 ≡ fμ id n → ∞ ! C. , fnf −1T (fnf −1)−1 → T n → ∞ ! T ∈ G, [Gn] → [G] !

τ n → ∞.

n

−1

fn f −1

fn f −1

/ + ! ϕ−1 1 "! $ ! # ϕ1 ! (V σ , dT ) (Vσ , dT ) " " - ϕ 1 : (Vσ , dT ) → (Vσ , τ ),

ϕ1 2 ! ! Vσ , 1# ! [Gn] → [G] !

τ n → ∞, !) ! < T1n, ..., Vh−g,n >∈ [Gn], < T1, ..., Vh−g >∈ [G] Tjn → Tj , j = 1, ..., g, Uin → Ui, Vin → Vi, i = 1, ..., h − g, ! ! C n → ∞. -2 dT ([Gn], [G]) = inf lnK(fn) → 0 n → ∞, fn ! ! " *

" G ! Gn . % ;+ G ! 3 * " ) " G Gn !) fn0 ! " *

" G Gn K(fn0) ! * - ! [Gn] → [G] !

τ n → ∞ !+ [Gn] → [G] ! dT n → ∞. .! 2 ϕ−1 1 "! A " " 02 ! " !" ! # F h, ) "# ! W N t ! 2" ∗ ˜ Mh,t ˜ , h !# ! F/ < W >, Mh,t ˜ ! !) ! ! ! ,# ) Th. / < [F, α] >W W −! ! 2 # ) ! [F, α], ! ! Mh,t ˜ = ∪ < [F, α] >W = ∪ < [F1, α ] >W , ! K + ! ∗ F ∈ Mh,t ˜ ! α, ! α ! ! F1 < F1 ∗ ! ! Mh,t ˜ . $!+ " 2" #9 ;" ! M0 M1 2 ! Mh,t ˜ !+ Θ∗σ −! !" ! [F, α] ∈ M0 ψ ∈ Θσ < C CC [F, αψ] ∈ M1. , " 2 ! Mh,t ˜ K ) ! " Θ∗σ −! ! F ! ! ∗ K ) ! Θ∗σ − " # ! F ∈ Mh,t ˜ , [F, α] ! Θ∗σ −! !# [F, β], ! ψ ∈ Θσ β = αψ F0 : ! ! h). ?! ! ! Th # " ModTh ! [F, α] ! [F, β], α, β "

" ! : 6 F0 F. -! ! Γβ ([F, α]) = [F, αβ] ! # α ! F, β ! F0, 2) ! ) ModTh. $ < C ! ϕ0 ! (Vσ , τ ) Qσ ⊂ C3|σ|−3 . - ( CC C C "! 2 Φσ : Th → Qσ , h =| σ | . = !) "!" 2

−1 Φσ : Th → Vσ , Φσ = ϕ−1 0 Φσ , Φσ : Th → Vσ , Φσ = ϕ1 Φσ ,

˜ = Φ (M˜ ). I Th = Th(F0). / V˜h,t ˜ = Φσ (Mh,t ˜ ) Vh,t σ ˜ h,t t Qh˜ = Φσ (Mh,t ˜ ). , 02 ! Qth˜ ! " ˜ 3 +t, # 3h− + !" " 1/;=,7 H.,$/ ? !# ! 2 Φσ " Θσ Φσ ([F, α]) = Φσ ([F, β]), [F, α] ! Θ∗σ −! !# [F, β]. - ! " Θ∗σ −! ! 2 ! Mh,t ˜ 2) Φσ 2 2 ! ! V˜h,t ˜ . $ "! Φσ ! ∗ Mh,t ˜ , " Φσ ! Θσ −! ! ˜˜ . 2 ! Mh,t ˜ ! Vh,t ; Θ∗σ −! ! Mh,t ˜ , 2 # < [F, α] >W , < [F, α] >W,Θ . -2 " ! V˜h,t ˜ !" - [G] [G ] " V˜h,t ˜ !

τ, ! dT !! " C'L , [G] [G ] 2 # ! V˜h,t ˜ . 1# ! ! < [F, α] >W,Θ # [G] ∈ Φσ (< [F, α] >W,Θ ). - [F, α] 2 [G] f ! * G G K(f ) * ˜ = K(f ) , f˜ : [F, α] → [F , f˜α] K(f) * = f˜ * 2 f ! ) " G F = D /G D ! " G . F ! 2 f0 : [F, α] → [F , f˜α] !! ! ) 1 ≤ K(f0) ≤ K(f˜). .! [F , f˜α] 2 [G ] ! dT [F, α]. ? '( [F , f˜α] ∈< [F, α] >W , [G ] ∈ Φσ (< [F, α] >W,Θ ). , " V˜h,t ˜ !"

∗ σ

∗ σ

∗ σ

∗ σ

1 ! V˜h,t ˜ ! Vσ . $ ! V˜h,t ˜ " C'L ! [Gn ] → [G] ! dT n → ∞, [G] ∈ Vσ , [Gn] ∈ Φσ (< [F, α] >W,Θ ) < [F, α] >W,Θ "# Θ∗σ −! ! Mh,t ˜ . -2 [G] ∈ Φσ (< [F, α] >W,Θ ). - [F , α ] 2 [G] fn ! " *

" G Gn K(fn) → 1 n → ∞. 8 Fn = Dn /Gn [Fn, αn ], αn = fñα , fñ * ! 2 fn D, D Dn ! " " G Gn ! ! - ) Φσ (< [Fn , αn] >) Gn ! n. , ! [Gn] ! dT , ! [Fn, αn] ! dT n → ∞. $ ! Mh,t ˜ [Fn , αn ] ∈ M0 9 n, M0 ! < [F, α] >W,Θ . ? M0 [Fn, αn] ! dT [F0, α0] ∈ M0. A"! Φσ !+ [Gn] = Φσ ([Fn, αn]) ! dT n → ∞ Φσ ([F0, α0]) = [G] ∈ Φσ (< [F, α] >W,Θ ), V˜h,t ˜ !

∗ σ

∗ σ

∗ σ

∗ σ

Vσ .

∗ σ

-! Φσ M0 : Θ∗σ −! ! < [F, α] >W,Θ ) Φσ (M0) ≡ Φσ (< [F, α] >W,Θ ). A2 Φσ ! M0 (Φσ )−1 "! - [F, α] = [F , α ] ! Th, + [F, α]

[F , α ] 2 M0, [G] = [G ], [F, α] [F , α ] 2 [G] [G ] ! ! , ! "# q : F → F. , [F , α ] W −! ! [F, qα ], [F, qα ] ∈ M0 2 2 [G]. .! qα ! αψ, ψ ∈ Θσ . 7 ψ 2 ! ! F0, [F, α] [F, qα ] 2 ! " < [F, α] >W,Θ . / ) [F , α ] = [F, qα ] = [F, α] ! Th. , "! (Φσ )−1 "! 2 ! V˜h,t ˜ . , Φσ M0 . , " Mh,t ˜ ) ! ModTh, )" ! " V˜h,t ˜ " 1# ! V1 V2 " V˜h,t ˜ . $+ M1 M2 "

Mh,t ˜ , 2 V1 V2 ! ! . ! Γ ∈ ModTh , ! M1 M2. , Φσ Γ(Φσ )−1 2 V1 V2 ; (Φσ )−1 ! V1 M1 . $ '( M0 ! # 3h˜ − 3 + t. .!

∗ σ

∗ σ

∗ σ

Φσ (M0) V˜h,t ˜ , ! ˜˜ . V˜h,t ˜ , ! + Vh,t 1 ! F + ! ∗ ∗ ˜˜ Θσ −! ! # F ∈ Mh,t ˜ . .! Vh,t + t $ " C'L ! !2 V˜h,t ˜ !" Qh ˜ ! Qσ , 2 V˜h,t ˜ ! Vσ . , / Qnc,h˜ 2 ! ! ;+ Qσ ! !) " ;+ " σ, ) " !" ! F h, ) ) !)* ) J n ! 2" ) " !" = h˜ ! F/ < J >, < J > 2+ J, h = 2h˜ + n − 1, h ≥ 2. - 02 ! Qnc,h˜ ! ! !" 6h˜ − 6 + 3n, + !" " 1 ! ! ! " C'3

! !)) C'E

# (.* (

;" !" ! ) " ! " " !" G %! * L3< LJ 8 'E 8 8+ D3< DJ ; F 4(< 4' $% &! '( ?! " !" ! ) ! " " ! " !) " ! ! 8 ! # ! ! " !" ! # ! = !) ) ) ! ! [F0, {ak , bk }hk=1] h ≥ 2, {ak , bk }hk=1 F0 ak ∩ bk = O0 ∈ F0, k = 1, ..., h. ;2 * @ μ = μ(z) dzdz F0 ! ! ! 2 f ! F0 ) ) ! ! F 2 h, ) ! {f (ak ), f (bk)}hk=1 F. - ) ) ) ! ! ! #9 [F, f ]. $ ) ! ! Th,

!" " " !" ! h ≥ 2, ) ! " ! + ! ! !) ! ! & - 2 ! * " ! 2+" " ! ! # # !# ! $!" !" " 2 ! !" ! # ) " " " ! ! @ Nσ 9) ) ! # π1 (F0, O0 ) = a1 , b1, ..., ah, bh :

h

[aj , bj ] = 1,

j=1

2) " # b1 b2, ..., bg, [ag+1, bg+1], i i [ag+2, bg+2], ..., [ag+s, bg+s] j=1 [ah−i +j , bh−i +j ], j=1 [ag+s+j , bg+s+j ], ..., −1 −1 [aj , bj ] = aj bj aj bj , σ = (g, s, i1, ..., ip) !"# "# *" *" !!) ! G ik = 1, k = 1, ..., p g + s + i1 + ... + ip = |σ| = h. ;#! Gσ ∼ = π1 (F0, O0 )/Nσ , ! !) Nσ

) ! p

1

p

p

Gσ = T1 , ..., Tg , U1, V1 , ..., Uh−g, Vh−g : [U1, V1 ] = ... = [Us, Vs ] = =

i1

[Us+j , Vs+j ] =

j=1

i2

[Us+i1+j , Vs+i1+j ] = ... =

j=1

ip [Us+p−1 in+j , Vs+p−1 in +j ] = 1, = j=1

n=1

n=1

[Uk , Vk ] = Uk Vk Uk−1Vk−1, 2 ! ! C, # # ;+ " σ F ! !" ! # EST −" (g, s) p " !" Gσ = Gg,s ∗ Gi1 ∗ Gi2 ∗ ... ∗ Gip .

@ Θσ ! ψ ! F0 G 6 ψ(O0) = O0 , C6 ! π1 (F0, O0), * !"# ψ, ! ! Nσ , (6 " ;+ Gσ Gσ,ψ , ) ! F0 !) " "

Tj,ψ = Tj Ui,ψ = Ui, Vi,ψ = Vi , j = 1, ..., g, i = 1, ..., h−g, Gσ,ψ ) " ψ(Nσ ), Gσ Nσ .

1 # " !2 !" !" " ! ! ) " " !" ! ! ! ) " " EST −" "# # " ;+ " σ : p = 0). / ! " " " " # # - F ! ! h ≥ 2, ) ! " ! G W N ≥ 2,

) # ! 2" F ; M C 2k, k ≥ 2, ! 2 " F, + W M = MW N |k. - ! ! ! 2 ! ! 2" M !

! " # ! * # " < W > . 0 2 ! " !" ! # F h ≥ 2, ) ! " ! W M, ∗ Mh,2k , Mh,2k ! !) 2 ! ! ! ,# ) Th = Th(F0), F0 ! ! h. = h ! F/ < W, M >, < W, M > 2+ ! W M. = h 2 9 8 %! * 2h − 2 = (2 h − 2)2 + 2m

" π1 : F/ < W >→ F/ < W, M >, h ! F/ < W >, "# ! 2(g + s) + m + 1, k = N m. / ) h = [(h − m − 1)/2] + 1 = g + s + 1. .# "! !! # " 2 ! Mh,2k Q2kh 2 ! ! Qσ , ! !) " EST − σ = (g, s), (g + s = h), ) " !" ! ;

) !# ! 2 ! Q2kh ) ! σ. ,2 ! " !! Mh,2k 9 W −! ! 2 ! Mh,2k " W −! !" ! # ,# ) = 2"# W −! ! ! + " M−! !" ! # 1 (W, M)−! !

" # 1!

" ! :! ,# )6 [F, α] [F , α ] h "!) (W, M)−! !" ∗ F, F ∗ ∈ Mh,2k

f : [F, α] → [F ∗, α∗] ! * W ∗f W −1 M ∗f M −1, f ∈ α∗α−1, α α∗ " ! : 6 F0 F F ∗ ! ! A ! (W, M)−! ! ! # ! ! ) " " !" ! # 1# ! [F, α] [F ∗, α∗] ! (W, M)−! !" ! ,# ) q (q∗) "# [F, α] !# [q(F ), qα] :[F ∗, α∗] [q∗(F ∗), q∗α∗]) : ∗

∗

[F, α]

q

-

α∗ α−1 f

[q(F ), qα] f1 ∈ q ∗ α∗ α−1q −1

?

[F ∗ , α∗]

?

q

∗

-

[q ∗ (F ∗), q ∗α∗ ]

-2 " ! ,# ) [q(F ), qα] [q ∗(F ∗), q ∗α∗ ] 2 (W, M)−! !" 1 ) * ) (q ∗W ∗ (q ∗)−1)f1(qW q −1)−1 ≈ q ∗W ∗ (q ∗)−1q ∗ α∗ α−1q −1 qW −1q −1 = = q ∗ W ∗ α∗ α−1W −1q −1 ≈ q ∗ W ∗f W −1q −1 ≈ q ∗ f q −1 = f1

: ≈ )6 / ) ! ) W −! ! " ! # ! ! M. , ! + 2 ! Mh,2k " (W, M)−! !" ! # ,# ) M" * ) (W, M)−! ! ! ! Th !) ! !

,# ) ; F 12 ? 4(< 4E - X : C ∞) ! h ≥ 2. / Dif f +(X) ) ! ) * ) ! X C ∞− #< Dif f0(X) ! Dif f +(X), ! " " 2 ! X; M(X)

! " X, ! " # X * # $ " " " " " ! C ∞− X. $" ) ) μ0 X, ) ! ! Xμ . - z "# +"# ! U (p) p ∈ Xμ . , z = ϕ(q), q ∈ U (p) ⊂ Xμ , z ∈ D ⊂ C, 2) p ! ) ?! " !# ! ! ) X. - 2# μ ∈ M(X) !"! ! ! 2 +" ! " ! ds2 = |dz + μ(z)dz|2, ! " ! !" * " = μ(z)

* ! " Xμ ! 0

0

0

0

μ∞ = esssup|μ(z)| < 1, z ∈ D.

(1)

; * μ = μ(z) dzdz ! # " Xμ "! * @ ! Xμ . - * ) μ(z), +) ! Xμ , # *

μ(z) U ! # "!)# Xμ . - ! # *

π : U → Xμ , * @ μ(z) dzdz z ! Xμ * @ μ(z ) d U, d z !!) 9 ) 0

0

0

0

0

0

μ(γ z)

γ ( z) = μ( z ), γ ∈ Γ, z ∈ U, γ ( z)

Γ ! ! # !) ! ! U U/Γ = Xμ : ) # ! ! 6 > ! :6 ! ! *

μ(z) ! " Xμ 2 !" + μ( z ) U, 2 " ! Xμ . 89 ! @ 0

0

0

wz − μ( z )wz = 0

(2)

! U, ! : ) *

# * #6 ! 2 wμ U !# wμ (U ). > ! :6 ! !! 9 !

:C6 ! 2 * ) ; ! 2 wμ *

:! ) * )6 2 !# # Γ,

! # !)# ! U, ! !# # Γμ, ! # !)# wμ(U ), Γ γ → wμ γ(wμ)−1 ∈ Γμ .

A! ! ! μ

μ

w (U )/Γ :

Xμ

!

U

w

-

μ

wμ (U )

π

πμ ?

? -

Xμ0

f

Xμ

- wμ ! Xμ , ! 2 f !" ! # Xμ → Xμ f π = πμwμ. , 2 * μ(z) Xμ ! ! 2 f ! Xμ Xμ, μ(z) = μf (z) = ff

w = f (z) 1 ! " X !) ) !) # |dz+μ(z)dz|2, 9 ! @ wz − μ(z)wz = 0 ! # "!)# Xμ

! 2 f ! Xμ !) ! ! Xμ : # μ = μf ). 7 ! # ! Dif f0(X) M(X) 4( G 0 0

0

0

z

z

0

0

μf · g = μf ◦g , g ∈ Dif f0(X).

8 ! " " H < Dif f +(X), " # H < Dif f +(X). - # " H ! " ! ! Xμ h, # H ! # " ! ! 8 2 ! ! " X, " H 2 # " ! ! !" ! F 2 ! H− ! " M(X)H , M(X)H = {μ1 ∈ M(X) : μ1 · h = μ1 , h ∈ H}.

@ ! M(X)H N2N # μ. = ) ) μ ∈ M(X), ! ! μ, 2 ! H < Dif f +(X),

H, ! # " ! ! !# ! Xμ . , + 2 ! ! " " 2 " H :

M(X)H = {μ1 ∈ M(X) : μ1 · h = μ1 , h ∈ H },

N2+N # μ. , ! + ! ! M(X), ) " H < Dif f +(X), " " ! ! " " # # H < Dif f +(X), ! ! ! K 2 ! H − ! " " 8 2 ! ! 2" : 6 " H : M(X)H = {μ ∈ M(X) : μ · h = μ, h ∈ H}.

7 μ ∈ M(X)H , # !

Dif f0(X) μ !2" ! G μ·f0 ∈ M(X)H , μ·f0 2 M(X)H , f0 ∈ Dif f0(X). $ ! μ · f0 !#+ ! 2 " " ! 2 ! M(X)H *

# ! ) Dif f0(X). M" 2 ! Dif f0 (X) !" 2 f0 " " !"! " 2 ! M(X)H . 1 ! / 4( - f0 ∈ Dif f0 (X), μ ∈ M(X)H . , μ · f0 2 2 2 ! M(X)H , f0 ∈ C0(H) = Dif f0(X) ∩ C +(H), C + (H) = {f ∈ Dif f +(X) : f h = hf, h ∈ H} * " H ! Dif f +(X). 8 !G Th = Th(X) = M(X)/Dif f0(X) ! ,# ) h T (X, H) = M(X)H /C0(H) ! ,# ) / i ! ! 2 T (X, H) ! Th(X). 0 4( $2 i : T (X, H) → Th(X) K ! 1/;=,7 H.,$/ / ! - μ1, μ2 ∈ M(X)H . - 2 !) ! " 2 C0(H), [μ1 ]C0 (H) = [μ2 ]C0 (H) ,

(3)

i([μ1]C0 (H) ) = i([μ2]C0 (H) ) = [μ]Dif f0(X) , μ ∈ M(X). / ) μ1 = μ2 · f0 f0 ∈ Dif f0(X). , μ2 ∈ M(X)H , f0 ∈ Dif f0(X) μ1 = μ2 · f0 ∈ M(X)H , "# ) f0 ∈ C0(H), [μ1]C0(H) = [μ2]C0(H). - !

:(6 . ! 1 ! !2 i # *

Φ : M(X) → M(X)/Dif f0(X) = Th (X)

!"!)) G M(X)H μ

id

-

μ ∈ M(X) Φ

C0(H) ?

Dif f0(X) ?

-

T (X, H) [μ]C0 (H)

i

[μ]Dif f0(X) ∈ Th(X)

? ! # " Φ(M(X)H ) = i(T (X, H)).

/ +

Dif f (X)/Dif f0(X) :

!) * )

Θ

(4) Dif f +(X)

Θ : f → f Dif f0(X), f ∈ Dif f +(X).

/ 2 ! Θ(H)− ! T (X)Θ(H) = {Φ(μ) ∈ Th(X) : Φ(μ) · Θ(H) = Φ(μ)},

! T (X)Θ(H) = {[μ]Dif f0(X) ∈ Th : [μ]Dif f0(X) · [h]Dif f0(X) = [μ]Dif f0(X) , h ∈ H}.

= μ ∈ M(X)H , Φ(μ) ∈ T (X)Θ(H), 2 ! M(X)H 2 ) Φ ! T (X)Θ(H) : Φ(M(X)H ) ⊆ T (X)Θ(H)

+ :'6

i(T (X, H)) ⊆ T (X)Θ(H) ⊂ Th(X).

02 ! T (X)Θ(H) ! K 2 ! Φ(M(X)H ) ! " H < Dif f +(X), " H, Θ(H ) = Θ(H). ; Φ(M(X)H ) Φ(M(X)H ) ) ) ! !2 H = f Hf −1 ! 2 f ∈ Dif f0(X) ! Xμ Xμ , μ μ ! !" μ ∈ M(X)H , μ ∈ M(X)H . 1# ! μ = μf = μ f, f ∈ Dif f0(X). A Xμ # ! " ! ! H, Xμ * ! H = f Hf −1 ! " ! ! 7 H 2 # " ! ! Xμ , ) μ · h = μ, h ∈ H , μ ∈ M(X)H . -! μ ∈ M(X)H , !" ! ! " H : μ · h = μ, h ∈ H. 1 H = f −1H f. / ) h ∈ H ! !

μ · h = μf · h = μf ◦h = μf ◦f −1h f = μh ◦f = μf = μ ,

μ ∈ M(X)H . :; F 4' 6 1 ) h ≥ 2 ! G 6 ! T (X)Θ(H) ! " !" 2 ! ! Th(X), " ! ! !< 6 ! T (X)Θ(H) ! Th(X) Φ : M(X)H → T (X)Θ(H)

! " 2 ? " !" Φ(M(X)H ) = i(T (X, H)) ! ! "" " ! T (X)Θ(H), ! ! 2 ! T (X)Θ(H) ) Φ(M(X)H ) = T (X)Θ(H). / ) 0 # 4' /2 Φ : M(X)H → T (X)Θ(H) )K ! ? ! + :'6 ! i(T (X, H)) = T (X)Θ(H). , ! 2 ! T (X)Θ(H) ! # Φ 2 ! M(X)H 2 ! ! H − ! " " H = f Hf −1 ! 2 f ∈ Dif f0(X).

. 6 ! " (W, M)−! ! " " ! 2 ! Mh,2k , ! ) !" " 2 ! ! Th(F ), " ! ! ! " @ [F, α]W W −! !" ! # ,# ) 2 # ! [F, α] : ! 6 [F, α](W,M ) (W, M)−! ! 2 ! [F, α]. 02 ! Mh,2k ! ! ! ! K # Mh,2k = ∪[F, α](W,M ) = ∪[F1, α ](W,M ) , ! K + ∗

! α ! ! F ∈ Mh,2k α ! F1 < F1 ! ∗ ! Mh,2k . " ; " D0 D1 (W, M)−! !" ! # ,# ) 2 ! Mh,2k "!) Θ∗σ −! !" ! [F, α] ∈ D0 ψ ∈ Θσ [F, αψ] ∈ D1. , ! " 2 ! Mh,2k K ) ! " Θ∗σ −! ! F ! ! K ) ! ∗ Θ∗σ − " # ! F ∈ Mh,2k , [F, α] Θ∗σ −! ! [F, β], ! ψ ∈ Θσ , β = αψ F0. $ C( ! ϕ0 ! (Vσ , τ ) ! : ##6 ! ! " ;+ σ = (g, s, i1, ..., ip) :! EST − i1 = i2 = ... = ip = 0) Qσ ⊂ C3|σ|−3 . , 2 "! 2 Φσ : Th → Qσ , h = |σ|. = ! "! 2 Φσ : Th → Vσ , Φσ = ϕ−1 0 Φσ , 2k Th = Th(F0). -2 Vh,2k = Φσ (Mh,2k ) ⊂ Vσ , Qh = Φσ (Mh,2k ) ⊂ Qσ . 6 02 ! Mh,2k ! " ! Th # 3h − 3 + 2m (m = Nk ), + !" " < C6 02 ! Q2kh ! " # 3h − 3 + 2m, + !" " 1/;=,7 H.,$/ ? !# ! 2 Φσ " Θσ Φσ ([F, α]) = Φσ ([F, β]), [F, α] ! Θ∗σ −! !# [F, β]. , ! " % &

#

Θ∗σ −! ! 2 ! Mh,2k 2) Φσ 2 2 ! ! Vh,2k . $ "! Φσ ! Mh,2k " Φσ ! Θ∗σ −! ! Mh,2k ! Vh,2k . ; Θ∗σ −! ! Mh,2k 2 # [F, α](W,M ),

[F, α](W,M ),Θ∗σ .

-! " 2 ! Mh,2k : " (W, M)−! ! 6 !" " - [F, α] ∈ Mh,2k

[F ∗, α∗] 2 ! Mh,2k , [F, α] ! ,# ) dT . >!2 [F ∗, α∗] (W, M)−! ! ! [F, α]. , " [F, α]W 2 ! Mh,2k ∗ ) F ∈ Mh,2k

!" '( [F, α] [F ∗, α∗] 2 [F, α]W . -2 2 2 [F, α](W,M ) , [F, α] [F ∗, α∗ ] M−! !" - f 2 ,# ) ! [F, α] ! [F ∗, α∗], K0 , 1 ≤ K0 [F, α] [F ∗, α∗] ,# ) dT . 8 2 f −1M ∗ f : [F, α] → [F, f −1M ∗ f (α)], ) K ≤ K02 : !# ! ! " 2 # K(f1f2) ≤ K(f1)K(f2), K(f −1) = K(f )). - K02 dT ([F, α], [F, f −1M ∗f (α)]) A ! ,# ) !) ! ! F : " 6 !" ! ,# ) K02 ≥ 1 ! ! [F, α] = [F, f −1M ∗ f (α)]. / ) f −1M ∗f 2 - ! ! ! M−! ! ) ! [F, α] [F, f −1M ∗f (α)] : " # !) ! " M M ∗∗ = f −1M ∗f ! ! 6 M−! !" ? M−! ! f −1M ∗ f ≈ M ∗∗ f −1M ∗ f M −1, ) f −1M ∗ f ≈ (f −1M ∗ f )f −1M ∗ f M −1 M ≈ f −1M ∗ f. $! ! f −1M ∗ f, %! * ) f −1M ∗ f = M. , [F, α] (W, M)−! ! [F ∗, α∗], (W, M)−" 2 ! Mh,2k !" -2 1 ! ) !) D0 = [F , α ](W,M ) ! ,# ) [Fn, αn], 2) D0 )

,# ) [F, α] ∈ Th. 12 [F, α] 2 # 2 !# D0. = D0 = [F , α ]W ∩ [F , α ]M . - ! [Fn, αn] 2 [F , α ]W [F, α] ∈ Th, !! 2 ! [F , α ]W [F, α] ∈ [F , α ]W . . # " ! [Fn, αn] 2 2 2 ! [F , α ]M . .! + :! ,# )6 !) [F, α] ! ! 2 2 ! [F , α ]M . ,

[F, α] ∈ [F , α ]W ∩ [F , α ]M = D0.

- D0 2 ! ! ,# ) ! Th. , ! ! ! !) ! Vh,2k Vh,2k ! Vσ ! 2 ! A ! ! ! Vh,2k !+ [G] [G ] " Vh,2k !

τ, ! ,# ) dT !! " C'L , [G] [G ] 2 # ! Vh,2k . 1# ! ! [F, α](W,M ),Θ # [G] ∈ Φσ ([F, α](W,M ),Θ ). - [F, α] 2 [G] :G [F, α]) f ! * G G , K(f ) * , f : [F, α] → [F , fα] = K(f ). = f * 2 f K(f) Ω(G) "! EST −" G F = Ω(G )/G , Ω(G ) "! " G . F ! 2 f0 : [F, α] → [F , fα] !! ! ) 1 ≤ K(f0) ≤ K(f). .! ) [F , fα] 2 [G ] ! dT [F, α] : dT ([F, α], [F , fα]) = lnK(f0)). - " 2 ! Mh,2k : " (W, M)−! ! 6

!" [F , fα] ∈ [F, α](W,M ), [G ] = Φσ ([F , fα]) ∈ Φσ ([F, α](W,M ),Θ ). , " Vh,2k !" 1 ! Vh,2k ! Vσ . $ ! Vh,2k " C'L ! [Gn], ) [G] ! ,# ) dT n → ∞, [Gn] ∈ Φσ ([F, α](W,M ),Θ ), [G] ∈ Vσ [F, α](W,M ),Θ "# Θ∗σ −! ! Mh,2k . -2 [G] ∈ Φσ ([F, α](W,M ),Θ ). - [F , α ] 2 [G] fn ! "

∗ σ

∗ σ

∗ σ

∗ σ

∗ σ

∗ σ

*

" G Gn, K(fn) → 1, n → ∞. 8 Fn = Ω(Gn)/Gn [Fn , αn ], αn = fn α , fn 2 fn Ω(Gn )/Gn. ? ) Φσ ([Fn, αn]) = Gn ! n. , ! [Gn] ! dT , ! [Fn, αn] 2 ! dT Th n → ∞. $ ! Mh,2k [Fn, αn] ∈ M0 9 n, M0 2 ! Mh,2k ! [F, α](W,M ),Θ . ? M0 [Fn, αn] ! dT [F0, α0] ∈ M0. A"! Φσ !+ [Gn] = Φσ ([Fn, αn ]) 2 ! dT n → ∞ Φσ ([F0, α0]) = [G] ∈ Φσ ([F, α](W,M ),Θ ), Vh,2k ! Vσ . -2 Φσ 2 " M0 :)# Θ∗σ −! ! [F, α](W,M ),Θ ) Φσ (M0) ≡ Φσ ([F, α](W,M ),Θ ). A2 Φσ ! M0 (Φσ )−1 : Φσ (M0) → M0 "! - [F, α] = [F , α ] ! ! ,# ) Th, [F, α] [F , α ] 2 M0 [G] = [G ], [F, α] [F , α ] 2 [G] [G ] ! ! , ! "# q : F → F. , [F , α ] ! ! [F, qα ], [F, qα ] 2 M0 2 [G]. .! qα = αψ, ψ ∈ Θσ . 7 ψ 2 ! 2 ) F0, ,# ) [F, α] [F, qα ] 2 ! " [F, α](W,M ),Θ . - ! ! ) 2 M0. / ) [F , α ] = [F, qα ] = [F, α] ! Th . , Φσ ! M0 . 1 ! "! (Φσ )−1 ! ! ! Vh,2k . / ) Φσ M0. ; " !"9 " Mh,2k : " (W, M)−! ! ! # ,# )6 " / ) ! # " ModTh :! " " W −! ! 6 ) " ! " 2 # :,# ) 6 fφ, +" ! W − ! " " ! ∗ :! " * ! φ(z)dz2 F ∈ Mh,2k " 2 W −! ! 6 / ) ) " ! " Vh,2k 2 " 1# ! V1

V2 )" ! " Vh,2k . $+ M1 M2 " Mh,2k , 2 V1 V2 ! ! - g

∗ σ

∗ σ

∗ σ

∗ σ

∗ σ

2 " M1 M2. , V1 2 V2 ! Φσ g(Φσ )−1; (Φσ )−1 ! V1 M1 . 1 Vh,2k , ! !! # " Vh,2k . ; ! ! ,# ) Th,2m, ! ! ,# ) h 2m = 2kN " :!"" 6 !) ) ,2 ! '( 2 ! !2 im Th,2m !) M0 2 ! Mh,2k , ! ! 3 h − 3 + 2m. 1# ! !2 ! ) - [F, α] ! ,# ) Th,2m, # ξ1, ..., ξ2m !) " :!"" 6 α : F0 → F = F0 = F0/W0 , M0 0 ! ! Th,2m ξ10, ..., ξ2m , " !) * π0 : F0 → F0/W0, M0 ! 2" ! M0 F0. -! [F, α] ! ! ! ! ! [F , α], !)) ! # "!)# F !! ξ j , j = 1, ..., 2m, ! ξj , j = 1, ..., 2m. A ! # "!)# F # ! ! " ! M ! "# * !# " F . / α "!)# ! F ! π1α = απ10, π1 : F → F, π10 : F0/W0 → F0/W0, M0. , [F , α] !!+) N − ) "!)) [F, α], # * ! ! M, # ! * W , 2+ ! W. ! W, !) # " ! N − "

F F , 2 " : N ) * !# " F. 02 ! ! 2" M F 2 " ! ! ) 2 ! 2k = N 2m G ξ1 ∈ π−1(ξ1), ..., ξ2m ∈ π−1(ξ2m), W l {ξ1 , ..., ξ2m}, l = 1, ..., N − 1, π : F → F. - ! ! ! 2 ! ! 2" M F # ! " W . - " ! " M W ) F ) / α F ! ! ! ) πα = απ0. / ) [F, α] ∈ Mh,2k . / ! [F, α] !# !# " M0 = [F, α](W,M ) ! ! [F/W, M, π(ξj ), j = 1, ..., 2k; α1] Th,2m,

α1π0 = πα * # π : F → F/W, M π0 : F0 → F0/W0, M0; M(ξj ) = ξj , j = 1, ..., 2k, F. $ !# " Mh,2k !2 im ! 1# ! !) ! " ! ,# ) [F, α] [F1, α1] Th,2m, " im([F, α]) = im([F1, α1]) = [F, α], ) !2

! ! F = F/ < W, M >= F1. /" ! F F1 2 ! *

! 2" M F. - α F !! ! ! ! G πα = α π 0 , πα = α 1 π 0 .

/ ) α = α1 ! [F, α] = [F1, α1] ! Th. - ! !" " 2 ! Mh,2k " + ! 2 ! Mh,2k . ; Φσ M0, 2 Φσ (M0) Vh,2k ! ∗ Vh,2k , ! + Vh,2k . 1 ! F ∈ Mh,2k

+ ! Θ∗σ −! ! # F. .! Vh,2k + $ !2 Vh,2k !" 2 Q2k ! Qσ ! ϕ0. h 1 ! Mh,2k ! 2 " ! Th, ! ! !) ! ! C B # # G ! ! ΩW,M (F ), (W, M)− ! " " ! " * ! F. - Mh,2k ! Th . , % & .! ! ! C'' !2 2 ! 4' !+ !"9 / !2 2 ! T (X)Θ(H), !) ! 9 # " 2 ! Mh,2k H =< W, M >, ! " !" ! ! ,# ) Th. % & $ 8 " "

Q2kh 2 2 ! " EST −" G, )# ! [F, α] ∈ Mh,2k . ; ! C( G

) ! G G =< W1N ; T1, ..., T2g; U1, V1, ..., U2s, V2s ; A1, ..., Am; T1,2, ..., T2g,N ; U1,2, V1,2, ..., U2s,N , V2s,N ; A1,2, ..., Am,N : [Uj , Vj ] = 1, [Ujn, Vjn ] = 1, j = 1, ..., 2s, n = 2, ..., N >,

M1(z) = −z, W1(z) = αz, α ∈ C, |α| > 1, Tg+i = M1TiM1, i = 1, ..., g, −(n−1)

Us+j = M1 Uj M1 , Vs+j = M1 Vj M1 , j = 1, ..., s, Ti,n = W1n−1TiW1 ,i = −(n−1) −(n−1) n−1 n−1 , Vj,n = W1 Vj W1 ,j = 1, ..., 2g, n = 2, ..., N, Uj,n = W1 Uj W1 −(n−1) n−1 , l = 1, ..., m, n = 2, ..., N. 1, ..., 2s, n = 2, ..., N, Al,n = W1 Al W1 $ ## ! ! " EST − = BGB −1 , B [G] !" # ! G

# 2 ! # " ! 2" ) T1 g = 0 : ! ))6 ! ! 2" ) W1 ! D ∞ 2 ) B # ! D ∞ ! ! A !"# ) " G 2 2 ! ) " 3N (2(g + s) + m) " :# : " G) 1 2 1 2 , ξg+1 , kg+1, ..., ξ2g , ξ2g , k2g ; (α; ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; ξg+1

1 1 , p2s+1, ..., p12s, q2s , p22s; ξˆ11, ξˆ12, kˆ1, ..., p11, q11, p21, ..., p1s , qs1, p2s ; p1s+1, qs+1 1 ˆ2 ˆ 1 2 1 2 1 , ξm, km ; ξ1,2 , ξ1,2 , k1,2, ..., ξ2g,N , ξ2g,N , k2g,N ; p11,2, q1,2 , p21,2, ..., ξˆm p1 , q 1 , p2 ; ξˆ1 , ξˆ2 , kˆ1,2, ..., ξˆ1 , ξˆ2 , kˆm,N ), 2s,N

2s,N

2s,N

1,2

1,2

m,N

m,N

(5)

* 2 W1< " " ! 2" Ti, Ti,k , Al, Al,k ! ! i = 1, ..., 2g, k = 2, ..., N, l = 1, ..., m, ) T1 g = 0 + ! ξ11 = 1; ki 2 Ti , i = 1, ..., 2g; ki , |ki| = 1, ! z+b k + k1 = (aj + dj )2 − 2 ! |k| < 1 :Ti (z) = ac z+d , ai di − bi ci = 1, ci = 0, i = 1, ..., 2g); kˆl , ki,n , kˆl,n 2 Al , Ti,n, Al,n ! ! 1 l = 1, ..., m, i = 1, ..., 2g, n = 2, ..., N ; p1j , qj1 , p2j , p1j,n , qj,n , p2j,n *" −1 2 # Uj , Uj−1, Vj , Uj,n, Uj,n , Vj,n ! ! α

∈

C, |α|

>

1,

1 2 1 ˆ2 (ξi1, ξi2), (ξi,k , ξi,k ), (ξˆl1, ξˆl2), (ξˆl,k , ξl,k )

i

i

i

i

j = 1, ..., 2s, n = 2, ..., N.

M" !" :E6 ! " ) " G !" ! " ,!" W1; T1, ..., Tg ; U1, V1 , ..., Us, Vs ; A1, ..., Am.

.! ! ! ! " # " G !" ) # 3g + 3s + 2m " (α, ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; p11, q11, p21, ..., 1 ˆ p1s , qs1, p2s ; ξˆ11, kˆ1, ..., ξˆm , km).

(6)

, + Q2kh ! 3g + 3s + 2m = 3h − 3 + 2m. % & ' $ " : 6 ! :E6 ! ! :46 !"2) ! " $" 9 ! 9 2 g = 0 : EST −" G 1 2 6 ξg+i = M1 (ξi1) = −ξi1, ξg+i = M1 (ξi2) = −ξi2 , kg+i = ki , i = 1, ..., g; C6 " G N N 1 M1, p1s+j = −p1j , qs+j = −qj1 , p2s+j = −p2j , j = 1, ..., s; (6 ξˆl2 = −ξˆl1, l = 1, ..., m; k k '6 ξi,n = αn−1ξik , ξg+i,n = −αn−1ξik , i = 1, ..., g, n = 2, ..., N, k = 1, 2; ki,n = ki, i = 1, ..., 2g, n = 2, ..., N ;

E6 " G N N2 1 1 W1, pkj,n = αn−1pkj , qj,n = αn−1qj1 , pks+j,n = −αn−1 pkj , qs+j,n =

−αn−1qj1 , j = 1, ..., s, n = 2, ..., N, k = 1, 2; 1 2 1 46 ξˆl,n = αn−1ξˆl1 , ξˆl,n = −ξˆl,n = −αn−1ξˆl1 , kˆl,n = kˆl , l = 1, ..., m, n = 2, ..., N.

- # :46 "# ! " ) " G ! ! ! !" G W1N (z) = αN z, ξj1 (z − ξj2 ) − kj ξj2 (z − ξj1 ) , j = 1, ..., g, Tj (z) = z − ξj2 − kj (z − ξj1 ) 1

1 2

i) qi1 z − (pi +q 4 , i = 1, ..., s, Ui(z) = z − p1i 1

1 2

i) (p1i + qi1 − p2i )z − (pi +q 4 Vi (z) = , i = 1, ..., s, z − p2i ˆ1 ˆ ˆ1 1 z + ξl + kl (z − ξl ) ˆ Al (z) = ξl , l = 1, ..., m. z + ξˆl1 − kˆl (z − ξˆl1 )

0 Q2kh ! # 3h − 3 + 2m !

Qσ ⊂ C3h−3, σ = (N (2g + m) + 1, 2N s, 0, ..., 0), h = |σ|. 1/;=,7 H.,$/ , ! C'' Q2kh ! 2 ! !)# " ! Qσ . A " :# 6 ! :E6 ) ) EST −" G ! Qσ . 02 ! Q2kh ! 2 !

Qσ , 9 2 EST −" G, !" " !

C'E ) ! ) *

# " Qσ , ! - :46 3h − 3 + 2m ! " " # ! # ! :E6 ! " ) ) Q2k ! Qσ . ; 2 !2" !"9 :! h C''

C'E6 Q2kh ! 3h − 3 + 2m. , - F ! ! h ≥ 2, ) ) ! " ! G P N1 ≥ 2 ! ! 2" F ; M C 2k, k ≥ 2, ! 2" F, ! " !) ! 2" P, + P l = M ! N1 = 2l, l ∈ N. $ " ! "# " < W, M >, + # " " ! ! < P, M > . ,2 h ! 1 F/ < P, M > . ; ! ! 2 ! Mh,2k 2 ! ! ! ,# ) Th, " " !" ! # h ≥ 2, ) ! " ! P M, ! !!+ 9 P − M−! ! , (P, M)−! ! ! # ! ! ) " " !" ! 1 # : ! 6 ! + 2 ! Mh,2k + " (P, M)−! !" ! # ,# ) 1 ' 6 02 ! Mh,2k ! " ! Th # 3h+2m− 1 = 3 h−3+(2m+2), m = k−1 l , + !"

$

" < 1 C6 02 ! Q2kh,1 = Φσ (Mh,2k ) ⊂ Qσ ! " # 3h + 2m − 1, + !" " 1/;=,7 H.,$/ 1 ! " ! # ) * !# ! !2+# " ! ! ! ! C'' C'E - 9 " ! ! " "!) ) !# ! " ! ! P M. > ! !! 1 # " Mh,2k . ; ! ! ,# ) Th,t, ! ! ,# ) h t = 2m + 2 " :!"" 6 ! ) ) ,2 ! '( 2 ! !2 Th,t !) M0 1 2 ! Mh,2k :"# (P, M)−! !" ! # ,# )6 ! ! 3h − 3 + t = 3h − 1 + 2m. - [F, α] ! ,# ) Th,2m+2 2m + 2 , " α : F0 → ξ1 , ..., ξ2m, ξ2k−1, ξ2k α F. = F0 = F0/ < P0 , M0 > ! ! Th,2m+2. -! F ! ! ! ! ! F , !)) ! # "!)# F !! ξ1, ..., ξ2m, ξ2k−1, ξ2k ! ξ1, ..., ξ2m, ξ2k−1, ξ2k . A ! # ! F # ! ! M :! 6 !) # " "# * !# " , F l− ) "!)) F, π1 : F → F , !! ξ2k−1, ξ2k l − 1 ξ2k−1, ξ2k . 02 ! ξ1 ∈ π1−1 (ξ 1 ), ..., ξ2m ∈ π1−1 (ξ 2m ), P i{ξ1 , ..., ξ2m}, i = 1, ..., l − 1, ξ2k−1, ξ2k + 2ml + 2 = 2k ! 2" * ! ! M F. $! 2 ! ! 2" M F

! # ! # ! P, ! ) " ! l− "

F F . - ! " M P, " * !# " F, ! - + α ! F !! ! ) πα = απ0 , π0 : F0 → F0/ < P0, M0 >, π : F → F. / )

1 [F, α] ∈ Mh,2k .

/ ! [F, α] !# !# " M0 = [F, α](P,M ) ! ! [F/ < P, M >, π(ξj ), j = 1, ..., 2k; α1] Th,2m+2, M(ξj ) = ξj , j = 1, ..., 2k, P (ξ2k−1) = ξ2k−1, P (ξ2k ) = ξ2k F α1 !! ! ) πα = α1 π 0 , π : F → F/ < P, M >, π0 : F0 → F0/ < P0 , M0 > . $ !# 1 " Mh,2k !2 ! > ! M0 ! 1 2 ! Mh,2k + ! 2 ! 1 Mh,2k . , 2 ! ! " C'' 2 1 1 2 ! Vh,2k = Φσ (Mh,2k ), Φσ : Th → Vσ , +

!" " " 2 # 1 3 h−1+2m. $! 2 ! Vh,2k

Q2kh,1, ! !2 1 Vh,2k ! !" 2 Q2kh,1 ! Qσ . 8 Q2kh,1 2 2 ! ! ! " # 1 EST −" G, )# ! [F, α] ∈ Mh,2k . , G ) !

G =< T1, ..., T2g; U1, V1, ..., U2s, V2s; A1, ..., Am; T1,2, ..., T2g,l; U1,2, V1,2, ..., U2s,l, V2s,l ; A1,2, ..., Am,l :

[Uj , Vj ] = 1, [Ujn, Vjn] = 1, j = 1, ..., 2s, n = 2, ..., l >, M1 (z) = −z, P1 (z) = z exp( 2πi N1 ), Tg+i = M1 Ti M1 , i = 1, ..., g, −(n−1)

Us+j = M1 Uj M1, Vs+j = M1 Vj M1 , j = 1, ..., s; Ti,n = P1n−1 TiP1 ,i = −(n−1) −(n−1) n−1 n−1 , Vj,n = P1 Vj P1 , j = 1, ..., 2s, At,n = 1, ..., 2g, Uj,n = P1 Uj P1 −(n−1) n−1 , t = 1, ..., m, n = 2, ..., l. P1 AtP1

$!+ ! 2 ! # " ! 2" ) T1 g = 0 : !) )6 !# - ! 2" P1 2 !" D ∞. $ ! ) " G !" ! " T1, ..., Tg; U1, V1, ..., Us, Vs; A1, ..., Am. F 2 2 ! ) " 3g + 3s + 2m − 1 ! " " :#6 1 ˆ (ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; p11, q11 , p21, ..., p1s , qs1 , p2s ; ξˆ11, kˆ1, ..., ξˆm , km).

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