t ∈ R ƕƦƷˊᘙႎƳ̊ưƋǔŵ (2)2 ቇҥƳ๫፼բ᫆ƩƕŴP SL2( R ) Ʒ ɶ࣎Ƹᐯଢ ưƋǔŵƜǕƴǑǓŴγ ∈ P SL2( R ) ƴݣƠ ƯܭLJǔ ϋᢿᐯࠁӷ׹ P SL2( R ) g → γ · g · γ−1 ∈ P SL2( R ) ǛᎋƑǔƜƱƴǑƬƯŴҥݧ Ƴแӷ׹ P SL2( R

IV.

חᛯƔǒᙸƨȪȸȞȳ᩿Ʒ٭࢟

§ 1.

_{ˮႻ፭Ʒᐯࠁӷ׹}

ԡ᫆ᲴˮႻ፭

R

^{Ʒᐯࠁӷ׹፭Ƹ}

R

^×

= R\{ 0 }

ƴǑǔNjƷƠƔƳƍŵ

ᚰଢᲴ

α : R → R

ƕᐯࠁӷ׹ưƋǔƱƢǔŵ

α(1) = 1

ƱƠƯNjǑƍŵƢǔƱŴž

n

_̿ſǛ ᎋƑǔƜƱƴǑƬƯŴ

α(λ) = λ, ∀ λ ∈ Q

^ƕ ЎƔǔŵஇࢸƴŴໜЗƷಊᨂǛᎋƑǔƜƱ ƴǑƬƯŴ

α = id

_{ƱƳǔƜƱƕࢼƏŵ}

ഏƴŴ

P SL

₂

( R )

Ʒᐯࠁӷ׹፭ƴƭƍƯᎋƑ ǑƏŵˮႻ፭

R

ǑǓᢕƔƴᙐᩃƳನᡯǛƠƯ ƍǔŵƠƔƠŴƦƷɶƴಮŷƳž

R

^Ʒ΂ſᲷ žɟࢲૠᢿЎ፭ſƕλƬƯƍǔŵ̊ƑƹŴЭ

ׅᙸƨׅ᠃፭

cos(t) sin(t)

− sin(t) cos(t)

| t ∈ R

ƕƦƷˊᘙႎƳ̊ưƋǔŵ

2

ቇҥƳ๫፼բ᫆ƩƕŴ

P SL

₂

( R )

_{Ʒ ɶ࣎Ƹᐯଢ} ưƋǔŵƜǕƴǑǓŴ

γ ∈ P SL

₂

( R )

_ƴݣƠ ƯܭLJǔ ϋᢿᐯࠁӷ׹

P SL

₂

( R ) g → γ · g · γ

⁻¹

∈ P SL

₂

( R )

ǛᎋƑǔƜƱƴǑƬƯŴҥݧ Ƴแӷ׹

P SL

₂

( R ) → Aut(P SL

₂

( R ))

ᲢƨƩƠŴž

Aut( − )

ſƸˮႻ፭Ʒᐯࠁӷ׹Უƕ ưƖǔŵɟ૾ưŴ

1 0 0 − 1

ưσࢫƢǔƜ ƱƴǑǔ

ι ∈ Aut(P SL

₂

( R ))

_NjƋǔŵƜƷ ᐯࠁӷ׹

ι

_{ƸŴɥҞ࠯᩿Ʒ}

z → z

⁻¹ _ƱƍƏ ӒദЩᐯࠁӷ׹ Ჷ ᙐእσࢫ ƴݣࣖƠƯƍ ǔŵƪǐƬƱᩊƠƍܭྸƩƕŴɟࢲૠᢿЎ፭ ሁǛᎋƑǔƜƱƴǑƬƯഏƷǑƏƳƜƱƕ ᚰଢưƖǔŵ

ܭྸᲴɥƷแӷ׹

P SL

₂

( R ) → Aut(P SL

₂

( R ))

Ʒ΂ƷਦૠƸ

2

_ưƋǓŴ

ι

_{Ʒ΂ƴǑƬƯՠ}

Aut(P SL

₂

( R ))/P SL

₂

( R )

_{Ƹဃ঺ƞǕǔŵ}

3

§ 2.

_{ɥҞ࠯᩿Ʒᐯࠁӷ׹፭}

ɥҞ࠯᩿

H

Ʒᛅƴ৏ǖƏŵȪȸȞȳ᩿

H

_Ʒ ɦᢿˮႻᆰ᧓ Ǜ

T

_Ʊ୿Ƙŵ

ኒᲴᢿЎ፭

Aut(H ) ⊆ Aut(T )

_ƴƭƍƯᲴ

(i) Aut(T )

_ϋƷ

Aut(H )

_{Ʒ ɶ࣎҄ᢿЎ፭}

Z

Ƹ ᐯଢ ưƋǔŵ

(ii) Aut(T )

_ϋƷ

Aut(H )

_{Ʒ ദᙹ҄ᢿЎ፭}

N

ƸŴ

Aut(H )

_Ʊž

ι : z → z

⁻¹_{ſưဃ঺ƞǕ} ǔŵཎƴŴ

[N : Aut(H )] = 2

_ŵ

ᚰଢᲴ

(i) α ∈ Z

_ƱƢǔŵ

α

_Ƹ

§ 1

_Ʒׅ᠃፭ Ʊӧ੭Ƣǔŵɟ૾ŴƜƷׅ᠃፭Ʒ՗ɟƷɧ ѣໜƸҾໜưƋǔŵࢼƬƯ

α

_{ƸҾໜǛ׍ܭ} Ƣǔŵഏƴ

Aut(H ) ∼ = P SL

₂

( R )

_ƕ

H

_ƴ ਖ਼ᆆႎ ƴ˺ဇƢǔƜƱǛ࣬ƍЈƦƏŵ

α

_ƕ

Aut(H )

ƷƢǂƯƷΨƱӧ੭ƢǔƨNJŴƜǕ

ư

α

_ƕ

H

ƷƢǂƯƷໜǛ׍ܭƢǔƜƱƕ ЎƔǔŵ

(ii) α ∈ N

ƱƢǔŵܾତƴᄩᛐưƖǔǑƏ ƴŴˮႻ፭

Aut(H ) ∼ = P SL

₂

( R )

_ƷˮႻǛŴ

4

H

ǁƷ˺ဇƔǒဃƣǔNjƷƱᙸǔƜƱƕư ƖǔŵࢼƬƯŴ

α

ưσࢫƢǔƜƱƴǑƬƯ ˮႻ፭

Aut(H )

Ʒ ᐯࠁӷ׹ ƕܭLJǔŵ

§ 1

_Ʒ ܭྸǑǓŴဃơƏǔ

Aut(H )

_{Ʒᐯࠁӷ׹ƕŴ}

(ii)

ƴ୿ƔǕƨNjƷƴᨂǔƜƱƕЎƔǔŵLJ ƨ

(i)

ǑǓžσࢫſƢǔƜƱƴǑƬƯڂǘǕ ǔ

α

_{ƕƳƍƜƱƕЎƔǔŵ}

חᛯႎƳᇌئ ƔǒLjǔƱŴƜƷኒƷॖ፯ Ƹ ഏƷᚐ᣷ƴƋǔᲴኒƷ

(ii)

_ƴǑǓŴ

H

_{Ʒ ദЩ} ನᡯ ǛŴƲƜƔƷ ӋᎋȢȇȫ

C

^ƴǑƬƯܭ

፯ƞǕǔNjƷưƸƳƘŴɦᢿˮႻᆰ᧓

T

_Ʒᐯ ࠁӷႻϙ΂፭

Aut(T )

_{ϋƷžཎКƳᢿЎ፭ſ}

A

^def

= Aut(H ) ⊆ Aut(T )

ƷཎܭƴǑǔNjƷƱᙸǔƜƱƕưƖǔŵƦƏ ƢǔƱᐯࠁӷႻϙ΂

T →

^∼

T

_{ƕžᲢӒᲣദЩſ} ƱƸŴžƜƷᢿЎ፭

A

_{Ǜ̬ƭſƱƍƏவˑ} ưܭ፯ƢǔƱŴᲢኒƷ

(ii)

_{ƴǑǓᲣƜǕƸ}

୍ᡫƷܭ፯ Ʊ ɟᐲ ƢǔŵƭLJǓŴӋᎋȢȇȫ

C

Ɣǒᚐ્ ƞǕƨƜƱƴƳǔŵ

5

§ 3.

_{ȪȸȞȳ᩿ɥƷࣇЎ}

ȪȸȞȳ᩿ƷஜஹƷܭ፯ƴ৏ǖƏŵȪȸȞ ȳ᩿

X

_{ƷŴƋǔ᧏ᨼӳƨƪƷ}

C

^ǁƷ؈NJ ᡂLjǛ̅ƏƜƱƴǑƬƯŴ

C

^{ϋƷ؈NJᡂLj} Ʒ΂ƷɥƷ ࣇЎ

f (z ) dz

ᲢƨƩƠ

f (z)

ƸദЩ᧙ૠᲣǛᎋƑǔƜƱƕ ưƖǔŵƜƷǑƏƳࣇЎƨƪƕŴžദЩ Ƴ

ᝳǓӳǘƤƷӷ׹ƨƪſ

z

₂

= h

₂₁

(z

₁

)

_Ʊɲ ᇌႎưƋǔƱƖŴұƪ

f

₂

(z

₂

)dz

₂

= (dh

₂₁

(z

₁

)/dz

₁

) · (f

₂

◦ h

₂₁

)(z

₁

)dz

₁

= f

₁

(z

₁

)dz

_{1 ųųųųųų}

Ǜ฼ƨƢƱƖŴƜǕǒƷžޅ৑ႎƳࣇЎƨ ƪſǛŴȪȸȞȳ᩿μ˳ƷɥƷࣇЎ Ʊᙸǔ ƜƱƕưƖǔŵƜƷǑƏƳȪȸȞȳ᩿μ˳

ƷɥƷࣇЎƨƪƸŴᚡӭ

Γ(X, ω

_X

)

ưᘙƞǕǔ

C

^{șǯȈȫᆰ᧓Ǜ঺Ƣŵ}

6

ȪȸȞȳ᩿

X

ƕ dzȳȑǯȈ ưƋǔƱˎܭ ƠǑƏŵƢǔƱŴᆔૠ

g

_{ᲷžȉȸȊȄཞƷᆭ} ƷૠſƱƍƏ᣻ᙲƳ ˮႻႎɧ٭᣽ ƕƋǔŵ ഏƷܭྸƸȪȸȞȳ᩿ƷྸᛯƷؕஜႎƳኽ ௐưƋǔŵ

ܭྸᲴ

dim

_C

(Γ(X, ω

_X

)) = g

ӞχႎƳᇌئƴᇌƭƱŴžࣇЎſ

θ

_ƕƋǔƱŴ ƦǕǛ࢘໱ ᆢЎ ƠƨƘƳǔŵƜƷئӳŴ୺

᩿ɥƷᛅƳƷưŴؕໜ

p

_{Ǜ׍ܭƠƯƓƍƯƦ} ƷໜƔǒКƷ˓ॖƷໜ

x

_{ǁƷ ᢊᲷžȑǹſ}

γ

ƴඝƬƯᆢЎƢǔƜƱƴƳǔŵ

7

ƜƷǑƏƴಮŷƳ

x

_Ǎ

θ

_{ƴݣƠƯᆢЎǛ}

γ

θ ∈ C

ᚘምƢǔƱŴϙ΂

X → V

^def

= Γ(X, ω

_X

)

^∗

= Hom

_C

(Γ(X, ω

_X

), C )

ᲢƨƩƠŴž

∗

ſƸӑݣșǯȈȫᆰ᧓ǛᘙƢᲣ

ƕࢽǒǕƦƏƩƕŴNjƏݲƠɠݗƴᎋƑǔ ƱŴƦƏҥኝƳཞඞưƸƳƍƜƱƕЎƔǔŵ

p

_Ʊ

x

_{ƕൿLJƬƯƍƯNj}

γ

_{ƴƸಮŷƳӧᏡࣱ}

ƕƋǔŵཎƴŴ

x = p

_{ƷƱƖᲢƭLJǓŴž᧍}

ơƨȑǹſƷƱƖᲣ

θ

_{ƷᆢЎƸ࣏ƣƠNj}

0

_ƴ ƳǔƱƸᨂǒƳƍŵƜƷǑƏƳž᧍ơƨȑ ǹſ

γ

ƴඝƬƯᆢЎƠƯࢽǒǕǔΨ

∈ V

_Ǜ ԗ஖ ƱԠƿŵ

8

ԗ஖ƨƪƸŴ

V

_ϋƷ

఍܇ ų

Λ ⊆ V

ᲢᲷ

Z

^2g ƴӷ׹ƳNjƷᲣǛܭ፯ƠƯƍƯŴұ ƪՠ

J

^def

= V /Λ

Ƹ ᭗ഏΨƷᙐእȈȸȩǹ ƴ ƳǔŵഏƷኽௐᲷžǢȸșȫƷܭྸſƸŴȪ ȸȞȳ᩿ƷྸᛯƴƓƚǔ᣻ᙲƳӞχႎኽௐ ưƋǔŵ

ܭྸᲴ

g ≥ 1

ƷƱƖŴᆢЎƢǔƜƱƴǑƬƯ ࢽǒǕǔݧ

X → J = V /Λ

Ƹ ദЩƳ؈NJᡂLj ƴƳǔŵ

9

§ 4.

ȪȸȞȳ᩿ɥƷʚഏࣇЎ

§ 3

ƷǑƏƳžɟഏࣇЎſƩƚưƳƘŴ˓ॖƷ

n ∈ Z

ƴݣƢǔž᭗ഏƷࣇЎſǛᎋƑǔƜƱ NjӧᏡưƋǔŵƭLJǓŴȪȸȞȳ᩿ƷӲŷƷ ޅ৑ႎƳ

C

ǁƷ؈NJᡂLjǛ̅ƏƜƱƴǑƬ ƯŴ

C

ϋƷ؈NJᡂLjƷ΂ƷɥƷ

n

_ഏࣇЎ

f (z ) dz

ⁿ

ᲢƨƩƠ

f (z)

ƸദЩ᧙ૠᲣǛᎋƑǔŵƜƷǑ ƏƳ

n

_{ഏࣇЎƨƪƕŴ}žദЩ Ƴ ᝳǓӳǘƤƷ ӷ׹ƨƪſ

z

₂

= h

₂₁

(z

₁

)

_{ƴݣƠƯவˑ}

f

₂

(z

₂

)dz

₂ⁿ

= (dh

₂₁

(z

₁

)/dz

₁

)

ⁿ

· (f

₂

◦ h

₂₁

)(z

₁

)dz

₁ⁿ

= f

₁

(z

₁

)dz

₁ⁿ _ųųųųųų

Ǜ฼ƨƢƱƖŴƜǕǒƷžޅ৑ႎƳ

n

_ഏࣇЎ ƨƪſǛŴȪȸȞȳ᩿μ˳ƷɥƷ

n

_ഏࣇЎ ƱᙸǔƜƱƕưƖǔŵƜƷǑƏƳȪȸȞȳ

᩿μ˳ƷɥƷ

n

_{ഏࣇЎƨƪƸŴᚡӭ}

Γ(X, ω

_X^⊗n

)

ưᘙƞǕǔ

C

^{șǯȈȫᆰ᧓Ǜ঺Ƣŵ}

10

ȪȸȞȳ᩿

X

ƕ dzȳȑǯȈ ưƠƔNj ӑ୺ႎ ưƋǔƱˎܭƠǑƏŵƜƷӑ୺ࣱƷவˑƸŴ

ܱƸ

g ≥ 2

ƱƍƏவˑƱӷ͌ưƋǔƜƱƸŴ ɟॖ҄ܭྸ ǑǓႺƪƴࢼƏŵᲢƭLJǓŴ

g = 0

ƩƱŴ

X

_{Ƹ ྶ᩿ ƴƳǓŴ}

g = 1

_ƩƱŴ

X

_Ƹ ɟഏΨᙐእȈȸȩǹ ƴƳǔƨNJŴ୍ᢄᘮᙴ Ƹ

C

ƱദЩƴӷ׹ƴƳǔŵᲣ

ƢǔƱŴƍǘǏǔ ȪȸȞȳȷȭȃț ƱƍƏŴ ȪȸȞȳ᩿ƷӞχႎƳྸᛯƴƓƚǔؕஜႎ ƳܭྸǑǓഏƷኽௐƕࢼƏŵ

ܭྸᲴ

(i) n < 0

_ƷƱƖŴ

dim

_C

(Γ(X, ω

_X^⊗n

)) = 0 (ii) n = 0

_ƷƱƖŴ

dim

_C

(Γ(X, ω

_X^⊗n

)) = 1 (iii) n = 1

_ƷƱƖŴ

dim

_C

(Γ(X, ω

_X^⊗n

)) = g (iv) n ≥ 2

_ƷƱƖŴ

ųų

dim

_C

(Γ(X, ω

_X^⊗n

)) = (2n − 1)(g − 1)

ཎƴŴ

dim

_C

(Γ(X, ω

_X^⊗2

)) = 3(g − 1)

11

n

_{ഏࣇЎƷɶưNjŴ}žʚഏࣇЎſƸཎƴ᣻ᙲ ưƋǔŵƦǕƸɟᚕưƍƏƱŴʚഏࣇЎƸŴ

ȪȸȞȳ᩿ƷȢǸȥȩǤ

Ʊ݅੗ƴ᧙̞ƠƯƍǔƔǒưƋǔŵƜƜư ƍƏžȪȸȞȳ᩿ƷȢǸȥȩǤſƱƸŴɦ ᢿˮႻ୺᩿Ǜ׍ܭƠƨƱƖŴദЩನᡯ ƕƲ ƷƘǒƍѣƖࢽǔƔŴƱƍƏƜƱưƋǔŵ

̊ƑƹŴ

0 = θ ∈ Γ(X, ω

_X^⊗2

)

_{ƱƠǑƏŵƢǔ} ƱŴ

θ

ƕ ᩐໜǛਤƨƳƍŴȪȸȞȳ᩿

X

_Ʒ җЎƴݱƞƍ᧏ᨼӳ

U

_Ʒໜ

p ∈ U

_{Ǜ ؕໜ} ƴᢠƿƱŴ

x ∈ U

_ƴݣƠƯŴ

p

_Ɣǒ

x

_ǁƷȑ ǹ

γ

_ƴඝƬƯ

θ

_{Ʒ ࠯૾ఌ}

± √

θ

_ǛᆢЎƢǔ

γ

± √ θ

ƜƱƴǑƬƯŴ

U

ɥƷ ޅ৑ႎƳദЩࡈ೅ ƕ ưƖǔŵƜƷǑƏƳࡈ೅Ǜ ٭࢟ ƢǔƜƱƴ ǑƬƯ

X

ƷȢǸȥȩǤǛᎋݑƢǔƜƱƕŴ Ӟχႎ

Teichm¨ uller

ྸᛯ ƷЈႆໜưƋǔŵ

12

§ 5.

ȪȸȞȳ᩿ɥƷ࠯ᘍׄᡀ࢟ሁ

ȪȸȞȳ᩿

X

_{ƷҗЎƴݱƞƍ᧏ᨼӳ}

U ⊆ X

ɥƴʚഏࣇЎƷ࠯૾ఌƷᆢЎ

z

^def

=

γ

± √ θ

ƴǑǔ ࡈ೅

z = x + iy

_{ƴƭƍƯᎋƑǔŵLJ} ƣŴ

λ ∈ R

^>₀ ^ƴݣƠƯ

z

_λ ^def

= λ · x + iy

ƱƍƏ КƷࡈ೅ ǛᎋƑǔƜƱƕưƖǔŵƜ ƷૼƠƍࡈ೅

z

_{λ ƴǑƬƯŴ}

ૼƠƍ ദЩನᡯ

ƕൿLJǔŵ࢘ƨǓЭưƸƳƍƕŴƜƷǑƏ ƴ ޅ৑ႎ ƴ˺ƬƨૼƠƍദЩನᡯƨƪƸଓ ƘᝳǓӳƏƨNJŴ

X

_{ƷɦᢿˮႻ୺᩿}

T

_μ˳

ƷɥƷૼƠƍദЩನᡯǛܭ፯ƠƯƍǔŵ

13

ƜƷǑƏƴƠƯưƖǔȪȸȞȳ᩿Ǜ

X

_{λ Ʊ}

୿ƘƱŴ

X

_Ʊ

X

_λ ƕӷɟƷɦᢿˮႻ୺᩿Ǜ σஊƠƯƍǔƜƱƴǑƬƯŴᲢᲢӒᲣദЩᲷ ሁᚌưƸƳƍƕᲣલሁᚌ ƳӷႻϙ΂

X → X

_λ

ƕưƖǔŵƜƷǑƏƴƠƯ˺Ƭƨϙ΂Ƹ

Teichm¨ uller

_ϙ΂

ƱԠƹǕǔNjƷưŴಮŷƳཎКƳࣱឋǛਤƬ ƯƍǔŵƳƓŴ

λ ∈ R

^>₀ ^ǛѣƔƢ

ƜƱƴǑƬƯŴ

X

_{ƷɦᢿˮႻ୺᩿}

T

_ɥƴദ ЩನᡯƷ ɟࢲૠଈ ƕưƖǔŵ

ഏƴŴɥƷᛅƷǑƏƳ ࡈ೅ƨƪ Ǜ̅ƬƯ

C ∈ { Squr, Rect, Para }

ƴݣƠƯŴ

X

ɥƷ ׄᚌŴᧈ૾࢟ŴƋǔƍƸ

࠯ᘍׄᡀ࢟ ƔǒƳǔח

C (X )

_Ǜ˺ǔƜƱƕ ưƖǔƜƱƴදႸƠǑƏŵ

14

ܭྸᲴ

X

_Ʊ

Y

Ƹӑ୺ႎȪȸȞȳ᩿ƱƢǔŵ

(i) C ∈ { Squr, Rect }

^ƷƱƖŴ

חӷ͌

C (X ) → C

^∼

(Y )

_Ʒӷ׹᫏

Ʊ

ᲢӒᲣദЩƳӷ׹

X →

^∼

Y

Ƹ

1

_ݣ

1

_{ƴݣࣖƠƯƍǔŵ}

(ii) C = Para

_ƷƱƖŴ

חӷ͌

C (X ) → C

^∼

(Y )

_Ʒӷ׹᫏

Ʊ

ᲢӒᲣ

Teichm¨ uller

_ϙ΂

X →

^∼

Y

Ƹ

1

_ݣ

1

_{ƴݣࣖƠƯƍǔŵ}

15

ᚰଢƸƦǕDŽƲᩊƠƘƳƍƕɦᚡƷᛯ૨ƴ ᜯǔŵ

S. Mochizuki, Conformal and Quasiconfor-

mal Categorical ..., Hiroshima Math. J. 36

(2006), pp. 405-441.