On Some Problems of Kobayashi and Lang

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On some problems of Kobayashi and Lang

Claire Voisin

Contents

CONTENTS
CONTENTS

Chapter 0

Introduction

≥

CHAPTER 0. INTRODUCTION

∈

∗

→
∈∈
→
→

→∞
∗

∞

→∞
→∞

2
2
1

CHAPTER 0. INTRODUCTION

≤

≥

⊂
≥

≥

21
2
3

CHAPTER 0. INTRODUCTION

Chapter 1

The analytic setting

1.1 Basic results and deﬁnitions

1.1.1 Deﬁnition of the pseudo-metrics/volume forms

{|
|
}

∗

∈

{|
|
∧
∧
}

∗

CHAPTER 1. THE ANALYTIC SETTING

∈
∈

1
1

∀ ≥
⊂
→

−
|| ||

∧

− |
|

|| ||
⊂

|| ||

∈

1.1. BASIC RESULTS AND DEFINITIONS

→

[0 1]
(0)= (1)=
[0 1]

|| ||

1.1.2 First properties

→

∗

≤
∧

∗

{
}

+1
+1

+1
2
1

CHAPTER 1. THE ANALYTIC SETTING

≤

00
0

≤

→

2
+

{
}
t

∈

1.1. BASIC RESULTS AND DEFINITIONS

|
|≤

≤
| · |
|

−1

|
| · |
|≥

∈

∧
{|
|
∧
}

∗

≤

∗

≤

CHAPTER 1. THE ANALYTIC SETTING

1.1.3 Brody’s theorem, applications

→
→
≥

|
|
∀ ∈

→

→∞

→

∈

|| ||

( )

|| ||
⊂
|| ||

→

1.1. BASIC RESULTS AND DEFINITIONS

|| ||

( )

∈

|| ||
≥
∈
∈

∗

⊂
→

||
|| ≤

→

CHAPTER 1. THE ANALYTIC SETTING

∈
→
∀ ∈
∈

{ ∈
}

{ }

≥

1.1. BASIC RESULTS AND DEFINITIONS

→∞
→∞

→∞

≤

→∞
→∞

◦

→∞
→∞

→
X →
∈

X
∈

→

||
|| ||
||

∈C

⊂

CHAPTER 1. THE ANALYTIC SETTING

→

1.1.4 Relation between volume and metric

→∞
→∞

→

→∞

→∞
∗

→

1.2 Curvature arguments

1.2. CURVATURE ARGUMENTS

1.2.1 Hyperbolic geometry and the Ahlfors-Schwarz lemma

∧

1
2 2

− |
|

∧

2 2

− |
|

∧

==1
2 2

− |
|

−
− |
|

− |
|

∧

2 2

− |
|

≥
≥

CHAPTER 1. THE ANALYTIC SETTING

∞
1−
1−

1−
∞
1−

−

∈

≤

−1

∗

≤

1.2. CURVATURE ARGUMENTS

→

∗
∗

≤≥

∗

O_P(

∗

)

O_P(

−

)

→
−
→

∗∗

^∗O_P(

)

→

^∗O_P(

−

1
)

∗

≥

∗

→

∗

≤

≥| |

O_P(

−

)

CHAPTER 1. THE ANALYTIC SETTING

≥
≥
→

∗
∗

≤
≥

1.2.2 Another deﬁnition of the Kobayashi inﬁnitesimal pseudometric

→
→
↓

1
−1

−1

◦

1.2. CURVATURE ARGUMENTS

→

|| ||
|| ||

≥
|| ||

|| ||

∗

|| ||
≤

≤

−1

−1
−1

|| ||
||
||
|
|

|| ||
∈
≥

∗

−1

◦

∗
∗

0
∗

|| ||
||
≤|| ||
≤

CHAPTER 1. THE ANALYTIC SETTING

|| ||
≤|
|≤

|
| − |
|

| |

≥

1.2.3 Jet diﬀerentials

→
→

∗

−
→→
→
O

∗

| |

|
| | ||
|
∈
∈

1.2. CURVATURE ARGUMENTS

O −
| |

|
·
| | ||
|
∈
∈

O −

1
2

→
→

1
2

∞

−

◦

∗

∈

≤ −
→

2
−1

1
1

≤ −

|| ||

CHAPTER 1. THE ANALYTIC SETTING

∗

−1

→
→

0
2

1.3 Conjectures, examples

1.3.1 Kobayashi’s conjecture on volumes

1.3. CONJECTURES, EXAMPLES

−

→
→

≥

⊂

1
−1

∗

≤

×P¹

×
⊂
×

∗

A →
A

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