Cover

Title page

Contents

Preface

Expository and survey articles

Some applications of 𝑝-adic uniformization to algebraic dynamics

1. A motivation: potential density

2. Near a fixed point

3. Uniformization of orbits and applications

4. Acknowledgments

References

Special manifolds, arithmetic and hyperbolic aspects: a short survey

1. Introduction

2. Special manifolds: Bogomolov sheaves

3. Special manifolds: orbifold base

4. The orbifold version 𝐶^{𝑜𝑟𝑏}_{𝑛,𝑚} of the 𝐶_{𝑛,𝑚} conjecture

5. The core fibration

6. The decomposition 𝑐=(𝐽𝑟)ⁿ of the core

7. Conjectures for smooth and integral orbifolds

8. Examples of fibrations of general type on simply-connected manifolds

9. Orbifold cotangent sheaves: generic semi-positivity, applications

10. H-principle and specialness

References

Invitation to integral and rational points on curves and surfaces

1. Introduction

2. Some Diophantine equations

3. Diophantine geometry - rational and integral points on curves

4. Rational and integral points on surfaces

References

Roth’s theorem: an introduction to diophantine approximation

0. Introduction

1. Liouville’s theorem and beyond

2. Roth’s theorem: an overview

3. Controlling the index: Roth’s lemma

4. Controlling the index: Dyson’s lemma

5. Extensions and questions

References

The Thue-Siegel method in diophantine geometry

1. The Thue-Siegel method

2. Basic constructions in Bombieri’s proof

3. Divisors, heights, and Jacobians

4. An upper bound for the height

5. A lower bound

6. Construction of a global section

7. The index

8. The end of the proof

References

Research articles

Optimal pinching for the holomorphic sectional curvature of Hitchin’s metrics on Hirzebruch surfaces

1. Introduction

2. Basic definitions and description of the family of metrics under consideration

3. Proof of Theorem 1.1

4. Geometric interpretation of our computations

5. Proof of Theorem 1.2

References

The Lefschetz property for families of curves

References

Separable rational connectedness and stability

References

Curve classes on rationally connected varieties

1. Introduction

2. Preliminaries

3. Proof of the Main Theorem

References

Back Cover