We say that a complex manifold is hyperbolic if there is no nonconstant holomorphic map from ℂ to it. This paper discusses the new techniques in hyperbolicity problems introduced in recent years in a series of joint papers which I wrote with Sai-Kee Yeung [Siu and Yeung 1996b; 1996a; 1997] and in a series of papers by Michael McQuillan [McQuillan 1996; 1997]. The goal is to explain the motivations and the main ideas of these techniques. In the process we examine known results using new approaches, providing streamlined proofs for them. The paper consists of three parts: an Introduction, Chapter 1, and Chapter 2. The Introduction provides the necessary background, states the main problems, and discusses the motivations and the main ideas of the recent new techniques. Chapter 1 presents a proof of the following theorem, using techniques from diophantine approximation.

Introduction

0.1. Statement of Hyperbolicity Problems. Hyperbolicity problems have two aspects, the qualitative aspect and the quantitative aspect. The easier qualitative aspect of the hyperbolicity problems is to prove that certain classes of complex manifolds are hyperbolic in the following sense. A complex manifold is hyperbolic if there is no nonconstant holomorphic map from ℂ to it. There are two classes of manifolds which are usually used to test techniques introduced to prove hyperbolicity.