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.

This volume consists of sixteen articles written by participants of the 1995-96 Special Year in Several Complex Variables held at the Mathematical Sciences Research Institute in Berkeley, California.

The field of Several Complex Variables is a central area of mathematics with strong interactions with partial differential equations, algebraic geometry and differential geometry. The 1995-96 MSRI program on Several Complex Variables emphasized these interactions and concentrated on developments and problems of current interest that capitalize on this interplay of ideas and techniques.

This collection provides a picture of the status of research in these overlapping areas at the time of the conference, with some updates. It will serve as a basis for continued contributions from researchers and as an introduction for students. Most of the articles are surveys or expositions of results and techniques from these overlapping areas in several complex variables, often summarizing a vast amount of literature from a unified point of view. A few articles are more oriented toward researchers but nonetheless have expository sections.

On August 29, 1997 Michael Schneider, one of the two editors of this volume, died in a rock-climbing accident in the French Alps. This volume is dedicated to his memory. The front matter includes his portrait, a listing of the major events in his mathematical career, and a selection of his mathematical contributions.