Introduction

We consider here some recent results concerning local biholomorphisms which map one real analytic (or real algebraic) subset of ℂN

into another such subset of the same dimension. One of the general questions studied is the following. Given M, M’ c ℂN, germs of real analytic subsets at p and p’ respectively with diniR M = dimℝ M', describe the (possibly empty) set of germs of biholomorphisms H : (ℂN,p) → (ℂN,p') with H(M) ⊂ M'.

Most of the new results stated here have been recently obtained in joint work with Peter Ebenfelt. We shall give precise definitions and specific references in the text.

1. Basic Definitions

Let Z be a complex manifold; recall that an n-cycle in Z is a locally finite Sum

Topology of the cycle space. For simplicity we assume here that cycles are compact. The continuity of a family of cycles (Cs)s ∈ s consists of two conditions: - Geometric continuity of the supports: This is the fact that ﹛s ∈ 5/|Cs| ⊂ U﹜ is open in S when U is an open set in Z. - Continuity of the volume: For any choice of a continuous positive hermitian (1,1) form on Z, the volume function

For more information on the relationship between volume and intersection multiplicities, see [Barlet 1980c]. A main tool in the topological study of cycles is E. Bishop's compactness theorem (see [Bishop 1964; Barlet 1978a; Lieberman 1978; Fujiki 1978; SGAN 1982]): THEOREM 1. Let Z a complex analytic space and Cn(Z) the (topological) space of compact n-cycles of Z.

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.

1. Introduction

This article surveys some developments, which started almost twenty years ago, on the applications of harmonic mappings to the study of topology and geometry of Kahler manifolds. The starting point of these developments was the strong rigidity theorem of Siu [1980], which is a generalization of a special case of the strong rigidity theorem of Mostow [1973] for locally symmetric manifolds. Siu's theorem introduced for the first time an effective way of using, in a broad way, the theory of harmonic mappings to study mappings between manifolds. Many interesting applications of harmonic mappings to the study of mappings of Kahler manifolds to nonpositively curved spaces have been developed since then by various authors. More generally the linear representations (and other representations) of their fundamental groups have also been studied. Our purpose here is to give a general survey of this work. One interesting by-product of this study is that it has produced new results on an old an challenging question: what groups can be fundamental groups of smooth projective varieties (or of compact Kahler manifolds)? These groups are called Kahler groups for short, and have been intensively studied in the last decade. New restrictions on Kahler groups have been obtained by these techniques. On the other hand new examples of Kahler groups have also shown the limitations of some of these methods. We do not discuss these developments in much detail because we have nothing to add to the recent book [Amoros et al. 1996] on this subject.

Introduction

This article reports some of the recent developments in the classification theory of compact complex Kahler manifolds with special emphasis on manifolds of non-positive Kodaira dimension (vaguely: semipositively curved manifolds). In the introduction we want to give some general comments on classification theory concerning main principles, objectives and methods. Of course one could ask more generally for a classification theory of arbitrary compact manifolds but this seems hopeless as most of the techniques available break down in the “general” case (such as Hodge theory). Also there are a lot of pathologies which tell us to introduce some reasonable assumptions. From an algebraic point of view one will restrict to projective manifolds but from a more complex-analytic viewpoint, the Kahler condition is the most natural. Clearly manifolds which are only bimeromorphic to a projective or Kahler manifold are interesting, too, but these will be mainly ignored in this article and might occur only as intermediate products. The most basic questions in classification theory are the following.

(A) Which topological or differentiate manifolds carry a complex (algebraic or Kahler) structure? If a topological manifold carries a complex structure, try to describe them (moduli spaces, deformations, invariants).

1. Introduction

In this paper we discuss low-dimensional dynamical systems described by complex numbers. There is a parallel theory for real numbers. The real numbers have the advantage of being more directly tuned to describing real-life systems. However, complex numbers offer additional regularity and besides, real systems usually complexify in a way that makes phenomena more clear: for example, periodic points disappear under parameter changes in the real case, but remain in the complex case.

In the case of the solar system and other complicated systems, one has to resign oneself to studying the time evolution of a small number of variables, since if one wants to precisely predict long-term evolution one runs into unsurmountable computer problems. One cannot forget unavoidable errors that are just necessary limits of knowledge. And some knowledge is hence limited to a phenomenological type.

Here we give a brief overview of some of the open questions in the area of complex dynamics in dimension 2 or more. We also discuss some new results by the authors about symplectic geometry and Hamiltonian mechanics, belonging to higher-dimensional complex dynamics.

2. Questions in Higher-Dimensional Complex Dynamics

Complex dynamics in one complex dimension arose in the end of the last century as an outgrowth of studies of Newton's method and the three body problem in celestial mechanics. See [Alexander 1994] for a historical treatment.

2.1. Local Theory. In the local theory one studies the behavior near a fixed point, f(x) = x. This was the beginning of the theory in one complex variable: see [Schröder 1871].

There were antecedents. Barrett [1984] gave an example of a smoothly bounded, nonpseudoconvex domain for which the Bergman projection B fails to preserve C∞ (Ω) Kiselman [1991] showed that B fails to preserve C∞ (Ω) for certain bounded but nonsmooth pseudoconvex Hartogs domains. Barrett [1992] added a fundamental insight and deduced that for the so-called worm domains, which are smoothly bounded and pseudoconvex, B fails to map the Sobolev space Hs to itself, for large s. Finally Christ [1996b] proved an a priori Hs estimate for smooth solutions on worm domains, and observed that this estimate would contradict Barrett's result if global C∞ regularity were valid.

This article discusses background, related results, the proof of global irregularity, and open questions. It is an expanded version of lectures given at MSRI in the Fall of 1995. A brief report on analytic hypoellipticity is also included. I am indebted to Emil Sträube for useful comments on a preliminary draft.

Introduction

The goal of the present paper is to investigate some duality properties connecting pseudoconvexity and pseudoconcavity. Our ultimate perspective would be a geometric duality theory parallel to Serre duality, in the sense that Serre duality would be the underlying cohomological theory. Although similar ideas have already been used by several authors in various contexts — for example, for the study of direct images of sheaves [Ramis et al. 1971], or in connection with the study of Fantappie transforms and lineal convexity [Kiselman 1997], or in the study of Monge-Ampere equations [Lempert 1985]—we feel that the “convex-concave” duality theory still suffers from a severe lack of understanding.