Since the appearance of Kobayashi's book, there have been several re sults at the basic level of hyperbolic spaces, for instance Brody's theorem, and results of Green, Kiernan, Kobayashi, Noguchi, etc. which make it worthwhile to have a systematic exposition. Although of necessity I re produce some theorems from Kobayashi, I take a different direction, with different applications in mind, so the present book does not super sede Kobayashi's. My interest in these matters stems from their relations with diophan tine geometry. Indeed, if X is a projective variety over the complex numbers, then I conjecture that X is hyperbolic if and only if X has only a finite number of rational points in every finitely generated field over the rational numbers. There are also a number of subsidiary conjectures related to this one. These conjectures are qualitative. Vojta has made quantitative conjectures by relating the Second Main Theorem of Nevan linna theory to the theory of heights, and he has conjectured bounds on heights stemming from inequalities having to do with diophantine approximations and implying both classical and modern conjectures. Noguchi has looked at the function field case and made substantial progress, after the line started by Grauert and Grauert-Reckziegel and continued by a recent paper of Riebesehl. The book is divided into three main parts: the basic complex analytic theory, differential geometric aspects, and Nevanlinna theory. Several chapters of this book are logically independent of each other. Title page Foreword Contents CHAPTER 0 Preliminaries 1. Length Functions 2. Complex Spaces CHAPTER I Basic Properties 1. The Kobayashi Semi Distance 2. Kobayashi Hyperbolic 3. Complete Hyperbolic 4. Connection with Ascoli's Theorem CHAPTER II Hyperbolic Imbeddings 1. Definition by Equivalent Properties 2. Kwack's Theorem (Big Picard) on D 3. Some Results in Measure Theory 4. Noguchi's Theorem on D 5. The Kiernan-Kobayashi-Kwack (K3) Theorem and Noguchi's Theorem CHAPTER III Brody's Theorem 1. Bounds on Radii of Discs 2. Brody's Criterion for Hyperbolicity 3. Applications 4. Further Applications: Complex Tori CHAPTER IV Negative Curvature on Line Bundles 1. Royden's Semi Length Function 2. Chern and Ricci Forms 3. The Ahlfors-Schwarz Lemma 4. The Equidimensional Case 5. Pseudo Canonical Varieties CHAPTER V Curvature on Vector Bundles 1. Connections on Vector Bundles 2. Complex Hermitian Connections and Ricci Tensor 3. The Ricci Function 4. Garrity's Theorem CHAPTER VI Nevanlinna Theory 1. The Poisson-Jensen Formula 2. Nevanlinna Height and the First Main Theorem 3. The Theorem on the Logarithmic Derivative 4. The Second Main Theorem CHAPTER VII Applications to Holomorphic Curves in P 1. Borcl's Theorem 2. Holomorphic Curves Missing Hyperplanes 3. The Height of a Map into Prt 4. The Fermat Hypersurface 5. Arbitrary Varieties 6. Second Main Theorem for Hyperplanes CHAPTER VIII Normal Families of the Disc in P Minus Hyperplanes 1. Some Criteria for Normal Families 2. The Borel Equation on D for Three Functions 3. Estimates of Bloch-Cartan 4. Cartan's Conjecture and the Case of Four Functions 5. The Case of Arbitrarily Many Functions Bibliography Index
