Lang S. Introduction to Complex Hyperbolic Spaces

Springer, 2010. — 277 p.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 otherCHAPTER 0 Preliminaries
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.Foreword
Preliminaries
Length Functions
Complex Spaces
Basic Properties
The Kobayashi Semi Distance
Kobayashi Hyperbolic
Complete Hyperbolic
Connection with Ascoli's Theorem
Hyperbolic Imbeddings
Definition by Equivalent Properties
Kwack's Theorem (Big Picard) on D
Some Results in Measure Theory
Noguchi's Theorem on D
The Kiernan-Kobayashi-Kwack (K3) Theorem and Noguchi's Theorem
Brody's Theorem
Bounds on Radii of Discs
Brody's Criterion for Hyperbolicity
Applications
Further Applications: Complex Tori
Negative Curvature on Line Bundles
Royden's Semi Length Function
Chern and Ricci Forms
The Ahlfors-Schwarz Lemma
The Equidimensional Case
Pseudo Canonical Varieties
Curvature on Vector Bundles
Connections on Vector Bundles
Complex Hermitian Connections and Ricci Tensor
The Ricci Function
Garrity's Theorem
Nevanlinna Theory
The Poisson-Jensen Formula
Nevanlinna Height and the First Main Theorem
The Theorem on the Logarithmic Derivative
The Second Main Theorem
Applications to Holomorphic Curves in P
Borcl's Theorem
Holomorphic Curves Missing Hyperplanes
The Height of a Map into Prt
The Fermat Hypersurface
Arbitrary Varieties
Second Main Theorem for Hyperplanes
Normal Families of the Disc in P Minus Hyperplanes
Some Criteria for Normal Families
The Borel Equation on D for Three Functions
Estimates of Bloch-Cartan
Cartan's Conjecture and the Case of Four Functions
The Case of Arbitrarily Many Functions
Bibliography
Index