Introduction to Complex Hyperbolic Spaces

1. Front Matter

   Pages i-viii

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2. Preliminaries

   • Serge Lang

   Pages 1-10

3. Basic Properties

   • Serge Lang

   Pages 11-30

4. Hyperbolic Imbeddings

   • Serge Lang

   Pages 31-64

5. Brody’s Theorem

   • Serge Lang

   Pages 65-86

6. Negative Curvature on Line Bundles

   • Serge Lang

   Pages 87-123

7. Curvature on Vector Bundles

   • Serge Lang

   Pages 124-157

8. Nevanlinna Theory

   • Serge Lang

   Pages 158-183

9. Applications to Holomorphic Curves in P ⁿ

   • Serge Lang

   Pages 184-223

10. Normal Families of the Disc in P ⁿ Minus Hyperplanes

    • Serge Lang

    Pages 224-261

11. Back Matter

    Pages 263-271

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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.

• Diophantine approximation
• Finite
• Nevanlinna theory
• approximation
• boundary element method
• complex number
• curvature
• field
• function
• geometry
• theorem
• vector bundle

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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