Algebraic Surfaces and Holomorphic Vector Bundles

1. Front Matter

   Pages i-ix

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2. Introduction

   • Robert Friedman

   Pages 1-5

3. Curves on a Surface

   • Robert Friedman

   Pages 7-24

4. Coherent Sheaves

   • Robert Friedman

   Pages 25-58

5. Birational Geometry

   • Robert Friedman

   Pages 59-83

6. Stability

   • Robert Friedman

   Pages 85-112

7. Some Examples of Surfaces

   • Robert Friedman

   Pages 113-139

8. Vector Bundles over Ruled Surfaces

   • Robert Friedman

   Pages 141-165

9. An Introduction to Elliptic Surfaces

   • Robert Friedman

   Pages 167-195

10. Vector Bundles over Elliptic Surfaces

    • Robert Friedman

    Pages 197-243

11. Bogomolov’s Inequality and Applications

    • Robert Friedman

    Pages 245-276

12. Classification of Algebraic Surfaces and of Stable Bundles

    • Robert Friedman

    Pages 277-314

13. Back Matter

    Pages 315-328

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This book is based on courses given at Columbia University on vector bun­ dles (1988) and on the theory of algebraic surfaces (1992), as well as lectures in the Park City lIAS Mathematics Institute on 4-manifolds and Donald­ son invariants. The goal of these lectures was to acquaint researchers in 4-manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic bundles on algebraic surfaces with a view toward computing Donaldson invariants. Since that time, the focus of 4-manifold topology has shifted dramatically, at first be­ cause topological methods have largely superseded algebro-geometric meth­ ods in computing Donaldson invariants, and more importantly because of and Witten, which have greatly sim­ the new invariants defined by Seiberg plified the theory and led to proofs of the basic conjectures concerning the 4-manifold topology of algebraic surfaces. However, the study of algebraic surfaces and the moduli spaces ofbundles on them remains a fundamen­ tal problem in algebraic geometry, and I hope that this book will make this subject more accessible. Moreover, the recent applications of Seiberg­ Witten theory to symplectic 4-manifolds suggest that there is room for yet another treatment of the classification of algebraic surfaces. In particular, despite the number of excellent books concerning algebraic surfaces, I hope that the half of this book devoted to them will serve as an introduction to the subject.

• Blowing up
• differential equation
• minimum
• moduli space
• vector bundle

• Department of Mathematics, Columbia University, New York, USA

  Robert Friedman

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