Differential Topology of Complex Surfaces

1. Front Matter

   Pages I-VII

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2. Introduction

   • John W. Morgan, Kieran G. O’Grady

   Pages 1-11

3. Unstable polynomials of algebraic surfaces

   • John W. Morgan, Kieran G. O’Grady

   Pages 12-32

4. Identification of δ_3,r (S, H) with γ₃(S)

   • John W. Morgan, Kieran G. O’Grady

   Pages 33-56

5. Certain moduli spaces for bundles on elliptic surfaces with p _g = 1

   • John W. Morgan, Kieran G. O’Grady

   Pages 57-98

6. Representatives for classes in the image of the ν-map

   • John W. Morgan, Kieran G. O’Grady

   Pages 99-111

7. The blow-up formula

   • John W. Morgan, Kieran G. O’Grady

   Pages 112-166

8. The proof of Theorem 1.1.1

   • John W. Morgan, Kieran G. O’Grady

   Pages 167-210

9. Back Matter

   Pages 211-224

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This book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donalson's polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both thebasic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduliof sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

• Blowing up
• diffeomorphism
• differential topology
• elliptic surfaces
• four-manifolfds
• moduli space

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