Differential Topology of Complex Surfaces: Elliptic Surfaces with p g =1: Smooth Classification

John W. Morgan, Kieran G. O’Grady (auth.)9780387566740, 0-387-56674-0

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.

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Table of contents :
Introduction….Pages 1-11
Unstable polynomials of algebraic surfaces….Pages 12-32
Identification of δ 3,r ( S, H ) with γ 3 ( S )….Pages 33-56
Certain moduli spaces for bundles on elliptic surfaces with p g = 1….Pages 57-98
Representatives for classes in the image of the ν -map….Pages 99-111
The blow-up formula….Pages 112-166
The proof of Theorem 1.1.1….Pages 167-210