Differential Topology of Complex Surfaces

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.