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-...

This book applies the recent techniques of gauge theory to study the smooth classification of compact complex surfaces. The study is divided into four main areas: Classical complex surface theory, gauge theory and Donaldson invariants, deformations of holomorphic vector bundles, and explicit calculations for elliptic sur faces. The book represents a marriage of the techniques of algebraic geometry and 4-manifold topology and gives a detailed exposition of some of the main themes in this very active area of current research.

This text is part of the IAS/Park City Mathematics series and focuses on gauge theory and the topology of four-manifolds.