Preface

Summary

This book is based on one-semester graduate courses I gave at Michigan in 1994 and 1998, and at Harvard in 1999. A part of the book is borrowed from an earlier version of my lecture notes which were published by the Seoul National University [22]. The main changes consist of including several chapters on algebraic invariant theory, simplifying and correcting proofs, and adding more examples from classical algebraic geometry. The last Lecture of [22], which contains some applications to construction of moduli spaces, has been omitted. The book is literally intended to be a first course in the subject to motivate a beginner to study more. A new edition of D. Mumford's book Geometric Invariant Theory with appendices by J. Fogarty and F. Kirwan [74] as well as a survey article of V. Popov and E. Vinberg [90] will help the reader to navigate in this broad and old subject of mathematics. Most of the results and their proofs discussed in the present book can be found in the literature. We include some of the extensive bibliography of the subject (with no claim for completeness). The main purpose of this book is to give a short and self-contained exposition of the main ideas of the theory. The sole novelty is including many examples illustrating the dependence of the quotient on a linearization of the action as well as including some basic constructions in toric geometry as examples of torus actions on affine space. We also give many examples related to classical algebraic geometry. Each chapter ends with a set of exercises and bibliographical notes.