[内容简介]
The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients. In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map. Via the universal Kobayashi-Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces. The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles. The text is fairly self-contained (e.g., the necessary background from the theory of principal bundles is included) and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry.
[目次]
Introduction 1
1 Geometric Invariant Theory 17
1.1 Algebraic Groups and their Representations 17
1.2 Geometric Invariant Theory for Affine Varieties - A First Encounter 30
1.3 Examples from Classical Invariant Theory 34
1.4 Mumford's Geometric Invariant Theory 49
1.5 Criteria for Stability and Semistability 66
1.6 The Variation of GIT-Quotients 92
1.7 The Analysis of Unstable Points 95
2 Decorated Principal Bundles 103
2.2 Rudiments of the Theory of Vector Bundles 118
2.3 Decorated Vector Bundles: Projective Fibers 135
2.4 Principal Bundles as Swamps 174
2.5 Decorated Tuples of Vector Bundles: Projective Fibers 210
2.6 Principal Bundles as Tumps 257
2.7 Decorated Principal Bundles: Projective Fibers 266
2.8 Decorated Principal Bundles: Affine Fibers 288
2.9 More Generalizations 321
Bibliography 363
Index 379