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Chapter 0．Preliminaries 1

1．Definitions 2

2．First properties 4

3．Good and bad actions 9

4．Further properties 13

5．Resum6 of some results of GROTHENDIECK 19

Chapter 1．Fundamental theorems for the actions of reductive groups 24

1．Definitions 24

2．The affine case 27

3．Linearization of an invertible sheaf 30

4．The general case 36

5．Functional properties 44

Chapter 2．Analysis of stability 48

1．A numeral criterion 48

2．The flag complex 55

3．Applications 63

Chapter 3．An elementary example 67

1．Pre-stability 67

2．Stability 72

Chapter 4．Further examples 76

1．Binary quantics 76

2．Hypersurfaces 79

3．Counter-examples 83

4．Sequences of linear subspaces 86

5．The projective adjoint action 88

6．Space curves 89

Chapter 5．The problem of moduli—1st construction 96

1．General discussion 96

2．Moduli as an orbit space 98

3．First chern classes 104

4．Utilization of 4.6 109

Chapter 6．Abelian schemes 115

1．Duals 115

2．Polarizations 120

3．Deformations 124

Chapter 7．The method of covariants一2nd construction 127

1．The technique 127

2．Moduli as an orbit space 129

3．The covariant 138

4．Application to curves 142

Chapter 8．The moment map 144

1．Symplectic geometry 144

2．Symplectic quotients and geometric invariant theory 148

3．K？hler and hyperk？hler quotients 152

4．Singular quotients 156

5．Geometry of the moment map 160

6．The cohomology of quotients:the symplectic case 164

7．The cohomology of quotients:the algebraic case 172

8．Vector bundles and the Yang-Mills functional 181

9．Yang-Mills theory over Riemann surfaces 185

Appendix to Chapter 1 191

Appendix to Chapter 2 202

Appendix to Chapter 3 205

Appendix to Chapter 4 206

Appendix to Chapter 5 217

Appendix to Chapter 7 234

References 253

Index of definitions and notations 291