几何不变量理论

Chapter 0.Preliminaries1.Definitions2.First properties3.Good and bad actions4.Further properties5.Resume of some results of GRorrHENDIECKChapter 1.Fundamental theorems for the actions of reductivegroups1.Definitions2.The affine case3.Linearization of an invertible sheaf4.The general case5.Functional propertiesChapter 2.Analysis of stability1.A numeral criterion2.The fiag complex3.ApplicationsChapter 3.An elementary example1.Pre-stability2．StabilityChapter 4.Further examples1.Binary quantics2.Hypersurfaces3.Counter-examples4.Sequences of linear subspaces5.The projective adjoint action6.Space curvesChapter 5.The problem of moduli-18t construction1.General discussion2.Moduli as an orbit space3.First chern classes4.Utilization of 4.6Chapter 6.Abelian， schemes1．Duals2.Polarizations3.DeformationsChapter 7.The method of covan：ants-2nd construction1.The technique2.Moduli as an orbit space3.The covariant4.Application to curvesChapter 8.The moment map1.Symplectic geometry2.Symplectic quotients and geometric invariant theory3.Kahler and hyperkahler quotients4.Singular quotients5.Geometry of the moment map6.The cohomology of quotients： the symplectic case7.The cohomology of quotients： the algebraic case8.Vector bundles and the Yang-Mills functional9.Yang-Mills theory over Riemann surfacesAppendix to Chapter 1Appendix to Chapter 2Appendix to Chapter 3Appendix to Chapter 4Appendix to Chapter 5Appendix to Chapter 7ReferencesIndex of definitions and notations

　　This edition of the book has been extended to take account ofone of these developments， one which was just hinted at in thesecond edition. A close and very fruitful relationship has beendiscovered between geometric invariant theory for quasi projectivecomplex varieties and the moment map in Symplectic geometry， and achapter has been added describing this relationship and some of itsapplications. In an infinite-dimensional setting the moment maplinks geometric invariant theory and Yang-Mills theory， which hasof course been the focus of much attention among mathematiciansover the last fifteen years.　　In style this extra chapter is closer to the appendices added inthe second edition than to the original text. In particular noproofs are given where satisfactory references exist.