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This standard reference on applications of invariant theory to the construction of moduli spaces is a systematic exposition of the geometric aspects of classical theory of polynomial invariants. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by Professor Frances Kirwan. It includes a fully updated bibliography of work in this area.

Geometric Invariant Theory

0. Preliminaries.- 1. Definitions.- 2. First properties.- 3. Good and bad actions.- 4. Further properties.- 5. Resumé of some results of Grothendieck.- 1. Fundamental theorems for the actions of reductive groups.- 1. Definitions.- 2. The affine case.- 3. Linearization of an invertible sheaf.- 4. The general case.- 5. Functional properties.- 2. Analysis of stability.- 1. A numeral criterion.- 2. The flag complex.- 3. Applications.- 3. An elementary example.- 1. Pre-stability.- 2. Stability.- 4. Further examples.- 1. Binary quantics.- 2. Hypersurfaces.- 3. Counter-examples.- 4. Sequences of linear subspaces.- 5. The projective adjoint action.- 6. Space curves.- 5. The problem of moduli - 1st construction.- 1. General discussion.- 2. Moduli as an orbit space.- 3. First chern classes.- 4. Utilization of 4.6.- 6. Abelian schemes.- 1. Duals.- 2. Polarizations.- 3. Deformations.- 7. The method of covariants - 2nd construction.- 1. The technique.- 2. Moduli as an orbit space.- 3. The covariant.- 4. Application to curves.- 8. The moment map.- 1. Symplectic geometry.- 2. Symplectic quotients and geometric invariant theory.- 3. Kähler and hyperkähler quotients.- 4. Singular quotients.- 5. Geometry of the moment map.- 6. The cohomology of quotients: the symplectic case.- 7. The cohomology of quotients: the algebraic case.- 8. Vector bundles and the Yang-Mills functional.- 9. Yang-Mills theory over Riemann surfaces.- Appendix to Chapter 1.- Appendix to Chapter 2.- Appendix to Chapter 3.- Appendix to Chapter 4.- Appendix to Chapter 5.- Appendix to Chapter 7.- References.- Index of definitions and notations.