Geometric Invariant Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (34))

"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants. Ch. 0. Preliminaries. 1. Definitions. 2. First Properties. 3. Good And Bad Actions. 4. Further Properties. 5. Resumé Of Some Results Of Grothendieck -- Ch. 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 -- Ch. 2. Analysis Of Stability. 1. A Numeral Criterion. 2. The Flag Complex. 3. Applications -- Ch. 3. An Elementary Example. 1. Pre-stability. 2. Stability -- Ch. 4. Further Examples. 1. Binary Quantics. 2. Hypersurfaces. 3. Counter-examples. 4. Sequences Of Linear Subspaces. 5. The Projective Adjoint Action. 6. Space Curves -- Ch. 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 -- Ch. 6. Abelian Schemes. 1. Duals. 2. Polarizations. 3. Deformations -- Ch. 7. The Method Of Covariants -- 2nd Construction. 1. The Technique. 2. Moduli As An Orbit Space. 3. The Covariant. 4. Application To Curves -- Ch. 8. The Moment Map. 1. Symplectic Geometry. 2. Symplectic Quotients And Geometric Invariant Theory. 3. Kahler And Hyperkahler 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. D. Mumford, J. Fogarty, F. Kirwan. Includes Bibliographical References (p. [253]-290) And Index. 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.