几何不变量理论
2012-8
世界图书出版公司
芒福德
292
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This edition of the book has been extended to take account of
one of these developments， one which was just hinted at in the
second edition. A close and very fruitful relationship has been
discovered between geometric invariant theory for quasi projective
complex varieties and the moment map in Symplectic geometry， and a
chapter has been added describing this relationship and some of its
applications. In an infinite-dimensional setting the moment map
links geometric invariant theory and Yang-Mills theory， which has
of course been the focus of much attention among mathematicians
over the last fifteen years.
In style this extra chapter is closer to the appendices added in
the second edition than to the original text. In particular no
proofs are given where satisfactory references exist.
Chapter 0.Preliminaries
1.Definitions
2.First properties
3.Good and bad actions
4.Further properties
5.Resume of some results of GRorrHENDIECK
Chapter 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
Chapter 2.Analysis of stability
1.A numeral criterion
2.The fiag complex
3.Applications
Chapter 3.An elementary example
1.Pre-stability
2．Stability
Chapter 4.Further examples
1.Binary quantics
2.Hypersurfaces
3.Counter-examples
4.Sequences of linear subspaces
5.The projective adjoint action
6.Space curves
Chapter 5.The problem of moduli-18t construction
1.General discussion
2.Moduli as an orbit space
3.First chern classes
4.Utilization of 4.6
Chapter 6.Abelian， schemes
1．Duals
2.Polarizations
3.Deformations
Chapter 7.The method of covan：ants-2nd construction
1.The technique
2.Moduli as an orbit space
3.The covariant
4.Application to curves
Chapter 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
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
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