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《几何不变量理论(第3版)(英文)》讲述不变量理论是数学的一个分支，它研究群在代数簇上的作用。不变量理论的古典课题是研究在线性群作用下保持不变的多项式函数。对于有限群，不变量理论与伽罗瓦理论有密切联系，一个较早的结果涉及了对称群Sn在多项式环上的作用：Sn作用下的不变量构成一个子环，由基本对称多项式生成，由于基本对称多项式彼此代数独立，此不变量环本身也同构于另一多项式环。Chevalley—Shephard—Todd定理刻划了其不变量环同构于多项式环的有限群。最近的研究则更关切算法问题，例如计算不变量环的生成元，或给出其次数的上界。
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| 目錄：
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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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