4.1 Geometric invariant theory

This chapter is in the nature of a digression to discuss the formation of quotients in algebraic geometry and its rela­tion to corresponding notions in classical and quantum mechanics. In the next chapter we shall apply these ideas in an infinite-dimensional context in order to get a better under­standing of the moduli spaces discussed in the last chapter.

We begin by reviewing classical invariant theory and its geometric interpretation as developed by Mumford [21].

If A is a polynomial algebra (over C) and G is a compact group of automorphisms then the algebra A8 of invariants is finitely generated. More generally the same applies if A is replaced by a finitely generated algebra, i.e. a quotient of a polynomial algebra.

There are graded and ungraded versions of invariant theory. Geometrically these correspond to affine and projective geometry respectively. We shall be interested in the graded projective case.

If A is the graded coordinate ring of a projective variety X then its subring of invariants Ac should be the coordinate ring of some quotient projective variety. This quotient should be approximately the space of Gc-orbits in X, where Gc is the complexification of G. However, since Gc is non-compact its orbit structure can be bad and the precise nature of the quotient construction is slightly subtle. Mumford's geometric invariant theory makes this precise, and we shall now rapidly review the main features.

Abstractly we start from a smooth projective variety X with an ample line-bundle if, i.e. such that some power of if defines a projective embedding of X. We assume G (or G°) acts on X and on if. Mumford then defines a Zariski open set Xs of X consisting of stable points. The Gc-orbits in Xs are closed and the quotient space Ys = XJ Gc is a well-defined smooth quasi-projective variety. To get a natural projective compac- tification Y of Ys Mumford defines the subset Xss of semi- stable points in X, and Y is obtained from an equivalence relation on the Gc-orbits in Xss. In fact Y can be identified with the closed Gc-orbits in Xss.

In the best cases stable points have a trivial isotropy group. In such cases the line-bundle if descends naturally to a line-bundle L on Ys and then can be extended to Y. Almost equally good is the case when the isotropy groups are finite. Then, for a suitable integer k, iffc descends to give a bundle on rs and then on Y.

In the free case (trivial isotropy on Xs) a Gc-invariant section of if on Xs descends to give a section of L on Ys. Moreover sections which extend to X correspond to sections which extend to Y.