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 AG 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 afline 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 AG 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 L, i.e. such that some power of L defines a projective embedding of X.