4 - Geometric Invariant Theory

Summary

These notes are based on lectures given at the CIMAT College on Vector Bundles and describe a method of constructing quotients in algebraic geometry. Geometric Invariant Theory (GIT) is due originally to Mumford [GIT], but some of the ideas go back to 19th century invariant theory, especially the work of Hilbert in the 1890s. The lecture notes are essentially unchanged from those given out when the lectures were given and are intended to be reasonably self-contained, although some proofs are omitted; further details, if needed, can be found in [Tata]. I have added an appendix describing the application of GIT to moduli problems and particularly to the construction of moduli spaces for vector bundles on algebraic curves; this material is adapted from my “Polish” notes [Pol].

The natural context for the construction is that of schemes, but for simplicity (except to a limited extent in the appendix) we work always with varieties defined over an algebraically closed field k; this field can have any characteristic. All topological terms refer to the Zariski topology.

The references are divided into two sections. The first lists works directly connected with the material described in the lectures and served as the reference list for those lectures. The second lists some other key works; it is not by any means a comprehensive list.

For the construction of quotients, we make the standard assumption that the group G is reductive.