Geometric invariant theory

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In mathematics, geometric invariant theory in algebraic geometry is a (technically complex) development building on nineteenth century invariant theory. It was founded by David Mumford in an eponymous book from 1965. Mumford's motivation was to develop a concrete, geometrical theory of moduli spaces of algebraic varieties. The book makes intensive use both of scheme theory and of computational techniques available in examples. The theory has been very influential, and the technical concept of stability used has been basic in much later research, for example on moduli spaces of vector bundles.

The abstract setting used is of group actions of a group G (taken to be a reductive algebraic group) on a scheme X. The simple-minded idea of an orbit space

G\X,

i.e. the quotient of X by the group action, runs into difficulties in algebraic geometry, for reasons that are explicable in abstract terms. There is in fact no general reason why equivalence relations should interact well with the (rather rigid) regular functions (polynomial functions), such as are at the heart of algebraic geometry. The functions on the orbit space G\X that should be considered are those on X that are invariant under the action of G. The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the Preface to the book:

The problem is, within the set of all models of the resulting birational class, there is one model whose geometric points classify the set of orbits in some action, or the set of algebraic objects in some moduli problem.

In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classical type — classify the big 'set' of all algebraic varieties subject only to being non-singular (and a requisite condition on polarization). The moduli are supposed to describe the parameter space. For example for curves it has been known from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9, 12, …

by the genus, and the moduli are functions on each component. In the coarse moduli problem Mumford considers the obstructions to be:

• non-separated topology on the moduli space (i.e. not enough parameters in good standing)
• infinitely many irreducible components (which isn't avoidable, but local finiteness may hold)
• failure of components to be representable as schemes, although respectable topologically.

It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved

[the third question] becomes essentially equivalent to the question of whether an orbit space of some locally closed subset of the Hilbert or Chow schemes by the projective group exists.

To deal with this he introduced a notion (in fact three) of stability. This enabled him to open up the previously treacherous area — much had been written, in particular by Francesco Severi, but the methods of the literature had limitations. The birational point of view can afford to be careless about subsets of codimension 1. To have a moduli space as a scheme is on one side a question about characterising schemes as representable functors (as the Grothendieck school would see it); but geometrically it is more like a compactification question, as the stability criteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as moduli space: varieties can degenerate to having singularities. On the other hand the points that would correspond to highly singular varieties are definitely too 'bad' to include in the answer. The correct middle ground, of points stable enough to be admitted, was isolated by Mumford's work. The concept was not entirely new, since certain aspects of it were to be found in David Hilbert's final ideas on invariant theory, before he moved on to other fields.

The book's Preface also enunciated the Mumford conjecture, later proved by Haboush.

In many contexts where both theories apply, geometric invariant theory quotients are equivalent to symplectic reduction.