Geometric invariant theory

From Wikipedia, the free encyclopedia

Geometric invariant theory, or GIT, is a modern development in mathematics 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 in algebraic geometry. The basic situation considered in GIT is an action of a group G on an algebraic variety X. It is desirable to have a good notion of a 'quotient' of X by G, and Mumford's theory provides the techniques for addressing this wish. Throughout 1970s and 1980s the theory developed in the directions of analyzing the fine geometry of the action and benefitted from interactions with symplectic geometry and equivariant topology. Geometric invariant theory was successfully applied in its turn to constructions of moduli spaces of objects of differential-geometric nature, for example, instantons and monopoles.

[edit] Background: invariant theory

Main article: Invariant theory

Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classical invariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of the classical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomial functions R(V) on V by the formula

g\cdot f(v)=f(g^{-1}v), \quad g\in G, v\in V.

The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the 'change of variables' due to the action of the group, so that g·f = f for all g in G. They form a commutative algebra A = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In the language of modern algebraic geometry,

V//G=\operatorname{Spec} A=\operatorname{Spec} R(V)^G.

Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotient to be an affine algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course of development of representation theory in the first half of the twentieth century, a large class of groups for which the answer is positive was identified; these are called reductive groups and include all finite groups and all classical groups.

The finite generation of the algebra A is but the first step towards the complete description of A, and the progress in resolving this more delicate question was rather modest. The invariants had classically been described only in a restricted range of situations, and the complexity of this description beyond the first few cases held out little hope for full understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariants f take the same value on a given pair of points u and v in V, yet these points are in different orbits of the G-action. A simple example is provided by the multiplicative group C* of non-zero complex numbers that acts on an n-dimensional complex vector space Cn by scalar multiplication. In this case, every polynomial invariant is a constant, but there are many different orbits of the action. The zero vector forms an orbit by itself, and the non-zero multiples of any non-zero vector form an orbit, so that non-zero orbits are paramatrized by the points of the complex projective space CPn−1. If this happens, one says that "invariants do not separate the orbits", and the algebra A reflects the topological quotient space X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In 1893 Hilbert formulated and proved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials. Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra, this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invariant theory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate, followed the logic of algebra rather than geometry.

[edit] Mumford's book Geometric invariant theory

Geometric invariant theory was founded and developed by Mumford in a wildly acclaimed monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some nearly forgotten results of Hilbert, to modern algebraic geometry questions. The book makes intensive use both of scheme theory and of computational techniques available in examples. The abstract setting used is that of a group action on a scheme X. The simple-minded idea of an orbit space

G\X,

i.e. the quotient space 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 algebraic curves it has been known from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9, …

according to the genus g =0, 1, 2, 3, 4, … , 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 William Haboush.

[edit] References