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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, used to construct
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
(or
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
and equivariant topology, and was used to construct moduli spaces of objects in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, such as instantons and
monopoles Monopole may refer to: * Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole * Monopole (mathematics), a connection over a principal bundle G with a section (the Higgs field) ...
.


Background

Invariant theory is concerned with a group action of a group on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
(or a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
) . Classical invariant theory addresses the situation when is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and is either a finite group, or one of the classical Lie groups that acts linearly on . This action induces a linear action of on the space of polynomial functions on by the formula : g\cdot f(v)=f(g^v), \quad g\in G, v\in V. The polynomial invariants of the -action on are those polynomial functions on which are fixed under the 'change of variables' due to the action of the group, so that for all in . They form a commutative
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, and this algebra is interpreted as the algebra of functions on the ' invariant theory quotient' because any one of these functions gives the same value for all points that are equivalent (that is, for all ). In the language of modern
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, : V/\!\!/G=\operatorname A=\operatorname 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 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 was the subject of
Hilbert's fourteenth problem In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that ''k'' is a field and let ''K'' be a subfield o ...
, and
Nagata Nagata is a surname which can be either of Japanese (written: 永田 or 長田) or Fijian origin. Notable people with the surname include: * Akira Nagata (born 1985), Japanese vocalist and actor * Alipate Nagata, Fijian politician * Anna Nagata (bo ...
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 is but the first step towards the complete description of , and 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 any polynomial invariant takes the same value on a given pair of points and in , yet these points are in different orbits of the -action. A simple example is provided by the multiplicative group of non-zero complex numbers that acts on an -dimensional complex vector space 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 parametrized by the points of the complex projective space . If this happens (different orbits having the same function values), one says that "invariants do not separate the orbits", and the algebra reflects the topological quotient space rather imperfectly. Indeed, the latter space, with the quotient topology, is frequently non-separated (non- Hausdorff). (This is the case in our example – the null orbit is not open because any neighborhood of the null vector contains points in all other orbits, so in the quotient topology any neighborhood of the null orbit contains all other orbits.) 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, 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.


Mumford's book

Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometry questions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford, and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniques available in examples. The abstract setting used is that of a group action on a scheme . The simple-minded idea of an
orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
:G \setminus X i.e. the quotient space of 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 relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
s should interact well with the (rather rigid) regular functions (polynomial functions), which are at the heart of algebraic geometry. The functions on the orbit space that should be considered are those on that are invariant under the action of . The direct approach can be made, by means of the function field of a variety (i.e.
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s): 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, \dots according to the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
,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 representable topologically. It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved
he third question''becomes essentially equivalent to the question of whether an orbit space of some
locally closed In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in ...
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 Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...
, but the methods of the literature had limitations. The birational point of view can afford to be careless about subsets of
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
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 mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
'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.


Stability

If a reductive group acts linearly on a vector space , then a non-zero point of is called *unstable if 0 is in the closure of its orbit, *semi-stable if 0 is not in the closure of its orbit, *stable if its orbit is closed, and its stabilizer is finite. There are equivalent ways to state these (this criterion is known as the Hilbert–Mumford criterion): *A non-zero point is unstable if and only if there is a 1-parameter subgroup of all of whose weights with respect to are positive. *A non-zero point is unstable if and only if every invariant polynomial has the same value on 0 and . *A non-zero point is semistable if and only if there is no 1-parameter subgroup of all of whose weights with respect to are positive. *A non-zero point is semistable if and only if some invariant polynomial has different values on 0 and . *A non-zero point is stable if and only if every 1-parameter subgroup of has positive (and negative) weights with respect to . *A non-zero point is stable if and only if for every not in the orbit of there is some invariant polynomial that has different values on and , and the ring of invariant polynomials has transcendence degree . A point of the corresponding projective space of is called unstable, semi-stable, or stable if it is the image of a point in with the same property. "Unstable" is the opposite of "semistable" (not "stable"). The unstable points form a Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets (possibly empty). These definitions are from and are not equivalent to the ones in the first edition of Mumford's book. Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projective space by some group action. These spaces can often be compactified by adding certain equivalence classes of semistable points. Different stable orbits correspond to different points in the quotient, but two different semistable orbits may correspond to the same point in the quotient if their closures intersect. Example: A
stable curve In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connected curve whose only singularities are ordin ...
is a reduced connected curve of genus ≥2 such that its only singularities are ordinary double points and every non-singular rational component meets the other components in at least 3 points. The moduli space of stable curves of genus is the quotient of a subset of the
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a ...
of curves in with Hilbert polynomial by the group . Example: A vector bundle over an algebraic curve (or over a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
) is a stable vector bundle if and only if :\displaystyle\frac < \frac for all proper non-zero subbundles of and is semistable if this condition holds with < replaced by ≤.


See also

*
GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring ...
* Geometric complexity theory *
Geometric quotient In algebraic geometry, a geometric quotient of an algebraic variety ''X'' with the action of an algebraic group ''G'' is a morphism of varieties \pi: X \to Y such that :(i) For each ''y'' in ''Y'', the fiber \pi^(y) is an orbit of ''G''. :(ii) The t ...
*
Categorical quotient In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism \pi: X \to Y that :(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the g ...
* Quantization commutes with reduction * K-stability * K-stability of Fano varieties *
Bridgeland stability condition In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particula ...
*
Stability (algebraic geometry) In mathematics, and especially algebraic geometry, stability is a notion which characterises when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some desirable properties for the purpose of classif ...


References

* * * Kirwan, Frances, ''Cohomology of quotients in symplectic and algebraic geometry''. Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984. i+211 pp. * Kraft, Hanspeter, ''Geometrische Methoden in der Invariantentheorie''. (German) (Geometrical methods in invariant theory) Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp. * *; (1st ed 1965); (2nd ed) * V. L. Popov, E. B. Vinberg, ''Invariant theory'', in ''Algebraic geometry''. IV. Encyclopaedia of Mathematical Sciences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994. vi+284 pp. {{ISBN, 3-540-54682-0 Moduli theory Scheme theory Algebraic groups Invariant theory