Geometric invariant theory
Encyclopedia
In mathematics
Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry
, used to construct moduli spaces. 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
G on an algebraic variety
(or scheme
) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli space
s in algebraic geometry
as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed
interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instanton
s and monopoles
.
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
The polynomial invariant
s 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
,
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 group
s 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.
, 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 X.
The simple-minded idea of an orbit space
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 relation
s should interact well with the (rather rigid) regular function
s (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 function
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:
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 curve
s it has been known from the time of Riemann that there should be connected component
s of dimensions
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:
It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved
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 functor
s (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
.
There are equivalent ways to state these:
A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the
image of a point in V 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 by 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
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 g is the quotient of a subset of the Hilbert scheme of curves in P5g-6 with Hilbert polynomial (6n−1)(g−1) by the group PGL5g−5.
Example:
A vector bundle W over an algebraic curve
(or over a Riemann surface
) is a stable vector bundle
if and only if
for all proper non-zero subbundles V of W
and is semistable if this condition holds with < replaced by ≤.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, used to construct moduli spaces. It was developed by David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
in 1965, using ideas from the paper in classical invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
.
Geometric invariant theory studies an action of a group
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
G on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
(or scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed
interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
s and monopoles
Monopole (mathematics)
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. The connection and Higgs field should satisfy the Bogomolnyi equation and be of finite action....
.
Background
Invariant theory is concerned with a group actionGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
(or a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
) X. Classical invariant theory addresses the situation when X = V is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
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
The polynomial invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
s 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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
,
Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, 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
Hilbert's fourteenth problem
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated....
, and Nagata
Masayoshi Nagata
Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....
demonstrated that the answer was negative in general. On the other hand, in the course of development of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
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 group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
s 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
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
CPn−1. If this happens, one says that "invariants do not separate the orbits", and the algebra A reflects the topological quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
X /G rather imperfectly. Indeed, the latter space is frequently non-separated
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
. 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
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, 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 HilbertDavid Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, 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
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on a scheme X.
The simple-minded idea of an orbit space
- G\X,
i.e. the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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 relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
s should interact well with the (rather rigid) regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...
s (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
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under the action of G. The direct approach can be made, by means of the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of a variety (i.e. rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
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
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
— 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 curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s it has been known from the time of Riemann that there should be connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
s 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 HilbertHilbert schemeIn 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 , refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials...
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
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
, but the methods of the literature had limitations. The birational point of view can afford to be careless about subsets of codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
1. To have a moduli space as a scheme is on one side a question about characterising schemes as representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s (as the Grothendieck school would see it); but geometrically it is more like a compactification
Compactification
Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...
question, as the stability criteria revealed. The restriction to non-singular varieties will not lead to a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
'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
William Haboush
William Joseph Haboush is an American mathematician at the University of Illinois at Urbana-Champaign who is best known for his 1975 proof of one of David Mumford's conjectures, known as the Haboush's theorem.-References:...
.
Stability
If a reductive group G acts linearly on a vector space V, then a non-zero point of V 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:
- A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights with respect to x are positive.
- A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.
- A non-zero point x is semistable if and only if there is no 1-parameter subgroup of G all of whose weights with respect to x are positive.
- A non-zero point x is semistable if and only if some invariant polynomial has different values on 0 and x.
- A non-zero point x is stable if and only if every 1-parameter subgroup of G has positive (and negative) weights with respect to x.
- A non-zero point x is stable if and only if for every y not in the orbit of x there is some invariant polynomial that has different values on y and x, and the ring of invariant polynomials has transcendence degree dim(V)−dim(G).
A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the
image of a point in V 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 by 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
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 ordinary double points and whose automorphism group is...
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 g is the quotient of a subset of the Hilbert scheme of curves in P5g-6 with Hilbert polynomial (6n−1)(g−1) by the group PGL5g−5.
Example:
A vector bundle W over an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
(or over a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
) is a stable vector bundle
Stable vector bundle
In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. They were defined by .-Stable vector bundles over curves:...
if and only if
for all proper non-zero subbundles V of W
and is semistable if this condition holds with < replaced by ≤.