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 ...
, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme
with an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
by a
group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
''G'' is the affine scheme
, the
prime spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
of the
ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is,
:R^f = \.
More generally, if ''G'' is a group acting on ''R'', then the subring of ''R''
:R^G = \
is called the ...
of ''A'', and is denoted by
. A GIT quotient is a
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 ...
: any invariant morphism uniquely factors through it.
Taking
Proj
PROJ (formerly PROJ.4) is a library for performing conversions between cartographic projections. The library is based on the work of Gerald Evenden at the United States Geological Survey (USGS), but since 2019-11-26 is an Open Source Geospatial Fo ...
(of a
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
) instead of
, one obtains a projective GIT quotient (which is a quotient of the set of
semistable points.)
A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a
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 ...
, then the two notions coincide: for example, one has
:
for an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
''G'' over a field ''k'' and closed subgroup ''H''.
If ''X'' is a complex
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
and if ''G'' is a reductive
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
, then the GIT quotient of ''X'' by ''G'' is homeomorphic to the
symplectic quotient In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
of ''X'' by a
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups.
Maximal compact subgroups play an important role in the class ...
of ''G'' (
Kempf–Ness theorem
In algebraic geometry, the Kempf–Ness theorem, introduced by , gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal comp ...
).
Construction of a GIT quotient
Let ''G'' be a
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
acting on a quasi-projective scheme ''X'' over a field and ''L'' a
linearized ample line bundle on ''X''. Let
:
be the section ring. By definition, the semistable locus
is the complement of the zero set
in ''X''; in other words, it is the union of all open subsets
for global sections ''s'' of
, ''n'' large. By ampleness, each
is affine; say
and so we can form the affine GIT quotient
:
Note that
is of finite type by
Hilbert's theorem on the ring of invariants. By universal property of
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 ...
s, these affine quotients glue and result in
:
which is the GIT quotient of ''X'' with respect to ''L''. Note that if ''X'' is projective; i.e., it is the Proj of ''R'', then the quotient
is given simply as the Proj of the
ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is,
:R^f = \.
More generally, if ''G'' is a group acting on ''R'', then the subring of ''R''
:R^G = \
is called the ...
.
The most interesting case is when the stable locus
is nonempty;
is the open set of semistable points that have finite stabilizers and orbits that are closed in
. In such a case, the GIT quotient restricts to
:
which has the property: every fiber is an orbit. That is to say,
is a genuine quotient (i.e.,
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 ...
) and one writes
. Because of this, when
is nonempty, the GIT quotient
is often referred to as a "compactification" of a geometric quotient of an open subset of ''X''.
A difficult and seemingly open question is: which geometric quotient arises in the above GIT fashion? The question is of a great interest since the GIT approach produces an ''explicit'' quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following: let
be a
locally factorial algebraic variety (for example, a smooth variety) with an action of
. Suppose there are an open subset
as well as a geometric quotient
such that (1)
is an
affine morphism In algebraic geometry, a sheaf of algebras on a ringed space ''X'' is a sheaf of commutative rings on ''X'' that is also a sheaf of \mathcal_X-modules. It is quasi-coherent if it is so as a module.
When ''X'' is a scheme, just like a ring, one ca ...
and (2)
is quasi-projective. Then
for some linearlized line bundle ''L'' on ''X''. (An analogous question is to determine which subring is the ring of invariants in some manner.)
Examples
Finite group action by
A simple example of a GIT quotient is given by the
-action on