Let us suppose that a complex reductive group acts on a projective variety , with a linearization (the pullback of the hyperplane bundle). Since the quotient is supposed to be described by -invariant functions on , in the projective case it makes sense to set ,
where is standard notation for global sections and .
Define a rational map by . Recall that
is finitely generated, in particular we may assume it is generated in some degree
construction is unchanged under taking the subring
); evaluate invariant sections of
to obtain a map
This gives a rational map from
, which is not defined on points for which
. This motivates the following
A point is called unstable if every vanishes on . It is called semi-stable if it is not unstable. The locus of semi-stable points in is denoted by .
Remark that this is a sensible definition because is generated in degree ; if there is one section (of any degree) not vanishing at , then there will also be a non-vanishing section of degree . Furthermore, is open in and we have a morphism .
The next definition is motivated by the following: we want to pick those orbits which are separated from nearby ones by invariant functions; morally, we want to see around which points the space of orbits embeds into the GIT quotient.
A point is stable if it is semi-stable, acts on with closed orbits (for some -invariant function s.t. ) and the stabilizer of in is finite.
The second part of this definition should mean that:
- if we restrict to an affine open inside (and trivialize ), for any two orbits and we can find -invariant functions s.t. and for ;
- this condition holds infinitesimally as well, i.e. for any vector we can find a -invariant function that vanishes on but s.t. .
Remark that also the locus of stable points is open. Unfortunately it might happen to be empty.
Let be a reductive group acting on a projective variety with a linearization of the action.
- The map is a good quotient.
- There exists an open subset s.t. and the restriction of to is a geometric quotient.
- For any , we have if and only if . A point is stable if and only if is closed in and has finite stabilizer.
The following proposition asserts that we can distinguish (semi-)stable points by looking at the orbits of their liftings in .
Let act on as above. Let and be any lifting of to . Then
- is semi-stable if and only if lies in the closure of the -orbit of .
- is stable if and only if it is semi-stable, is closed in and has finite stabilizer.
Observe that the action being linearized induces an action on .
A one-parameter subgroup (1PS) in is a homomorphism of algebraic groups , i.e. a cocharacter. Any 1PS in induces an action of on and it is clear from the above that (semi-)stable points for are (semi-)stable for as well. A non-trivial and useful result is that the behaviour under all the 1PS's determines stability under the action of the whole group.
Theorem 7.2 (Hilbert-Mumford criterion)
Let and be as above. A point is (semi-)stable under the action of if and only if it is (semi-)stable under the action of induced by every 1PS of .
The action of on can be diagonalized in some basis ; suppose that acts with weight on each . We may take a lifting of and express it in this basis, ; then . Define and . Then does not exist and exists and is not . Analogous statements hold with and . Therefore
A point is semi-stable for the action of if and only if and . It is stable iff strict inequalities hold.
It may be useful to notice that, for a 1PS , and , one has . Therefore one can change the 1PS by conjugation in whenever this makes the computations easier; in particular any 1PS can be conjugated into a chosen maximal subtorus of (e.g. diagonal matrices will work for the special linear group).
From the above it follows that the Hilbert-Mumford criterion may be expressed in the following numerical form.
Let . For any 1PS , set . Then is a fixed point for the 1PS, therefore acts on with a certain weight, say . Then is stable iff for any 1PS and semi-stable iff .
Let us go back to the example of acting on via . One possibility is to consider as acting on the trivial line bundle over but with a non-trivial character, say with weight . Then it is an exercise to check that are homogeneous polynomials of degree ; one concludes that with the line bundle - this is obvious for , for prove that homogeneous primes of a graded ring correspond one-to-one to homogeneous primes of for any fixed integer .
This can also be seen as follows: we can embed as an affine chart of , mapping to . The action of can be extended to . Show that unstable points are and all those of the form for ; all other points are stable and the quotient is .
Example 7.4 (Configurations of $n$ points in $\mathbb P^1$)
A length subscheme of is (semi-)stable for the natural action of iff each point has multiplicity ().
Consider a torus and fix a basis in which it is diagonal . Length subschemes are represented by degree homogeneous polynomials in two variables, i.e. . Write in the chosen basis. Observe that , so for unless , i.e. unless has multiplicity .
Alternatively, we could have said that tends to for the smallest s.t. ; and acts on the fiber over with weight .
Exercise Consider the case of , configurations of points in . We can see them as union of hyperplanes in the dual projective space.
Exercise Consider the action of on for . Show that semi-stable points correspond to injective homomorphisms. There are no strictly semi-stable points. The GIT quotient is .