Introduction to GIT
We start with some facts about algebraic groups we will need later on.
1 Algebraic groups
• Let X be a variety over K = K with char(K) = 0. • Let G be an a ne algebraic group acting on X:
σ : G × X → X with g.x := σ(g, x)
• Denote by XG := {x ∈ X | g.x = x ∀g ∈ G} the G-invariant points and by Gx := {g ∈ G | g.x = x} the stabiliser of x. • The set G.x := {g.x | ∀g ∈ G} is the orbit of x.
• Let Ψ: G × X → X × X be de ned as Ψ(g, x) = (σ(g, x), x).
• A map f : X → Y is called G-invariant if f(g.x) = f(x) ∀g ∈ G, x ∈ X.
De nition 1.1. An a ne algebraic group G over K is an a ne algebraic variety over K with a group structure such that the inversion map β : G → G and the group operation γ : G × G → G are regular maps.
Example. A torus T = (K∗)m is an a ne algebraic group. The general linear group GL(n, K) is an a ne algebraic group over K with matrix multiplication.
Remark. A ne algebraic groups can be embedded into GL(n, K) and are therefore linear algebraic groups.
De nition 1.2. A linear algebraic group G is called linearly reductive if for any rational representation ρ : G → GL(V ) and any nonzero G-invariant vector v ∈ V there exists a linear G-invariant function f on V such that f(v) 6= 0.
De nition 1.3. A linear algebraic group G is called geometrically reductive if for any rational representation ρ : G → GL(V ) and any nonzero G-invariant vector v ∈ V there exists a homogeneous G-invariant polynomial f on V such that f(v) 6= 0. Remark. Linearly reductive implies geometrically reductive.
1 Examples of geometrically reductive groups are (C∗)k and GL(n) for all k and n. Products of geometrically reductive groups are geometrically reduc- tive. For further details see [Hum75, ch. VII] and [MFK65, p. 191-193]. The reductiveness of a group gives a useful property for a group action which we will need later.
Lemma 1.1. Let G be a geometrically reductive group acting on some a ne variety X. Let Z1 and Z2 be two closed G-invariant (g.Zi ⊆ Zi for all g ∈ G) subsets of X with Z1 ∩ Z2 = ∅. Then there exists a G-invariant G funcion Ψ ∈ O(X) such that Ψ(Z1) = 1 and Ψ(Z2) = 0.
Proof.
Z1 ∩ Z2 = ∅ ⇒ (1) = I(Z1 ∩ Z2) = I(Z1) + I(Z2)
⇒ ∃f1 ∈ I(Z1), f2 ∈ I(Z2): f1 + f2 = 1
Let W ⊆ O(X) be the linear subspace spanned by {g.f2|g ∈ G} which is nite dimensional with some basis {ϕ1, ..., ϕn}. Set n . Since the are -invariant, we have f : X → A , x 7→ (ϕ1(x), ..., ϕn(x)) Zi G (g.f2)(Zi) = −1 f2(g .Zi) = f2(Zi).
⇒ ϕj(Z1) = f2(Z1) = 1 and ϕj(Z2) = f2(Z2) = 0
⇒ f(Z1) = (1, ..., 1) and f(Z2) = (0, ..., 0) is geometrically reductive homogeneous such that G ⇒ ∃F ∈ K[x1, ..., xn] F (1, .., 1) = 1 ⇒ Ψ := f ∗(F ) has the desired properties.
Just identifying points of the same orbit gives only a set-theoretical quotient. If we want to construct the quotient of a variety by the group action as a variety and the map to this quotient such that it be a morphism, we need to make the notion of quotients more precise. For this we follow [Dol03].
2 Quotients
De nition 2.1. A G-invariant map p : X → Y is called a categorical quotient (of the G action), if for every G-invariant f : X → Z there exists a unique f : Y → Z, such that f ◦ p = f. De nition 2.2.
• A G-equivariant map p : X → Y is called a good categorical quotient (of the G-action), if p satis es:
2 i) For all U ⊆ Y open, p∗ : O(U) → O(p−1(U)) is an isomorphism onto the subring O(p−1(U))G of G-invariant functions. ii) If W ⊆ X is closed and G-invariant, then p(W ) ⊆ Y is closed.
iii) If V1,V2 ⊆ X are closed, G-invariant, and V1 ∩ V2 = ∅, then p(V1) ∩ p(V2) = ∅. • A (good) categorical quotient is called a (good) geometric quotient, if Im(Ψ) = X ×Y X.
Remark. Satisfying the property for being a geometric quotient means that the preimage under p of one point in Y is exactly one G-orbit in X. Example. Let ∗ act on 2 by . Then G = C X = C \{0} g.(x1, x2) = (gx1, gx2) the map p : X → 1 , p(x , x ) = (x : x ) is a good geometric quotient, PC 1 2 1 2 because the smallest closed and G-invariant subsets are exactly the orbits and two dierent orbits are mapped to two dierent points.
Theorem 2.1. Let X be a ne, G geometrically reductive and acting on X. Then O(X)G ⊆ O(X) is a nitely generated K-algebra and the canonical map p : X → Y := Spec(O(X)G) is a good categorical quotient.
Proof. Let R := O(X). Then RG is nitely generated because of Nagata's theorem [Dol03, p. 41-45]. i) We show it for basic open sets D(f) ⊆ Y for f ∈ RG. g ∈ (RG) ⇔ g ∈ RG f n f g ⇔ G-invariant f n g ⇔ ∈ (R )G f n f
G G ⇒ (R )f = (Rf ) ii) Let W ⊆ X be closed and G-invariant and assume that p(W ) is not closed. ∃y ∈ p(W ) \ p(W ) with Z := p−1(y) ⊆ X closed, G-invariant, and Z ∩ W = ∅.
lemma 1.1 ⇒ ∃Φ ∈ O(X)G : Φ(W ) = 1 and Φ(Z) = 0 p∗ isomorphism ⇒ Φ = p∗(φ) for some φ ∈ O(Y ) ⇒ φ(y) = 0 and φ(p(W )) = 1, which is a contradiction.
iii) Let V1,V2 ∈ X closed, G-invariant, and V1 ∩ V2 = ∅.
lemma 1.1 ⇒ ∃φ ∈ O(Y ): φ(p(V1)) = 1 and φ(p(V2)) = 0
⇒ p(V1) ∩ p(V2) = ∅
3 Remark. If G is not reductive but O(X)G is nitely generated, then theorem 2.1 still gives a categorical quotient.
Example. Let the additive group act on 2 by C C g.(c1, c2) = (c1, c2 + gc1) with and 2. This corresponds to an action of on g ∈ C (c1, c2) ∈ C C C[x1, x2] by . Thus the invariant functions are C g.p(x1, x2) = p(x1, x2 +gx1) C[x1, x2] = and therefore the projection onto the rst coordinate is a categorical C[x1] quotient, but not a good one, because (0, 0) and (0, 1) are two closed C- invariant points but are projected onto the same point.
We are now able to construct good categorical quotients of a ne varieties, so we will approach projective varieties. For projective X we will embed X,→ Pn. This is done by choosing a (very ample) line bundle L over X. The idea is then to cover the image of X with open, a ne, G-invariant subsets. To do so, we need an extension of the G-action on X to a G-action on Pn.
3 Linearisation and stability
De nition 3.1. Let L be a line bundle on X with projection π : L → X.A G-linearisation of L is an extension of the action σ on X to an action σ on L, such that
σ G × L / L
i) id×π π commutes. σ G × X / X ii) The zero-section is G-invariant.
Remark. Every line bundle L on X is associated to an invertible sheaf F on X (see [Har77, exercise 5.18]). So in terms of invertible sheafs the G- linearisation σ on L is the same as an isomorphism
∗ ∗ Φ: σ F→˜ pr2F satisfying the co-cycle condition (see [MFK65, de nition 1.6]).
Example 3.1. Let L := X×C be the trivial line bundle. Then a linearisation can be given by choosing a character χ ∈ Hom(G, C∗). An action σ is then given by σ(g, (x, z)) := (σ(g, x), χ(g)z) for g ∈ G and (x, z) ∈ X × C. Therefore a G-linearisation is not unique.
4 Remark. Let V := Γ(X,L) be the global sections of L. Then the linearisation induces an action on V: (g.s)(x) = σ(g, s(σ(g−1, x))) = g.(s(g−1.x)) (1) for s ∈ V, g ∈ G, and x ∈ X. The next problem which occurs is that a cover of open, a ne, and G-invariant subsets of X,→ Pn where we can nd good categorical quotients need not exist, or we do not get a good categorical quotient by gluing them together.
Example 3.2. Let C∗ act on P1 by with ∗ and 1 t.(z1 : z2) = (z1 : tz2) t ∈ C (z1 : z2) ∈ P . This action has three orbits {(1 : t), (1 : 0), (0 : 1)}. To be a good categorical quotient a morphism from P1 to some Y has to take the orbit (1 : t) to one point. As the other two points are in the closure of this orbit, they have to go to the same point. Therefore the only possible quotient is p : P1 7→ pt, which does not satisfy property iii) of Def 2.2. This means that no good categorical quotient exists.
The solution is to take an open G-invariant subset which has a good categorical quo- tient. De nition 3.2. Let L be a G-linearised line bundle on X. We de ne ss ⊗d G i) X (L) := {x ∈ X | ∃d > 0, ∃f ∈ Γ(X,L ) : f(x) 6= 0 ∧ Xf is a ne} the set of semistable points with respect to L;
s ii) X (L) := {x ∈ X | ∃f as in i): G.y ⊆ Xf closed ∀y ∈ Xf ∧Gxis nite} the set of stable points with respect to L; iii) Xus(L) := X \ Xss(L) the set of unstable points with respect to L; where Xf = {x ∈ X | f(x) 6= 0}. Remark. • Taking L0 := L⊗n instead of L does not change the set of semistable ⊗d G points and the set of stable points. If f ∈ Γ(X,L ) has an a ne Xf , 0 n 0⊗d G then f := f ∈ Γ(X,L ) and Xf n = Xf . • If G is irreducible and X quasi-projective and normal, then there exists a G-linearised line bundle.
In example 3.2 we could take O(1) with the linearisation t.x1 = x1 and t.x2 = tx2. Then ss , which means that we take out the point . Now property X (O(1)) = Xx1 (0 : 1) iii) of de nition 2.2 is satis ed and hence p : Xss(O(1)) → pt is a good categorical quotient.
5 4 Construction of the GIT quotient
With the de ntion of the open subset of semistable points we can now construct the so- called GIT (geometric invariant theory) quotient. Theorem 4.1 (Mumford). Let X be irreducible and L a G-linearised line bundle. There exists a good categorical quotient
ss ss π : X (L) → X (L) G. There is an open subset ss such that −1 s and U ⊆ X (L) G π (U) = X (L) is a good geometric quotient. Furthermore ss is a quasi- π|π−1(U) X (L) G projective variety.
Proof. The idea of the proof is to cover the semistable points with a nite number of open sets with ⊗di G for and (open subsets of are Xsi si ∈ Γ(X,L ) i = 1, ..., r di > 0 X quasi-compact). The are a ne and we can apply theorem 2.1 to get good categorical Xsi quotients G π : Xsi 7→ Spec(O(Xsi ) ) i = 1, ..., r.
These can be glued via open subsets de ned by si/sj to get a good categorical quotient. For the complete proof see [Dol03, p. 118-120] or [MFK65, p. 38-39]. Proposition 4.1 (Mumford). Let X be projective and L be ample. Let M R = Γ(X,L⊗d). d≥0 Then ss ∼ G . X (L) G = Proj(R )
Proof. As in theorem 2.1, by Nagata's theorem [Dol03, p. 41-45] RG is nitely generated. Take a set of generators s0, ..., sn . Without loss of generality si ∈ Γ(X,L) otherwise we take ⊗n instead of . Then the are an a ne cover of ss . As in the proof of L L Xsi X (L) theorem 4.1 we can take the G and glue them together to get the good Yi := Spec(O(Xsi ) ) categorical quotient ss . On the other hand take the surjective map Y = X (L) G G (2) K[T0, ..., Tn] → R ,Ti 7→ si with the kernel . Then G ∼ n and the open subsets I ⊆ K[T0, ..., Tn] Z := Proj(R ) = V (I) ⊆ P Zi := Z ∩ D(Ti) cover Z. From the proof of theorem 2.1 and equation (2) it follows that ∼ ∼ G ∼ G ∼ G O(Zi) = (K[T0, ..., Tn]/I)(Ti) = (R )(si) = (R(si)) = O(Xsi ) . This gives maps which are categorical quotients as well. By the uniqueness fi : Xsi → Zi ∼ of the good categorical quotient we have that Zi = Yi for all i and as the Zi glue in the same way as the Yi it follows that G ∼ ss Proj(R ) = Z = Y = X (L) G.
6 5 Hilbert-Mumford-criterion
Let X ⊂ Pn via a very ample G-linearised line bundle L. We denote by x∗ ∈ Kn+1 a representative of x ∈ X. Then x is unstable i 0 ∈ G.x∗.A one-parameter subgroup of G, , acts in appropiate coordinates λ : Gm → G by the formula ∗ m0 mn λ(t).x = (t x0, ..., t xn). ∗ If all mi for which xi 6= 0 are strictly positive then 0 ∈ G.x and if all such −1 ∗ mi are strictly negative, take λ to see that 0 ∈ G.x . De ne µ(x, λ) := min(mi | xi 6= 0). Thus we get x ∈ Xss(L) ⇒ µ(x, λ) ≤ 0 ∀λ ∈ χ(G)∗.
ss If x ∈ X (L) and ∃λ : µ(x, λ) = 0 set I := {i | xi 6= 0, mi > 0} and x i∈ / I de ne y = (y , ..., y ) by y = i . Then y ∈ λ( ).x ⊆ G.x and 0 n i 0 i ∈ I Gm . Thus either and hence is not closed and therefore λ(Gm) ⊆ Gy y∈ / G.x G.x x is not stable. Or y ∈ G.x and thus in every Xf that contains x and has a non- nite stabiliser and hence x is again not stable. Therefore we get x ∈ Xs(L) ⇒ µ(x, λ) < 0 ∀λ ∈ χ(G)∗. Remark. One can show that µ(x, λ) is independent of the choice of coordi- nates. Theorem 5.1. Let G be a reductive group acting on a projective variety X. Let L be an ample G-linearised line bundle on X and let x ∈ X. Then x ∈ Xss(L) ⇔ µ(x, λ) ≤ 0 for all λ ∈ χ(G)∗, x ∈ Xs(L) ⇔ µ(x, λ) < 0 for all λ ∈ χ(G)∗. Proof. See [Dol03, 9.1 - 9.3].
References
[Dol03] Igor V. Dolgachev. Lectures on Invariant Theory. Cambridge Uni- versity Press, August 2003. [Har77] Robin Hartshorne. Algebraic Geometry. Springer, 1977. [Hum75] James E. Humphreys. Linear Algebraic Groups. Springer, 1975. [MFK65] David Mumford, James Fogarty, and Frances Kirwan. Geometric Invariant Theory. Springer, 1965.
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