STABLE AFFINE MODELS FOR ALGEBRAIC GROUP ACTIONS

June 26, 2003

Z. REICHSTEIN AND N. VONESSEN

Abstract. Let G be a reductive linear algebraic group defined over an algebraically closed base field k of characteristic zero. A G-variety is an algebraic variety with a regular action of G, defined over k. An affine G-variety is called stable if its points in general position have closed G-orbits. We give a simple necessary and sufficient condition for a G-variety to have a stable affine birational model.

1. Introduction

Let G be a linear algebraic group, defined over an algebraically closed base field k of characteristic zero. We shall refer to a reduced but not necessarily irreducible algebraic variety X (defined over k), with a regular action of G (also defined over k) as a G-variety. By a morphism X ⟶ Y of G-varieties, we shall mean a G-equivariant morphism. The notions of isomorphism, rational map, birational isomorphism, etc. of G-varieties are defined in a similar manner. As usual, given a G-action on X, we shall denote the orbit of x ∈ X by Gx and the stabilizer subgroup of x by G_{x} ⊆ G. Finally, we shall say that a property holds for x ∈ X in general position if it holds for every point x of some dense open subset of X.

In this note we will be interested in studying G-varieties up to birational isomorphism. In this context it is natural to ask whether or not a given G-variety X has an affine model. Indeed, there are numerous results and constructions in invariant theory that are available for affine G-varieties but not in general, especially if G is reductive; cf. [PV].

Recall that an affine G-variety X is called stable, if the orbit Gx is closed for x ∈ X in general position. If G is reductive, these varieties have many nice properties; for a summary, see, e.g., [V, Section 8]. The question we will address in this note is: Which G-varieties have a stable affine birational model? Our main result is the following:

2000 Mathematics Subject Classification. 14L30.

Key words and phrases. Algebraic group, group action, stable action, affine model.

Z. Reichstein was supported in part by an NSERC research grant.

N. Vonessen gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when this research was done.

Theorem 1. Let G be a reductive linear algebraic group and X be a Gvariety. Then the following are equivalent:

(a) X is birationally equivalent to a stable affine G-variety.

(b) The stabilizer G_{x} is reductive for x in general position in X.

In the case where X = G/H is a homogeneous space, Theorem 1 reduces to a theorem of Matsushima [Ma] which says that G/H is affine if and only if H is reductive. Moreover, the implication (a) ⟹ (b) of Theorem 1 is an immediate consequence of Matsushima’s theorem. Indeed, after replacing X by a stable affine model, we see that for x ∈ X in general position the orbit Gx ≃ G/G_{x} is affine, so that G_{x} is reductive.

Our proof of the implication (b) ⟹ (a) will be based on the following more general result: Theorem 2. Let G be a linear algebraic group and X a G-variety. Denote by G_{x} the stabilizer of x ∈ X in G. Assume that either

(i) G_{x} = {1} for x ∈ X in general position (i.e., the G-action on X is generically free), or

(ii) the normalizer N_{G}(G_{x}) is reductive for x ∈ X in general position. Then X is birationally isomorphic to a stable affine G-variety.

Note that if G and G_{x} are both reductive then so is the normalizer N_{G}(G_{x}); see [LR, Lemma 1]. Thus Theorem 2(ii) proves the implication (b) ⟹ (a) of Theorem 1.

The rest of this note will be devoted to proving Theorem 2. Our proof of part (ii) will be based on part (i) and a theorem of Richardson [Ri, Theorem 9.3.1] about the existence of stabilizers in general position.

We remark that the theorems of Matsushima and Richardson mentioned above were originally proved only for k = ℂ (by analytic methods). An algebraic proof of Matsushima’s theorem over an algebraically closed field k of characteristic zero can be found in [L, Section 2]. Richardson’s theorem is also valid over such k by the Lefschetz principle; it is stated in this form in [PV, Theorem 7.1]. Nevertheless, it would be interesting to find a direct algebraic proof.

2. Proof of Theorem 2(i)

We begin with a simple lemma. Lemma 3. Every linear algebraic group G has a stable generically free linear representation.

Proof. After embedding G as a closed subgroup in SL_{n} for some n ≥ 1, we may assume that G = SL_{n}. The action of SL_{n} on M_{n}(k) by left multiplication is easily seen to be generically free and stable. □

We are now ready to proceed with the proof of Theorem 2(i). Recall that a G-variety is called primitive if G transitively permutes the irreducible components of X. It is easy to see that every X is birationally isomorphic to a disjoint union of primitive G-varieties; cf. [Re, Lemma 2.2]. Hence we may assume that X is primitive.

By a theorem of Rosenlicht there exists a rational quotient map

π_{rat} : X −−→ Z ,

separating the G-orbits in general position in X; see [Ro_{1}] (for the case where X is irreducible) and [Ro_{2}] (for general X). Here Z is only defined up to birational isomorphism, so we may assume without loss of generality that Z is affine. After replacing X by a dense open G-invariant subset, we may assume that G_{x} = {1} for every x ∈ X, and that π_{rat} is regular and separates the G-orbits in X. Since X is primitive, Z is irreducible.

By Lemma 3 there exists a stable generically free linear representation V of G. Let V_{0} be a G-stable dense open subset of V such that every point v ∈ V_{0} has a closed orbit (in V) and trivial stabilizer. By [Re, Proposition 7.1] there is a G-equivariant rational map f: X −−→ V whose image contains a point v ∈ V_{0}. Let Y be the closure of the image of f × π_{rat} : X −−→ V × Z. Note that Y is G-primitive and affine. Moreover, U = Y ∩ (V_{0} × Z) is a G-invariant non-empty (and hence, dense) open subset of Y, and every point of U has a trivial stabilizer in G and a closed G-orbit in V × Z. Thus Y is a stable affine generically free G-variety.

It remains to show that f × π_{rat} is a birational isomorphism between X and Y. Since we are working in characteristic zero, since X is primitive, and since f × π_{rat} : X −−→ Y is dominant, it suffices to check that f × π_{rat} is injective on a dense open subset of X. Indeed, let W = (f × π_{rat})^{−1}(U). Then W is a G-stable nonempty (and thus dense) open subset of X. Now assume that y = (f × π_{rat})(x_{1}) = (f × π_{rat})(x_{2}) for some x_{1}, x_{2} ∈ W. Since π_{rat} separates the orbits in X, x_{2} = g(x_{1}) for some g ∈ G. But then g ∈ G_{y} = {1}. We conclude that x_{1} = x_{2}. □

3. Proof of Theorem 2(ii)

We begin with several preliminary reductions. First note that if N_{G}(G_{x}) is reductive then G_{x} itself must be reductive. Indeed, the unipotent radical R_{u}(G_{x}) is trivial, because it is a normal unipotent subgroup of N_{G}(G_{x}).

Secondly, we may assume, as we did in the previous section, that X is primitive, i.e., G transitively permutes the irreducible components of X.

Thirdly, by a theorem of Richardson (see [Ri, Theorem 9.3.1] or [PV, Theorem 7.1]), we may assume that X has a stabilizer S ⊆ G in general position. In other words, after replacing X by a G-invariant dense open subset, we may assume that G_{x} is conjugate to S for every x ∈ X. As we remarked above, S is reductive. Set N = N_{G}(S), and denote by X^{S} the set of S-fixed points in X. By comparing stabilizers, we see that

(4)  GX^{S} = X

and

(5)  if gx_{1} = x_{2} for some x_{1}, x_{2} ∈ X^{S} and g ∈ G, then g ∈ N.

Now let Y be the union of irreducible components of X^{S} of maximal dimension. Since S acts trivially on Y, we can think of Y as an N/S-variety. By our assumption G_{x} is conjugate to S for every x ∈ X. In particular, G_{x} = S for every x ∈ X^{S}. Hence, the N/S-action on Y is generically free and, by Theorem 2(i), there is a stable affine N/S-variety Z, birationally equivalent to Y.

Our goal is to show that X is birationally isomorphic to the G-variety X′ = G ∗_{N} Z. The remainder of the proof will amount to checking that X′ is affine and stable and constructing a birational isomorphism between X and X′. Some of our arguments are closely related to those in [P, 1.7]; however, for the sake of completeness (and because we are assuming that X is primitive but not necessarily irreducible), our proof will be self-contained.

First we observe that

(6)  GY is dense in X.

Indeed, consider the map f: G × X^{S} ⟶ X given by (g, x) ⟶ gx. By (4), f is surjective. By (5), the fibers of f are precisely the N-orbits in G × X^{S}, where N acts by n · (g, x) ⟶ (gn^{−1}, nx). Since this action is free (i.e., the stabilizer of every point is trivial), every fiber has the same dimension dim N, and (6) follows from the fiber dimension theorem.

Next we recall the definition of the G-variety G∗_{N} Z; cf., e.g., [PV, Section 4.8]. Consider the action of G × N on the affine variety G × Z given by

(7)  (g, n) · (h, z) ↦ (ghn^{−1}, nz) .

The variety X′ = G ∗_{N} Z is, by definition, the categorical quotient of G × Z for the N-action given by the above formula (where we identify N with the subgroup {1} × N of G × N). In particular, X′ is an affine variety. For future reference, we denote the categorical quotient map for the action of N by

π_{cat} : G × Z ⟶ X′ .

Since the actions of G and N on G × Z commute, the G-action on G × Z descends to a G-action on X′, thus giving X′ the structure of a G-variety. Theorem 2(ii) is now a consequence of the following: Lemma 8. (a) The G × N-action on G × Z, given by (7) is stable.

(b) The G-action on X′ is stable.

(c) Every N-orbit in G×Z is closed. Here we identify N with the subgroup {1} × N of G × N, and the N-action on G × Z is given by (7).

(d) X and X′ are birationally isomorphic as G-varieties.

Proof. (a) The G×N-orbit of (g, z) ∈ G×Z is G×(Nz). Since the N-action on Z is stable, this orbit is closed for z in general position in Z.

(b) G-orbits in X′ are images, under π_{cat}, of G × N-orbits in G × Z. The desired conclusion now follows from part (b) and the fact that π_{cat} maps N-invariant closed sets in G × Z to closed sets in X′; cf. [PV, p. 188, Corollary].

(c) Assume the contrary: there is a non-closed N-orbit in G × Z. Then the closure of this orbit contains an orbit of lower dimension. On the other hand, it is easy to see that the stabilizer N_{(g,z)} = {1} for every (g, z) ∈ G×Z. Consequently, every orbit has dimension dim N, a contradiction.

(d) Let ϕ: Z −−→^{≃} Y be a birational isomorphism between the N/Svarieties Z and Y. Define a G-equivariant rational map Ψ: G × Z −−→ X by Ψ(g, z) ⟶ gϕ(z). The N-action on G × Z is stable by part (c); hence, the categorical quotient map π_{cat} : G × Z ⟶ X′ separates closed orbits in G × Z; see, e.g., [PV, p. 189, Corollary]. This implies that π_{cat} is the rational quotient map for the N-variety G×Z (see, e.g, [Re, Remark 2.5]). Since Ψ sends N-orbits in G × Z to points in X, the universal property of rational quotients of N-varieties (see e.g., [Re, Remark 2.4]) says that Ψ descends to a rational map ψ: X′ −−→ X of G-varieties. We claim that ψ is a birational isomorphism.

To prove the claim, first observe that Ψ (and hence, ψ) is dominant by (6). Secondly, since the irreducible components of Y have the same dimension and the N-action on G×Z is stable and free (i.e., the stabilizer N_{(g,z)} = {1} for every (g, z) ∈ G × Z), we conclude that the irreducible components of G ∗_{N} Z are also of the same dimension (namely, of dimension, dim G + dim Y − dim N). Thus in order to show that ϕ is a birational isomorphism, we only need to check that ψ is generically one-to-one. More precisely, we will show that if z_{1} and z_{2} belong to a dense open subset of Z on which ϕ is defined and one-to-one, and if Ψ(g_{1}, z_{1}) = Ψ(g_{2}, z_{2}), then (g_{1}, z_{1}) and (g_{2}, z_{2}) lie in the same N-orbit in G × Z.

Indeed, Ψ(g_{1}, z_{1}) = Ψ(g_{2}, z_{2}) can, by definition, be rewritten as ϕ(z_{1}) = g_{1}^{−1}g_{2}ϕ(z_{2}). By (5), g_{1}^{−1}g_{2} ∈ N. Setting n = g_{1}^{−1}g_{2}, we see that (g_{1}, z_{1}) = (g_{2}n^{−1}, nz_{2}), so that (g_{1}, z_{1}) and (g_{2}, z_{2}) are, indeed, in the same N-orbit. This completes the proof of Lemma 8 and thus of Theorem 2(ii). □

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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca
URL: www.math.ubc.ca/^{∼}reichst

Department of Mathematical sciences, University of Montana, Missoula, MT 59812-0864, USA

E-mail address: Nikolaus.Vonessen@umontana.edu
URL: www.math.umt.edu/^{∼}vonessen