ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 63–71 (July 31, 1997) S 1079-6762(97)00025-5

ℂ^{∗}-ACTIONS ON ℂ^{3} ARE LINEARIZABLE

S. KALIMAN, M. KORAS, L. MAKAR-LIMANOV, AND P. RUSSELL

(Communicated by Hyman Bass)

Abstract. We give the outline of the proof of the linearization conjecture: every algebraic ℂ^{∗}-action on ℂ^{3} is linear in a suitable coordinate system.

1. Introduction

The purpose of this note is to outline the main ingredients in a proof of the following

Linearization Theorem. Every algebraic action of the torus T = ℂ^{∗} on affine space X = ℂ^{3} is linearizable, that is linear in suitably chosen coordinates for X.

It is known that the action has a fixpoint 0 ∈ X ([B-B]). The weights of the action are the weights

a_{1}, a_{2}, a_{3}

of the (diagonalized) action on the tangent space T_{0}X. (They are independent of the choice of fixpoint [KbR].) We will assume tacitly that the action is effective, or, equivalently, that GCD(a_{1}, a_{2}, a_{3}) = 1. Put

δ = dim X//T, τ = dim X^{T} .

Then 2 ≥ δ ≥ τ ≥ 0.

It is known that fixpointed actions, that is those for which all weights have the same sign, are linearizable [KbR]. This settles the following cases:

δ = 0 = τ, or three nonzero weights of the same sign;

δ = 1 = τ, or one zero weight, two nonzero weights of the same sign; δ = 2 = τ, or two zero weights, one nonzero weight.

The case

δ = 2, τ = 1, or one zero weight, two nonzero weights of opposite sign, was settled in [KR1].

It remains to consider the

Hyperbolic Case. δ = 2, τ = 0, or three nonzero weights, not all of the same sign.

A program to settle this case was proposed in [KR2]. It has two quite distinct components.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors March 5, 1997.

1991 Mathematics Subject Classification. Primary 14L30.

The first author was partially supported by an NSA grant.

Step I, the quotient. Show that X//T is as expected for a linear action, i.e.

X//T ≃ T_{0}X//T.

Let ω_{α} ⊂ ℂ^{∗} be the group of α-roots of 1. Linearizability follows from Step I (see 1.4 below) in the case dim X^{w_{α}} ≤ 1 for all α > 1, or equivalently, if the weights are pairwise relatively prime. This leads to

Step II, reduction of weights. Reduction of the proof to the case of pairwise relatively prime weights.

If α > 1 and α divides two weights, then X′ = X/ω_{α} is a smooth, affine threefold, but only after linearizability has been established is it at all clear that X′ ≃ ℂ^{3}. We are therefore led to study more general ℂ^{∗}-threefolds.

1.1. Standard conditions. Let X be a ℂ^{∗}-threefold. We consider the following conditions.

(i) X is smooth and the action of T = ℂ^{∗} is hyperbolic,

i.e. there is a unique fixpoint 0 and dim X//T = 2.

(ii) X is contractible.

(iii) \overline{κ}(X) = −∞ (\overline{κ} = logarithmic Kodaira dimension).

If we have 1.1 (i), the weights of the action are defined as above for X = ℂ^{3}, and we assume

a_{1} < 0, a_{2} > 0, a_{3} > 0, GCD(a_{1}, a_{2}, a_{3}) = 1.

We put

α_{i} = GCD({a_{1}, a_{2}, a_{3}} − {a_{i}}).

Then

−a_{1} = aα_{2}α_{3}, a_{2} = bα_{1}α_{3}, a_{3} = cα_{1}α_{2}

with a, b, c > 0 and reduced (pairwise relatively prime).

1.2. Proposition ([KR3], 2.5). Let X satisfy 1.1 (i).

(i) Suppose α_{i} > 1. Then dim X^{ω_{α_{i}}} = 2 and

X′ = X/ω_{α_{i}}

satisfies 1.1 (i) for T′ = T/ω_{α_{i}} ≃ ℂ^{∗} with weights a_{i} and a_{j}/α_{i} for j ≠ i.

(ii) X^{#} = X/ω_{α_{1}α_{2}α_{3}} satisfies 1.1 (i) for T^{#} = T/ω_{α_{1}α_{2}α_{3}} ≃ ℂ^{∗} and reduced weights −a, b, c.

(iii) X//T = X′//T′ = X^{#}//T^{#}.

(iv) If X satisfies 1.1 (ii) or (iii), then so do X′ and X^{#}.

Let X satisfy 1.1 (i) and (ii). We put ([KR3], 1.4)

X^{+} = {x ∈ X| lim_{t→0} t · x = 0}.

Then X^{+} ≃ ℂ^{2} and X^{+} = F^{−1}(0), where F is semiinvariant of weight a_{1}. ω_{a_{1}} acts on

X_{1} = F^{−1}(1)

and we have ([KR1], Lemma 2)

ℂ^{∗}-ACTIONS ON ℂ^{3} ARE LINEARIZABLE 65

1.3. X//T ≃ X_{1}/w_{a_{1}}.

The reduction of the proof to Steps I and II is now contained in

1.4. Proposition ([KR3], 2.3, 2.8, and 1.10). Let X satisfy 1.1 (i) and (ii) and suppose the weights are reduced. If

X//T ≃ T_{0}X//T,

or equivalently

X_{1}/ω_{a} ≃ ℂ^{2}/ω_{a},

where ω_{a} acts diagonally on ℂ^{2} with weights ≡ b, c mod a, then

X_{1} ≃ ℂ^{2},

and

X ≃_{e} ℂ^{3}

(X is equivariantly isomorphic to ℂ^{3} = T_{0}X).

2. The quotient

2.1. Theorem ([KR4], 1.2). Suppose X satisfies all conditions of 1.1. Then

S′ = X//ℂ^{∗} ≃ T_{0}X//ℂ^{∗}.

By 1.2, we may assume the weights are reduced when studying the quotient. Also, 2.1 is known ([KR2]) when S′ is smooth, or equivalently a = 1. So we assume a > 1. Then by [KR4], 2.4

2.2. S′ is contractible, \overline{κ}(S′) = −∞, S′ has a unique singular point q, q is analytically of the type of the origin in ℂ^{2}/ω_{a}, and Pic(S′ − q) ≃ ℤ/aℤ.

2.3. Theorem ([K]). If S′ is as in 2.2, then (i) if \overline{κ}(S′ − q) = −∞, then S′ ≃ ℂ^{2}/ω_{a}, (ii) \overline{κ}(S′ − q) ≠ 0, 1.

It remains to rule out \overline{κ}(S′ − q) = 2 to complete Step I.

2.4. Theorem ([KR4], 1.1). Let

S′ = X//T

with X satisfying all conditions of 1.1. Then

\overline{κ}(S′ − q) < 2.

The proof is rather involved. It relies in a crucial way on the theory of open algebraic surfaces, in particular the inequalities of Miyaoka [M] and Kobayashi [Ko] and the results on the existence of affine rulings of Miyanishi and Tsunoda [MT].

2.5. Proposition ([KR4], 2.8). Let S′ be as in 2.4. There exists a desingularization S of S′ admitting an 𝔸^{1}-ruling with all but one component E of the exceptional locus Ê in fibres. Moreover, S − Δ is simply connected, where Δ = Ê − E.

The proof of 2.4 proceeds by a detailed analysis of such “good” rulings under the conditions of 2.2.

3. Reduction of weights and “exotic affine spaces”

Step II, the reduction of weights, is achieved in a roundabout way. In [KR3], an explicit construction is given of a class of smooth, contractible ℂ^{∗}-threefolds that encompasses, in the equivariant sense, all possible counterexamples to linearization. It is then shown in [KM-L] that only the “obviously” equivariantly trivial threefolds in the class are isomorphic to ℂ^{3} (without reference to the ℂ^{∗}-action). The others are in themselves interesting examples of “exotic affine spaces” (algebraic varieties homeomorphic to ℂ^{3}). They include the threefolds described in [D], 4.36.

3.1. Theorem ([KR3], 4.1). The threefolds

X = Spec A

satisfying 1.1 (i) and (ii) and

X//ℂ^{∗} ≃ T_{0}X//ℂ^{∗}

are precisely the ones obtained as follows.

(1) Let

−a = a′_{1}, b = a′_{2}, c = a′_{3}

be a triple of reduced weights with a, b, c > 0. (These define a hyperbolic ℂ^{∗}-action on

W = Spec B ≃ ℂ^{3}

with B = ℂ[η, ξ, ζ] and η, ξ, ζ homogeneous of weight −a, b, c).

(2) Let

α_{1}, α_{2}, α_{3}

be a reduced triple of positive integers with GCD(α_{i}, a′_{i}) = 1, i = 1, 2, 3.

(3) Let C_{2} and C_{3} be ω_{a}-homogeneous “lines” (curves isomorphic to ℂ) in W_{1} = Spec k[ξ, ζ] ≃ ℂ^{2}, identified with η^{−1}(1) ⊂ W, such that

(i) C_{2} and C_{3} meet normally in r ≥ 1 points, including the origin,

(ii) U_{i} = \overline{ℂ^{∗} · C_{i}} ⊂ W is smooth, i = 2, 3.

(4) Let U_{1} = W^{+} = η^{−1}(0).

Then X is the “tri-cyclic” cover of W ramified to order α_{i} over U_{i}, i = 1, 2, 3, that is,

A = B[z_{1}, z_{2}, z_{3}],

where z_{i}^{α_{i}} = u_{i} with u_{1} = η and for i = 2, 3, u_{i} is an equation for U_{i} and uniquely determined by

s^{−a′_{i}}f_{i}(ξs^{a′_{2}}, ζs^{a′_{3}}) = u_{i}(s^{−a′_{1}}, ξ, ζ),

where f_{i} is an equation for C_{i} ⊂ W_{1}.

Moreover,

B = ℂ[u_{1}, u_{2}, u_{3}^{∗}] = ℂ[u_{1}, u_{2}^{∗}, u_{3}],

with u_{i} and u_{i}^{∗} homogeneous of weight a′_{i} and if

u_{2} = G_{2}(u_{1}, u_{2}^{∗}, u_{3}) and u_{3} = G_{3}(u_{1}, u_{2}, u_{3}^{∗}),

then the equations

z_{2}^{α_{2}} = G_{2}(z_{1}^{α_{1}}, z_{2}^{∗}, z_{2}^{α_{3}}) and z_{3}^{α_{3}} = G_{3}(z_{1}^{α_{1}}, z_{2}^{α_{2}}, z_{3}^{∗})

describe X (in two ways) as a hypersurface in ℂ^{4}.

3.1.1. Remark. 1) (3)(ii) imposes a rather mild restriction that can be made quite explicit ([KR3], 1.11.1).

2) Possibilities for f_{2}, f_{3}, and hence for G_{2}, G_{3}, can be worked out explicitly with the help of the epimorphism theorem of Abhyankar, Moh and Suzuki [AM], [S].

The key to 3.1 is the following observation.

3.2. Proposition ([KR3], 2.6, 2.7). Suppose X is as in 3.1 and α_{2} = 1, α_{3} > 1. Then

X/ω_{α_{3}} ≃_{e} ℂ^{3} implies X ≃_{e} ℂ^{3}.

A similar result holds if α_{2} > 1, α_{3} = 1. Also,

X/ω_{α_{1}} ≃_{e} ℂ^{3} implies X ≃_{e} ℂ^{3}.

In view of 1.4 we obtain a commutative diagram X

↙  ↘
ℂ^{3} ≃_{e} X/ω_{α_{2}}  X/ω_{α_{3}} ≃_{e} ℂ^{3}
↘  ↙
X/ω_{α_{2}α_{3}} ≃_{e} ℂ^{3}
↓
X/ω_{α_{1}α_{2}α_{3}} ≃_{e} ℂ^{3}

3.1 is an elaboration of the possibilities for such a diagram. It is not difficult to decide when X is equivariantly isomorphic to ℂ^{3} (see 3.4). The question of just isomorphism with ℂ^{3}, on the other hand, proved to be much more elusive.

3.3. Let us for instance choose a = b = c = 1, α_{2} = 2 and α_{3} = 3 and a parabola and straight line for C_{2} and C_{3}. Then in suitable coordinates X is defined in ℂ^{4} by

x + x^{2}y + z^{2} + t^{3} = 0.

X is dominated birationally by ℂ^{3} and there exists a surjective quasi-finite map ℂ^{3} → X ([KR3], 7.7 and 7.8). It is shown in [M-L1] that, nevertheless, X is not isomorphic to ℂ^{3}. The proof is based implicitly on the computation of the following invariant:

AK(X) = ⋂_{∂∈LND(X)} Ker ∂

where LND(X) is the set of locally nilpotent derivations on the ring ℂ[X] of regular functions on X. For this hypersurface AK(X) ≠ ℂ, but clearly AK(ℂ^{3}) = ℂ.

Let X be as in 3.1. We define

ε = (r − 1)(α_{2} − 1)(α_{3} − 1)

(ε = rank π_{2}(X − X^{+}) is an invariant of the higher-dimensional knot (X, X^{+}) ([KR3], 4.8)).

3.4. Theorem ([KR3], remark following 5.1). Let X be as in 3.1. Then X ≃_{e} ℂ^{3} if and only if ε = 0.

3.5. Theorem ([KM-L]). Let X be as in 3.1. If ε > 0, then X ≄ ℂ^{3}.

If now X is ℂ^{3} with a hyperbolic ℂ^{∗}-action, then by 2.1 it is one of the X in 3.1 and hence X ≃_{e} ℂ^{3} by 3.4 and 3.5.

4. The computation of AK(X)

4.1. Theorem 3.5 is again the consequence of the fact that AK(X) ≠ ℂ [KM-L]. More precisely, AK(X) = ℂ[X] unless X is isomorphic to a hypersurface in ℂ^{4} given by one of the following equations:

\begin{matrix}(i) & x + x^{k}y + z^{α_{2}} + t^{α_{3}} = 0 or \\ (ii) & x + y(x^{k} + z^{α_{2}})^{l} + t^{α_{3}} = 0 \end{matrix}

where k ≥ 2, l ≥ 1, and in the second equation (kl, α_{3}) = 1. In case (i) AK(X) is the restriction of ℂ[x] to X and in case (ii) AK(X) is the restriction of ℂ[x, z] to X.

4.2. The scheme of the computation of AK(X) is discussed below. Every X from 3.1 is the hypersurface P (x, y, z, t) = 0 where

(x, y, z, t) = (z_{3}^{∗}, z_{1}, z_{2}, z_{3}) and P (x, y, z, t) = t^{α_{3}} − G_{3}(y^{α_{1}}, z^{α_{2}}, x).

The polynomials from 4.1 (i) and (ii) are examples of such P. A derivation ∂ on ℂ[X] is said to be of Jacobian type if ∂(f) coincides with the restriction of J_{x,y,z,t}(P, φ_{1}, φ_{2}, φ) to X where φ_{1}, φ_{2} ∈ ℂ[x, y, z, t] are fixed and the restriction of φ ∈ ℂ[x, y, z, t] to X coincides with f ∈ ℂ[X].

4.3. Proposition ([KM-L]). Let δ ∈ LND(X) be nontrivial and let φ_{1}, φ_{2} be such that φ_{1}|_{X}, φ_{2}|_{X} ∈ Ker δ and P, φ_{1}, φ_{2} are algebraically independent. Then ∂ has the same kernel as δ.

Furthermore, since the transcendence degree of the field of fractions of Ker δ is 2 [M-L1], one can always find φ_{1}, φ_{2}, and therefore ∂ as above.

4.4. We consider degree functions L on ℂ[x, y, z, t] obtained by assigning real weights to the variables. The L-quasi-leading part φ^{L} of a nonzero polynomial φ is the sum of the terms from φ whose L-degree coincides with L(φ). Suppose, given φ_{1}, there exists a degree function L_{1} with positive values such that for any other degree function L_{2} with positive values each nonzero monomial from φ^{L_{2}} is also present in φ^{L_{1}}. We then call

φ̂ := φ^{L_{1}}

the quasi-leading part of φ. In cases 4.1 (i) and (ii) P̂ coincides with x^{k}y +z^{α_{2}} +t^{α_{3}} and y(x^{k} + z^{α_{2}})^{l} + t^{α_{3}} respectively. In all other cases P̂ also exists and can be computed explicitly by virtue of the Abhyankar-Moh-Suzuki theorem (see 3.1.1 (ii)). Consider further only those degree functions (may be with negative values) which satisfy the condition

P^{L} = P̂.

4.5. Proposition ([KM-L]). Let ∂ be a nontrivial locally nilpotent derivation of Jacobian type on ℂ[X]. Then polynomials φ_{1}, φ_{2} can be chosen so that φ_{1}|_{X}, φ_{2}|_{X} ∈ Ker ∂ and P̂, φ_{1}^{L}, φ_{2}^{L} are algebraically independent.

4.6. With ∂ as in 4.5, suppose that X̂ is the hypersurface P̂(x, y, z, t) = 0 in ℂ^{4} and that ∂^{L} is the derivation on ℂ[X̂] such that ∂^{L}(f) coincides with restriction of J_{x,y,z,t}(P̂, φ_{1}^{L}, φ_{2}^{L}, φ) to X̂, where the restriction of φ ∈ ℂ[x, y, z, t] to X̂ coincides with f ∈ ℂ[X̂]. Then ∂^{L} is nontrivial and also locally nilpotent [M-L1].

4.7. Since P̂ is known explicitly, we can find all nontrivial locally nilpotent derivations of Jacobian type on ℂ[X̂]. If P is not as in 4.1 (i) or (ii) there are no such derivations. By 4.3 and 4.6 there is no nontrivial locally nilpotent derivation on ℂ[X], that is, AK(X) = ℂ[X].

4.8. In case 4.1 (ii) the kernel of any nontrivial locally nilpotent derivation ∂^{L} on ℂ[X̂] is contained in ℂ[x, z]|X̂. Since this is true for every L satisfying condition 4.4, it follows ([KM-L], Theorem 8.4) that the kernel of the corresponding nontrivial locally nilpotent derivation ∂ on ℂ[X] is contained in ℂ[x, z]|_{X}. The transcendence degree of the field of fractions of Ker ∂ is 2 and Ker ∂ is algebraically closed in ℂ[X] [M-L1]. Hence Ker ∂ = ℂ[x, z]. In case 4.1 (ii) nontrivial locally nilpotent derivations on ℂ[X] exist, for instance J_{x,y,z,t}(P, x, z, φ)|_{X}. This yields AK(X) = ℂ[x, z]|_{X}.

4.9. In case 4.1 (i) nontrivial locally nilpotent derivations on ℂ[X] exist as well. Examples are J_{x,y,z,t}(P, x, z, φ)|_{X} and J_{x,y,z,t}(P, x, t, φ)|_{X}. The intersection of the kernels of these locally nilpotent derivations is ℂ[x]|_{X}, whence it suffices to show that x ∈ Ker ∂ for every nontrivial ∂ ∈ LND(X). It can be shown that Ker ∂^{L} ⊂ ℂ[x, z, t]|X̂ [KM-L]. Varying L under the condition 4.4 we prove that Ker ∂ ⊂ ℂ[x, z, t]|_{X}. From this we deduce that ∂(ℂ[x, z, t]|_{X}) ⊂ x^{k}ℂ[x, z, t]|_{X} with k as in 4.1. Since ℂ[x, z, t]|_{X} ⊄ Ker ∂, there exists f ∈ Ker ∂ \ 0 which is divisible by x, and then x ∈ Ker ∂ by [FLN], that is, AK(X) = ℂ[x].

5. Further results

Once 2.5 and 2.2 are established, the fact that S′ = X//T can be forgotten in the proof of 2.4. In special cases, a geometric characterization of ℂ^{2}/ω_{a} is obtained.

5.1. Theorem ([KR4], 10.1). Suppose S′ is as in 2.2. If either q is an ordinary a-fold point, that is b ≡ c mod a, or the minimal resolution of q is a single (−a)- curve, or q is a rational double point, that is, b ≡ −c mod a, or the minimal resolution of q is a chain of a − 1 (−2)-curves, then

S′ ≃ ℂ^{2}/ω_{a}.

We do not know whether the restriction on the analytic type of q is needed in 5.1.

Extending the arguments of [KP], Popov [P] recently proved that any effective action of a noncommutative, connected reductive group on ℂ^{3} is linearizable. Since effective actions of (ℂ^{∗})^{r}, r > 1, are linearizable by [B-B], we obtain

5.2. Theorem. Any action of a connected, reductive group G on ℂ^{3} is linearizable.

It is an open question whether the connectedness assumption in 5.2 can be removed, and in particular, whether finite group actions on ℂ^{3} are linearizable.

It is reasonable to expect that our methods and results will shed some light in general on codimension 2 torus actions on ℂ^{n}. As an illustration, consider the possibility that X × ℂ ≃ ℂ^{4}, where X is a ℂ^{∗}-threefold. For linearizability of the obvious (ℂ^{∗})^{2}-action the following weak cancellation conjecture is required: Let X be an affine threefold such that X × ℂ ≃ ℂ^{4}. Then X ≃ ℂ^{3} or X does not admit an effective ℂ^{∗}-action.

This is known for nonhyperbolic actions. For hyperbolic actions, one would have to show that X ×ℂ ≄ ℂ^{4} for the threefolds in 3.1. This is true in the case when X is not isomorphic to a hypersurface of the form 4.1 (i) or (ii) since AK(Y ×ℂ) = ℂ[Y] for every algebraic manifold Y with AK(Y) = ℂ[Y] [M-L2].

We remark that linearizability of 𝔾_{m}-actions on 𝔸^{3} in positive characteristic is an open question, even in certain nonhyperbolic cases.

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Department of Mathematics & Computer Science, University of Miami, Coral Gables, FL 33124

E-mail address: kaliman@paris-gw.cs.miami.edu

Institute of Mathematics, Warsaw University, Ul. Banacha 2, Warsaw, Poland

E-mail address: koras@mimuw.edu.pl

Department of Mathematics & Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel, and Department of Mathematics, Wayne State University, Detroit, MI 48202

E-mail address: lml@bimacs.cs.biu.ac.il; lml@math.wayne.edu

Department of Mathematics & Statistics, McGill University, Montreal, QC, Canada, and Centre Interuniversitaire, en Calcul Mathématique, Algébrique (CICMA)

E-mail address: russell@Math.McGill.CA