×

Algebraic group actions on affine spaces. (English) Zbl 0587.14031

Group actions on rings, Proc. AMS-IMS-SIAM Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 43, 1-23 (1985).
[For the entire collection see Zbl 0563.00007.]
Let the reductive linear algebraic group G act on the affine scheme X, both defined over the characteristic zero algebraically closed field k. Then there is an affine categorical quotient scheme \(Y=X/G\) and a projection \(p:\quad X\to Y\) each of whose fibers contains a unique closed orbit. This article surveys some of the theory of G, p, X and Y related to the ”linearization problem”: if X is affine n-space, is the action conjugate, by an automorphism of X, to a linear one ?
This question clearly depends on some knowledge of the automorphism group of affine n-space, the affine Cremona group \(GA_ n(k)\). This group is known for \(n=2\) from which it is possible to deduce that reductive group actions on the plane are linear. The question also depends on fixed points, since linear actions fix zero. In some cases, such as finite p- groups or \(SL_ 2(k)\) (with \(n\leq 7)\), fixed points are known to exist. The structure of X over Y is relevant: when the only closed orbits are fixed points, X is a G-vector bundle over Y, which means that if there is a unique fixed point the action is isomorphic to the linear action on the tangent space at that point. The structure of X over Y in general is given by Luna’s étale slice theorem. This implies that if Y is zero- dimensional then the action is linearizable. Non-linearizable actions are unknown, but a possible approach to finding some is given in terms of a G-equivariant K-theory problem. Since general linearization would imply, in particular, that cartesian factors of affine spaces (with the other factor an affine space) are themselves affine spaces (the cancellation problem), and the cancellation problem is felt to be difficult, it is expected that the linearization problem is also difficult.
Reviewer: A.R.Magid

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
14E07 Birational automorphisms, Cremona group and generalizations

Citations:

Zbl 0563.00007