Smooth conjugacy of algebraic actions on affine varieties

Abstract

The motivation for this result is the linearity conjecture which asserts that any algebraic action of a reductive algebraic group on affine n-space is conjugate to a linear action. This topic has been popularized by H. Bass and H. Kraft. See for example [4], [2] and [S] for the current status of the conjecture. In addition to being interesting, the conjecture is tough. It is not even known whether the set of algebraic conjugacy classes of algebraic actions of a reductive group of affine n-space is countable, as implied by the linearity conjecture. Theorem 1 is a version of this countability question. We shall see in the course of the proof of Theorem 1 that there are an uncountable number of conjugacy classes of smooth actions of a nontrivial compact Lie group on real n-space. Theorem 1 shows that only countably many of these are conjugate to algebraic actions on real n-space. The main tools in the proof are the compactification theorem of Corollary 4 and the Palais Rigidity Theorem [lo]. Whether this rigidity theorem has an algebraic analog is an interesting question whose consequences we briefly explore at the end. Before giving the proof, we introduce some notation. In this discussion varieties are affine. The ground field is the reals [w or the complex numbers @. An algebraic map between affine spaces (over [w or C) is a map whose coordinate functions are polynomials. An algebraic map between affine varieties is a map which extends to an algebraic map of affine spaces containing the varieties as subvarieties. Let W be a nonsingular variety resp. a smooth manifold and Aut( W, c) for c=a or s denote the group of algebraic resp. smooth automorphisms of W with the C” topology. Let Hom(G, Aut( W, c)) denote the space of c actions of G on W. A point d of this space is a homomorphism d:BG+Aut( W, c) such that the induced map from G x W to W is a c-map. Topologize Hom(G, Aut( W, c)) as the subspace of the space of all smooth maps of G x W to Win the C” topology. The group Aut( W,c) acts by conjugation on the space of c-actions and the orbit space is denoted by Hom(G, Aut( W, c))/Aut( W, c). In the case W is a nonsingular variety V, weakening structure defines a continuous function from the space of a-actions to the space of s-actions and induces a map of corresponding orbit spaces. We are interested in the cardinality of the image. The idea is to factor this map through a countable set. In the case V is simply connected at infinity (i.e. given any compact set C in Vthere is a compact set D containing C