Equivariant affine line bundles and linearization

Abstract. We show that every algebraic action of a linearly reductive group on a–ne n-space Cn which is given by Jonquiµere automorphisms is linearizable. Simi-larly, every holomorphic action of a compact group K by (holomorphic) Jonquiµere automorphisms is linearizable. Moreover, any holomorphic action of K on C2 by overshears is linearizable, too. These results are based on the fact that equivariant algebraic or holomorphic a–ne line bundles over Cn are trivial. x1. Algebraic affine line bundles Let G be an algebraic group and let X be a variety with an algebraic action of G. We assume that the base fleld k is algebraically closed of arbitrary characteristic (cf. Remark 3 at the end of x1). Consider the trivial a–ne line bundle …:X £A1! X over X. We want to study the possible lifts of the G-action to X £ A1. Clearly, every character ´ of G deflnes such a lift: g(x; z): = (gx; ´(g)¢z) for g 2 G; x 2 X; z 2 A1: We call this a trivial lift (with character ´). Proposition 1. Assume that G is linearly reductive and that G acts trivially o