Linearization of holomorphic families of algebraic automorphisms of the affine plane

Abstract

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on ${\mathbb{C}}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on ${\mathbb{C}}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of ${\mathbb{C}}^2$.

14 pages, to appear in Transformation Groups