Linearization of holomorphic families of algebraic automorphisms of the affine plane

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 $\mathbb{ C}^2$.

Categories: math.AG math.CV