Linearization of holomorphic families of algebraic automorphisms of the affine plane

TL;DR

This paper proves that holomorphically parametrized families of polynomial automorphisms of the affine plane are linearizable, and applies this to certain reductive group actions on three-dimensional complex space, using a special Oka property.

Contribution

It establishes the linearizability of holomorphic families of algebraic automorphisms of the affine plane and applies this to specific reductive group actions on low space.

Findings

Holomorphic families of polynomial automorphisms of low space are linearizable.

A class of reductive group actions on low space are shown to be linearizable.

A restrictive Oka property for automorphism groups is proven.

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 $\mathbb{C}^2$.