Automorphisms of $\mathbb{C}^3$ Commuting with a $\mathbb{C}^+$-Action

Abstract

Let $ρ$ be an algebraic action of the additive group $\mathbb{C}^+$ on the three-dimensional affine space $\mathbb{C}^3$. We describe the group $\textrm{Cent}(ρ)$ of polynomial automorphisms of $\mathbb{C}^3$ that commute with $ρ$. A particular emphasis lies in the description of the automorphisms in $\textrm{Cent}(ρ)$ coming from algebraic $\mathbb{C}^+$-actions. As an application we prove that the automorphisms in $\textrm{Cent}(ρ)$ that are the identity on the algebraic quotient of $ρ$ form a characteristic subgroup of $\textrm{Cent}(ρ)$.