Equivariant Jacobian Conjecture in dimension two

TL;DR

This paper proves that any étale endomorphism of the complex affine plane commuting with an even-order finite group action is necessarily an automorphism, advancing understanding of symmetry and invertibility in algebraic geometry.

Contribution

It establishes a new result linking group actions and étale endomorphisms, specifically proving automorphism conditions under symmetry constraints in dimension two.

Findings

Étale endomorphisms commuting with even-order group actions are automorphisms.

Provides conditions under which symmetries imply invertibility of polynomial maps.

Enhances understanding of the Jacobian Conjecture in the presence of symmetries.

Abstract

Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an \'etale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.