The Jacobian Conjecture fails for pseudo-planes

TL;DR

This paper investigates the failure of the Jacobian Conjecture for certain complex surfaces, classifies specific counterexamples, and constructs families of non-proper étale endomorphisms with high dimension and degree.

Contribution

It provides a classification of G-equivariant counterexamples for the Jacobian Conjecture on pseudo-planes and constructs explicit families of non-proper étale endomorphisms.

Findings

G-equivariant counterexamples exist only for G=ℂ*

Classification relates counterexamples to Belyi-Shabat polynomials

Constructs families of non-proper étale endomorphisms with arbitrarily high dimension and degree

Abstract

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its \'etale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant \'etale endomorphisms. We prove also that for every integers $r\geq 1, k\geq 2$ the $\mathbb{Q}$-acyclic rational hyperplane $u(1+u^{r}v)=w^k$, which has fundamental group $\mathbb{Z}_k$ and negative Kodaira dimension, admits families of non-proper \'etale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.