The Jacobian Conjecture fails for pseudo-planes

Abstract

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale 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 $G={\mathbb{C}}^{\ast }$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected ${\mathbb{C}}^{\ast }$-surfaces of negative Kodaira dimension which admit non-proper ${\mathbb{C}}^{\ast }$-equivariant étale endomorphisms. We prove also that for every integers $r\ge 1,k\ge 2$ 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 étale 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.

26 pages, 1 figure

References in corpus (2)

• Revêtements étales et groupe fondamental (SGA 1)
• Lectures on Geometry and Topology of Polynomials - Surrounding the Jacobian Conjecture

Cited by in corpus (1)

• Jacobian conjecture in ${\mathbb{R}}^2$