Proof of Miyanishi's conjecture on endomorphisms of varieties

TL;DR

This paper proves Miyanishi's conjecture by showing that certain birational endomorphisms of varieties are automorphisms, extending previous results and establishing a new finiteness property of class groups.

Contribution

It generalizes Miyanishi's conjecture and introduces a finiteness result on class groups relevant to algebraic geometry.

Findings

Birational endomorphisms with injectivity outside codimension ≥ 2 are automorphisms.

Established a finiteness result on class groups.

Extended classical theorems to broader classes of varieties.

Abstract

If $X$ is a quasi-projective variety over a field $k$ and $\phi$ a birational endomorphism of $X$ that is injective outside a closed subset of codimension $\geq 2$, we prove that $\phi$ is an automorphism. This generalizes an old theorem of Ax and proves a conjecture of Miyanishi. A key step in our proof is a finiteness result on class groups, which is of interest in its own right.