On Miyanishi conjecture for quasi-projective varieties

TL;DR

This paper proves Miyanishi conjecture for certain quasi-projective varieties with specific conditions on their compactifications and singularities, using minimal model program techniques.

Contribution

It establishes Miyanishi conjecture for quasi-projective varieties with ample canonical or anti-canonical divisors and extends results to cases with canonical singularities.

Findings

Proved Miyanishi conjecture for varieties with ample divisors.

Extended conjecture validity to varieties with canonical singularities.

Utilized minimal model program to relax conditions.

Abstract

Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least $2$ is bijective. We prove Miyanishi conjecture for any quasi-projective variety $X$ which is a dense open subset of a $\mathbb{Q}$-factorial normal projective variety $\overline{X}$ such that codim $(\overline{X} \setminus X) \ge 2$ with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that $\overline{X}$ has canonical singularities and $\overline{X}$ has the canonical model which is obtained by divisorial contractions.