On Miyanishi conjecture for quasi-projective varieties

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.