On the injective self-maps of algebraic varieties

A conjecture of Miyanishi says that an endomorphism of an algebraic variety, defined over an algebraically closed field of characteristic zero, is an automorphism if the endomorphism is injective outside a closed subset of codimension at least $2$. We prove the conjecture in the following cases: (1) The variety is non-singular. (2) The variety is a surface. (3) The variety is locally a complete intersection that is regular in codimension $2$. We also discuss a few instances where an endomorphism of a variety, satisfying the hypothesis of the conjecture of Miyanishi, induces an automorphism of the non-singular locus of the variety. Under additional hypotheses, we prove that the conjecture holds when the variety has only isolated singularities.