On the injective self-maps of algebraic varieties

Abstract

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.