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.

arXiv commentFinal version. To appear in J. Pure Appl. Algebra