On the injective self-maps of algebraic varieties

TL;DR

The paper proves Miyanishi’s conjecture in several key cases by developing an algebraic, differential-geometry–inspired framework. It reduces to normal varieties and uses Kähler differentials, submersive morphisms, and homological criteria to show that endomorphisms injective off a codimension $\ge 2$ subset are automorphisms in: (i) non-singular $X$, (ii) $X$ a surface, and (iii) $X$ locally a complete intersection with regularity in codimension $2$; it also analyzes the behavior on the non-singular locus and under isolated singularities. Central to the approach are the exact sequences of differentials, the notion of submersive morphisms via the ramification module, and depth/pd arguments ensuring reflexivity and torsion-free properties. The results connect algebro-geometric and homological methods to extend automorphism properties from open loci to entire varieties, providing a robust framework for Miyanishi-type questions in broader settings.

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.

Theorems & Definitions (40)

• Conjecture 1.1: Miyanishi
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• Remark 2.2
• Lemma 2.4
• proof
• Remark 2.5