On Endomorphism of Algebraic Varieties

TL;DR

The paper proves that certain injective endomorphisms of algebraic varieties over characteristic zero fields are actually automorphisms, especially when injectivity holds outside a small subvariety.

Contribution

It establishes conditions under which endomorphisms become automorphisms, extending previous results to broader classes of algebraic varieties.

Findings

Injective endomorphisms outside a closed subvariety are automorphisms over characteristic zero fields.

Injectivity outside a codimension at least 2 subvariety implies automorphism for complex algebraic varieties.

Results generalize known automorphism criteria for algebraic varieties.

Abstract

We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism of complex algebraic variety that is injective on the complement of a closed subvariety of codimension at least $2$, is an automorphism.