Varieties Characterized by their Endomorphisms

TL;DR

The paper proves that for certain affine varieties containing a line, isomorphic endomorphism semigroups imply the varieties are isomorphic up to field automorphism, extending a holomorphic result.

Contribution

It establishes a new characterization of affine varieties based on their endomorphism semigroups, linking algebraic and holomorphic perspectives.

Findings

Isomorphic endomorphism semigroups imply isomorphic varieties up to automorphism.

The result applies specifically to affine varieties containing a line.

Extension of a holomorphic version to algebraic varieties.

Abstract

We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.