A surface with discrete and non-finitely generated automorphism group

TL;DR

This paper constructs a smooth complex projective variety in any dimension ≥2 with a discrete, non-finitely generated automorphism group, and demonstrates it has infinitely many non-isomorphic real forms, answering open questions.

Contribution

It provides the first example of such a variety with a non-finitely generated automorphism group and multiple real forms, advancing understanding of automorphism groups in algebraic geometry.

Findings

Existence of a smooth complex projective variety with a non-finitely generated automorphism group

The variety admits infinitely many non-isomorphic real forms

Answers to open questions by Dolgachev, Esnault, and Lesieutre

Abstract

We show that there is a smooth complex projective variety, of any dimension greater than or equal to two, whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are mutually non-isomorphic over the real number field. Our result is inspired by the work of Lesieutre and answers questions by Dolgachev, Esnault and Lesieutre.