Smooth rational projective varieties with non-finitely generated discrete automorphism group and infinitely many real forms

TL;DR

This paper constructs smooth rational projective varieties of any dimension at least 3 that have a discrete, non-finitely generated automorphism group and infinitely many non-isomorphic real forms, expanding understanding of automorphism groups and real forms.

Contribution

It provides explicit examples of high-dimensional rational varieties with complex automorphism groups and infinitely many real forms, inspired by prior research in the field.

Findings

Existence of such varieties for all dimensions n ≥ 3

Automorphism groups are discrete and non-finitely generated

Infinitely many mutually non-isomorphic real forms

Abstract

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.