Finiteness of Klein actions and real structures on compact hyperk\"ahler manifolds

TL;DR

This paper proves that compact hyperk"ahler manifolds have finitely many real structures and finite group actions up to conjugacy, and also shows their automorphism groups are finitely presented, advancing understanding in real algebraic geometry.

Contribution

It establishes finiteness of real structures and Klein actions on compact hyperk"ahler manifolds, and proves the automorphism group is finitely presented, addressing key questions in the field.

Findings

Finitely many real structures up to equivalence.

Finitely many Klein actions up to conjugacy.

Automorphism group is finitely presented.

Abstract

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperk\"ahler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperk\"ahler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperk\"ahler manifold is finitely presented.