Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2020; 57(1): 109-115

Online first article August 6, 2019      Printed January 31, 2020

https://doi.org/10.4134/BKMS.b190084

Copyright © The Korean Mathematical Society.

On the finiteness of real structures of projective manifolds

Jin Hong Kim

Chosun University

Abstract

Recently, Lesieutre constructed a $6$-dimensional projective variety $X$ over any field of characteristic zero whose automorphism group ${\rm Aut}(X)$ is discrete but not finitely generated. As an application, he also showed that $X$ is an example of a projective variety with infinitely many non-isomorphic real structures. On the other hand, there are also several finiteness results of real structures of projective varieties. The aim of this short paper is to give a sufficient condition for the finiteness of real structures on a projective manifold in terms of the structure of the automorphism group. To be more precise, in this paper we show that, when $X$ is a projective manifold of any dimension$ \ge 2$, if ${\rm Aut}(X)$ does not contain a subgroup isomorphic to the non-abelian free group $\mathbb{Z}\ast \mathbb{Z}$, then there are only finitely many real structures on $X$, up to $\mathbb{R}$-isomorphisms.

Keywords: Real structures, projective manifolds, automorphism groups, entropy, theorem of Tits type

MSC numbers: 14J26, 14J50

Supported by: This study was supported by research fund from Chosun University, 2019.