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.
Jin Hong Kim
Chosun University
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.
-0001; 31(1): 95-100
2015; 52(4): 1321-1325
1998; 35(1): 173-187
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd