On the finiteness of real structures of projective manifolds
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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.
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.
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