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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.
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참고문헌 (15)
참고문헌 신청
Mohamed Benzerga / 2016 / Real structures on rational surfaces and automorphisms acting trivially on Picard groups / Mathematische Zeitschrift 282(3-4):1127~1136
M. Benzerga / 2017 / Finiteness of real structuctures on KLT Calabi-Yau regular smooth pairs of dimension 2 / arXiv:1702.08808v1
A. Borel / 1964 / Théorèmes de finitude en cohomologie galoisienne / Commentarii Mathematici Helvetici 39(1):111~164
A. Degtyarev / 2000 / Real Enriques surfaces / Springer-Verlag
T.-C. Dinh / A surface with discrete and non-finitely generated automorphism group / arXiv:1710.07019v3
M. Benzerga / 2017 / Finiteness of real structuctures on KLT Calabi-Yau regular smooth pairs of dimension 2 / arXiv:1702.08808v1
A. Borel / 1964 / Théorèmes de finitude en cohomologie galoisienne / Commentarii Mathematici Helvetici 39(1):111~164
A. Degtyarev / 2000 / Real Enriques surfaces / Springer-Verlag
T.-C. Dinh / A surface with discrete and non-finitely generated automorphism group / arXiv:1710.07019v3
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