Comptes Rendus
Théorie des nombres
Projective varieties have countably many real forms
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 863-867.

In this note, we check that a complex projective algebraic variety has (at most) countably many real forms. We more generally prove it when the field of reals is replaced with a field that has only countably many finite extensions up to isomorphism. The verification consists in gathering known results about automorphism groups and Galois cohomology. This contrasts with the recent discovery by A. Bot of an affine real variety with uncountably many real forms.

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DOI : 10.5802/crmath.441
Classification : 12G05, 12F10, 11S25, 11E72, 14J50
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Timoth\'ee L. Labinet},
     title = {Projective varieties have countably many real forms},
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     pages = {863--867},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.441},
     language = {en},
}
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Timothée L. Labinet. Projective varieties have countably many real forms. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 863-867. doi : 10.5802/crmath.441. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.441/

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