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Realisation of Abelian varieties as automorphism groups
Mathieu Florence1
1 Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 4, pp. 623-638.

Let A be an Abelian variety over a field F. We show that A is isomorphic to the automorphism group scheme of a smooth projective F-variety if, and only if, Aut gp (A ¯) is finite. This result was proved by Lombardo and Maffei [6] in the case F=ℂ, and recently by Blanc and Brion [1] in the case of an algebraically closed F.

Soit A une variété abélienne sur un corps F. On montre que A est isomorphe au schéma en groupes des automorphismes d’une F-variété projective et lisse, si et seulement si le groupe des F ¯-automorphismes de A est fini. Ce résultat est dû à Lombardo et Maffei [6] lorsque F=ℂ. Il est dû à Blanc et Brion [1] lorsque F=F ¯.

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DOI: 10.5802/afst.1747

Mathieu Florence 1

1 Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Mathieu Florence. Realisation of Abelian varieties as automorphism groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 4, pp. 623-638. doi : 10.5802/afst.1747. https://afst.centre-mersenne.org/articles/10.5802/afst.1747/

[1] Jérémy Blanc; Michel Brion Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics (preprint, http://www-fourier.univ-grenoble-alpes.fr/~mbrion/abelian.pdf)

[2] Michel Brion Homomorphisms of algebraic groups: representability and rigidity, Mich. Math. J., Volume 72 (2022), pp. 51-76 | MR | Zbl

[3] François Charles; Bjorn Poonen Bertini irreducibility theorems over finite fields, J. Am. Math. Soc., Volume 29 (2016) no. 1, pp. 81-94 | DOI | MR | Zbl

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[5] Everett W. Howe; Hui June Zhu On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field, J. Number Theory, Volume 92 (2002) no. 1, pp. 139-163 | DOI | MR | Zbl

[6] Davide Lombardo; Andrea Maffei Abelian varieties as automorphism groups of smooth projective varieties, Int. Math. Res. Not., Volume 2020 (2020) no. 7, pp. 1921-1932 | MR | Zbl

[7] Gebhard Martin Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces (2020) | arXiv

[8] Hideyuki Matsumura; Frans Oort Representability of group functors, and automorphisms of algebraic schemes, Invent. Math., Volume 4 (1967), pp. 1-25 | DOI | MR | Zbl

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[10] Bjorn Poonen Bertini theorems over finite fields, Ann. Math., Volume 160 (2005) no. 3, pp. 1099-1127 | DOI | MR | Zbl

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