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Classification of Bagnera–de Franchis Varieties in Small Dimensions
Andreas Demleitner1
1 Andreas Demleitner, Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, D-95447 Bayreuth, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 1, pp. 111-133.

A Bagnera–de Franchis variety X=A/G is the quotient of an Abelian variety A by a free action of a finite cyclic group G=〈g〉⊂Bihol(A), which does not contain only translations. Constructing explicit polarizations and using a method introduced by F. Catanese, we classify split Bagnera–de Franchis varieties (i.e., there is a direct sum decomposition of A into G-invariant Abelian subvarieties, indexed by k|#G, on which the linear part of g acts with eigenvalues of order k) up to complex conjugation in dimensions ≤4.

Une variété de Bagnera–de Franchis X est un quotient A/G d’une variété abélienne A par un groupe cyclique fini G=〈g〉⊂Bihol(A) agissant librement et ne consistant pas uniquement en des translations. Par une construction explicite des polarisations, et en suivant une méthode introduite par F. Catanese, nous donnons une classification exhaustive à conjugaison complexe près et au sens suivant des variétés de Bagnera–de Franchis scindées jusqu’en dimension 4 : A admet une décomposition en somme directe de sous-variétés abéliennes G-invariantes indexées par k|#G sur lesquelles la partie linéaire de g agit avec des valeurs propres d’ordre k.

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Accepted:
Published online:
DOI: 10.5802/afst.1626

Andreas Demleitner 1

1 Andreas Demleitner, Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, D-95447 Bayreuth, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Classification of {Bagnera{\textendash}de} {Franchis} {Varieties} in {Small} {Dimensions}},
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Andreas Demleitner. Classification of Bagnera–de Franchis Varieties in Small Dimensions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 1, pp. 111-133. doi : 10.5802/afst.1626. https://afst.centre-mersenne.org/articles/10.5802/afst.1626/

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