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Fact :

  • Let $B$ an abelian subvariety of an abelian variety over a field $K$. We know that there exist an abelian subvariety $C$ of $A$ such that the restriction of addition gives an isogeny $B\times_K C\to A$.

  • The analogous result is true for an linear torus.

Questions :

1)Is this result is true in the category of semiabelians varieties ?

2)Is there a good reference on semi-abelians varieties ?

Thanks !

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1 Answer 1

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No. Take an extension of an elliptic curve by a torus which is of infinite order in the Ext group and take B the torus. (Reference: Serre, Groupes algébriques et corps de classes, Ch VII).

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