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We study finiteness problems for isogeny classes of abelian varieties over an algebraic function field K in one variable over the field of complex numbers. In particular, we construct explicitly a non‐isotrivial absolutely simple abelian fourfold X over a certain K such that the isogeny class of X×X contains infinitely many mutually non‐isomorphic principally polarized abelian varieties. (Such examples do not exist when the ground field is finitely generated over its prime subfield.)

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