Abelian varieties with no power isogenous to a Jacobian

Olivier de Gaay Fortman; Stefan Schreieder

doi:10.1112/s0010437x25007171

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arXiv · 2401.06577

Abelian varieties with no power isogenous to a Jacobian

Olivier de Gaay Fortman·Stefan Schreieder

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Abstract

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.

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Olivier de Gaay Fortman, Stefan Schreieder. 2024-01-12. Abelian varieties with no power isogenous to a Jacobian. https://doi.org/10.1112/s0010437x25007171

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