Abelian varieties with no power isogenous to a Jacobian
arXiv:2401.06577 · doi:10.1112/S0010437X25007171
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.
61 pages, final version, to appear in Compositio Mathematica