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Abelian varieties with no power isogenous to a Jacobian
- Part of
-
- Journal:
- Compositio Mathematica / Volume 161 / Issue 6 / June 2025
- Published online by Cambridge University Press:
- 30 October 2025, pp. 1404-1457
- Print publication:
- June 2025
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- Article
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Let
$X$ be a curve of genus at least 4 that is very general or very general hyperelliptic. We classify all the ways in which a power 
$(JX)^k$ of the Jacobian of 
$X$ can be isogenous to a product of Jacobians of curves. As an application, we show that if 
$A$ is a very general principally polarized abelian variety of dimension at least 4 or the intermediate Jacobian of a very general cubic threefold, then no power 
$A^k$ is isogenous to a product of Jacobians of curves. This confirms various cases of the Coleman–Oort conjecture. We further deduce from our results some progress on the question of whether the integral Hodge conjecture fails for 
$A$ as above.
