Matroids and the integral Hodge conjecture for abelian varieties
Algebraic Geometry
2026-03-31 v3 Combinatorics
Abstract
We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids.
Cite
@article{arxiv.2507.15704,
title = {Matroids and the integral Hodge conjecture for abelian varieties},
author = {Philip Engel and Olivier de Gaay Fortman and Stefan Schreieder},
journal= {arXiv preprint arXiv:2507.15704},
year = {2026}
}
Comments
85 pages, v3: Corollary 1.2 improved; Definition 3.2 corrected; results in Sections 5-8 slightly extended (to the extent needed in arXiv:2512.04902); references added