Matroids and the integral Hodge conjecture for abelian varieties

TL;DR

This paper proves that certain cohomology classes on very general abelian varieties and cubic threefolds are multiples of minimal classes, disproving the integral Hodge conjecture and impacting rationality questions, using tropical geometry and matroid theory.

Contribution

It demonstrates the failure of the integral Hodge conjecture for abelian varieties and links tropical geometry with classical algebraic geometry through matroids.

Findings

Cohomology class of any curve on a very general abelian variety is an even multiple of the minimal class.

Disproves the integral Hodge conjecture for abelian varieties.

Shows very general cubic threefolds are not stably rational.

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.