Matroids and the Integral Hodge Conjecture for Abelian Varieties

Philip Engel explains a counterexample to the integral Hodge conjecture for very general abelian varieties of dimension at least four. Degenerations of principally polarized abelian varieties are associated with regular matroids, and a new matroid invariant obstructs algebraicity of the minimal curve class on a very general fiber. Combining this obstruction with a theorem of Voisin and the intermediate Jacobian gives stable irrationality of a very general cubic threefold. The work is joint with Olivier de Gaay Fortman and Stefan Schreieder.