Goethe-Universität — Stefan Schreieder: Matroids and the integral Hodge conjecture for abelian varieties

Frankfurter Seminar - Kolloquium des Instituts für Mathematik

Ginkgo Seminar

Vorkolloquium für Doktoranden, Post-Docs und interessierte Studierende

An den jeweiligen Veranstaltungstagen wird mit dem Ginkgo-Seminar (Gingko - Grundlagen, Intuition, Neugier für und auf das Kolloquium) gestartet, einer exklusiven Veranstaltung von Doktoranden für Doktoranden, Post-Docs und interessierte Studierende.

Ziel ist es, die Vorträge "aus der anderen Ecke des Instituts" für alle Interessierten zugänglicher zu machen.

Das Vorkolloquium findet immer ab 15.00 (c.t.) in Raum 711 groß vor dem jeweiligen Vortrag statt.

Frankfurter Seminar

Jan 14 2026

16:45

Stefan Schreieder: Matroids and the integral Hodge conjecture for abelian varieties

(Leibniz Universität Hannover)

Abstract: Hodge originally formulated his conjecture — one of the seven Millennium Prize Problems — with integral coefficients.

Atiyah and Hirzebruch showed in 1962 that it fails for torsion classes, and later Kollár produced non-torsion counterexamples. Since then, the conjecture has been understood with rational coefficients, while its integral form is regarded as a property of individual varieties rather than a universal statement. A central and longstanding open case concerned the integral Hodge conjecture for abelian varieties.

In this talk, I will explain how ideas from the combinatorial theory of regular matroids allow us to prove that it fails for large classes of abelian varieties. As a consequence, building on work of Voisin, we show that very general cubic threefolds are not stably rational, strengthening the classical result of Clemens and Griffiths (1972) that smooth cubic threefolds are not rational.

This is joint work with Philip Engel and Olivier de Gaay Fortman.