Hodge theory for combinatorial projective bundles

Published: April 23, 2026PDF arXiv

Abstract

We prove the Hard Lefschetz theorem and Hodge-Riemann relations for certain rings which resemble the cohomology rings of projectivizations of globally generated vector bundles over toric varieties. This proves new cases of the standard conjecture of Hodge type and gives Bloch-Gieseker-type results for tautological classes of matroids.

Hodge theory for combinatorial projective bundles

Matt Larson, Ethan Partida

math.AG math.CO

Published: April 23, 2026PDF arXiv

Abstract

We prove the Hard Lefschetz theorem and Hodge-Riemann relations for certain rings which resemble the cohomology rings of projectivizations of globally generated vector bundles over toric varieties. This proves new cases of the standard conjecture of Hodge type and gives Bloch-Gieseker-type results for tautological classes of matroids.