Hodge theory for combinatorial projective bundles

TL;DR

This paper establishes Hodge-theoretic properties for rings akin to cohomology rings of projective bundles over toric varieties, advancing conjectures in algebraic geometry and matroid theory.

Contribution

It proves the Hard Lefschetz theorem and Hodge-Riemann relations for new classes of rings, extending Hodge theory to combinatorial and algebraic structures.

Findings

Proved Hard Lefschetz theorem for certain combinatorial rings.

Established Hodge-Riemann relations in new algebraic contexts.

Derived Bloch-Gieseker-type results for matroid tautological classes.

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.