Hodge Theory for Combinatorial Geometries

TL;DR

This paper establishes Hodge-theoretic properties for matroids, proving log-concavity of their characteristic polynomial coefficients and independence complex f-vectors, thereby resolving longstanding conjectures in combinatorics.

Contribution

It introduces a novel Hodge theory framework for matroids, proving key conjectures about their algebraic and combinatorial properties.

Findings

Proved the hard Lefschetz theorem for matroid-associated rings.

Established the Hodge-Riemann relations for these rings.

Confirmed the log-concavity of matroid characteristic polynomial coefficients.

Abstract

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.