Singular Hodge theory for combinatorial geometries

TL;DR

This paper develops a new intersection cohomology theory for matroids, establishing key geometric properties and applying them to prove conjectures and nonnegativity results in combinatorics.

Contribution

It introduces the intersection cohomology module for matroids and proves fundamental geometric theorems, leading to solutions of longstanding combinatorial conjectures.

Findings

Proves Poincaré duality, hard Lefschetz, and Hodge-Riemann relations for matroid cohomology.

Provides proofs for Dowling and Wilson's Top-Heavy conjecture.

Demonstrates nonnegativity of Kazhdan-Lusztig polynomial coefficients for all matroids.

Abstract

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.