Hodge Theory for Polymatroids

TL;DR

This paper develops a Hodge-theoretic framework for discrete polymatroids, establishing algebraic and topological properties analogous to classical Hodge theory, and generalizes key formulas and dualities.

Contribution

It constructs a Leray model for polymatroids with arbitrary building sets and proves Hodge-theoretic properties for their Chow rings, extending classical geometric results.

Findings

Proved Poincaré duality for the Chow ring of polymatroids

Established Hard Lefschetz and Hodge-Riemann theorems in this context

Derived a generalized Goresky-MacPherson formula

Abstract

We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky-MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincar\'e duality, Hard Lefschetz, and Hodge-Riemann theorems for the Chow ring. Furthermore we provide a relative Lefschetz decomposition with respect to the deletion of an element.