The Bergman fan of a polymatroid

TL;DR

This work defines the Bergman fan of a polymatroid and proves a natural isomorphism between its Chow ring $A(\Sigma_{P,\mathscr G})$ and the polymatroid Chow ring $\mathrm{DP}(P,\mathscr G)$, thereby providing a tropical, combinatorial model for the De Concini–Procesi compactification of subspace arrangements in the polymatroid setting. The authors introduce multisymmetric lifts to reduce polymatroidal statements to matroidal ones, and develop geometric building sets to generalize Bergman fans, obtaining a full Gröbner-basis description and monomial basis for the Chow rings. They prove that the Kähler package (Poincaré duality, Hard Lefschetz, and Hodge–Riemann) holds for these Chow rings with respect to any strictly convex piecewise linear function on the Bergman fan, using the tropical Hodge theory of ADH20 and the matroidal case AHK18. The framework recovers and strengthens results of Pagaria–Pezzoli and connects to polypermutohedra in the Boolean polymatroid case, thereby unifying combinatorics, tropical geometry, and intersection theory for both realizable and non-realizable subspace arrangements.

Abstract

We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish the Kähler package for the Chow ring of the polymatroid, recovering and strengthening a result of Pagaria-Pezzoli.

Theorems & Definitions (71)

• Example 1.1
• Definition 1.2
• Definition 1.3
• Example 1.4
• Example 1.5
• Example 1.6
• Definition 1.7
• Definition \ref{thm:isomorphism}
• Example \ref{thm:isomorphism}
• Example \ref{thm:isomorphism}