Tropical fans and normal complexes

TL;DR

This paper introduces normal complexes associated with divisors on tropical fans, establishing their properties and connections to volume calculations, and links them to the combinatorial Hodge theory.

Contribution

It develops a new polytopal model called normal complexes for divisors on tropical fans, extending the theory of normal polytopes and connecting to Hodge theory.

Findings

Normal complexes are constructed for divisors on tropical fans.

Certain cones of divisors have volumes equal to associated normal complexes.

Existence of open families of divisor cones with nonempty interiors for Bergman fans.

Abstract

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the "volume" of each divisor in the cone - that is, the degree of its top power - is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.