Homological smoothness and Deligne resolution for tropical fans

TL;DR

This paper introduces the concept of homological smoothness for tropical fans, proves its stability, and develops a tropical analogue of Deligne's spectral sequence to analyze tropical homology and cohomology.

Contribution

It defines homological smoothness for tropical fans, proves its stability, and establishes a tropical Deligne weight spectral sequence, advancing tropical Hodge theory.

Findings

Homological smoothness is T-stable in tropical fans.

Quasilinear fans are homologically smooth.

A tropical Deligne spectral sequence is constructed.

Abstract

We say that a tropical fan is homologically smooth if each of its open subsets verify tropical Poincare duality. A tropical homology manifold is a tropical variety that is locally modelled by open subsets of homologically smooth tropical fans. We show that homological smoothness is a T-stable property in the category of tropical fans. This implies in particular that quasilinear fans are homologically smooth, and tropical varieties locally modelled by them are tropical homology manifolds. Previously, this was known only for locally matroidal tropical varieties. In order to show the above results, we prove a tropical analogue of the Deligne weight spectral sequence for homologically smooth tropical fans. This allows to describe the cohomology of tropical modifications, and will be of importance in our companion work which develops a Hodge theory in the tropical setting.