A Poincar{\'e}-Lefschetz Theorem for Cellular Cosheaves and an Application to the Tropical Homology of Orbifold Toric Varieties

TL;DR

This paper extends the Poincaré-Lefschetz theorem to cellular cosheaves and applies this to generalize tropical Lefschetz theorems for singular tropical toric varieties and hypersurfaces.

Contribution

It introduces a new version of the Poincaré-Lefschetz theorem for cellular cosheaves and applies it to tropical geometry, specifically for singular tropical toric varieties.

Findings

Established a Poincaré-Lefschetz theorem for cellular cosheaves.

Generalized tropical Lefschetz hyperplane section theorem to singular cases.

Connected cellular cosheaf cohomology with homology of initial cosheaves.

Abstract

In a first time we present a version of the Poincar{\'e}-Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex K. To that end we construct a cellular sheaf on K whose cohomology with compact support is isomorphic to the homology of the initial cosheaf. In a second time we use the first result to generalise the tropical version of the Lefschetz hyperplane section theorem to singular tropical toric varieties and singular tropical hypersurfaces.