Lefschetz (1,1)-theorem in tropical geometry

TL;DR

This paper establishes a tropical analogue of the Lefschetz (1,1)-theorem, characterizing certain tropical homology classes via the kernel of the eigenwave map, and relates tropical line bundles to cohomological kernels.

Contribution

It introduces a tropical Lefschetz (1,1)-theorem for rational polyhedral spaces, linking tropical line bundles to the kernel of the wave homomorphism, and extends this to tropical manifolds.

Findings

Tropical homology classes of degree (n-1, n-1) as fundamental classes are characterized by the eigenwave map kernel.

A tropical version of the Lefschetz (1,1)-theorem is established for rational polyhedral spaces.

The result connects tropical line bundles with the kernel of the wave homomorphism on cohomology.

Abstract

For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar\'e duality for integral tropical homology.