Tropical eigenwave and intermediate Jacobians

TL;DR

This paper introduces the concept of tropical eigenwaves and intermediate Jacobians, establishing their role in understanding tropical manifolds and their relation to complex manifold degenerations.

Contribution

It defines tropical eigenwaves and intermediate Jacobians, linking tropical geometry with classical complex geometry through monodromy and cohomology.

Findings

Eigenwave records monodromy in degenerating complex families

Tropical intermediate Jacobians serve as analogs of classical Jacobians

Provides a new framework for tropical and complex geometry connection

Abstract

Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of similar type are responsible for deformations of the tropical structure. If a tropical manifold is approximable by a 1-parametric family of complex manifolds then the eigenwave records the monodromy of the family around the tropical limit. With the help of tropical homology and the eigenwave we define tropical intermediate Jacobians which can be viewed as tropical analogs of classical intermediate Jacobians.