# Towards a tropical Hodge bundle

## TL;DR

This paper introduces a tropical analogue of the Hodge bundle on the moduli space of tropical curves, exploring its combinatorial structure and providing explicit examples and computations.

## Contribution

It proposes a novel tropical Hodge bundle on $M_g^{trop}$ and analyzes its combinatorial properties, extending classical algebraic geometry concepts to tropical geometry.

## Key findings

- Defined a tropical Hodge bundle structure
- Analyzed combinatorial properties of the bundle
- Provided explicit examples and computations

## Abstract

The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.