Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles

TL;DR

This paper develops a tropical analogue of higher Abel-Jacobi invariants using tropical Hodge structures, providing new tools to study tropical cycles and tautological classes in tropical moduli spaces.

Contribution

It introduces a formal framework for higher Abel-Jacobi invariants in tropical geometry, linking tropical Hodge theory with tropical Chow groups and tautological classes.

Findings

Defines tropical normal functions and their derivatives via Gauss-Manin connection.

Establishes the connection between tropical Abel-Jacobi invariants and the tropical Bloch-Beilinson filtration.

Proposes an approach to study tautological classes in tropical moduli spaces.

Abstract

We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical category. They naturally identify the tropical Bloch-Beilinson filtration on the tropical Chow group. We examine this construction on the moduli of tropical curves with marked points, in order to study the tropical tautological classes in the tautological ring of $\mathcal{M}_{g,n}^{\text{trop}}$. The expectation is the nontriviality of these cycles could be examined with less complexity in the tropical category. The construction is compatible with the tropicalization functor on the category of schemes, and the aforementioned procedure will also provide an alternative way to examine the relations in the tautological ring of $\mathcal{M}_{g,n}$ in the schemes category.