Tropical Abel-Jacobi theory

TL;DR

The paper develops a tropical analogue of Griffiths' intermediate Jacobians for tropical varieties of arbitrary dimension by defining $\mathrm{JH}_{p,q}(X)=\mathrm{H}_{p,q}(X,\mathbb{R})/\mathrm{L}_{p,q}(X)$ via tropical homology and monodromy, and introduces a functorial Abel–Jacobi map $\mathrm{AJ}: \mathrm{A}_{p}^{\circ}(X) \to \mathrm{JH}_{p+1,p}(X)$. It defines tropical Albanese varieties, establishes obstructions to algebraic equivalence through monodromy, and shows the 1-dimensional case recovers the tropical curve Abel–Jacobi theory; a Ceresa class is given an explicit combinatorial formula in terms of the tropical curve's graph. The framework clarifies when $\mathrm{JH}_{p,q}(X)$ are compact tori (via the weight–monodromy property) and connects to the Morita/Johnson picture through a tropical Ceresa class, with concrete computations for graphs such as $K_4$ and $TL_3$. The work unifies tropical geometry with classical Abel–Jacobi theory, providing tools to study torsion phenomena, algebraic equivalence obstructions, and potential tropical Roitman-type results, along with explicit combinatorial examples illustrating the theory.

Abstract

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves. As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve.

Figures (6)

• Figure 1.1: For this arrangement of curves, the values of $\mathrm{sgn} \sb{\space\scaleobj{.8}{\gamma,\gamma'}}$ on $\beta_1$, $\beta_2$, $\beta_3$, $\beta_4$, $\beta_5$ are $-1$, $1$, $1$, $0$, $0$, respectively.
• Figure 4.1: An illustration of the proof of Proposition \ref{['prop:monodromyAbelianVariety']}
• Figure 6.1: A genus 2 tropical curve $\Gamma$ and the trees $S$ and $S'$ in the universal cover of $\mathrm{Jac}(\Gamma)$
• Figure 6.2: The values of $\mathrm{sgn} \sb{\space\scaleobj{.8}{T}} \sp{\space\flat} (e,\varepsilon)$ are, left to right: $2$, $-2$, $1$, $-1$, $0$.
• Figure 6.3: The values of $\overline{\mathrm{sgn}} \sb{\space\scaleobj{.8}{T}} (e,\varepsilon)$ are, left to right: $1$, $-1$, $0$

Theorems & Definitions (62)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof : Proof of Proposition \ref{['prop:minimum_sedenarity']}
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Remark 3.2