Metric properties of the tropical Abel–Jacobi map

Abstract

Let Γ be a tropical curve (or metric graph), and fix a base point p?Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel---Jacobi map ? p :Γ?J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ?. As an application of our direct limit theorem, we derive some local comparison formulas between ? and ${\varPhi}_{p}^{*}(\rho)$ for three different natural metrics ? on J(Γ). One of these formulas implies that ? p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ Zh? on a metric graph Γ, defined by S. Zhang, measures lengths on ? p (Γ) with respect to the sup-norm on J(Γ).