Metric properties of the tropical Abel–Jacobi map

By Matthew Baker, Xander Faber. 2009

90 citations recorded by the source corpus.

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 Φp: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 ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and ${\varPhi}_{p}^{*}(\rho)$ for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φp is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μZh on a metric graph Γ, defined by S. Zhang, measures lengths on Φp(Γ) with respect to the “sup-norm” on J(Γ).

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/67c6cee2fed5f85feded762144e65b9c1c677c12

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In Geometry and Topology

Citations in the corpus

Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “5 , we will first define the analogue of the Zhang measure on Berkovich curves following [BF11].”
A tropical proof of the Brill-Noether Theorem (2010, background): “is an extension of a real torus of dimension g by Z [25,3].”