Metric properties of the tropical Abel–Jacobi map |
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Authors: | Matthew Baker Xander Faber |
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Institution: | (1) Department of Mathematics, U.C.L.A, CA 90024 Los Angeles, USA |
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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 Φ
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 ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between
ρ and
\varPhip*(r){\varPhi}_{p}^{*}(\rho) for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φ
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 Φ
p
(Γ) with respect to the “sup-norm” on J(Γ). |
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Keywords: | |
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