Metric properties of the tropical Abel-Jacobi map

Baker, Matthew ; Faber, Xander

Journal of Algebraic Combinatorics, Tome 33 (2011) no. 3, pp. 349-381.

Résumé

Summary: Let $\Gamma $ be a tropical curve (or metric graph), and fix a base point $p\in \Gamma $. We define the Jacobian group $J( G)$ of a finite weighted graph $G$, and show that the Jacobian $J( \Gamma )$ is canonically isomorphic to the direct limit of $J( G)$ over all weighted graph models $G$ for $\Gamma $. This result is useful for reducing certain questions about the Abel-Jacobi map $\Phi _{ p }: \Gamma \rightarrow J( \Gamma )$, 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 $\Bbb Q$. As an application of our direct limit theorem, we derive some local comparison formulas between $\rho $ and $\varPhi _{ p} ^{*}( r)$ varPhi_p^*(rho) for three different natural "metrics" $\rho $ on $J( \Gamma )$. One of these formulas implies that $\Phi _{ p }$ is a tropical isometry when $\Gamma $ is 2-edge-connected. Another shows that the canonical measure $\mu _{Zh }$ on a metric graph $\Gamma $, defined by S. Zhang, measures lengths on $\Phi _{ p }( \Gamma )$ with respect to the "sup-norm" on $J( \Gamma )$.

Détail

Keywords: keywords tropical curve, tropical Jacobian, Picard group, Abel-Jacobi, metric graph, foster's theorem