Metric properties of the tropical Abel-Jacobi map
Matthew Baker
and Xander Faber
DOI: 10.1007/s10801-010-0247-3
Abstract
Let Γ be a tropical curve (or metric graph), and fix a base point p\in 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 \rightarrow 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 \Bbb Q. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and \varPhi p *( r) {\varPhi}_{p}^{*}(ρ) 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\thinspace on a metric graph Γ , defined by S. Zhang, measures lengths on Φ p ( Γ ) with respect to the “sup-norm” on J( Γ ).
Pages: 349–381
Keywords: keywords tropical curve; tropical Jacobian; Picard group; Abel-Jacobi; metric graph; foster's theorem
Full Text: PDF
References
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2. Baker, M., Faber, X.: Metrized graphs, Laplacian operators, and electrical networks. In: Quantum Graphs and Their Applications. Contemp. Math., vol. 415, pp. 15-33. Am. Math. Soc., Providence (2006)
3. Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766-788 (2007)
4. Baker, M., Rumely, R.S.: Harmonic analysis on metrized graphs. Can. J. Math. 59(2), 225-275 (2007)
5. Balacheff, F.: Invariant d'Hermite du réseau des flots entiers d'un graphe pondéré. Enseign. Math. (2) 52(3-4), 255-266 (2006)
6. Biggs, N.: Algebraic potential theory on graphs. Bull. Lond. Math. Soc. 29(6), 641-682 (1997)
7. Coyle, L.N., Lawler, G.F.: Lectures on Contemporary Probability. Student Mathematical Library, vol.
2. American Mathematical Society, Providence (1999)
8. Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. Carus Mathematical Monographs, vol.
22. Mathematical Association of America, Washington (1984)
9. Faber, X.W.C.: The geometric Bogomolov conjecture for curves of small genus. Exp. Math. 18(3), 347-367 (2009)
10. Flanders, H.: A new proof of R. Foster's averaging formula in networks. Linear Algebra Appl. 8, 35-37 (1974)
11. Foster, R.M.: The average impedance of an electrical network. In: Reissner Anniversary Volume. Contributions to Applied Mechanics, pp. 333-340. Edwards, Ann Arbor (1949)
12. Haase, C., Musiker, G., Yu, J.: Linear systems on tropical curve (2009). [math.AG]
13. Kotani, M., Sunada, T.: Jacobian tori associated with a finite graph and its abelian covering graphs. Adv. Appl. Math. 24(2), 89-110 (2000)
14. Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. In: Alexeev, V. et al.
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