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Metric properties of the tropical Abel-Jacobi map. (English) Zbl 1215.14060

In algebraic geometry, we have the Abel–Jacobi map \(\Phi_p\) from a Riemann surface \(X\) to \(J(X)\), the Jacobian of \(X\), depending on a base point \(p\in X\). A somewhat parallel combinatorial theory is developed by the authors for finite metric weighted graphs, following previous works by various authors (Bacher, de la Harpe, Nagnibeda, Baker, Norine, Balacheff, Kotani, Sunada, Mikhalkin, Zharkov). Some results in this combinatorial theory have no clear analog in the classical theory, though.
The link connecting these two theories are tropical curves. In this paper, a tropical curve is a compact metric graph \(\Gamma\) of finite total length. It is proved that \(J(\Gamma)\) is canonically isomorphic to the direct limit of \(J(G)\), where \(G\) runs through all models of \(\Gamma\). A characterization of those \(G\) having finite Jacobian \(J(G)\) is given. This allows reducing questions about the tropical Abel–Jacobi map \(\Phi_p\) (defined by Mikhalkin and Zharkov) to simpler questions about weighted graphs.
The authors consider a number of natural “metrics” on \(J(\Gamma)\) (quotation marks meaning here that the triangle inequality is not always satisfied): Foster–Zhang’s metric, Euclidean “metric” and tropical “metric”. For each such“metric” and for an edge \(e\) of \(\Gamma\), the authors give formulas relating the length of \(\Phi_p(e)\) with the length \(l(e)\) of \(e\) in some model for \(\Gamma\). In particular, the authors prove that \(\Phi_p\) is a tropical isometry onto the image, away from bridges. An interpretation of the Foster coefficients is given in terms of weighted spanning trees, etc.

MSC:

14T05 Tropical geometry (MSC2010)
05C22 Signed and weighted graphs
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