Torelli theorem for graphs and tropical curves

Abstract

Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Finally we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

Final version incorporating the referee's suggestions. To appear in DMJ. 30 pages

References in corpus (3)

• Tropical Geometry and its applications
• Enumerative tropical algebraic geometry in R2
• Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)

Cited by in corpus (29)

• The tropicalization of the moduli space of curves
• On the tropical Torelli map
• Equations of tropical varieties
• Algebraic and tropical curves: comparing their moduli spaces
• A moduli stack of tropical curves
• Alternating links and definite surfaces
• Geometry of tropical moduli spaces and linkage of graphs
• A note on Brill-Noether thoery and rank determining sets for metric graphs
• Hodge theory for tropical varieties
• Invariant Differential Forms on Complexes of Graphs and Feynman Integrals
• Tropicalization of the moduli space of stable maps
• The geometry and combinatorics of cographic toric face rings
• Towards a tropical Hodge bundle
• Maximal harmonic group actions on finite graphs
• Lattice of integer flows and poset of strongly connected orientations
• Combinatorics of the tropical Torelli map
• Canonical systems and their limits on stable curves
• A construction of the graphic matroid from the lattice of integer flows
• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values
• Matrix polynomials, generalized Jacobians, and graphical zonotopes
• Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves
• The tropical Manin-Mumford conjecture
• Voronoi tilings, toric arrangements and degenerations of line bundles III
• The Abel Jacobi map for Mumford Curves via Integration
• Combinatorial Iterated Integrals and the Harmonic Volume of Graphs
• A Torelli Theorem for Graph Isomorphisms
• A Torelli theorem for graphs via quasistable divisors
• The Laplacian lattice of a graph under a simplicial distance function
• Torelli theorem for stable curves