On the tropical Torelli map

Abstract

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.

36 pages. Final section updated. Final version, to appear in Adv. Math

References in corpus (7)

• Enumerative tropical algebraic geometry in R2
• Moduli spaces of rational tropical curves
• Geometry of tropical moduli spaces and linkage of graphs
• Intersecting Psi-classes on tropical M_{0,n}
• Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
• A Note on Space Tiling Zonotopes
• Labeled floor diagrams for plane curves

Cited by in corpus (28)

• The tropicalization of the moduli space of curves
• Tropical curves, graph complexes, and top weight cohomology of M_g
• Tropical Amplitudes
• Algebraic and tropical curves: comparing their moduli spaces
• A moduli stack of tropical curves
• Moduli of Tropical Plane Curves
• Geometry of tropical moduli spaces and linkage of graphs
• The tropicalization of the moduli space of curves II: Topology and applications
• The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian
• Tropical complexes
• Schottky Algorithms: Classical meets Tropical
• Combinatorial tropical surfaces
• The Brill-Noether rank of a tropical curve
• Tropical moduli spaces as symmetric Delta-complexes
• A note on Brill-Noether thoery and rank determining sets for metric graphs
• Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus
• Riemann-Roch theory for weighted graphs and tropical curves
• Harmonic tropical morphisms and approximation
• Invariant Differential Forms on Complexes of Graphs and Feynman Integrals
• Tropicalization of the moduli space of stable maps
• Towards a tropical Hodge bundle
• Tropicalizing the space of admissible covers
• Tropical Gaussians: A Brief Survey
• Combinatorics of the tropical Torelli map
• A universal tropical Jacobian over $M_g^{\trop}$
• Tropical hyperelliptic curves
• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values
• Unstable cohomology of ${\mathsf{GL}}_{2n}(\mathbb{Z})$ and the odd commutative graph complex