On the tropical Torelli map

TL;DR

This paper constructs and analyzes the tropical Torelli map between moduli spaces of tropical curves and abelian varieties, establishing its properties and solving the tropical Schottky problem with implications for classical Torelli questions.

Contribution

It introduces the tropical Torelli map in the stacky fan setting, studies its fibers and image, and addresses the tropical Schottky problem, providing new insights into tropical and classical Torelli theory.

Findings

Construction of tropical moduli spaces in stacky fans

Identification of the tropical Torelli map's fibers and image

Solution to the tropical Schottky problem and classical Torelli question

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.