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On the tropical Torelli map. (English) Zbl 1218.14056
This paper studies the tropical analogue of the Torelli map and the Schottky problem. The tropical Torelli map which sends a tropical curve to a tropical abelian variety has been defined and studied by other authors [G. Mikhalkin and I. Zharkov, “Tropical curves, their Jacobians and theta functions”, Contemporary Mathematics 465, 203–230 (2008; Zbl 1152.14028)]; however, some questions remained open. The authors prove tropical analogues of the Torelli theorem and solve the tropical Schottky problem. They apply their tropical techniques to answer a question posed by Namikawa about the compactified Torelli map.
In Chapter 3, tropical curves of genus \(g\) are defined and the moduli space of abstract tropical curves is constructed. The space also contains tropical curves with genus in vertices. The space is equipped with the structure of a stacky fan. The definition containes a small mistake which is corrected in a recent preprint by M. Chan [“Combinatorics of the tropical Torelli map”, arXiv:1012.4539].
Chapter 4 is concerned with tropical abelian varieties, i.e. real tori \(\mathbb R^g/\Lambda\) together with a positive semi-definite quadratic form \(Q\) on \(\mathbb R^g\). The moduli space of tropical abelian varieties thus coincides with the space parametrising positive semi-definite quadratic forms up to equivalence. The definition of tropical abelian variety was given by G. Mikhalkin and I. Zharkov [loc. cit.], but has been generalized by the authors to the situation of a tropical curve with genus in vertices. The space of positive semi-definite quadratic forms has been studied by other authors [Y. Namikawa, Toroidal compactification of Siegel spaces. Lect. Notes Math. 812. Berlin-Heidelberg-New York: Springer (1980; Zbl 0466.14011)].
In Chapter 5, the authors show that the tropical Torelli map is a map of stacky fans, and describe the image of this map, thus solving the tropical Schottky problem. They prove that the tropical Torelli map is of degree one onto its image. In Chapter 6, a planar analogue of the tropical Torelli map is studied and used to answer a question by Y. Namikawa [loc. cit.] concerning the image of curves whose dual graph is planar under the compactified Torelli map.

MSC:
14T05 Tropical geometry (MSC2010)
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