Tropical Geometric Compactification of Moduli, I - $M_g$ case -

Abstract

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which ${\bigsqcup }_g{M}_g$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_g$ (cf., [ACP]).

v2: Revision of the FORMER half (curve, $M_g$ case) of version 1. v3: revised exposition of v2. Re-submitted