Our paper [OO] is a sequel to a series by the first author [Od14, Od16], which compactified both the moduli space of compact Riemann surfaces $M_{g}(g\geq 2)$ and that of principally polarized abelian varieties $A_{g}$. In each case, as we actually expect an analogue for any moduli of general polarized Kähler-Einstein varieties with non-positive scalar curvatures, we introduce and study two similar (non-variety) compactifications of the moduli space $\mathcal{M}$, which we denote by $\overline{\mathcal{M}}^{\rm GH}$ and $\overline{\mathcal{M}}^{\rm T}$. The former $\overline{\mathcal{M}}^{\rm GH}$ is the Gromov-Hausdorff compactification with respect to rescaled Kähler-Einstein metrics of fixed diameters and the latter “tropical geometric compactification” $\overline{\mathcal{M}}^{\rm T}$ should dominate the former $\overline{\mathcal{M}}^{\rm GH}$ as its boundary $\partial\overline{\mathcal{M}}^{\rm T}$ encodes more structure of the Gromov-Hausdorff limits (collapses) rather than just distance structure. For a precise definition of $\overline{\mathcal{M}}^{\rm GH}$ we employ the same definition as [Od14, §2.3], [Od16, §2.2].¹¹ 1 However, its compactness is unknown at least to the authors in higher dimensional negative scalar curvature case. For $\overline{\mathcal{M}}^{\rm T}$, we have a case by case definition for only particular classes of varieties. Here, we recall the structure theorem of $\overline{A_{g}}^{\rm GH}$ from [Od16, Theorems 2.1 2.3 and Corollary 2.5].

In the above case, we simply set $\overline{A_{g}}^{\rm T}:=\overline{A_{g}}^{\rm GH}$. On the other hand, in the analogue for $M_{g}$ [Od14], we distinguish $\overline{M_{g}}^{\rm GH}$ and $\overline{M_{g}}^{\rm T}$, where the boundaries of $\overline{M_{g}}^{\rm GH}$ (resp., $\overline{M_{g}}^{\rm T}$) parametrize metrized graphs (resp., metrized graphs with integer weights on the vertices). We refer the details to [Od14].