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We compactify the classical moduli variety $A_g$ of principally polarizedabelian varieties of complex dimension $g$ by attaching the moduli of flat toriof real dimensions at most $g$ in an explicit manner. Equivalently, weexplicitly determine the Gromov-Hausdorff limits of principally polarizedabelian varieties. This work is analogous to the first of our series (availableat arXiv:1406.7772v2), which compactified the moduli of curves by attaching themoduli of metrized graphs.  Then, we also explicitly specify the Gromov-Hausdorff limits alongholomorphic family of abelian varieties and show that they form specialnon-trivial subsets of the whole boundary. We also do it for algebraic curvescase and observe a crucial difference with the case of abelian varieties.

Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -