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