There is a continuous map $\Psi$ (call “geometric realization map”) from the Satake compactification $(M\subset)\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ with respect to the adjoint representation to the Gromov-Hausdorff compactification of $M$, extending the identity map on $M$. The $(b_{2}(X)-4)$-dimensional boundary strata of $\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ parametrize via $\Psi$ the projective space $\mathbb{P}^{n}$ with special Kähler metrics in the sense of [Freed99] and the metric space parametrized by $0$-dimensional cusps are all homeomorphic to the closed ball of dimension $n$.