Degeneration of Kahler-Einstein manifolds of negative scalar curvature

Let $\pi :{\mathcal{X}}^{\ast }\to B^{\ast }$ be an algebraic family of compact K"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^{\ast }\in \mathbb{C}$. Let $g_t$ be the unique K"ahler-Einstein metric on ${\mathcal{X}}_t={\pi }^{-1}(t)$. We show that as $t\to 0$, $({\mathcal{X}}_t,g_t)$ converges in pointed Gromov-Hausdorff topology to a unique finite disjoint union of complete metric length spaces ${\coprod }_{\alpha =1}^{\mathcal{A}}(Y_{\alpha },d_{\alpha })$ without loss of volume. Each $(Y_{\alpha },d_{\alpha })$ is a smooth open K"ahler-Einstein manifold of complex dimension n outside its closed singular set of Hausdorff dimension no greater than $2n-4$. Furthermore, ${\coprod }_{\alpha =1}^{\mathcal{A}}{Y}_{\alpha }$ is a quasi-projective variety isomorphic to $$\mathcal{X}$_0 \setminus LCS($\mathcal{X}$_0)$,where$$\mathcal{X}$_0$ is a projective semi-log canonical model and $LCS({\mathcal{X}}_0)$ is the non-log terminal locus of ${\mathcal{X}}_0$. This is the first step of our approach toward compactification of the analytic geometric moduli space of K"ahler-Einstein manifolds of negative scalar curvature.