Degeneration of Kahler-Einstein manifolds of negative scalar curvature

Jian Song

Published 2017-06-05Version 1

Let $\pi: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact K\"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique K\"ahler-Einstein metric on $\mathcal{X}_t= \pi^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g_t)$ 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.