Degeneration of Kahler-Einstein manifolds of negative scalar curvature

Abstract

Let $\pi :{\mathcal{X}}^{\ast }\to B^{\ast }$ be an algebraic family of compact Kähler 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 Kähler-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 Kähler-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 Kähler-Einstein manifolds of negative scalar curvature.

References in corpus (3)

• Riemannian geometry of Kahler-Einstein currents
• Ricci flow and birational surgery
• Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata's base point free theorem

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• Kahler-Einstein metric near an isolated log canonical singularity
• Existence of Kahler-Ricci solitons on smoothable Q-Fano varities
• The K"ahler-Ricci flow with Log Canonical Singularities
• On the construction of a complete Kahler-Einstein metric with negative scalar curvature near an isolated log-canonical singularity
• Global pluripotential theory on hybrid spaces
• An Analytic Proof of the Stable Reduction Theorem