Kähler--Einstein metrics on quasi-projective manifolds

TL;DR

The paper constructs an almost-complete singular Kähler--Einstein metric $\omega_D$ on the quasi-projective manifold $X\setminus D$ when $K_X+D$ is big and nef, showing it extends to a current of full Monge--Ampère mass with $\mathrm{Ric}\,\omega_D=-\omega_D+[D]$ and coincides with the BEGZ metric. It also proves that conic KE metrics $\omega_\epsilon$ with negative curvature converge weakly to $\omega_D$ as $\epsilon\to 0^+$ under the weaker hypothesis that $K_X+D$ is merely big. A central technical contribution is a robust stability analysis for complex Monge--Ampère equations using a quantitative domination principle, which yields capacity-based monotonicity and $L^1$ convergence of potentials even when the right-hand side can have unbounded mass. The results advance the understanding of degeneration of canonical metrics on quasi-projective varieties and establish a framework that could apply to broader degeneration problems in complex geometry.

Abstract

Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular Kähler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X \backslash D$ in the sense of Tian--Yau. In our second main result, we establish the weak convergence of conic Kähler--Einstein metrics of negative curvature to the above-mentioned metric when $K_X+D$ is merely big, answering partly a recent question posed by Biquard--Guenancia. Potentials of low energy play an important role in our approach.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: Non-quantitative domination principle
• Theorem 2.2: Do-Vu_quantitative
• Definition 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4