K\"ahler--Einstein metrics on quasi-projective manifolds

TL;DR

This paper proves the existence and convergence of certain singular K"ahler--Einstein metrics on quasi-projective manifolds, advancing understanding of their geometric properties and limits.

Contribution

It establishes the almost-completeness of Berman--Guenancia's metric and demonstrates the weak convergence of conic K"ahler--Einstein metrics under broader conditions.

Findings

Singular K"ahler--Einstein metric is almost-complete on $X ackslash D$.

Conic K"ahler--Einstein metrics converge weakly to the singular metric.

Results partly answer a recent question by Biquard--Guenancia.

Abstract

Let $X$ be a compact K\"ahler 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 K\"ahler--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 K\"ahler--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.