On the boundary behavior of K\"ahler-Einstein metrics on log canonical pairs

TL;DR

This paper investigates the boundary behavior of negatively curved K"ahler-Einstein metrics on log canonical pairs, providing precise estimates and describing singularities near the boundary, with implications for stable varieties.

Contribution

It offers new detailed estimates of KE metrics near boundaries and characterizes their singularities in general singular cases, advancing understanding of their boundary behavior.

Findings

Precise potential estimates near boundary divisors.

KE metrics exhibit mixed cone and cusp singularities.

Behavior of KE metrics on stable varieties in codimension one.

Abstract

In this paper, we study the boundary behavior of the negatively curved K\"ahler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the potential of the KE metric near the boundary $D$. In the more general singular case ($D$ being assumed effective though), we show that the KE metric has mixed cone and cusp singularities near $D$ on the snc locus of the pair. As a corollary, we derive the behavior in codimension one of the KE metric of a stable variety.