K\"ahler-Einstein metrics: from cones to cusps

TL;DR

This paper proves that K"ahler-Einstein cusp metrics can be obtained as limits of conic metrics with shrinking cone angles on certain compact K"ahler manifolds, and studies their boundary behavior.

Contribution

It establishes the convergence of conic K"ahler-Einstein metrics to cusp metrics as the cone angle approaches zero, and analyzes the rescaled boundary limits.

Findings

Conic metrics converge to cusp metrics as cone angle tends to zero.

Rescaled metrics near the boundary converge to cylindrical metrics.

Provides a strong sense convergence result for these metrics.

Abstract

In this note, we prove that on a compact K\"ahler manifold $X$ carrying a smooth divisor $D$ such that $K_X+D$ is ample, the K\"ahler-Einstein cusp metric is the limit (in a strong sense) of the K\"ahler-Einstein conic metrics when the cone angle goes to $0$. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on $\mathbb C^*\times \mathbb C^{n-1}$.