Local models for conical K\"ahler-Einstein metrics

TL;DR

This paper constructs new local models for conical Kähler-Einstein metrics using the Calabi ansatz, revealing detailed structures of singularities and cusps in complex hyperbolic moduli spaces.

Contribution

It introduces a method to produce regular Calabi-Yau cones and negative Ricci Kähler-Einstein metrics with cuspidal points, advancing understanding of singularities in complex geometry.

Findings

Construction of Calabi-Yau cones with conical singularities

Description of cuspidal ends in complex hyperbolic metrics

Application to moduli spaces of points in projective line

Abstract

In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and K\"ahler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.