The Calabi metric and desingularization of Einstein orbifolds

TL;DR

This paper investigates the process of desingularizing Einstein orbifolds using Calabi metrics, identifying obstructions to perturbing approximate metrics into true Einstein metrics, and establishing conditions for existence or non-existence of such metrics.

Contribution

It extends Calabi's construction to higher dimensions, identifying the first obstruction to Einstein metric perturbation and providing existence and non-existence results in asymptotically hyperbolic cases.

Findings

Identifies the first obstruction to perturbing approximate Einstein metrics.

Provides conditions under which desingularizations admit Einstein metrics.

Extends Biquard's 2D construction to higher dimensions.

Abstract

Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat K\"ahler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate Einstein metric to an actual Einstein metric. If $M_0$ is compact, we can use this to produce examples of Einstein orbifolds which do not admit any Einstein metric in a neighborhood of the natural approximate Einstein metric on the desingularization. In the case that $(M_0,g_0)$ is asymptotically hyperbolic Einstein and non-degenerate, we show that if the first obstruction vanishes, then there does in fact exist an asymptotically hyperbolic Einstein metric on the desingularization. We also obtain a non-existence result in the asymptotically hyperbolic Einstein case, provided that the obstruction does not vanish. This work extends a construction of Biquard in the case $n =2$, in which case the Calabi metric is also known as the Eguchi-Hanson metric, but there are some key points for which the higher-dimensional case differs.