Higher order obstructions to the desingularization of Einstein metrics

TL;DR

This paper identifies new obstructions to desingularizing Einstein orbifolds into smooth Einstein metrics, especially in the compact case, and explores their implications for Ricci-flat metrics and Kähler structures.

Contribution

It introduces 84 new obstructions specific to compact Einstein orbifolds and analyzes their impact on desingularization possibilities, including Ricci-flat and Kähler metrics.

Findings

Most flat orbifold metrics on T^4/Z_2 are unlikely to be limits of Ricci-flat metrics with generic holonomy.

A 14-dimensional family of desingularizations satisfies all 84 obstructions in symmetric cases.

New obstructions challenge previous assumptions about desingularization and metric limits.

Abstract

We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be K\"ahler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not K\"ahler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that almost all flat orbifold metrics on $\mathbb{T}^4/\mathbb{Z}_2$ should not be limit of Ricci-flat metrics with generic holonomy while bubbling out Eguchi-Hanson metrics. Perhaps surprisingly, in the most symmetric situation, we also identify a $14$-dimensional family of desingularizations satisfying all of our $84$ obstructions.