Higher order obstructions to the desingularization of Einstein metrics

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 Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler 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.

Keywords

Einstein $4$-manifolds, desingularization, reduced holonomy

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The author would like to thank his PhD advisor, Olivier Biquard, for his help and interest in the early stages of this project.

Received 31 March 2021

Published 22 March 2022