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We study the question of the integrability of Einstein deformations andrelate it to the question of the desingularization of Einstein metrics. Ourmain application is a negative answer to the long-standing question of whetheror not every Einstein $4$-orbifold (which is an Einstein metric space in asynthetic sense) is limit of smooth Einstein $4$-manifolds. We more preciselyshow that spherical and hyperbolic $4$-orbifolds with the simplestsingularities cannot be Gromov-Hausdorff limits of smooth Einstein $4$-metricswithout relying on previous integrability assumptions. For this, we analyze theintegrability of deformations of Ricci-flat ALE metrics through variations ofSchoen's Pohozaev identity. Inspired by Taub's preserved quantity in GeneralRelativity, we also introduce preserved integral quantities based on thesymmetries of Einstein metrics. These quantities are obstructions to theintegrability of infinitesimal Einstein deformations "closing up" inside ahypersurface - even with change of topology. We show that many previouslyidentified obstructions to the desingularization of Einstein $4$-metrics areequivalent to these quantities on Ricci-flat cones. In particular, all of theobstructions to desingularizations bubbling out Eguchi-Hanson metrics arerecovered. This lets us further interpret the obstructions to thedesingularization of Einstein metrics as a defect of integrability.

Integrability of Einstein deformations and desingularizations