Integrability of Einstein deformations and desingularizations

We study the question of integrability Einstein deformations and relate it to desingularization metrics. Our main application is a negative answer long-standing whether or not every 4-orbifold (which an metric space in synthetic sense) limit smooth 4-manifolds. more precisely show that spherical hyperbolic 4-orbifolds with simplest singularities cannot be Gromov-Hausdorff limits 4-metrics without relying on previous assumptions. For this, we analyze Ricci-flat ALE metrics through variations Schoen's Pohozaev identity. Inspired by Taub's conserved quantity General Relativity, also introduce integral quantities based symmetries These are obstructions infinitesimal “closing up” inside hypersurface – even change topology. many previously identified equivalent these cones. In particular, all desingularizations bubbling off Eguchi-Hanson recovered. This lets us further interpret as defect integrability.