Higher order obstructions to the desingularization of Einstein metrics

We exhibit new obstructions to the desingularization of Einstein metrics in dimension 4. These obstructions are specific to the compact situation and raise the question of whether or not a sequence of Einstein metrics degenerating while bubbling out gravitational instantons has to be Kähler-Einstein. We then test these obstructions to discuss the possibility of producing a Ricci-flat but not Kähler metric by the promising desingularization configuration proposed by Page in 1981.