Zoom link: https://princeton.zoom.us/j/594605776

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.