Event description
Abstract: Complete scalar-flat Kähler ALE spaces appear as bubbles in degenerations of cscK manifolds. Conversely, they serve as building blocks in gluing constructions such as blow-ups and resolutions of orbifolds. After recalling this picture, I will present two opposite features of their geometry. On the flexible side, inserting a Burns–Simanca model far out in the ALE end lets point blow-ups realise any ADM mass not smaller than the original one, in a fixed asymptotic chart. This yields, for instance, zero-mass non-Ricci-flat metrics from negative-mass ones. On the rigid side, a Gromov–Hausdorff compactness theorem for holomorphic isometries shows that complete non-compact Kähler manifolds with asymptotically non-negative Ricci curvature admit no holomorphic isometry into a finite-dimensional CP^N. In addition, a Pringsheim–Calabi obstruction rules out every rescaling of Gibbons–Hawking, Hwang–Singer and other Ricci- and scalar-flat models, even into CP^∞. The Burns–Simanca metric is the only exception.The final part of this talk is based on joint works with Loi, Placini and Zedda.