Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy

Abstract

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mni, of riemannian manifolds with special holonomy. We show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension ≥4. Additional results on the the detailed structure of the singular set support our main conjecture that if the Mni are compact and a certain characteristic number, C(Mni), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.