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

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mⁿ_i, 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 Mⁿ_i are compact and a certain characteristic number, C(Mⁿ_i), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.The first author was partially supported by NSF Grant DMS 0104128 and the second by NSF Grant DMS 0302744.