Spin(7)-Orbifold Resolutions

Abstract

We develop an analytic and geometric framework for resolving compact Spin(7)-orbifolds by smooth torsion-free Spin(7)-manifolds. These orbifolds arise naturally as boundary points in the Gromov--Hausdorff compactification of the moduli space of exceptional holonomy metrics, and smooth Gromov--Hausdorff resolutions can be viewed as paths from the boundary back into the smooth part of the moduli space. Our construction replaces the singular strata by adiabatic torsion-free asymptotically conically fibred spaces. The local resolution data are encoded by McKay-type correspondences and Chen--Ruan local systems, while the global deformation problem is controlled by the uniform elliptic theory for Dirac-type operators on orbifold resolutions developed in the author's previous work. In particular, the obstruction map and the associated isentropicity condition from that theory provide the criterion for whether the local harmonic resolution data glue to global harmonic forms on the smooth resolution. In this paper, we link the vanishing of the resulting obstruction map to the string cohomology of the orbifold. When this obstruction vanishes, we deform the preglued Spin(7)-structure to a genuine torsion-free Spin(7)-structure. This extends Joyce's resolution theorem to the nonflat case and yields new families of compact Spin(7)-manifolds. By dimensional reduction, the same framework recovers and extends the Joyce--Karigiannis theory of G2-orbifold resolutions.