Spin(7)-Orbifold Resolutions

We establish a general analytic and geometric framework for resolving Spin(7)–orbifolds. These spaces arise naturally as boundary points in the moduli space of exceptional holonomy metrics, and smooth Gromov–Hausdorff resolutions can be viewed as paths from the boundary to the interior of the moduli space. Our construction combines algebraic and symplectic techniques with analytic gluing methods: singular strata are replaced by families of adiabatic torsion–free, asymptotically conical fibred spaces whose existence is controlled by the analysis of Dirac–type operators on orbifold resolutions. We formulate obstruction conditions in terms of string cohomology, prove existence results for torsion–free Spin(7)–structures extending Joyce’s theorem arXiv:math/9910002 to the setting of orbifold resolutions, and construct new families of compact Spin(7)–manifolds. By dimensional reduction, our results also recover, generalise and extend the results of Joyce and Karigiannis arXiv:1707.09325 on $G_2$–orbifold resolutions. This unifies and extends all known resolution methods for exceptional holonomy orbifolds.