Constructing compact 8-manifolds with holonomy Spin(7) from Calabi-Yau orbifolds

Compact Riemannian 7- and 8-manifolds with holonomy G(2) arid Spin(7) were first constructed by the author in 1994-5, by resolving orbifolds T-7/Gamma and T-8/Gamma. This paper describes a new construction of compact 8-manifolds with holonomy Spin(7). We start with a Calabi-Yau 4-orbifold Y with isolated singularities of a special kind. We divide by an antiholomorphic involution a of Y to get a real 8-orbifold Z = Y/. Then we resolve tire singularities of Z to get a compact 8-manifold M, which has metrics with holonomy Spin(7). Manifolds constructed in this way typically have large fourth Betti number b(4)(M).