A new construction of compact 8-manifolds with holonomy Spin(7)

Abstract

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. In a previous paper (Invent. math. 123 (1996), 507-552) the author constructed the first examples of compact 8-manifolds with holonomy Spin(7), by resolving orbifolds T^8/G, where T^8 is the 8-torus and G a finite group of automorphisms of T^8. This paper describes a different construction of compact 8-manifolds with holonomy Spin(7). We start with a Calabi-Yau 4-orbifold Y with isolated singularities, and an isometric, antiholomorphic involution \sigma of Y fixing only the singular points. Let Z=Y/<\sigma>. Then Z is an orbifold with isolated singularities, and a natural Spin(7)-structure. We resolve the singular points of Z to get a compact 8-manifold M, and show that M has holonomy Spin(7). Taking Y to be a hypersurface in a complex weighted projective space, we construct new examples of compact 8-manifolds with holonomy Spin(7), and calculate their Betti numbers b^k. The fourth Betti number b^4 tends to be rather large, as high as 11,662 in one example.