[Paper Review] A new construction of compact 8-manifolds with holonomy Spin(7)

Dominic Joyce|arXiv (Cornell University)|Oct 1, 1999

Geometry and complex manifolds9 references19 citations

TL;DR

This paper presents a new construction of compact 8-manifolds with G2 holonomy by starting from Calabi–Yau 4-orbifolds with isolated singularities and an antiholomorphic isometric involution that fixes these singular points. The involution reduces the structure group from SU(4) to Spin(7), and the quotient orbifold is resolved using Asymptotically Locally Euclidean Spin(7)-manifolds to yield new compact 8-manifolds with full Spin(7) holonomy. The key contribution is the discovery of examples with middle Betti number b⁴ as large as 11,662, significantly exceeding previous constructions.

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 σof Y fixing only the singular points. Let Z=Y/. 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.

Motivation & Objective

• To develop a new method for constructing compact 8-manifolds with holonomy Spin(7), distinct from prior torus-based constructions.
• To explore the role of antiholomorphic involutions on Calabi–Yau 4-orbifolds in reducing structure group from SU(4) to Spin(7).
• To resolve quotient orbifolds Z = Y/⟨σ⟩ arising from such involutions using Asymptotically Locally Euclidean (ALE) Spin(7)-manifolds.
• To compute and analyze the Betti numbers of the resulting compact Spin(7)-manifolds, particularly the middle Betti number b⁴.
• To identify topological invariants that distinguish these new examples from previously known compact Spin(7)-manifolds.

Proposed method

• Start with a Calabi–Yau 4-orbifold Y with isolated singularities p₁,…,pₖ, modeled on C⁴/ℤ₄ or C⁴/{±1}.
• Introduce an antiholomorphic, isometric involution σ: Y → Y that fixes the singular points and preserves the induced Spin(7)-structure.
• Form the quotient orbifold Z = Y/⟨σ⟩, which inherits a torsion-free Spin(7)-structure with isolated singularities.
• Resolve the singularities of Z using ALE Spin(7)-manifolds asymptotic to R⁸/G, ensuring the resolution preserves the Spin(7) holonomy.
• Use analytic methods to deform the resulting family of nearly-parallel Spin(7)-structures to ones with zero torsion, yielding metrics with full Spin(7) holonomy.
• Compute Betti numbers via algebraic geometry techniques, particularly using Hodge theory and the topology of the resolution.

Experimental results

Research questions

• RQ1 Can compact 8-manifolds with Spin(7) holonomy be constructed from Calabi–Yau 4-orbifolds via antiholomorphic involutions?
• RQ2 What conditions on the involution and singularities ensure that the quotient orbifold can be resolved within the Spin(7) holonomy class?
• RQ3 How do the Betti numbers of the resulting manifolds compare to those from previous constructions based on torus quotients?
• RQ4 Are there Calabi–Yau 4-orbifolds with singularities that admit a crepant resolution in the Spin(7) setting, and which types of singularities allow this?
• RQ5 What is the range of possible values for the middle Betti number b⁴ in such constructions?

Key findings

• The construction yields at least 14 topologically distinct compact 8-manifolds with Spin(7) holonomy, none of which have the same Betti numbers as those from earlier torus-based constructions.
• The middle Betti number b⁴ reaches values as high as 11,662 in one example, far exceeding the previous upper bound of 162 observed in earlier constructions.
• For a family of hypersurfaces in weighted projective spaces, the Betti numbers are computed explicitly: b² = 4−k, b³ = 33, b⁴ = 200+2k for k = 0,…,4.
• The dimension of the moduli space of Spin(7) metrics on the resulting manifolds is 69+k, indicating a rich family of distinct holonomy metrics.
• Not all singularities of type C⁴/{±1} admit a crepant resolution within the Spin(7) holonomy class; only certain actions of the involution on the tangent space allow such a resolution.
• Examples with b⁴ = 11,662 arise from a Calabi–Yau 4-orbifold with 12 singularities, each modeled on C⁴/ℤ₄, and an involution with specific action on the tangent space.