2002/03/15 by Dominic Joyce, Joyce, Dominic · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG
paper · pdf · doi:10.48550/arxiv.math/0203158
17 pages. Lecture for Clay Institute School on Geometry and String Theory, Cambridge, March 2002
arxiv created 2002/03/15 · openalex publication_date 2002/03/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T7/G or T8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T7 or T8. There are also more complicated constructions which begin with a Calabi-Yau manifold or orbifold. All the material in this paper is covered in much more detail in the author's book, "Compact manifolds with special holonomy", Oxford University Press, 2000.