GÖKOVA GEOMETRY / TOPOLOGY PROCEEDINGS

Published in Proceedings of Gökova Geometry-Topology Conference 2005
Title The exceptional holonomy groups and calibrated geometry
Author Dominic Joyce
Abstract
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 exceptional holonomy, in two parts. Part I introduces the exceptional holonomy groups, and explains 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/ Γ or T8/ Γ, for Γ 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. Part II discusses the calibrated submanifolds of G2 and Spin(7)-manifolds: associative 3-folds and coassociative 4-folds for G2, and Cayley 4-folds for Spin(7). We explain the general theory, following Harvey and Lawson, and the known examples. Finally we describe the deformation theory of compact calibrated submanifolds, following McLean.
Pages110-139
Download PDF
 2005 Proceedings main page

Last updated: January 2007
Wed address: GokovaGT.org/proceedings/2005