ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53, 55, 58 Joyce, Dominic, Lincoln College, Oxford, England Compact Riemannian manifolds with exceptional holonomy

In the classification of Riemannian holonomy groups, the most mysterious cases are the exceptional holonomy groups G_{2} in 7 dimensions, and Spin(7) in 8 dimensions. We outline the construction of the first known examples of compact 7- and 8-manifolds with holonomy G_{2} and Spin(7). In the case of G_{2}, we first choose a flat G_{2}-structure on the torus T^{7}, preserved by a finite group G of diffeomorphisms of T^{7}. Then T^{7}/G is an orbifold. Secondly, we resolve the singularities of T^{7}/G to get a compact 7-manifold M. Finally we use analysis, and an understanding of Calabi-Yau metrics, to construct a family of metrics with holonomy G_{2} on M, which converge to the singular metric on T^{7}/G.

ICM 1998 Berlin · Plenary and Invited Lectures