Riemannian Holonomy and Algebraic Geometry

Details Riemannian Holonomy and Algebraic Geometry Riemannian Holonomy and Algebraic Geometry

1999-02-18

arxiv.org/abs/math/9902110v1

Mathematics

Algebraic Geometry

27 pages, Plain TeX with xypic

Scientific paper

This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives a complete (and rather small) list of the groups which can appear. Surprisingly, the compact manifolds with holonomy smaller than SO(n) are all related in some way to Algebraic Geometry. This leads to the study of special algebraic varieties (Calabi-Yau, complex symplectic or complex contact manifolds) for which Riemannian geometry rises interesting questions.

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