Riemannian Holonomy and Algebraic Geometry

TL;DR

This survey explores the connection between Riemannian manifolds with special holonomy and algebraic geometry, highlighting how certain geometric structures relate to special algebraic varieties like Calabi-Yau and symplectic manifolds.

Contribution

It provides a comprehensive overview of the relationship between special holonomy groups in Riemannian geometry and algebraic geometric structures, emphasizing their interplay.

Findings

Compact manifolds with reduced holonomy relate to algebraic varieties.

Special holonomy groups correspond to important classes of algebraic varieties.

The study reveals deep links between Riemannian geometry and algebraic geometry.

Abstract

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.