Graduate lecture course, 16 lectures, Michaelmas Term 2006.

Professor Joyce

Riemannian holonomy groups and calibrated geometry

Wednesdays 11am in L3 and Fridays 11am in SR1, the Maths Institute, Oxford.

Riemannian holonomy groups is an area of Riemannian geometry, in the field of Differential Geometry. The holonomy group Hol(g) of a Riemannian manifold (M,g) determines the geometrical structures on M compatible with g. Thus, Berger's classification of Riemannian holonomy groups gives a list of interesting geometrical structures compatible with a Riemannian metric. Most of  the holonomy groups on Berger's list are important in String Theory.

Given some class of mathematical objects, there is often a natural class of subobjects living inside them, such as groups and subgroups for instance. The natural subobjects of Riemannian manifolds (M,g) with special holonomy are calibrated submanifolds --- lower-dimensional, volume-minimizing submanifolds N in M compatible with the geometric structures coming from the holonomy reduction. Calibrated submanifolds are also important in String
Theory, as 'supersymmetric cycles' or 'branes'.

This lecture course is aimed at graduate students in Geometry, Mathematical Physics, or String Theory, and should be accessible to new graduates. A major focus of the course will be Calabi-Yau manifolds. It should be a good preparation for Dr de la Ossa's graduate lecture course on Calabi-Yau Manifolds and Mirror Symmetry in Hilary Term 07. The main prerequisites are knowledge of Differentiable Manifolds and Lie Groups.


Synopsis

Lectures 1-2: Background material. Smooth manifolds. Tensors and exterior forms. The exterior derivative and de Rham cohomology. Riemannian metrics. Connections and curvature. Lie groups.

Lectures 3-4: Riemannian holonomy groups. The holonomy group of a Riemannian metric. Relation with constant tensors. Berger's classification, sketch of proof. Discussion of each case.

Lectures 5-7: Complex and Khler geometry. Complex manifolds and holomorphic functions. Complex submanifolds of CPn and complex algebraic geometry. Holomorphic line bundles and sections. Khler metrics on complex manifolds. De Rham cohomology of Khler manifolds, the Khler class.

Lectures 8-10: The Calabi Conjecture and Calabi-Yau manifolds. The Conjecture and its proof. Ricci-flat Khler manifolds and Calabi-Yau manifolds. Finding examples of Calabi-Yau manifolds using algebraic geometry. String Theory, and Mirror Symmetry of Calabi-Yau 3-folds.

Lectures 11-12: The holonomy groups Sp(m), Sp(m)Sp(1), G2 and Spin(7). Brief introduction to the geometry of each holonomy group; compact manifolds with these holonomy groups.

Lectures 13-14: Introduction to calibrated geometry. Submanifolds. Minimal submanifolds of Riemannian submanifolds. Calibrations and calibrated submanifolds. Natural calibrations on manifolds with special holonomy. Geometric Measure Theory and calibrated currents.

Lectures 15-16: Special Lagrangian submanifolds of Calabi-Yau manifolds. Symplectic manifolds and Lagrangian submanifolds. McLean's Theorem on deformations of compact special Lagrangians. The SYZ Conjecture. Singularities of special Lagrangians.