Notes on GIT and symplectic reduction for bundles and varieties

Abstract

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.

Expanded version of talk at JDG 2005 conference, for "Surveys in Differential Geometry". 51 pages, 15 figures. Final version

References in corpus (2)

• An obstruction to the existence of constant scalar curvature Kähler metrics
• Lower bounds on the Calabi functional

Cited by in corpus (10)

• Numerical Calabi-Yau metrics
• Extremal metrics on blow ups
• Exactly marginal deformations from exceptional generalised geometry
• Extremal metrics and K-stability (PhD thesis)
• On energy functionals, Kahler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood
• Generalising G${}_{\text{2}}$ geometry: involutivity, moment maps and moduli
• Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics
• Eulerian digraphs and toric Calabi-Yau varieties
• The Hitchin-cscK system
• Cotangent bundles of toric varieties and coverings of toric hyperkähler manifolds