Moment Maps and Geometric Invariant Theory

These are expanded notes from a set of lectures given at the school “Actions Hamiltoniennes: leurs invariants et classification” at Luminy in April 2009. The topics center around the theorem of Kempf and Ness [58], which describes the equivalence between the notion of quotient in geometric invariant theory introduced by Mumford in the 1960’s [80], and the notion of symplectic quotient introduced by Meyer [79] and Marsden-Weinstein [77] in the 1970’s. Infinite-dimensional generalizations of this equivalence have played an increasingly important role in geometry, starting with the theorem of Narasimhan and Seshadri [81] connecting unitary structures on a bundle with holomorphic stability, which by historical accident preceded the finite-dimensional theorem. The proof of the Kempf-Ness theorem depends on the convexity of certain Kempf-Ness functions whose minima are zeros of the moment map. The convexity also plays an important role in the relation to