The Moment-Weight Inequality and the Hilbert–Mumford Criterion: GIT from the Differential Geometric Viewpoint

Description

This book provides an introduction to geometric invariant theory from a differential geometric viewpoint.  It is inspired by certain infinite-dimensional analogues of geometric invariant theory that arise naturally in several different areas of geometry. The central ingredients are the moment-weight inequality relating the Mumford numerical invariants to the norm of the moment map, the negative gradient flow of the moment map squared, and the Kempf--Ness function. The exposition is essentially self-contained, except for an appeal to the Lojasiewicz gradient inequality. A broad variety of examples illustrate the theory, and five appendices cover essential topics that go beyond the basic concepts of differential geometry. The comprehensive bibliography will be a valuable resource for researchers. The book is addressed to graduate students and researchers interested in geometric invariant theory and related subjects.  It will be easily accessible to readers with a basic understanding of differential geometry and does not require any knowledge of algebraic geometry.  Preface Contents 1 Introduction 2 The Moment Map 3 The Moment Map Squared 4 The Kempf–Ness Function 5 μ-Weights 6 The Moment-Weight Inequality 7 Stability in Symplectic Geometry 8 Stability in Algebraic Geometry 9 Rationality 10 The Dominant μ-Weight 11 Torus Actions 12 The Hilbert–Mumford Criterion 13 Critical Orbits 14 Examples A Nonpositive Sectional Curvature B The Complexified Group C The Homogeneous Space M=Gc/G D Toral Generators E The Partial Flag Manifold Gc/PG/C References Index