The Moment-Weight Inequality and the Hilbert–Mumford Criterion

Beschreibung

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.
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. 

Provides the first complete and thorough treatment of GIT from a differential geometric viewpoint Treats Hamiltonian group actions on general, not necessarily projective, compact Kähler manifolds Presents another route to many of the main results of Mumford's theory which will have long lasting significance

Autor*in

Valentina Georgoulas

Themen in »The Moment-Weight Inequality and the Hilbert–Mumford Criterion«

Symplectic Geometry Kähler Manifold Hamiltonian Group Action Moment Map Mumford Weights Kempf-Ness Function Moment-weight Inequality Hilbert-Mumford Criterion Geometric Invariant Theory

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“The book is carefully written with detailed proofs and should be accessible to anyone with a basic background knowledge of (real and complex) differential geometry. … This book is an excellent addition to the literature on GIT and symplectic reduction and could serve as an ideal reference for a graduate lecture course or seminar, but also as a valuable source for researchers interested in GIT or in the problems in geometry where the ideas appear in an infinite-dimensional context.” (Markus Röser, Mathematical Reviews, May, 2023)
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