The Moment-Weight Inequality and the Hilbert-Mumford Criterion : GIT from the Differential Geometric Viewpoint (Lecture Notes in Mathematics)

Georgoulas, Valentina/ Robbin, Joel W./ Salamon, Dietmar Arno

Springer Nature Switzerland AG（2021/11発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 192 p.
• 言語 ENG
• 商品コード 9783030892999

Full 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. 

Contents

- Introduction. - The Moment Map. - The Moment Map Squared. - The Kempf-Ness Function. - μ-Weights. - The Moment-Weight Inequality. - Stability in Symplectic Geometry. - Stability in Algebraic Geometry. - Rationality. - The Dominant μ-Weight. - Torus Actions. - The Hilbert-Mumford Criterion. - Critical Orbits.