Moment Maps and Combinatorial Invariants of Hamiltonian Tn-Spaces
The action of a compact Lie group, G, on a compact sympletic
manifold gives rise to some remarkable combinatorial invariants.
The simplest and most interesting of these is the moment polytope,
a convex polyhedron which sits inside the dual of the Lie algebra
of G. One of the main goals of this monograph is to describe what
kinds of geometric information are encoded in this polytope. For
instance, the first chapter is largely devoted to the Delzant
theorem, which says that there is a one-one correspondence between
certain types of moment polytopes and certain types of symplectic
G-spaces. (One of the most challenging unsolved problems in
symplectic geometry is to determine to what extent Delzant s
theorem is true of every compact symplectic G-Space.)
The moment polytope also encodes quantum information about the
actions of G. Using the methods of geometric quantization, one can
frequently convert this action into a representations, p , of G on
a Hilbert space, and in some sense the moment polytope is a
diagrammatic picture of the irreducible representations of G which
occur as subrepresentations of p. Precise versions of this item of
folklore are discussed in Chapters 3 and 4. Also, midway through
Chapter 2 a more complicated object is discussed: the
Duistermaat-Heckman measure, and the author explains in Chapter 4
how one can read off from this measure the approximate
multiplicities with which the irreducible representations of G
occur in p. This gives an excuse to touch on some results which are
in themselves of great current interest: the Duistermaat-Heckman
theorem, the localization theorems in equivariant cohomology of
Atiyah-Bott and Berline-Vergne and the recent extremely exciting
generalizations of these results by Witten, Jeffrey-Kirwan,
Lalkman, and others.
The last two chapters of this book are a self-contained and
somewhat unorthodox treatment of the theory of toric varieties in
which the usual hierarchal relation of complex to symplectic is
reversed. This book is addressed to researchers
"...carefully written book [on] the so-called Delzant spaces... Many of the statements in the body of the book are left as exercises (with copious hints). At the end of each chapter there are very interesting bibliographical notes. The reader who works through this book will acquire a solid knowledge of this beautiful and elegant part of mathematics." --Mathematical Reviews
Blick ins Buch "Moment Maps and Combinatorial Invariants of Hamiltonian Tn-Spaces"
Inhaltsangabe
Preface.-Introduction.-Basic Definitions and Examples.-The Duistermaat-Heckman Theorem.-Multiplicities as Invariants of Reduced Spaces.-Partition Functions.-Appendix 1: Toric Varieties.-Appendix 2. Kaehler Structures on Toric Varieties.-References.
