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2004 | Buch | 2. Auflage

Introductory Preface Kapitel lesen Erstes Kapitel lesen

Torus Actions on Symplectic Manifolds

verfasst von: Michèle Audin

Verlag: Birkhäuser Basel

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

How I have (re-)written this book The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work-the first edition has been sold out for a few years. There was absolutely no question of just correcting numerous misprints and a few mathematical errors. When I wrote the first edition, in 1989, the convexity and Duistermaat-Heckman theorems together with the irruption of toric varieties on the scene of symplectic geometry, due to Delzant, around which the book was organized, were still rather recent (less than ten years). I myself was rather happy with a small contribution I had made to the subject. I was giving a post-graduate course on all that and, well, these were lecture notes, just lecture notes. By chance, the book turned out to be rather popular: during the years since then, I had the opportunity to meet quite a few people(1) who kindly pretended to have learnt the subject in this book. However, the older book does not satisfy at all the idea I have now of what a good book should be. So that this "new edition" is, indeed, another book.

Inhaltsverzeichnis

Frontmatter
Introductory Preface
Abstract
The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work—the first edition has been sold out for a few years.
Michèle Audin
Chapter I. Smooth Lie Group Actions on Manifolds
Abstract
In this chapter, we list the basic definitions and properties of Lie group actions. Then we investigate the actions of the circle S1 on surfaces and 3-manifolds.
Michèle Audin
Chapter II. Symplectic Manifolds
Abstract
In this chapter, we define symplectic manifolds, give their basic properties, and give a few examples of such objects, among which the Hλis we have already met: Hλ is the set of all n × n Hermitian matrices with given spectrum \( \lambda = \left( {{\lambda _{1, \ldots ,}}{\lambda _n}} \right) \in {R^n}\).We already know that Hλ is indeed a manifold, as this is an orbit of the compact group U(n) (see Example 1.1.6) acting by conjugation on the vector space H of all Hermitian matrices. We will prove that they are symplectic manifolds.
Michèle Audin
Chapter III. Symplectic and Hamiltonian Group Actions
Abstract
In this chapter, we define symplectic and Hamiltonian actions. These are the main topics of this book, so that we spend some time on their properties. Hamiltonian actions of tori of maximal dimension are a special case of integrable systems. We prove (this is the Arnold―Liouville theorem), that they are also the local form of all integrable systems with compact level sets.
Michèle Audin
Chapter IV. Morse Theory for Hamiltonians
Abstract
This chapter is the heart of the book. Quite a few spectacular theorems will be proved. The main ones are
  • the famous convexity theorem of Atiyah [7] and Guillemin—Sternberg [63] which asserts that the image of a compact connected symplectic manifold under the momentum mapping of a Hamiltonian torus action is a convex polyhedron (this is Theorem IV.4.3)
  • the uniqueness theorem of Delzant [39], according to which, when the torus acting is half the dimension of the manifold, the polyhedron determines the manifold (this is Theorem IV.4.20).
Michèle Audin
Chapter V. Moduli Spaces of Flat Connections
Abstract
In this chapter, we consider the moduli space of flat connections on a (trivial) bundle over a surface (defined in § V.1). There is a Hamiltonian group action that allows us to endow this space with a Poisson structure, as we show in § V.2 (this is due to Atiyah and Bott [10] in the case of a closed surface and to Fock and Roslyi [50] in the general case of a surface with boundary). We then look at a special case in § V.3, in which there is an integrable system (due to Goldman [56]) on the moduli space. Following Jeffrey and Weitsman [77], we exhibit a torus action and its momentum mapping, that is, action-angle variables for this system.
Michèle Audin
Chapter VI. Equivariant Cohomology and the Duistermaat-Heckman Theorem
Abstract
Another famous and spectacular theorem in the theory of Hamiltonian torus actions is the Duistermaat-Heckman Theorem [44] (as we have learned since that time, similar results had been proved by Karasev [81] in 1981). This theorem has two versions, both asserting, once again, the importance of linear phenomena in the theory.
Michèle Audin
Chapter VII. Toric Manifolds
Abstract
The goal of this chapter is to present a very beautiful family of symplectic manifolds endowed with Hamiltonian torus actions, the toric varieties (1). Long before their symplectic aspects were understood, toric manifolds were introduced by Demazure as closures of complex torus orbits in algebraic manifolds. They have been investigated since then by numerous authors(2). These are algebraic varieties which can be defined over any field (of course we shall restrict ourselves to the field of complex numbers). The prototype is the closure of any orbit of a (complex) torus acting in a linear way on a projective space which we already met in Theorem IV.4.25. We shall present here an alternative description.
Michèle Audin
Chapter VIII. Hamiltonian Circle Actions on Manifolds of Dimension 4
Abstract
In this chapter, we will investigate symplectic 4-manifolds endowed with Hamiltonian circle actions. Together with what was done in the previous chapters, this will give us a rather complete description of all the compact symplectic manifolds endowed with a Hamiltonian group action.
Michèle Audin
Backmatter
Metadaten
Titel
Torus Actions on Symplectic Manifolds
verfasst von
Michèle Audin
Copyright-Jahr
2004
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-7960-6
Print ISBN
978-3-0348-9637-5
DOI
https://doi.org/10.1007/978-3-0348-7960-6