Torus Actions on Symplectic Manifolds

This is an extended second edition of "The Topology of Torus Actions on Symplectic Manifolds" published as Volume 93 in this series in 1991. The material and references have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains much more material, particularly lots of new examples and exercises.

Completely revised and enlarged second edition of PM 93, taking into account the state of the art in the subject

Autorentext
Michèle Audin; Professor of Mathematics at IRMA, Université de Strasbourg et CNRS, France.

Inhalt
Introductory preface.- How I have (re-)written this book.- Acknowledgements.- What I have written in this book.- I. Smooth Lie group actions on manifolds.- I.1. Generalities.- I.2. Equivariant tubular neighborhoods and orbit types decomposition.- I.3. Examples: S 1-actions on manifolds of dimension 2 and 3.- I.4. Appendix: Lie groups, Lie algebras, homogeneous spaces.- Exercises.- II. Symplectic manifolds.- II.1What is a symplectic manifold?.- II.2. Calibrated almost complex structures.- II.3. Hamiltonian vector fields and Poisson brackets.- Exercises.- III. Symplectic and Hamiltonian group actions.- III.1. Hamiltonian group actions.- III.2. Properties of momentum mappings.- III.3. Torus actions and integrable systems.- Exercises.- IV. Morse theory for Hamiltonians.- IV.1. Critical points of almost periodic Hamiltonians.- IV.2. Morse functions (in the sense of Bott).- IV.3. Connectedness of the fibers of the momentum mapping.- IV.4. Application to convexity theorems.- IV.5. Appendix: compact symplectic SU(2)-manifolds of dimension 4.- Exercises.- V. Moduli spaces of flat connections.- V.1. The moduli space of fiat connections.- V.2. A Poisson structure on the moduli space of flat connections.- V.3. Construction of commuting functions on M.- V.4. Appendix: connections on principal bundles.- Exercises.- VI. Equivariant cohomology and the Duistermaat-Heckman theorem.- VI.1. Milnor joins, Borel construction and equivariant cohomology.- VI.2. Hamiltonian actions and the Duistermaat-Heckman theorem.- VI.3. Localization at fixed points and the Duistermaat-Heckman formula.- VI.4. Appendix: some algebraic topology.- VI.5. Appendix: various notions of Euler classes.- Exercises.- VII. Toric manifolds.- VII.1. Fans and toric varieties.- VII.2. Symplectic reduction and convex polyhedra.- VII.3. Cohomology of X ?.- VII.4. Complex toric surfaces.- Exercises.- VIII. Hamiltonian circle actions on manifolds of dimension 4.- VIII.1. Symplectic S 1-actions, generalities.- VIII.2. Periodic Hamiltonians on 4-dimensional manifolds.- Exercises.