# Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory. So my problem is to find a detailed proof of this theorem:

1. What are the prerequisits: Morse-Bott functions, equivariant Darboux Theorem...?
2. Is the original proof by Atiyah different from the Guillemin-Sternberg's proof?
3. What is "the best reference" for a detailed treatment of this theorem?

Thanks for any help.

For this topic in general, I really recommend a book of Anna Cannas da Silva Lectures on Symplectic Geometry. It's wonderfully written and very clear.

You can read a proof of the theorem in the book of Michel Audin: Topology of torus actions on symplectic manifolds.

There are some more things here http://www.math.ucsd.edu/~alpelayo/Docs/torictalk.pdf

Thanks Thomas, Liviu and Olga, The Atiyah's proof is done by induction on the dimension of the torus. If $$\mu$$ is the moment map and

$$A_m$$: the level sets of $$\mu$$ are connected, for any $$\mathbb{T}^m$$ hamiltonian action.

$$B_m$$: the image of $$\mu$$ is convex, for any $$\mathbb{T}^m$$ hamiltonian action.

The hard part (for me) is $$A_1$$ which is based on the connectedness of the levels of a Morse-Bott function on a compact manifold!

The rest of the proof is very well explained in:

1. Ana Cannas da Silva, Lectures on Symplectic Geometry (as exercises),
2. Michèle Audin: Topology of torus actions on symplectic manifolds,
3. http://www.math.nyu.edu/~kessler/teaching/group/convexity.pdf

The book by Liviu Nicolaescu is very useful and the complete proof can be found in:

McDuff & Salamon, Introduction to Symplectic Topology.