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.


2 Answers 2


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.


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