Atiyah–Guillemin–Sternberg convexity theorem

Remarks

The theorem was proved independently by Atiyah and by Guillemin and Sternberg. The fixed set $M^T$ has finitely many connected components, and $\mu$ is constant on each of them, which gives part 1. The heart of the proof is that each component $\mu^\xi = \langle \mu, \xi \rangle$ is a Morse–Bott function whose critical submanifolds have even index and coindex. Such functions have connected level sets, and an induction on $k$ gives convexity.

Example: $T^n$ acts on $\mathbb{CP}^n$ by $(t_1, \ldots, t_n) \cdot [z_0 : z_1 : \cdots : z_n] = [z_0 : t_1 z_1 : \cdots : t_n z_n]$. A moment map is $\mu([z]) = -\frac{1}{2} \left( \abs{z_1}^2, \ldots, \abs{z_n}^2 \right) / \abs{z}^2$, for a suitable normalization of the Fubini–Study form, with image a simplex whose $n + 1$ vertices are the images of the $n + 1$ coordinate points.

Compactness is needed. For the circle acting on $\C$ by rotation with $\mu(z) = -\frac{1}{2}\abs{z}^2$, the image is the half-line $(-\infty, 0]$, which is not the hull of the single fixed value $0$.

Going deeper

For a student

The theorem is the start of the link between symplectic geometry and convex geometry. For toric manifolds the polytope determines the manifold completely, by Delzant's theorem. The Duistermaat–Heckman formula describes how the symplectic volume is spread over the polytope. Kirwan extended the convexity statement to actions of compact nonabelian groups, where one intersects the image with a positive Weyl chamber.