---
tag: 5CTHC6612N
kind: statement
role: theorem
name: Atiyah–Guillemin–Sternberg convexity theorem
msc: [53D20, 52B20]
uses: [ZARM34FZ6Z, CAN490EPRT, 0FQSDC2EQ2]
about: [ZARM34FZ6Z]
sources:
  - url: https://en.wikipedia.org/wiki/Momentum_map
---

## Statement

Let $(M, \omega)$ be a compact connected [[symplectic manifold|CAN490EPRT]].
Let a torus $T = (S^1)^k$ act on $M$ with a [[moment map|ZARM34FZ6Z]] $\mu \colon M \to \mathfrak{t}^*$, where $\mathfrak{t}^* \cong \R^k$ is the dual of the Lie algebra of $T$.
Let $M^T$ be the set of points fixed by every element of $T$.
Then:

1. $\mu(M^T)$ is a finite set.
2. $\mu(M)$ is the [[convex hull|0FQSDC2EQ2]] of $\mu(M^T)$, so it is a convex polytope.
3. Every fiber $\mu^{-1}(a)$, $a \in \mathfrak{t}^*$, is connected or empty.

## 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 $0$.

## Explanations

### L1 human

A torus symmetry of a compact symplectic manifold has a moment map with values in a vector space.
The theorem says that the image is always a convex polytope, and its corners are the images of the points that the torus does not move.
So a complicated manifold casts a very simple convex shadow.

### L2 human

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|WBKCGFMT0P]].
The [[Duistermaat–Heckman formula|EWJK7NY6ZB]] 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.