Atiyah–Guillemin–Sternberg convexity theorem

Statement

Let a compact torus $T$ act through a Hamiltonian torus action on a compact connected symplectic manifold $(M,\omega)$ , with moment map $\mu:M\to\mathfrak t^*$ . The Atiyah–Guillemin–Sternberg convexity theorem states that:

every fiber $\mu^{-1}(\alpha)$ is connected; and
the image is the convex hull of the fixed-point image,

\mu(M)=\operatorname{conv}\bigl(\mu(M^T)\bigr).

Consequently, $\mu(M)$ is a convex polytope. The moment map is constant on each connected component of $M^T$ , so the convex hull is generated by finitely many values. Neither effectiveness of the action nor isolatedness of the fixed points is required.

Proof idea

For each $\xi\in\mathfrak t$ , the component

\mu^\xi=\langle\mu,\xi\rangle

is a Morse–Bott function whose critical set is the fixed set of the subtorus generated by $\xi$ . Its critical submanifolds have even index and coindex. Morse-theoretic analysis of these components gives connected level sets and controls extrema in every direction. Separating-hyperplane arguments then identify the full image with the convex hull of fixed-point values Atiyah, §§2–3.

Guillemin and Sternberg obtain the convexity statement through closely related Hamiltonian and symplectic arguments Guillemin–Sternberg, §§1–3.

Consequences

Every linear functional on $\mathfrak t^*$ attains its maximum and minimum on $\mu(M)$ at images of torus-fixed components. Thus the fixed-point data determine the entire moment image. For an effective $T^n$ -action on a $2n$ -manifold, the resulting polytope is the starting point of Delzant’s classification of compact symplectic toric manifolds.

The image $\mu(M)$ , together with its integral-affine structure, is called the moment polytope of the Hamiltonian torus space.

Connectedness of the fibers is stronger than convexity of the image. It helps identify reduced spaces as parameters vary and rules out disconnected inverse images that a generic map onto the same polytope could have.

Scope and near misses

Compactness cannot simply be dropped: noncompact Hamiltonian actions can have nonclosed or nonconvex moment images unless the moment map satisfies appropriate properness hypotheses. Connectedness of $M$ is also needed for the single-polytope conclusion.

For a nonabelian compact group, the raw moment image in $\mathfrak g^*$ need not be convex. The corresponding theorem concerns its intersection with a positive Weyl chamber and requires a different formulation.

References

M. F. Atiyah, “Convexity and Commuting Hamiltonians,” Bulletin of the London Mathematical Society 14 (1982), 1–15. DOI record. Relevant: §§2–3, connected level sets and convexity.
V. Guillemin and S. Sternberg, “Convexity Properties of the Moment Mapping,” Inventiones Mathematicae 67 (1982), 491–513. DOI record. Relevant: the fixed-point convex-hull theorem for Hamiltonian torus actions.