Atiyah-Guillemin-Sternberg convexity theorem

Intuition [Beginner]

When a torus $T=T^k$ acts on a closed symplectic manifold $M$ in a way that respects the symplectic form, every choice of one-parameter subgroup of $T$ gives a function on $M$ whose flow is the corresponding rotation. Bundling these functions together gives the moment map $\mu :M\to {\mathbb{R}}^k$. The Atiyah-Guillemin-Sternberg theorem describes the image $\mu (M)$ as a remarkably rigid object: it is always a convex polytope, and that polytope is the convex hull of the images of the points fixed by the whole torus.

The simplest case is the rotation of the round sphere $S^2$ about a vertical axis. The height function is the moment map; its image is the interval $[-1,1]$, and the two endpoints are the two poles — the only fixed points. Three-dimensional examples like ${\mathbb{CP}}^2$ produce filled triangles, with the three vertices coming from three fixed points of the torus action.

The theorem is a bridge between symplectic geometry and the combinatorics of polytopes. Once you know the polytope, you know a great deal about the manifold; the Delzant classification later turns this into a full equivalence for toric manifolds.

Visual [Beginner]

The sphere $S^2$ rotating about a vertical axis, with horizontal arrows showing the orbits as latitude circles, and the moment-map image drawn beside it as the segment $[-1,1]$ with the north and south pole marked.

The picture you should keep is: the orbits of the torus collapse onto a flat polytope, and the corners of the polytope are exactly the fixed points.

Worked example [Beginner]

Take $M={\mathbb{CP}}^2$ with homogeneous coordinates $[z_0:z_1:z_2]$ and the action of the two-torus $T^2$ by $({\theta }_1,{\theta }_2)\cdot [z_0:z_1:z_2]=[z_0:e^{i{\theta }_1}{z}_1:e^{i{\theta }_2}{z}_2]$. The moment map is

$$\mu ([z_0:z_1:z_2])=\frac{1}{|z_0{|}^2+|z_1{|}^2+|z_2{|}^2}(|z_1{|}^2,|z_2{|}^2).$$

The fixed points of the $T^2$-action are the three coordinate vertices: $[1:0:0]$, $[0:1:0]$, and $[0:0:1]$. Their moment-map images are $(0,0)$, $(1,0)$, and $(0,1)$. Every other point of ${\mathbb{CP}}^2$ has positive entries summing to at most $1$, so the image $\mu ({\mathbb{CP}}^2)$ is the closed triangle with these three corners.

What this tells us: a complicated four-real-dimensional manifold collapses, under the moment map, to a flat triangle whose corners are the only fixed points. The convexity theorem promises this picture for every torus action of this type.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(M,\omega )$ be a closed connected symplectic manifold of dimension $2n$, and let $T=T^k$ be a torus acting on $M$ in a Hamiltonian fashion, with moment map $\mu :M\to {\mathfrak{t}}^{\ast }$, where $\mathfrak{t}=\mathrm{Lie}(T)\cong {\mathbb{R}}^k$ and ${\mathfrak{t}}^{\ast }\cong {\mathbb{R}}^k$ is its linear dual. For each $\xi \in \mathfrak{t}$ the component of the moment map in direction $\xi$ is the smooth function

$${\mu }^{\xi }:=⟨\mu ,\xi ⟩:M\to \mathbb{R},$$

whose Hamiltonian flow is the one-parameter subgroup $\exp (t\xi )\subset T$ acting on $M$. The element $\xi$ is generic when the closure of the one-parameter subgroup it generates is all of $T$; equivalently, $\xi$ has rationally independent coordinates with respect to a basis of the integer lattice $\Lambda \subset \mathfrak{t}$.

The fixed-point set of the $T$-action is denoted $M^T\subset M$. Each connected component is a closed symplectic submanifold of $M$. The image $\mu (M^T)\subset {\mathfrak{t}}^{\ast }$ is a finite set when $M^T$ has finitely many components.

A subset $P\subset {\mathfrak{t}}^{\ast }$ is a convex polytope when it is the convex hull of a finite set of points. Equivalently, $P$ is a bounded intersection of finitely many closed half-spaces.

Key theorem with proof [Intermediate+]

Theorem (Atiyah 1982; Guillemin-Sternberg 1982). Let $(M,\omega )$ be a closed connected symplectic manifold with a Hamiltonian action of a torus $T=T^k$ and moment map $\mu : M \to \mathfrak{t}^$. Then:*

1. For every $c \in \mathfrak{t}^$,thefibre$\mu^{-1}(c)$ is connected (or empty).*
2. The image $\mu (M)$ is a convex polytope.
3. That polytope is the convex hull of the images of the fixed-point components: $\mu (M)=\mathrm{conv}(\mu (M^T))$.

The proof rests on a Morse-theoretic study of the components ${\mu }^{\xi }$. The structure of the argument is the same in Atiyah's and in Guillemin-Sternberg's papers; the form below follows Atiyah's exposition.

Proof. The argument proceeds in five stages.

Stage 1 (Morse-Bott property). For any $\xi \in \mathfrak{t}$, the function ${\mu }^{\xi }$ is Morse-Bott, with critical set equal to the fixed-point set of the closed one-parameter subgroup $T_{\xi }=\overline{\exp (\mathbb{R}\xi )}\subseteq T$. When $\xi$ is generic, $T_{\xi }=T$, so the critical set of ${\mu }^{\xi }$ is exactly $M^T$. The Hamiltonian condition $d{\mu }^{\xi }={\iota }_{X_{\xi }}\omega$ identifies critical points of ${\mu }^{\xi }$ with zeros of the fundamental vector field $X_{\xi }$, and zeros of $X_{\xi }$ are exactly the fixed points of the flow of $\xi$. The Hessian of ${\mu }^{\xi }$ along the normal directions to $M^{T_{\xi }}$ is non-degenerate because $\omega$ is non-degenerate and the linearised action is a faithful representation of $T_{\xi }$ on the normal bundle.

Stage 2 (index parity). Every Morse-Bott index and coindex of ${\mu }^{\xi }$ is even. At a fixed point $p\in M^{T_{\xi }}$, the tangent space $T_{p}M$ decomposes under the linearised $T_{\xi }$-action into a direct sum $T_{p}M=T_p{M}^{T_{\xi }}\oplus {⨁}_{\alpha }{V}_{\alpha }$ of weight spaces, where each $V_{\alpha }$ is a real two-dimensional subspace on which $T_{\xi }$ acts by a non-zero weight $\alpha :{\mathfrak{t}}_{\xi }\to \mathbb{R}$. The Hessian of ${\mu }^{\xi }$ on $V_{\alpha }$ is a multiple of the standard rotation-invariant symplectic form, with sign equal to $\mathrm{sign}\alpha (\xi )$. Each weight space contributes a two-dimensional summand to either the negative or the positive eigenspace; both index and coindex are sums of these even-dimensional contributions.

Stage 3 (connectedness of level sets). Index parity implies that for every $\xi \in \mathfrak{t}$ and every $a\in \mathbb{R}$, the sublevel set $({\mu }^{\xi }{)}^{-1}((-\infty ,a])$ is connected, and so is the level set $({\mu }^{\xi }{)}^{-1}(a)$. The argument is the standard Morse-Bott handle decomposition: as $a$ increases through a critical value $c$ of ${\mu }^{\xi }$, the sublevel set changes by attaching handles along a sphere bundle of dimension $\mathrm{index}-1$. When the index is $0$, the change adds a new connected component, but in our setting any such component is attached to the existing manifold along a coindex-direction sphere bundle of dimension $\mathrm{coindex}-1\ge 1$ as soon as the coindex is at least $2$ — which is forced by index-plus-coindex equalling the codimension of the critical submanifold and both being even and the codimension being positive away from a global minimum. When the index is $\ge 2$, the attaching sphere is connected, so a connected sublevel set remains connected. A symmetric argument applies to superlevel sets, so level sets are connected. Connectedness of $M$ is the base case of the induction on $a$.

Stage 4 (convexity, by induction on rank). The image $\mu (M)$ is convex. Induct on $k=\dim T$. The case $k=0$ is empty. The case $k=1$: ${\mu }^{\xi }$ for the generator $\xi$ is a connected real-valued function on connected $M$, so its image is an interval — a convex polytope in $\mathbb{R}$. For the inductive step assume the statement for tori of rank $<k$. Pick any two points $p_0,p_1\in M$; one needs to produce a path from $\mu (p_0)$ to $\mu (p_1)$ in $\mu (M)$ that follows the straight line in ${\mathfrak{t}}^{\ast }$. Choose a generic direction $\xi \in \mathfrak{t}$ and use connectedness of the level sets of ${\mu }^{\xi }$ at every height between ${\mu }^{\xi }(p_0)$ and ${\mu }^{\xi }(p_1)$: each such level set $L_a:=({\mu }^{\xi }{)}^{-1}(a)$ is connected by Stage 3. Restrict $\mu$ to $L_a$; this restricted moment map takes values in the affine hyperplane $\{\eta \in {\mathfrak{t}}^{\ast }:⟨\eta ,\xi ⟩=a\}$, and after quotienting by the rank-$1$ subtorus $T_{\xi }$ — which acts as the identity on $L_a/T_{\xi }$ by construction — one obtains a Hamiltonian action of the rank-$(k-1)$ quotient torus $T/T_{\xi }$ on the symplectic-reduced quotient $L_a/T_{\xi }$ with moment map taking values in this hyperplane. The inductive hypothesis applied to the reduced action gives convexity of the image inside the hyperplane. Sweeping $a$ from ${\mu }^{\xi }(p_0)$ to ${\mu }^{\xi }(p_1)$ produces the desired straight-line path in $\mu (M)$, establishing convexity.

Stage 5 (vertices and the polytope description). The image is a polytope, and its vertices are images of fixed-point components. A point $v\in \mu (M)$ is an extreme point of the convex set $\mu (M)$ iff every ${\mu }^{\xi }$ attains its minimum or maximum on ${\mu }^{-1}(v)$ for some $\xi$; the critical-set characterisation of Stage 1 gives ${\mu }^{-1}(v)\cap M^T\ne \varnothing$ at every extreme point. Conversely, the moment-map image of any fixed-point component is a single point in ${\mathfrak{t}}^{\ast }$, and these finitely many points generate $\mu (M)$ by convexity. Hence $\mu (M)=\mathrm{conv}(\mu (M^T))$, a convex polytope with vertex set contained in $\mu (M^T)$. Connectedness of fibres in (1) follows from Stage 3 applied to a generic $\xi$ and the fact that two points in the same generic-${\mu }^{\xi }$-level set lying in ${\mu }^{-1}(c)$ are connected through the reduced-space convexity of Stage 4. $\square$

Bridge. The convexity theorem builds toward 05.04.04 (Delzant's theorem), where the polytope produced here appears again in the role of a complete invariant: when $\dim T=\frac{1}{2}\dim M$ and the action is effective, the Delzant polytope $\mu (M)\subset {\mathfrak{t}}^{\ast }$ determines $(M,\omega ,T)$ up to equivariant symplectomorphism. The Morse-Bott index-parity argument is exactly the mechanism that lets reduced-space symplectic geometry 05.04.02 inherit a smooth structure from the ambient $M$, because connectedness of ${\mu }^{-1}(c)$ is what makes the regular reduction ${\mu }^{-1}(c)/T$ a connected symplectic manifold rather than a disjoint union. Putting these together, the foundational reason convexity holds is exactly that the symplectic structure forces every Morse-Bott index to come in two-dimensional weight-space contributions — index parity is the bridge between the analytic non-degeneracy of $\omega$ and the combinatorial flatness of $\mu (M)$. The theorem identifies the Hamiltonian-action structure on $M$ with a piece of polyhedral data on ${\mathfrak{t}}^{\ast }$.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not currently support Hamiltonian torus actions or moment maps in a form that admits the Atiyah-Guillemin-Sternberg theorem. The skeleton below sketches the statement at the type level; the proof would require the Morse-Bott apparatus described in lean_mathlib_gap.

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Advanced results [Master]

The convexity theorem belongs to a family of results that combine Hamiltonian symmetry with convex-geometric output. Several distinct generalisations exist, each retaining the Morse-Bott / level-set-connectedness skeleton.

Non-abelian convexity (Kirwan 1984). When the symmetry group is a compact connected Lie group $G$ rather than a torus, the moment map $\mu :M\to {\mathfrak{g}}^{\ast }$ no longer has convex image directly; the image is $G$-invariant. Kirwan's theorem states that the intersection $\mu (M)\cap {\mathfrak{t}}_{+}^{\ast }$ with a closed positive Weyl chamber is a convex polytope. The proof reduces to the abelian case applied to the maximal torus, plus the fact that every $G$-orbit in ${\mathfrak{g}}^{\ast }$ meets ${\mathfrak{t}}_{+}^{\ast }$ in exactly one point.

Symplectic implosion (Guillemin-Jeffrey-Sjamaar 2002). A construction that turns a Hamiltonian $G$-manifold into a Hamiltonian $T$-manifold whose moment polytope is exactly the Kirwan polytope. The implosion machinery realises Kirwan's theorem as an instance of Atiyah-Guillemin-Sternberg applied to a derived $T$-manifold.

Singular convexity. Convexity persists when one drops compactness and replaces it by properness of the moment map: Lerman-Meinrenken-Tolman-Woodward 1998 and earlier Hilgert-Neeb-Plank 1994 established convexity of the image under properness assumptions.

Local convexity at fixed points. Atiyah's argument also yields a local statement: the image of $\mu$ near any fixed point $p$ is the cone on the polytope of the linearised torus action on $T_{p}M$, intersected with a translate of the fixed-point image $\mu (p)$. This is the seed of the local description used in Delzant's classification.

Morse theory of $|\mu {|}^2$. Kirwan's Cohomology of Quotients (1984) develops a Morse theory for the function $|\mu {|}^2:M\to \mathbb{R}$, whose critical sets stratify $M$ in a way compatible with the moment-polytope facets. This stratification is the key technical input for computing cohomology of symplectic quotients $M//G$.

Vertex-edge structure. When $T$ acts effectively with $\dim T=\frac{1}{2}\dim M$ — the toric case — the moment polytope is a Delzant polytope: simple, rational, smooth at each vertex (in the sense that the edge directions at each vertex form a $\mathbb{Z}$-basis of the integer lattice). The edges of the polytope correspond to one-dimensional fixed-component spheres in $M$, and their lengths are determined by the symplectic areas of those spheres.

Synthesis. The Atiyah-Guillemin-Sternberg theorem identifies a piece of symplectic geometry with a piece of polyhedral combinatorics, and the bridge is a single analytic fact: components of the moment map are Morse-Bott functions whose indices are constrained to be even by the weight-space decomposition of the symplectic structure. Read as a structural statement, the theorem generalises the elementary fact that a connected real-valued Morse function on a connected manifold has an interval as its image — index parity is what allows iterating to higher rank without breaking connectedness. Read in the opposite direction, the theorem is dual to a representation-theoretic fact: the multiplicity of a weight $\alpha \in {\Lambda }^{\ast }$ in a quantisation of $(M,\omega ,\mu )$ is supported on the lattice points of the polytope $\mu (M)$, and the polytope is therefore the support of the spectrum of a quantum operator.

Putting these together, the foundational reason that fixed points control the global image is exactly Stage 2 of the proof: the symplectic non-degeneracy forces the second-order behaviour of ${\mu }^{\xi }$ at any fixed point to be a sum of two-dimensional rotation Hessians, and this single linear-algebra ingredient is the bridge between the analytic moment map and the combinatorial polytope. The theorem appears again in Delzant's classification, where the polytope is upgraded from invariant to complete invariant — putting these together one sees that every compact symplectic toric manifold is exactly its polytope, in the precise categorical sense that the polytope-to-manifold construction is an equivalence of groupoids.

Full proof set [Master]

Lemma (weight-space decomposition). Let $T_{\xi }$ be a closed one-parameter subgroup of $T$ acting on a symplectic vector space $(V,{\omega }_V)$ by symplectomorphisms, and let $V^{T_{\xi }}\subseteq V$ be the fixed subspace. Then the symplectic complement $W=(V^{T_{\xi }}{)}^{{\omega }_V}$ admits an orthogonal direct-sum decomposition $W={⨁}_{\alpha }{W}_{\alpha }$ into real two-dimensional symplectic subspaces, indexed by a finite set of non-zero weights $\alpha :{\mathfrak{t}}_{\xi }\to \mathbb{R}$, such that $T_{\xi }$ acts on $W_{\alpha }$ by the rotation of weight $\alpha$.

Proof. The action of $T_{\xi }\cong S^1$ on $V$ commutes with ${\omega }_V$, so the complexification $V{\otimes }_{\mathbb{R}}\mathbb{C}$ decomposes into character spaces $V_{\alpha }^{\mathbb{C}}$ on which $T_{\xi }$ acts by the character $\theta \mapsto e^{i\alpha \theta }$. Pairs $(V_{\alpha }^{\mathbb{C}},V_{-\alpha }^{\mathbb{C}})$ for $\alpha \ne 0$ assemble into real two-dimensional summands $W_{\alpha }\subset V$; the symplectic form pairs each $W_{\alpha }$ with itself non-degenerately because $T_{\xi }$ preserves ${\omega }_V$. The fixed subspace $V^{T_{\xi }}$ is the $\alpha =0$ summand and is automatically symplectic. $\square$

Lemma (Hessian sign on weight spaces). In the setting of the previous lemma, the Hessian of ${\mu }^{\xi }$ at the fixed point, restricted to the weight space $W_{\alpha }$, equals $\mathrm{sign}\alpha (\xi )\cdot Q_{\alpha }$, where $Q_{\alpha }$ is the positive-definite rotation-invariant inner product on $W_{\alpha }$ induced by the symplectic structure and a choice of $T_{\xi }$-invariant complex structure.

Proof. On $W_{\alpha }$ choose a $T_{\xi }$-invariant complex structure $J$ with ${\omega }_V(\cdot ,J\cdot )>0$. Then $T_{\xi }$ acts on $W_{\alpha }\cong \mathbb{C}$ by $\theta \cdot z=e^{i\alpha \theta }z$, the moment map for this circle action is ${\mu }^{\xi }=-\frac{\alpha (\xi )}{2}|z{|}^2$ (the sign convention from $d{\mu }^{\xi }={\iota }_{X_{\xi }}\omega$). The Hessian at $z=0$ is $-\alpha (\xi )\cdot |\cdot {|}^2$, with sign $-\mathrm{sign}\alpha (\xi )$ relative to the positive-definite $|\cdot {|}^2$. Either sign of $\alpha (\xi )$ gives a definite quadratic form on $W_{\alpha }$, contributing $2$ to either index or coindex. $\square$

Lemma (connectedness of level sets). Let $f:M\to \mathbb{R}$ be Morse-Bott on a closed connected manifold $M$ with all indices and coindices even. Then for every $a\in \mathbb{R}$ the level set $f^{-1}(a)$ is empty or connected, and so is every sublevel set.

Proof. Connectedness of sublevel sets is by induction on critical values. Below the minimum, the sublevel set is empty. Just above the minimum, the sublevel set retracts onto the connected critical submanifold (assuming connected $M$ has a connected global-minimum critical set; if there are multiple minima, repeat the argument component by component, but on $M$ connected the global minimum must be a single connected submanifold by Morse-Bott + connectedness of $M$, since otherwise a path between two minima would have to cross a critical level of higher index whose attaching coindex sphere is connected, making the two pieces equal in ${\pi }_0$). At a critical value $c$ the sublevel set changes by attaching the disk bundle of the negative normal bundle of the critical submanifold $C_c$, glued along the sphere bundle of dimension $\mathrm{index}(C_c)-1$. Even index $\ge 2$ makes this attaching sphere bundle have connected fibres, hence the gluing keeps connectedness. Index $0$ would create a new component, but on $M$ connected and Morse-Bott, this only happens at the global minimum (already handled). Symmetric argument for superlevel sets via $-f$. The level set $f^{-1}(a)$ is a deformation retract of a thin shell between sublevel and superlevel, both of which are connected; so $f^{-1}(a)$ is connected. $\square$

Theorem (full statement, restated and proved). Let $(M,\omega )$ be closed connected symplectic with a Hamiltonian $T=T^k$-action and moment map $\mu$. Then ${\mu }^{-1}(c)$ is connected (or empty) for all $c \in \mathfrak{t}^$,and$\mu(M) = \mathrm{conv}\big(\mu(M^T)\big)$ is a convex polytope.*

Proof. Stages 1-5 of the proof in the Intermediate section, made rigorous by the three lemmas above. Stage 1 uses the Hamiltonian condition $d{\mu }^{\xi }={\iota }_{X_{\xi }}\omega$ and non-degeneracy of $\omega$ to identify critical points of ${\mu }^{\xi }$ with zeros of the fundamental vector field, hence with fixed points of the closure $T_{\xi }$ of $\exp (\mathbb{R}\xi )$. The Morse-Bott property along $M^{T_{\xi }}$ comes from the weight-space lemma applied to the linearised action on the normal bundle. Stage 2 is the Hessian-sign lemma: every weight-space contribution is even-dimensional, so total index and coindex are even. Stage 3 is the connectedness-of-level-sets lemma applied to ${\mu }^{\xi }$ for any $\xi$.

Stage 4 (convexity) inducts on $k$. The base case $k=1$: image is an interval. Inductive step: pick generic $\xi$, restrict to a level set $L_a=({\mu }^{\xi }{)}^{-1}(a)$, observe that $L_a$ inherits a Hamiltonian action of the rank-$(k-1)$ quotient torus $T/T_{\xi }$ via the symplectic reduction $L_a/T_{\xi }$ (regular reduction at the level $a$ of ${\mu }^{\xi }$ gives a smooth symplectic manifold $L_a/T_{\xi }$ when $a$ is a regular value of ${\mu }^{\xi }$, which it is for almost every $a$ in the image; the restricted moment map is the projection of $\mu {|}_{L_a}$ to the hyperplane). The inductive hypothesis gives convexity of the image inside the hyperplane. Sweeping $a$ from ${\mu }^{\xi }(p_0)$ to ${\mu }^{\xi }(p_1)$ produces the straight-line path. Stage 5 (vertex characterisation) follows because every extreme point of the image is the unique minimum or maximum of some ${\mu }^{\xi }$, which is a fixed-point image. Connectedness of fibres uses Stage 3 plus the inductive convexity. $\square$

Connections [Master]

• Moment map 05.04.01. Atiyah-Guillemin-Sternberg is the central global theorem about the moment map; the unit's three-condition definition is what enables Stage 1 of the proof, and the Hamiltonian sign convention used here is the same as that unit's.

• Symplectic reduction 05.04.02. The inductive step in Stage 4 of the proof uses regular Marsden-Weinstein-Meyer reduction at non-critical levels of ${\mu }^{\xi }$ to descend to a torus action of one rank lower. Index parity in Stage 2 also explains why regular reduction levels have connected fibres — a fact used implicitly in the smooth-quotient construction.

• Delzant's theorem 05.04.04. The downstream classification result: in the toric case ($\dim T=n=\frac{1}{2}\dim M$), the moment polytope produced by Atiyah-Guillemin-Sternberg is upgraded to a complete invariant, classifying compact symplectic toric manifolds up to equivariant symplectomorphism. The vertices and edge structure of the polytope encode the global manifold.

• Coadjoint orbit 05.03.01. The Schur-Horn theorem is the convexity theorem applied to a coadjoint orbit of $U(n)$ with the maximal-torus action — giving the permutohedron as moment polytope and the diagonal-of-Hermitian-matrix as moment map. This is the historical bridge between the convexity theorem and 1920s linear algebra.

• Symplectic manifold 05.01.02. The convexity statement crucially uses non-degeneracy of $\omega$ — both in the Morse-Bott property of ${\mu }^{\xi }$ at fixed points and in the weight-space sign argument. A degenerate two-form would not force index parity, and the conclusion would fail.

• Hamiltonian vector field 05.02.01. The components ${\mu }^{\xi }$ are the Hamiltonians whose flow is the corresponding one-parameter subgroup of $T$; the proof's analytic backbone is precisely the unit's identity $d{\mu }^{\xi }={\iota }_{X_{\xi }}\omega$.

Historical & philosophical context [Master]

Michael Atiyah's 1982 paper Convexity and commuting Hamiltonians (Bull. London Math. Soc. 14, 1-15) ^{[Atiyah 1982]} and Victor Guillemin and Shlomo Sternberg's Convexity properties of the moment mapping (Invent. Math. 67, 491-513) ^{[Guillemin-Sternberg 1982]} appeared independently and in the same year. Both proofs use the Morse-theoretic structure of moment-map components, with Atiyah's exposition emphasising the index-parity step and Guillemin-Sternberg's developing the level-set-connectedness machinery in greater generality. The two papers are joint originators of the theorem.

The result has a substantial linear-algebra prehistory. Issai Schur's 1923 paper Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie (Sitzungsber. Berlin. Math. Ges. 22) ^{[Schur 1923]} established that the diagonal entries of a Hermitian matrix with prescribed eigenvalues lie in the convex hull of the permutations of the eigenvalues — the "$\subseteq$" direction of what is now called the Schur-Horn theorem. Alfred Horn's 1954 paper Doubly stochastic matrices and the diagonal of a rotation matrix (Amer. J. Math. 76, 620-630) ^{[Horn 1954]} proved the converse direction, completing the equivalence. Atiyah and Guillemin-Sternberg's theorem reveals the Schur-Horn statement as the special case of moment-map convexity for the maximal-torus action on a unitary coadjoint orbit.

Frances Kirwan's 1984 monograph Cohomology of Quotients of Symplectic and Algebraic Varieties ^{[Kirwan 1984]} generalised the convexity theorem to non-abelian compact group actions, establishing that the intersection of the moment image with a closed positive Weyl chamber is convex polyhedral. Kirwan's argument also yields a Morse theory for $|\mu {|}^2$ that has become the standard tool for computing cohomology of symplectic quotients. Michèle Audin's Topology of Torus Actions on Symplectic Manifolds (1991) ^{[Audin 1991]} gave the canonical textbook treatment of the abelian convexity result and its local-cone refinements. Thomas Delzant's 1988 thesis ^{[Delzant 1988]} carried the convexity theorem to its sharpest form in the toric case, classifying compact symplectic toric manifolds by their moment polytopes.

Bibliography [Master]

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