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  "body": "\n## Statement\n\nLet $(M, \\omega)$ be a compact connected [[symplectic manifold|CAN490EPRT]].\nLet 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 $T$.\nLet $M^T$ be the set of points fixed by every element of $T$.\nThen:\n\n1. $\\mu(M^T)$ is a finite set.\n2. $\\mu(M)$ is the [[convex hull|0FQSDC2EQ2]] of $\\mu(M^T)$, so it is a convex polytope.\n3. Every fiber $\\mu^{-1}(a)$, $a \\in \\mathfrak{t}^*$, is connected or empty.\n\n## Remarks\n\nThe theorem was proved independently by Atiyah and by Guillemin and Sternberg.\nThe fixed set $M^T$ has finitely many connected components, and $\\mu$ is constant on each of them, which gives part 1.\nThe 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.\nSuch functions have connected level sets, and an induction on $k$ gives convexity.\n\nExample: $T^n$ acts on $\\mathbb{CP}^n$ by $(t_1, \\ldots, t_n) \\cdot [z_0 : z_1 : \\cdots : z_n] = [z_0 : t_1 z_1 : \\cdots : t_n z_n]$.\nA 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.\n\nCompactness is needed.\nFor 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 $0$.\n\n## Explanations\n\n### L1 human\n\nA torus symmetry of a compact symplectic manifold has a moment map with values in a vector space.\nThe 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.\nSo a complicated manifold casts a very simple convex shadow.\n\n### L2 human\n\nThe theorem is the start of the link between symplectic geometry and convex geometry.\nFor toric manifolds the polytope determines the manifold completely, by [[Delzant's theorem|WBKCGFMT0P]].\nThe [[Duistermaat–Heckman formula|EWJK7NY6ZB]] describes how the symplectic volume is spread over the polytope.\nKirwan extended the convexity statement to actions of compact nonabelian groups, where one intersects the image with a positive Weyl chamber.\n",
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