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  "body": "\n## Statement\n\nA *symplectic toric manifold* is a compact connected [[symplectic manifold|CAN490EPRT]] $(M, \\omega)$ of dimension $2n$ with two data.\nThese are an effective action of the torus $T^n = (S^1)^n$ and a [[moment map|ZARM34FZ6Z]] $\\mu \\colon M \\to \\R^n$ for it.\nA *Delzant polytope* in $\\R^n$ is a convex polytope $\\Delta$ such that at each vertex $v$:\n\n1. exactly $n$ edges meet at $v$;\n2. the edges at $v$ have the form $v + t u_i$, $t \\ge 0$, with $u_1, \\ldots, u_n \\in \\Z^n$;\n3. these vectors $u_1, \\ldots, u_n$ can be chosen to form a basis of the lattice $\\Z^n$.\n\nThen the image $\\mu(M)$ of a symplectic toric manifold is a Delzant polytope.\nThe map $(M, \\omega, \\mu) \\mapsto \\mu(M)$ is a bijection from symplectic toric manifolds onto Delzant polytopes in $\\R^n$.\nHere toric manifolds are taken up to $T^n$-equivariant symplectomorphisms that intertwine the moment maps.\n\n## Remarks\n\nThat $\\mu(M)$ is a convex polytope is the [[Atiyah–Guillemin–Sternberg convexity theorem|5CTHC6612N]]; the local normal form at fixed points gives the three conditions.\nFor surjectivity, Delzant builds $M$ from $\\Delta$ as a [[symplectic reduction|SBJVATVH44]] of $\\C^d$ by a subtorus of $T^d$, where $d$ is the number of facets of $\\Delta$.\nInjectivity uses the local structure near each orbit and a gluing argument.\n\nExample: the simplex with vertices $0, e_1, \\ldots, e_n$ is a Delzant polytope, and it corresponds to $\\mathbb{CP}^n$; the square $[0,1]^2$ corresponds to $S^2 \\times S^2$.\nThe triangle with vertices $(0,0)$, $(1,0)$, $(0,2)$ is not Delzant.\nAt $(1,0)$ the primitive edge vectors are $(-1,0)$ and $(-1,2)$, with determinant $-2$, so they do not form a basis of $\\Z^2$.\n\n## Explanations\n\n### L1 human\n\nA toric manifold is a symplectic manifold with as much torus symmetry as its dimension allows.\nDelzant's theorem says that such a manifold is completely described by a polytope with integer edge directions, its moment polytope.\nSo questions about these manifolds can be turned into questions about polytopes.\n\n### L2 human\n\nThe theorem gives a dictionary: faces of the polytope correspond to orbit closures, and vertices to fixed points of the torus.\nEvery symplectic toric manifold has a compatible complex structure that makes it a smooth toric variety.\nRational simple polytopes with a positive integer label on each facet correspond in the same way to toric orbifolds, by work of Lerman and Tolman.\n",
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