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  "body": "\n## Statement\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]] and let $L$ be a [[holomorphic line bundle|QAC7BYK9KX]] on $X$.\nSuppose that $L$ is [[positive|J45KWFHENP]]: it has a Hermitian metric whose Chern curvature form $\\frac{i}{2\\pi}\\Theta$ is a positive $(1,1)$-form.\nThen there is an integer $k_0$ with the following property.\nFor every $k \\ge k_0$, the holomorphic sections of $L^{\\otimes k}$ define a holomorphic embedding of $X$ into a complex [[projective space|ZT9A6JK8QF]] $\\mathbb{CP}^N$.\nIn particular, $X$ is projective.\n\n## Remarks\n\nProof idea: choose a basis $s_0, \\ldots, s_N$ of the space of holomorphic sections of $L^{\\otimes k}$ and consider $x \\mapsto [s_0(x) : \\cdots : s_N(x)]$.\nThis is an embedding as soon as the sections separate points and tangent vectors.\nTo get sections with prescribed values at one or two points, one blows up these points.\nThen the [[Kodaira vanishing theorem|NY77DQ2AX8]] on the blow-up kills the obstruction in $H^1$; for large $k$ the needed positivity holds.\n\nEquivalent form: a compact complex manifold is projective if and only if it has a [[Kähler|X60JG9AEQJ]] form $\\omega$ with a rational cohomology class.\nRational means that $[\\omega]$ lies in the image of $H^2(X; \\Q)$; after scaling, one may take it integral.\nSuch $X$ are called Hodge manifolds.\nThen $L$ is the line bundle with first Chern class $[\\omega]$.\nBy [[Chow's theorem|6EFEG5VJEF]], $X$ is then the zero set of homogeneous polynomials in $\\mathbb{CP}^N$.\n\nExample: on $\\mathbb{CP}^n$ the hyperplane bundle $\\mathcal{O}(1)$ is positive, with the Fubini–Study metric, and already $k = 1$ gives the identity embedding.\n\nThe positivity hypothesis cannot be dropped.\nA generic complex torus $\\C^2 / \\Lambda$ is Kähler, but its Kähler classes are not rational.\nIt has no positive line bundle and it is not projective; it even has no nonconstant meromorphic functions.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a test for when a compact complex manifold can be described by polynomial equations in projective space.\nThe test is the existence of a line bundle with positive curvature.\nIf there is one, high powers of it have enough holomorphic sections to serve as projective coordinates and to embed the manifold.\n\n### L2 human\n\nThe theorem is the bridge between complex differential geometry and algebraic geometry: a curvature condition implies an algebraic conclusion.\nCombined with the theory of Kähler manifolds it says that projective manifolds are exactly the compact Kähler manifolds with an integral Kähler class.\nA positive line bundle is thus [[ample|RRK8P6TXG2]], and the converse also holds.\nIt shows, for instance, that every compact Riemann surface is a projective algebraic curve.\nVia the Riemann bilinear relations, it also tells which complex tori are abelian varieties.\n",
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