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  "body": "\n## Statement\n\nLet $X \\subset \\mathbb{P}^N_{\\C}$ be a smooth complex [[projective variety|A4H4CZ2V35]] of dimension $n$.\nLet $\\omega \\in H^2(X, \\Q)$ be the class of a hyperplane section $X \\cap H$, in [[singular cohomology|6N7BC70CX0]].\nLet $L \\colon H^j(X, \\Q) \\to H^{j+2}(X, \\Q)$ be the map $\\alpha \\mapsto \\omega \\smile \\alpha$, given by the [[cup product|JX47MS1R1H]].\nThen for each integer $k$ with $0 \\le k \\le n$, the map\n$$L^k \\colon H^{n-k}(X, \\Q) \\to H^{n+k}(X, \\Q)$$\nis an isomorphism.\n\n## Remarks\n\nThe classical proof uses [[Hodge theory|32YJXG30MF]] on the [[Kähler manifold|X60JG9AEQJ]] $X(\\C)$.\nThe operator $L$, its adjoint $\\Lambda$ and the degree operator form a representation of $\\mathfrak{sl}_2$ on harmonic forms, and $L^k$ is an isomorphism by the representation theory of $\\mathfrak{sl}_2$.\nFor $X = \\mathbb{P}^n$, $H^{2i}(X, \\Q) = \\Q \\, \\omega^i$ for $0 \\le i \\le n$ and the odd groups vanish, so $L^k$ sends $\\omega^{(n-k)/2}$ to $\\omega^{(n+k)/2}$, which is a basis.\nThe theorem shows that $L \\colon H^j \\to H^{j+2}$ is injective for $j \u003c n$, so the Betti numbers satisfy $b_j \\le b_{j+2}$ for $j \u003c n$, and also $b_j = b_{2n-j}$.\nOver a field of positive characteristic, Deligne proved the analogue for $\\ell$-adic cohomology.\nThe hypothesis that $\\omega$ comes from an ample class matters: for a class $\\omega$ with $\\omega^n = 0$, such as $0$, the map $L^n \\colon H^0 \\to H^{2n}$ is zero.\n\n## Explanations\n\n### L1 human\n\nA smooth projective variety of dimension $n$ has cohomology groups in degrees $0$ to $2n$.\nCutting by a hyperplane again and again gives a map from degree $n-k$ to degree $n+k$.\nThe theorem says that this map loses nothing and adds nothing: it is a bijection.\nSo the cohomology is symmetric around the middle degree in a very rigid way.\n\n### L2 human\n\nThe theorem gives the Lefschetz decomposition of $H^*(X, \\Q)$ into primitive parts, and together with the Hodge–Riemann bilinear relations it shows that the cohomology carries a polarized Hodge structure.\nIt is one of the standard tools that restrict which manifolds can be projective varieties: for example, the sequences $b_0, b_2, b_4, \\ldots$ and $b_1, b_3, b_5, \\ldots$ cannot decrease before the middle degree.\nIt complements the [[Lefschetz hyperplane theorem|EX6YSYN7B4]], which compares the cohomology of $X$ with that of a hyperplane section below the middle degree.\nGrothendieck's Lefschetz-type standard conjecture asks that the inverse of $L^k$ come from an algebraic cycle.\n",
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