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  "body": "\n## Statement\n\nLet $X$ be a complex [[projective variety|A4H4CZ2V35]] of dimension $n$.\nLet $D \\subseteq X$ be an effective [[ample|RRK8P6TXG2]] [[Cartier divisor|FHV05ZK99F]], for example a hyperplane section, such that $X \\setminus D$ is smooth.\nThen, for [[singular cohomology|6N7BC70CX0]] with integer coefficients, the restriction map\n$$H^i(X, \\Z) \\to H^i(D, \\Z)$$\nis an isomorphism for $i \\le n - 2$ and injective for $i = n - 1$.\n\n## Remarks\n\nThe idea of the proof: $X \\setminus D$ is a smooth affine variety of dimension $n$.\nBy the Andreotti–Frankel theorem it has the homotopy type of a CW complex of real dimension at most $n$.\nLefschetz duality then gives $H^i(X, D; \\Z) \\cong H_{2n - i}(X \\setminus D; \\Z) = 0$ for $i \u003c n$, and the long exact sequence of the pair gives the claim.\nThe homotopy version says that $\\pi_i(D) \\to \\pi_i(X)$ is an isomorphism for $i \\le n - 2$ and surjective for $i = n - 1$.\n\nExample: a smooth hypersurface $X \\subseteq \\mathbb{P}^{n+1}_\\C$ of dimension $n$ is an ample divisor in $\\mathbb{P}^{n+1}$.\nSo $H^i(X, \\Z) \\cong H^i(\\mathbb{P}^{n+1}, \\Z)$ for $i \\le n - 1$, which is $\\Z$ for even $i$ and $0$ for odd $i$; in particular $X$ is simply connected when $n \\ge 2$.\n\nThe bound $n - 1$ is sharp.\nLet $X = \\mathbb{P}^1 \\times \\mathbb{P}^1$, of dimension $n = 2$, embedded in $\\mathbb{P}^3$ as a quadric by the Segre embedding.\nA hyperplane section is a smooth conic $D \\cong \\mathbb{P}^1$, and $H^2(X, \\Z) \\cong \\Z^2$ while $H^2(D, \\Z) \\cong \\Z$.\nSo the map in degree $n = 2$ is not injective.\n\n## Explanations\n\n### L1 human\n\nCut a projective variety by a hyperplane.\nThe theorem says that the slice has the same cohomology as the whole variety in all low degrees, below the middle dimension of the slice.\nSo the topology of a variety is largely determined by the topology of its hyperplane sections, and new topology can only appear in the middle degree.\n\n### L2 human\n\nThe theorem is the basis for induction on dimension in the topology of algebraic varieties, through Lefschetz pencils.\nTogether with Poincaré duality it shows that all the interesting cohomology of a smooth complete intersection sits in the middle degree.\nIt is complemented by the hard Lefschetz theorem, which describes the middle and upper degrees, and it has analogues in étale cohomology used in the proof of the Weil conjectures.\n",
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