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  "body": "\n## Statement\n\nLet $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] of dimension $n$ over a field of characteristic $0$, with [[canonical bundle|W05VYY4HBR]] $\\omega_X$.\nLet $L$ be an [[ample line bundle|RRK8P6TXG2]] on $X$.\nThen\n$$H^i(X, \\omega_X \\otimes L) = 0 \\quad \\text{for all } i \u003e 0,$$\nwhere $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].\nEquivalently, by Serre duality, $H^i(X, L^{-1}) = 0$ for all $i \u003c n$.\n\n## Remarks\n\nKodaira's proof over $\\C$ uses a Hermitian metric on $L$ with positive curvature and the Bochner–Kodaira–Nakano identity, which shows that harmonic forms representing the classes must vanish.\nDeligne and Illusie gave an algebraic proof by reduction to positive characteristic, using a lifting of $X$ modulo $p^2$.\n\nExample: on $X = \\mathbb{P}^n_\\C$ with $L = \\mathcal{O}(d)$, $d \u003e 0$, one has $\\omega_X \\otimes L = \\mathcal{O}(d - n - 1)$, and indeed $H^i(\\mathbb{P}^n, \\mathcal{O}(m)) = 0$ for $0 \u003c i \u003c n$ and all $m$, and for $i = n$ when $m \u003e -n-1$.\n\nThe ampleness hypothesis cannot be dropped: for a curve $X$ of genus $g \\ge 1$ and $L = \\mathcal{O}_X$, $H^1(X, \\omega_X) \\cong \\C \\ne 0$.\nThe characteristic hypothesis is also needed: Raynaud gave counterexamples in positive characteristic.\n\n## Explanations\n\n### L1 human\n\nThe Kodaira vanishing theorem says that certain cohomology groups of a positive line bundle, twisted by the canonical bundle, are all zero.\nWhen cohomology vanishes, global sections are easy to count, because the Euler characteristic, which Riemann–Roch computes, equals the dimension of the space of sections.\nSo the theorem turns an existence question into a numerical computation.\n\n### L2 human\n\nThe theorem is a key input of the Kodaira embedding theorem and of the theory of adjoint linear systems $\\abs{K_X + L}$.\nIts generalizations, the Nakano vanishing theorem for higher differential forms and the Kawamata–Viehweg vanishing theorem for nef and big bundles, are basic tools of the minimal model program.\nIt fails in positive characteristic, which makes it a test case for methods there.\n",
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