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  "body": "\n## Statement\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $L$ be a [[positive line bundle|J45KWFHENP]] on $X$.\nFor $p \\ge 0$ let $\\Omega_X^p$ be the sheaf of holomorphic $p$-forms on $X$.\nThen\n$$H^q(X, \\Omega_X^p \\otimes L) = 0 \\quad \\text{whenever } p + q \u003e n,$$\nwhere $H^q$ is [[sheaf cohomology|PQ6Q143ARJ]].\n\n## Remarks\n\nThe theorem is also called the Akizuki–Nakano vanishing theorem.\nA compact manifold with a positive line bundle is Kähler, with Kähler form $\\omega = i\\Theta_h$ given by the curvature of a positive metric $h$.\nBy Dolbeault's theorem, a class in $H^q(X, \\Omega^p_X \\otimes L)$ is represented by an $L$-valued harmonic [[$(p,q)$-form|DC81TCR98Y]] $u$.\nThe Bochner–Kodaira–Nakano identity gives $\\langle [i\\Theta_h, \\Lambda] u, u \\rangle \\le 0$ for such $u$, and with $\\omega = i\\Theta_h$ the operator $[i\\Theta_h, \\Lambda]$ is multiplication by $p + q - n$ on $(p,q)$-forms.\nSo $(p + q - n)\\norm{u}^2 \\le 0$, which forces $u = 0$ when $p + q \u003e n$.\n\nThe case $p = n$ is the [[Kodaira vanishing theorem|NY77DQ2AX8]]: $\\Omega^n_X = K_X$ and $H^q(X, K_X \\otimes L) = 0$ for $q \\ge 1$.\nBy Serre duality the theorem is equivalent to $H^q(X, \\Omega^p_X \\otimes L^{-1}) = 0$ for $p + q \u003c n$.\n\nPositivity of $L$ cannot be dropped.\nFor $X = \\mathbb{P}^1$, $n = 1$, $p = q = 1$ and $L$ trivial, one has $H^1(\\mathbb{P}^1, \\Omega^1) \\cong \\C \\ne 0$, though $p + q = 2 \u003e 1$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that many cohomology groups of a positive line bundle, twisted by holomorphic forms, are zero.\nZero cohomology means that local problems have global solutions, so it is a strong and useful conclusion.\nThe hypothesis is a positivity of curvature, the analytic form of ampleness.\n\n### L2 human\n\nThe proof is a model of the Bochner technique: a curvature identity shows that harmonic representatives must vanish.\nIt extends Kodaira vanishing, which is the case $p = n$, to all bundles of holomorphic forms.\nIt is used to prove the Lefschetz hyperplane theorem for cohomology and to control cohomology of $\\Omega^p$ on projective manifolds.\nFor vector bundles the analogous result needs Nakano positivity, not just Griffiths positivity.\n",
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