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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 a [[nef line bundle|RZ66X0NAM5]] on $X$ that is big, that is, whose top self-intersection number satisfies $(L^n) \u003e 0$.\nThen\n$$H^i(X, \\omega_X \\otimes L) = 0 \\quad \\text{for all } i \u003e 0,$$\nwhere $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].\n\n## Remarks\n\nFor a nef line bundle, bigness in the sense of maximal Iitaka dimension is equivalent to $(L^n) \u003e 0$.\nThe idea of the proof: by Kodaira's lemma, some multiple of $L$ is an ample divisor plus an effective divisor.\nA cyclic covering trick and a log resolution reduce the statement to the [[Kodaira vanishing theorem|NY77DQ2AX8]] on a covering variety.\n\nThere is a more general form for $\\Q$-divisors: if $D$ is a nef and big $\\Q$-divisor whose fractional part has simple normal crossing support, then $H^i(X, \\omega_X \\otimes \\mathcal{O}_X(\\lceil D \\rceil)) = 0$ for $i \u003e 0$.\n\nExample: let $n \\ge 2$, let $X \\to \\mathbb{P}^n$ be the blowing up of a point, and let $L$ be the pullback of $\\mathcal{O}(1)$.\nThen $L$ is nef and big but not ample, since it has degree $0$ on the lines in the exceptional divisor.\nThe theorem still gives $H^i(X, \\omega_X \\otimes L) = 0$ for $i \u003e 0$, which Kodaira's theorem does not cover.\n\nBigness is needed: on $X = C \\times \\mathbb{P}^1$ with $C$ a curve of genus $g \\ge 1$, the pullback $L$ of $\\mathcal{O}_{\\mathbb{P}^1}(2)$ is nef with $(L^2) = 0$, and $H^1(X, \\omega_X \\otimes L) \\cong H^1(C, \\omega_C) \\otimes H^0(\\mathbb{P}^1, \\mathcal{O}) \\ne 0$ by the Künneth formula.\n\n## Explanations\n\n### L1 human\n\nThe Kodaira vanishing theorem needs a positive (ample) line bundle.\nThe Kawamata–Viehweg theorem weakens this to a bundle that is never negative on curves and that has positive volume.\nSuch bundles appear naturally after blowing up, so the weaker hypothesis is much more flexible.\n\n### L2 human\n\nThis theorem is the main cohomological tool of the minimal model program.\nIt is used in the proofs of the basepoint-free theorem and of the cone theorem, where one works with nef and big divisors that are not ample.\nIts version for $\\Q$-divisors allows rounding of fractional divisors, which is needed when one works with pairs $(X, \\Delta)$.\n",
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