Kawamata–Viehweg vanishing theorem

Remarks

For a nef line bundle, bigness in the sense of maximal Iitaka dimension is equivalent to $(L^n) > 0$. The idea of the proof: by Kodaira's lemma, some multiple of $L$ is an ample divisor plus an effective divisor. A cyclic covering trick and a log resolution reduce the statement to the Kodaira vanishing theorem on a covering variety.

There 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 $i > 0$.

Example: let $n \ge 2$, let $X \to \mathbb{P}^n$ be the blowing up of a point, and let $L$ be the pullback of $\mathcal{O}(1)$. Then $L$ is nef and big but not ample, since it has degree $0$ on the lines in the exceptional divisor. The theorem still gives $H^i(X, \omega_X \otimes L) = 0$ for $i > 0$, which Kodaira's theorem does not cover.

Bigness 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.

Going deeper

For a student

This theorem is the main cohomological tool of the minimal model program. It 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. Its version for $\Q$-divisors allows rounding of fractional divisors, which is needed when one works with pairs $(X, \Delta)$.