Kodaira vanishing theorem

Remarks

Kodaira'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. Deligne and Illusie gave an algebraic proof by reduction to positive characteristic, using a lifting of $X$ modulo $p^2$.

Example: on $X = \mathbb{P}^n_\C$ with $L = \mathcal{O}(d)$, $d > 0$, one has $\omega_X \otimes L = \mathcal{O}(d - n - 1)$, and indeed $H^i(\mathbb{P}^n, \mathcal{O}(m)) = 0$ for $0 < i < n$ and all $m$, and for $i = n$ when $m > -n-1$.

The ampleness hypothesis cannot be dropped: for a curve $X$ of genus $g \ge 1$ and $L = \mathcal{O}_X$, $H^1(X, \omega_X) \cong \C \ne 0$. The characteristic hypothesis is also needed: Raynaud gave counterexamples in positive characteristic.

Going deeper

For a student

The theorem is a key input of the Kodaira embedding theorem and of the theory of adjoint linear systems $\abs{K_X + L}$. Its 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. It fails in positive characteristic, which makes it a test case for methods there.