(Kodaira vanishing)
Let \(X\) be a smooth projective variety of dimension \(n\) over a field of characteristic zero, and let \(L\) be an ample line bundle on \(X\). Then \[ H^i(X, \omega_X \otimes L) = 0 \quad \text{for } i > 0 . \] Equivalently, by Serre duality, \(H^i(X, L^{-1}) = 0\) for \(i < n\).
On a smooth projective curve of genus \(g\), an ample line bundle is one of positive degree, and \(H^1(C, \omega_C \otimes L) \cong \dualof{H^0(C, L^{-1})} = 0\) because \(L^{-1}\) has negative degree. On \(\PP^n\), with \(\omega = \OO(-n-1)\) and \(L = \OO(m)\) for \(m > 0\), the theorem says \(H^i(\PP^n, \OO(m-n-1)) = 0\) for \(i > 0\), which agrees with the Čech computation of the cohomology of projective space.
The characteristic-zero hypothesis is necessary: Raynaud constructed smooth projective surfaces in characteristic \(p\) with an ample line bundle \(L\) and \(H^1(X, L^{-1}) \neq 0\).