Two proofs of the Kodaira vanishing theorem

The Kodaira theorem ensures vanishing of higher cohomology for line bundles of the form $K_X + A$, where $A$ is ample, on a smooth complex projective manifold $X$. This fundamental result was initially shown using methods of complex analytic and differential geometry; a purely algebraic proof appeared much later. I will present two proofs of this theorem, assuming existence of the Hodge decomposition.