the Kodaira vanishing theorem
/ koh-DYE-rah /
Cohomology groups are obstructions: a nonzero higher cohomology group typically signals that some equation cannot be solved or some section cannot be extended. The Kodaira vanishing theorem is a powerful guarantee that, under a positivity hypothesis, those obstructing higher groups are simply zero. It is the engine that converts the geometric notion of a 'positive' line bundle into the concrete ability to produce holomorphic sections, and it underlies the whole projective-embedding story.
Precisely, let M be a compact Kähler (in particular projective) manifold of complex dimension n, let K_M be its canonical line bundle (the top exterior power of the holomorphic cotangent bundle), and let L be a positive line bundle — one admitting a Hermitian metric whose Chern curvature is a positive (1,1)-form, equivalently whose first Chern class contains a Kähler form. Then the cohomology groups H^q(M, K_M tensor L) vanish for all q >= 1. An equivalent and frequently used form: H^q(M, L) = 0 for q >= 1 whenever L tensor (dual of K_M) is positive (i.e. L is 'sufficiently positive'). The proof is a Bochner-type argument: the Kähler identities turn the relevant Laplacian into a sum of a manifestly nonnegative term and a curvature term that positivity makes strictly positive, forcing harmonic representatives — hence the cohomology — to vanish.
Why it matters: vanishing of higher cohomology is exactly what makes Riemann-Roch computable (the Euler characteristic collapses to the dimension of global sections) and what feeds the Kodaira embedding theorem — enough sections of a high power of a positive bundle to embed the manifold in projective space. The crucial honesty caveats: positivity is essential and cannot be dropped (a flat or negative bundle generally has nonvanishing higher cohomology), and the theorem genuinely fails in positive characteristic — Raynaud constructed counterexamples — so Kodaira vanishing is a theorem of complex (characteristic-zero, Kähler) geometry, not a formal algebraic identity.
On CP^n the bundle O(k) is positive precisely when k > 0. The canonical bundle is K = O(-n-1). Kodaira vanishing then gives H^q(CP^n, O(k)) = 0 for all q >= 1 whenever k > -n-1, which matches the explicit fact that the higher cohomology of O(k) on projective space vanishes except for very negative k.
On CP^n, positivity of O(k) (k > 0) forces all higher cohomology of O(k) to vanish.
Two non-negotiable caveats: positivity of the bundle is essential (drop it and higher cohomology reappears), and the theorem is false in positive characteristic (Raynaud's counterexamples). It is a complex/Kähler, characteristic-zero theorem.