Kodaira vanishing theorem

This is a vanishing theorem

Statement

Let $L$ be a holomorphic line bundle on a compact Kahler manifold $M$, whose topological Chern class can be represented by a closed $(1,1)$-form which is positive definite. Let $E$ be any other holomorphic vector bundle. Then the cohomology groups $H^{q}(M,E\otimes L)$ vanish for large enough $r$ whenever $q$ is at least 1.

Relation with other results

Kodaira embedding theorem

Further information: Kodaira embedding theorem

Lichnerowicz vanishing theorem

Further information: Lichnerowicz vanishing theorem