Kodaira vanishing theorem

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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