On a compact Kähler manifold , if is a positive holomorphic line bundle, then for . The canonical-bundle twist is part of the hypothesis and conclusion; positivity alone does not remove it on a general manifold. For a complex torus, the canonical bundle is trivial, so higher cohomology of a positive line bundle vanishes without an additional twist.
Let be positive and a closed smooth hypersurface in a complex torus. The semipositive canonical bundle of a submanifold of a complex torus makes positive. Since , the restriction of the -th ambient twist is when . Apply the Kodaira vanishing theorem on , and use and its long exact sequence to induct from the positive bundle on the torus. No unproved ambient positivity of is needed.
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The Kodaira vanishing theorem is a fundamental result in algebraic geometry, named after Kunihiko Kodaira. It provides important information about the cohomology of certain types of sheaves on smooth projective varieties. ### Statement of the Theorem In its classical form, the Kodaira vanishing theorem can be stated as follows: Let \( X \) be a smooth projective variety over the complex numbers, and let \( L \) be an ample line bundle on \( X \).