Kodaira vanishing theorem

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 \( X \).