Mumford vanishing theorem

In algebraic geometry, the Mumford vanishing theorem proved by Mumford^[1] in 1967 states that if L is a semi-ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then

${\displaystyle H^{i}(X,L^{-1})=0{\text{ for }}i=0,1.\ }$

The Mumford vanishing theorem is related to the Ramanujam vanishing theorem, and is generalized by the Kawamata–Viehweg vanishing theorem.