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In [[algebraic geometry]], the '''Mumford vanishing theorem''' {{harvtxt|Mumford|1967}} states that if ''L'' is a semi-ample invertible sheaf with [[Iitaka dimension]] at least 2 on a [[complex projective manifold]], then

:<math>H^i(X,L^{-1})=0\text{ for }i = 0,1.\ </math>

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

==References==

*{{Citation | last1=Kawamata | first1=Yujiro | title=A generalization of Kodaira-Ramanujam's vanishing theorem | url=http://dx.doi.org/10.1007/BF01456407 | doi=10.1007/BF01456407 | id={{MR|675204}} | year=1982 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=261 | issue=1 | pages=43–46}}
*{{Citation | doi=10.2307/2373099 | last1=Mumford | first1=David | author1-link=David Mumford | title=Pathologies. III | id={{MathSciNet | id = 0217091}} | year=1967 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=89 | pages=94–104 | issue=1 | publisher=The Johns Hopkins University Press | jstor=2373099}}

[[Category:Theorems in algebraic geometry]]