Editing Projective variety (section)

===Kodaira vanishing===
The fundamental [[Kodaira vanishing theorem]] states that for an ample line bundle <math>\mathcal{L}</math> on a smooth projective variety ''X'' over a field of characteristic zero,

:<math>H^i(X, \mathcal{L}\otimes \omega_X) = 0</math>

for ''i'' > 0, or, equivalently by Serre duality <math>H^i(X, \mathcal L^{-1}) = 0</math> for ''i'' < ''n''.<ref>{{harvnb|Hartshorne|1977|loc=Ch III. Remark 7.15.}}</ref> The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.<ref>{{citation| last1=Esnault| first1=Hélène| last2=Viehweg| first2=Eckart |year=1992|title=Lectures on vanishing theorems|publisher=Birkhäuser}}</ref>