nLab Kodaira vanishing theorem




The Kodaira vanishing theorem for complex geometry says that if XX is a Kähler manifold and LL a holomorphic line bundle on XX which is positive, then the abelian sheaf cohomology of XX with coefficients in the sheaf of sections of the tensor product

Ω X n,0(L)LΩ X n,0 \Omega^{n,0}_X(L) \simeq L \otimes \Omega^{n,0}_X

with the canonical line bundle Ω X n,0\Omega^{n,0}_X is concentrated in degree 0:

(Lpositive)H 1(X,Ω X n,0(L))=0. (L \; positive) \;\;\Rightarrow\;\; H^{\bullet \geq 1}(X, \Omega^{n,0}_X(L)) = 0 \,.


The statement is due to Kunihiko Kodaira.

Lecture notes include

Last revised on August 17, 2014 at 13:58:02. See the history of this page for a list of all contributions to it.