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Kodaira vanishing theorem

Contents

Idea

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 \,.

References

The statement is due to Kunihiko Kodaira.

Lecture notes include

Revised on August 17, 2014 13:58:02 by Urs Schreiber (89.204.155.233)