Contents

complex geometry

# Contents

## Idea

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

$\Omega^{n,0}_X(L) \simeq L \otimes \Omega^{n,0}_X$

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

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