# Contents

## Idea

A holomorphic line bundle on a complex manifold is called positive if its curvature differential 2-form is, after multiplication with $i = \sqrt{-1}$, a positive definite $(1,1)$-form.

## Properties

### Relation to ample line bundles

The Kodaira embedding theorem implies that a positive line bundle is an ample line bundle and conversely that any ample line bundle admits a Hermitean metric that makes it a positive line bundle.

### Sheaf cohomology of a positive line bundle

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

Last revised on August 17, 2014 at 21:32:51. See the history of this page for a list of all contributions to it.