nLab
positive line bundle

Contents

Idea

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

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