nLab positive line bundle

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:

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

Related concepts

• When a prequantum line bundle is positive, then traditional geometric quantization with Kähler polarization coincides with geometric quantization by push-forward, see this remark.

• stable vector bundle,

References

• Wikipedia, Positive line bundle

• Kefeng Liu, Xiaofeng Sun, Xiaokui Yang, Positivity and vanishing theorems for ample vector bundles (arXiv:1006.1465)