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.
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.
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
with the canonical line bundle $\Omega^{n,0}_X$ is concentrated in degree 0:
When a prequantum line bundle is positive, then traditional geometric quantization with Kähler polarization coincides with geometric quantization by push-forward, see this remark.
Wikipedia, Positive line bundle
Kefeng Liu, Xiaofeng Sun, Xiaokui Yang, Positivity and vanishing theorems for ample vector bundles (arXiv:1006.1465)
Last revised on August 17, 2014 at 21:32:51. See the history of this page for a list of all contributions to it.