Positive holomorphic line bundle

A holomorphic line bundle is positive when it admits a Hermitian metric whose Chern connection has positive curvature form $i{F}_{\nabla }$, equivalently when $i{F}_{\nabla }$ is a positive real (1, 1)-form.

The Kodaira embedding theorem says that a compact complex manifold carrying a positive holomorphic line bundle is projective: sufficiently large tensor powers of the bundle have global holomorphic sections defining a holomorphic embedding into Complex projective space. Conversely, the restriction of $\mathcal{O}(1)$ gives a positive line bundle on every projective complex manifold.