In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds.
In effect it says precisely which complex manifolds are defined by homogeneous polynomials.
Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N. The fact that M embeds as an algebraic variety follows from its compactness by Chow's theorem.
Let X be a compact Kähler manifold, and L a holomorphic line bundle on X.
Then L is a positive line bundle if and only if there is a holomorphic embedding