Kodaira embedding theorem

From Wikipedia, the free encyclopedia

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 cohomology class in degree 2 defined by the Kähler form ω that 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 by its compactness from Chow's theorem. The Kodaira result gives a sufficient condition; that the integrality is necessary is more elementary and was already known.