Kodaira embedding theorem

Kodaira embedding theorem

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.

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