# 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 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.
A Kähler manifold with a Hodge metric is occasionally called a **Hodge manifold** (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective.
The converse that projective manifolds are Hodge manifolds is more elementary and was already known.

## See also

## References

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 - Kodaira, Kunihiko (1954), "On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)",
*Annals of Mathematics. Second Series*,**60**(1): 28–48, doi:10.2307/1969701, ISSN 0003-486X, JSTOR 1969701, MR 0068871 - A proof of the embedding theorem without the vanishing theorem (due to Simon Donaldson) appears in the lecture notes here.