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 a Hodge metric, meaning that the 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. A Kahler 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.

[edit] See also

• Hodge structure
• Moishezon manifold

[edit] References

• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, MR0463157, ISBN 978-0-387-90244-9, OCLC 13348052 
• Kodaira, Kunihiko (1954), "On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)", Annals of Mathematics. Second Series 60: 28–48, MR0068871, ISSN 0003-486X