# Kodaira embedding theorem Kodaira embedding theorem In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, sur les nombres complexes, 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 (nommé d'après W. V. ré. 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.

Kodaira also proved (Kodaira 1963), by recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later simplified by Buchdahl to remove reliance on the classification (Buchdahl 2008).

Kodaira embedding 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 {style d'affichage varphi :Xrightarrow mathbb {P} } of X into some projective space such that {displaystyle varphi ^{*}{mathématique {O}}_{mathbb {P} }(1)=L^{otimes m}} for some m > 0.

See also Fujita conjecture Hodge structure Moishezon manifold References Buchdahl, Nicolas (2008), "Algebraic deformations of compact Kähler surfaces II", Journal mathématique, 258 (3): 493–498, est ce que je:10.1007/s00209-007-0168-6 Hartshorne, Rouge-gorge (1977), Géométrie algébrique, Berlin, New York: Springer Verlag, ISBN 978-0-387-90244-9, M 0463157, OCLC 13348052 Kodaira, Kunihiko (1954), "On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)", Annales de Mathématiques, Deuxième série, 60 (1): 28–48, est ce que je:10.2307/1969701, ISSN 0003-486X, JSTOR 1969701, M 0068871 Kodaira, Kunihiko (1963), "On compact analytic surfaces III", Annales de Mathématiques, Deuxième série, 78 (1): 1–40, est ce que je:10.2307/1970500, ISSN 0003-486X, JSTOR 1970500 A proof of the embedding theorem without the vanishing theorem (due to Simon Donaldson) appears in the lecture notes here. Catégories: Theorems in complex geometryTheorems in algebraic geometry

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