View source for Kodaira embedding theorem

{{short description|Characterises non-singular projective varieties amongst compact Kähler manifolds}}
{{More footnotes needed|date=September 2022}}
In [[mathematics]], the '''Kodaira embedding theorem''' characterises [[Algebraic curve#Singularities|non-singular]] [[projective varieties]], over the [[complex number]]s, amongst [[compact space|compact]] [[Kähler manifold]]s. In effect it says precisely which [[complex manifold]]s are defined by [[homogeneous polynomial]]s.

[[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 [[Algebraic geometry and analytic geometry#Chow.27s theorem|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.

Kodaira also proved (Kodaira 1963), by recourse to [[Enriques–Kodaira classification|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 form#Positive line bundles|positive line bundle]] if and only if there is a holomorphic embedding <math>\varphi:X\rightarrow \mathbb{P}^N</math> of ''X'' into some projective space such that <math>\varphi^*\mathcal O_{\mathbb{P}^N}(1)=L^{\otimes m}</math> for some&nbsp;''m''&nbsp;>&nbsp;0.

==See also==
*[[Fujita conjecture]]
*[[Hodge structure]]
*[[Moishezon manifold]]

==References==
* {{Citation | doi=10.1007/s00209-007-0168-6|last1=Buchdahl|first1=Nicholas|title=Algebraic deformations of compact Kähler surfaces II | year=2008|journal=Mathematische Zeitschrift | volume=258 | issue=3 | pages= 493–498|s2cid=122002987 }}
*{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 |mr=0463157 | year=1977}}
*{{Citation | doi=10.2307/1969701 | last1=Kodaira | first1=Kunihiko | title=On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties) |mr=0068871 | year=1954 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=60 | issue=1 | pages=28–48 | jstor=1969701}}
*{{Citation | doi=10.2307/1970500 | last1=Kodaira | first1=Kunihiko | title=On compact analytic surfaces III | year=1963 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=78 | issue=1 | pages=1–40 | jstor= 1970500}}
* A proof of the embedding theorem without the vanishing theorem (due to [[Simon Donaldson]]) appears in the lecture notes [http://ocw.mit.edu/courses/mathematics/18-966-geometry-of-manifolds-spring-2007/lecture-notes/ here].
*{{cite book |doi=10.1017/CBO9780511629327.004|chapter=Coherent Sheaves |title=Several Complex Variables and Complex Manifolds II |year=1982 |pages=127–198 |isbn=9780521288880 }}

[[Category:Theorems in complex geometry]]
[[Category:Theorems in algebraic geometry]]