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3 Added historical remark. Suppressed allusion to Donu Arapura's forthcoming book

Dear Nikita, Kodaira has proved the following fantastic embedding theorem for complex manifolds (conjectured by Hodge in 1950):

A compact complex manifold X is projective algebraic if and only if it has a closed positive (1,1) form whose cohomology class is rational.

Succinct explanation : The positivity is a complex differential geometric notion: locally a (1,1) form can be written $i \Sigma h_{jk} dz_j d\bar {z_k}$ and the matrix $(h_{jk})$ is required to be positive definite.This closed form represents a class in singular cohomology by De Rham's theorem and this class should actually be in $H^2(X,\mathbb Q)$

It follows easily from this that

Given a holomorphic finite unramified covering $\tilde X \to X$ of compact holomorphic manifolds, the manifold $X$ is projective algebraic if and only if $\tilde X$ is

You can read the proofs in Griffiths-Harris's Principles of Algebraic Geometry. I heard through the grapevine that Donu will soon publish a book on the subject: just to be on the safe side you might already order it...

2 Replaced implication by equivalence and added boldface + stylistic modifications

Dear Nikita, Kodaira proved the following fantastic embedding theorem for complex manifolds:

A compact complex manifold X is projective algebraic if and only if it has a closed positive (1,1) form whose cohomology class is rational.

Succinct explanation : The positivity is a complex differential geometric notion: locally a (1,1) form can be written $i \Sigma h_{jk} dz_j d\bar {z_k}$ and the matrix $(h_{jk})$ is required to be positive definite.This closed form represents a class in singular cohomology by De Rham's theorem and this class should actually be in $H^2(X,\mathbb Q)$

From this, it

It follows easily from this thatif the compact holomorphic manifold $\tilde X$ admits of

Given a finite (non-ramified) holomorphic finite unramified covering map $\tilde X \to X$ to our projective of compact holomorphic manifolds, the manifold $X$, then it X$is also projective . algebraic if and only if$\tilde X$is You can read about this result the proofs in Griffiths-Harris's Principles of Algebraic Geometry. I heard through the grapevine that Donu will soon publish a book on the subject: just to be on the safe side you might already order it... 1 Dear Nikita, Kodaira proved the following fantastic embedding theorem for complex manifolds: A compact complex manifold X is projective algebraic if and only if it has a closed positive (1,1) form whose cohomology class is rational. Succinct explanation : The positivity is a complex differential geometric notion: locally a (1,1) form can be written$ i \Sigma h_{jk} dz_j d\bar {z_k}$and the matrix$(h_{jk})$is required to be positive definite.This closed form represents a class in singular cohomology by De Rham's theorem and this class should actually be in$H^2(X,\mathbb Q)$From this, it follows easily that if the compact holomorphic manifold$\tilde X $admits of a finite (non-ramified) holomorphic covering map$\tilde X \to X$to our projective manifold$X\$, then it is also projective.

You can read about this result in Griffiths-Harris's Principles of Algebraic Geometry. I heard through the grapevine that Donu will soon publish a book on the subject: just to be on the safe side you might already order it...