the Kodaira embedding theorem
/ koh-DYE-rah /
Abstract compact complex manifolds are flexible and can be hard to get one's hands on. But the most concrete and best-understood ones are the projective manifolds — those that sit inside some complex projective space CP^N as the zero set of polynomials, where all the tools of algebraic geometry apply. The Kodaira embedding theorem gives a clean, checkable criterion for when an abstract compact complex manifold is secretly projective: it is so exactly when it carries a positive line bundle.
Precisely, a compact complex manifold M admits a holomorphic embedding into some CP^N if and only if M carries a positive holomorphic line bundle L (a bundle with a Hermitian metric of positive Chern curvature, equivalently whose first Chern class is represented by a Kähler form with integral periods). The embedding is built from sections: take a high power L^{tensor m}; for m large, Kodaira vanishing guarantees enough global holomorphic sections s_0, ..., s_N that the map p -> [s_0(p) : ... : s_N(p)] into CP^N is well defined, injective, and an immersion — hence an embedding. So 'has a positive line bundle' is equivalent to 'is a smooth projective variety'. This is the bridge that identifies the analytic category (compact Kähler manifolds with an integral Kähler class) with the algebraic category (projective varieties).
Why it matters: it makes precise which complex manifolds are algebraic, and it is the converse direction to the easy fact that projective varieties are Kähler. Combined with Chow's theorem (a closed analytic subset of projective space is algebraic), it shows that such manifolds are cut out by polynomials. The honesty caveats are sharp: a positive line bundle requires the Kähler class to be integral (lie in the image of H^2(M, Z)), so a generic compact Kähler manifold — for instance a generic complex torus of dimension >= 2 — is Kähler but NOT projective, having no integral Kähler class and hence no positive line bundle. Kähler is necessary but not sufficient for projectivity; positivity (integrality) is the extra ingredient.
A complex torus C^n / L (n >= 2) is always a compact complex manifold and is Kähler with a flat metric, but a generic lattice L gives no positive line bundle with integral Kähler class, so the generic torus does not embed in any CP^N — it is non-algebraic. Special tori whose periods satisfy the Riemann bilinear relations do carry positive bundles; those are the abelian varieties, and Kodaira embeds exactly them.
A generic complex torus is Kähler but not projective; only those with a positive (integral) line bundle embed.
Kähler is necessary but not sufficient for projectivity. The positive line bundle requires an INTEGRAL Kähler class; a generic compact complex torus of dimension >= 2 is Kähler yet has no such bundle and is not algebraic.