Kodaira's Projective Embedding Theorem

In this chapter we are going to prove a famous theorem due to Kodaira, which gives a characterization of which compact complex manifolds admit an embedding into complex projective space. In Sec. 1 we shall define Hodge manifolds as those which carry an integral (1, 1) form which is positive definite in local coordinates. We then give various examples of such manifolds. Kodaira's theorem asserts that a compact complex manifold is projective algebraic if and only if it is a Hodge manifold. This is a very useful theorem, as we shall see, since it is often easy to verify the criterion. Chow's theorem asserts that projective algebraic manifolds are indeed algebraic, i.e., defined by the zeros of homogeneous polynomials. Thus the combination of these two theorems allows one to reduce problems of analysis to ones of algebra (cf. Serre's famous paper [2] in which this program of comparison is carried out in great detail).