Embedding of compact complex manifolds - 4
In this post we finally state and prove the Kodaira embedding theorem that gives an equivalent condition for a compact complex manifold to be a submanifold of the projective space.
Suppose
is well-defined, i.e., there exists such that . This is equivalent to that is injective, i.e., whenever . We claim that this is equivalent to that there exist sections such that and , which happens if and only if Indeed, first suppose that . With loss of generality, we may assume that . If , we just take , and , where . Otherwise there must exists such that , and then we take Conversely, suppose . Then there exists such that for all . Thus such and cannot exist. This proves our assertion for the equivalence. Note that the equivalent condition for to be injective given above indeed implies the well-definedness of . is an immersion, i.e., for each the tangent map is injective. It is clear that this is equivalent to that the pullback is surjective. With loss of generality, suppose . Then , where is an open subset with coordinates . Hence the surjectivity of the pullback can be expressed as that for any , there exists such that If we let then and . This may be expressed in a more intrinsic way. Denote the sheaf of holomorphic sections of vanishing at by . Consider the map where satisfies that . This map is well-defined in view of the following computation: It is not hard to see that is an immersion if and only if
Theorem (Kodaira embedding theorem). Suppose
Proof: In view of the preceding discussions, we need the following two steps:
(1) there existssuch that for any and any distinct , the restriction is surjective;
(2) there existssuch that for any and any , the differential map is surjective. We may actually reduce the above assertions to the case when
and are fixed. This is because if , then for and near and , and similarly for the condition that is injective at . Hence we can take the same bounds and in a neighborhood, and then the compactness of implies that the global bounds can be taken. Step (1). Suppose
is the blow-up of at and , with the exceptional divisors and . Let . For each , the pullback of gives If , then clearly and is an isomorphism. Otherwise, for each section , by Hartogs’ theorem its restriction to can be extended to , and we can see that . Thus is again an isomorphism. Meanwhile, note that
is trivial along and , respectively, i.e., Since and are both compact, we see that We can also see that the following diagram, where horizontal arrows are restrictions, commutes: 
Hence it suffices to show that the restiction from
to is surjective. Note that the direct image of under the inclusion is a sheaf on (for which we use the same notation) with . Consider the following exact sequence of sheaves on
: Then we have the exact sequence It remains to show that . Since is compact, there exists such that is positive for any . Then the corresponding form of is positive semi-definite. Meanwhile, there exists such that is positive for . Thus for , we see that is positive. The Kodaira vanishing theorem then tells us Step (2). Suppose
is the blow-up of at , with the exceptional divisor . Similarly we have the isomorphism Moreover, note that vanishes at if and only if vanishes along , we obatin an isomorphism Since is trivial along , we have It is not hard to see that the following diagram commutes: 
It suffices to show that the restriction from
to in surjective. Consider the exact sequence of sheaves: which yields the exact sequence It remains to show that . There exists such that is positive for . Thus for , we see that is positive. By Kodaira vanishing theorem,
Corollary. Suppose