We introduce some applications of the Hodge theory on compact hermitian manifolds in the post, including the isomorphism of and and the Kodaira-Serre duality.
Relations of harmonic forms and Dolbeault cohomology
Suppose is a compact hermitian complex manifold and is a hermitian holomorphic vector bundle on . Since we have well-defined operator for each , the -valued Dolbeault cohomology on can be defined analogously to the ordinary Dolbeault cohomology on .
Proposition. Suppose is a compact hermitian complex manifold and is a hermitian holomorphic vector bundle on . Then we have the orthogonal direct sum decompostion
Proof: Let be the orthogonal projection of to . Then by Hodge theorem (and part of its proof), we see that , where is the Green operator. For each , there is a decomposition implying that Meanwhile, noting that we obtain the orthogonal relations of with the image of and . The images of the two operators are orthogonal since .
Proposition. Suppose is a compact hermitian complex manifold and is a hermitian holomorphic vector bundle on . Then there exists a unique harmonic form in each cohomology class in . Such harmonic form is the only element in with the minimal length.
Proof: Consider the cohomology class represented by , which satisfies . By the previous proposition, there exists a unique , together with such that We claim that is desired harmonic form.
First we need , and to show this it suffices to prove . Note that we have The direct sum deecompostion implies that .
If there is another harmonic , then showing the uniqueness of such .
For any other , there is such that It follows that
As harmonic forms are all -closed, we have a natural homomorphism
Corollary. The above homomorphism is an isomorphism. In particular, is finite dimensional.
Let be the sheaf of -valued holomorphic -forms on . In analogy of the ordinary Dolbeault theorem, we have the canonical isomorphism
Kodaira-Serre duality
The Hodge theory permits us to study Dolbeault cohomology by harmonic forms. Recall that when is -dimensional, we have the star operator As commutes with , there is a induced ismorphism Using the isomorphism of and , we see that This relation is usually called Kodaira-Serre duality.
There is another approach to the above equality, which actually identify with the dual of . Consider the pairing given by
We claim that this induces a well-defined pairing on the corresponding cohomology groups. Suppose and are -closed forms. Then for any we have implying that Similarly for any it holds that Thus the value of only depends on the cohomology class of and , yielding a pairing We see that this pairing is nondegenerate and hence gives a natural isomorphism In particular, let , we obtain
Another useful form of the Kodaira-Serre duality is obtained by taking . In this case we have and particularly Considering the generalized Dolbeault theorem, we further get the natural isomorphism
One thing to mention that the Kodaira-Serre duality only needs the condition that is a compact complex manifold and is a holomorphic vector bundle on . On the one hand, the pairing above has nothing to do with the hermitian structure on and . On the other hand, by considering a partition of unity, we see that and can always be equipped with hermitian structures.