For a complex manifold , we have two different cohomology theory: the de Rham cohomology and the Dolbeault cohomology. As they both come from the cochain of differential forms on , the relations between them are worth considering. In this post we demonstrate this relation on compact Kähler manifolds, which is given by the Hodge decompostion.
Fundamental identities
Before the Hodge decompostion, we need some preliminary fundamental identities on Kähler manifolds which are useful not only for the Hodge decompostion in this post but also the Lefschetz decomposition in the next post.
Suppose is an -dimensional complex manifold with a hermitian metric and is the associated fundamental form of . The hermitian structure induces a Riemannian structure on , which further induces a volume form on and a star operator on the space of differential forms on that maps into . We have differential operators , and on which are related by Consider the operators The hermitian structure of gives an inner product on the space consisting of -form on with compact support by Then an inner product on is constructed from these inner products on . With respect to this inner product, it verifies that and are the dual of each other, and and are the dual of each other.
Using the fundamental form , we can define the Lefschetz operator by It is direct to see that . Define another operator on by We see that is exactly the dual of on with respect to the inner product . As is a real form, and hence must be real operators, that is
For two operators and on a space, their commutator is defined by
Proposition. If is a Kähler manifold, then it holds on that i.e., commutes with and , and commutes with and .
Proof: As is a Kähler form, we have . Since is a -form, it follows that . Hence for any , Now consier any . We have and analogously for .
Proposition. For a Kähler manifold , it holds on that Moreover, we have on .
Proof: Let us first consider the case when . Define operators and on by We see that and are -linear, and hence we may consider their dual operators and . The following anti-commutative relations is clear from our definition: Define operators and on by It is direct to see that and commute with each , , and . We also have Note that for each with and , we have implying that . Similarly we have . It follows that
Consider the interaction of and . First note that if , then holds for any , implying that At the same time, we have implying that It follows that which implies that Now for , we have and hence Since this also holds when either or , we obtain By a similar arguments, we see that and that for Moreover, for any and , it always holds that Now we can compute that Taking conjugation, we obtain Moving to the dual operators, it follows that We also have For a form , we have It follows that
Now the identities have been proven for . For the general cases, we need the local description of a Kähler metric. Consider an arbitrary point . There exists a local orthogonal frame around such that for any . Replacing by and doing the same computation, we see that and agrees up to some terms involving and , which vanish at . Using the similar arguments, the above identities all hold at . Since they only involve global operators, we obtain the desired results.
Hodge decomposition
Now suppose is a compact Kähler manifold. Recall that we have the Laplace-Beltrami operators Denote the spaces of corresponding harmonic forms by , and , and let where is or . By the Hodge theorem, we have the natural isomorphism for each and . Meanwhile, by the Hodge theory on compact Riemannian manifolds, there is an analogous result that we have the natural isomorphism for each .
Proposition. Suppose is a compact Kähler manifold. Then
Proof: We will use the fundamental identities we have shown above. Since , we have , and hence Since and , we obtain It follows that Hence it remains to show that . Direct computation yields that
This relations of , and implies that and hence we may simply denote them by . Since is clearly a real operator, we see that for , implying that Moreover, for a harmonic , we can decompose it into Then with . Using the uniqueness of the decomposition of -forms, we see that and hence . It then follows that we have the direct sum decompostion of spaces Passing this to the cohomology, we obtain the Hodge decompostion.
Theorem (Hodge decomposition). Suppose is a compact Kähler manifold, then for each , we have the decomposition Under this isomorphism, each is identified with a subspace of , through which we have
Recall that we have the Betti numbers Similarly, we define the Hodge numbers by By the above Hodge decomposition theorem, we have the relations between Betti numbers and Hodge numbers given as A direct consequence is that must be even as we have Thus there is a topological constraint for a manifold to be Kähler, that is the Betti numbers of odd order should be even, and the Betti numbers of even order should be positive.
We may apply the above results to compute the Dolbeault cohomology of .
Proposition. The Dolbeault cohomology groups of is given by
Proof: The assertion is equivalent to that Note that by the singular cohomology, we have If we have any with , then a contradiction. Hence we must have for each and for other cases.