As another application of the Hodge theory on compact Kähler manifolds, the Lefschetz decomposition of the de Rham cohomology groups is of great importance as well as the Hodge decomposition. We will use the representation theory of the Lie algebra to give a proof of the hard Lefschetz theorem, and discuss the Hodge index theorem which is regared as a corollary of this decomposition in this post.
Representations of
Recall that the Lie algebra consists of complex matrices with trace zero, with the Lie bracket given by As a vector space, has a basis given by , and generates the whole Lie algebra with the relations
For a finite dimensional vector space over , we have the Lie algebra of endomorphisms on . A representation of is given as a Lie algebra homomorphism and then is called a -module. This is equivalent to give satisfying the same relations as . For , we may write (resp.) simply as (resp.) for short if is clear from the context. A subspace of is called a -submodule, or simply, a submodule, if is closed under the action of . is called irreducible if has no nontrivial submodule, i.e., if is a submodule, then either or .
A result in the representation theory of Lie algebras tells that for any submodule , there exists another submodule such that . It follows that any -module can be decomposed into the direct sum of irreducible submodules.
Now suppose is an irreducible -module. We want to study the further structure of . The eigenspaces of , which are also called weight spaces, are considered in our analysis.
Suppose is a nonzero eigenvector of with respect to the eigenvalue . Then since and are both eigenvectors of . Moreover, if (resp.) is nonzero, then and (resp.) are linearly independent as they are eigenvectors with respect to different eigenvalues. Since is finite dimensional, there exists such that .
We say a vector is primitive if is a nonzero eigenvector of with . By the above discussion we see that primitive vectors exist.
Proposition. Suppose is primitive. Then is generated (as a vector space) by .
Proof: Since is irreducible, it suffices to show that the subspace generated by is a submodule. As are all eigenvectors of , it is clear that is closed under the action of and . Suppose . We show by induction that which certainly implies that is closed under .
For , comes from the definition of the primitive property. Assume the above formula holds for . Then for , we have completes the induction.
Corollary. can be decomposed into the eigenspaces of as where each is the eigenspace of with respect to , and has dimensional one.
Corollary. Suppose the dimensional of is . Then
Proof: Take a primitive vector , with the corresponding eigenvalue of . Since is -dimensional, we have and . Then implying that .
We may generalize the above stucture theorem of irreducible -modules. Suppose is a (finite-dimensional) -module which is not necessarily irreducible. Consider the decomposition of into irreducible submodules: with each irreducible. Let . Then is exactly the -dimensional subspace of generated by a primitive element, and it is also clear that is the direct sum of such . Since each satisfies that we obtain the decompostion of as
As each is decomposed into eigensubspaces of , so is . The decomposition into weight spaces and the decomposition into irreducibles are directly compatible, implying that We can also see that
Since for each and , are both isomorphisms, it follows that are isomorphisms as well.
Lefschetz decompostion and Hodge index theorem
Suppose is a compact Kähler manifold of dimensional with the Kähler form . A action on the -dimensional space will be introduced in order to apply the above results.
Lemma. The operators and on commute with .
Proof: Recall that we have Thus Taking the dual operators, we see that
By this lemma, and give well-defined operators on . Using the natural isomorphism , we obtain operators It worth noting that can be expressed as
In view of the Hodge decomposition, is considered as a subspace of . Let be the projection, and consider We obtain in the previous post that As maps into and does the converse thing, it is also direct to see that Therefore we have a representation of on by with
Applying the description of the structure of -modules, and noting that is exactly the eigenspace of with respect to the eigenvalue , we obtain the following Lefschetz decompostion.
Theorem (Hard Lefschetz theorem). Suppose is an -dimensional compact Kähler manifold. Then the map is an isomorphism for . Define the primitive cohomology by then we have and the Lefschetz decomposition
By considering the bidegree of differential forms, we see that the Lefschetz decomposition is compatible with the Hodge decomposition, that is There is also an isomorphism for each and .
As an application of the Hodge decomposition and the Lefschetz decomposition, we introduce the Hodge-Riemann bilinear relations and the Hodge index theorem.
Consider the bilinear form on given by As is a real form, defined a real bilinnear form on . Comparing the bidegree of forms, we see that when and , we have unless and .
Lemma. Suppose satisfies that vanishes at , then holds at .
This lemma can be shown by direct computation in a local orthonormal frame.
Theorem (Hodge-Riemann bilinear relations). Suppose is an -dimensional compact Kähler manifold. Then for a nonzero , we have
Proof: Without loss of generality, we may assume that is a harmonic -form such that everywhere. Then by the lemma we have
Note that , we obtain from the Lefschetz decomposition that is nondegenerate on . Meanwhile, the bilinear relations also imply the positive definite property of the quadratic form on (or simply if ) when is even.
Recall that for a general -dimensional real compact oriented smooth manifold , the Poincaré duality yields a nondegenerate bilinear form on by The index of is defined to be the signature of the quadratic form on .
Specifying the case when is a -dimensional compact Kähler manifold, it is clear that agrees with on . The Lefschetz decompostion tells us and considering the real part, we obtain
By the corollary of the Hodge-Riemann bilinear relations, it holds that
Theorem (Hodge index theorem). Suppose is a -dimensional compact Kähler manifold, then
Proof: By the Lefschetz decompostion, we have Together with the above formula of and the symmetric properties of , direct computation yields that It is also direct that implying the final conclusion.