Hodge theory is an important tool used in the study of geometry and topology. Roughly speaking, Hodge theorem provides us with a unique harmonic form as a representative of each cohomology class, which enables us to use analytic methods to explore the topology of manifolds. Meanwhile, Hodge theory reveals an essential relation between the de Rham cohomology and the Dolbeault cohomology, which is almost indispensible in the further discussion of Kähler manifolds.
In this post we introduce some operators on a compact hermitian manifold with a hermitian bundle and formulate the Hodge theorem.
Inner product structure on the section space
Suppose is a compact hermitian manifold with hermitian metric on , and is a holomorphic hermitian vector bundle on with hermitian etric . Recall that we have the holomorphic cotangent bundle and the antiholomorphic cotangent bundle and of , whose direct sum yields the complexified cotangent bundle . Denote that
The sheaf of the sections of is denoted by and that of the tensor product is denoted by . For an open subset , denote the space of -forms and -valued -forms on by and , i.e., We also consider the direct sum
As we have hermitian metrics on and , the bundle can also admit a hermitian structure. Specifically, consider local orthonormal frames and of and . Suppose the dual frame of them are given by and , respectively. Then we have the local expression of and as
The fundamental form associated to the hermitian metric is and the volumn form is then
Requiring that form a local orthogonal frame of and that for each , we obtain a hermitian metric locally on . More generally, to introduce a hermitian structure on , we let be a local orthogonal frame on , with Meanwhile, we can also give a hermitian structure, with a local orthogonal frame and
We can verify that these hermitian metrics are independent of the choice of the local orthogonal frame of and , and hence are well-defined hermitian structure globally.
In order to pass the inner product at each point of the bundle to the inner product of global sections, we need to consider the integral. Suppose are two global -valued -forms on . For each point , the hermitian metric on gives an inner product on the fiber at . Note that is a smooth function in , we can define that where is the volumn form on given by the hermitian metric . It can be verified that this indeed gives an inner product on . Since is exactly the direct sum of these , we also obtain an inner product on . The norm on induced by this inner product is denoted by .
To discuss further properties of this inner product, the star operator is introduced. First let us consider defined by the relation Since we have the nondegenrate pairing the operator is well-defined.
We illustrate the local expression of in order to show that gives a bundle homomorphism Suppose with respect to the local frame we can express as
Direct computation yields that
where and are the complements of and in and is the sign of the permutation and analogously for .
This verifies that depends smoothly on , implying that yields the desired bundle homomorphism. This further induces a homomorphism The inner product on can then be expressed as We can see the property of that and that
The star operator can also be defined on the tensor bundle . Consider the bundle homomorphism given by We see that this is a bundle isomorphism whose inverse is often denoted by . Since induces a hermitian metric on and is identified with , we also have . It is direct to check that is nothing else than . We define the star operator on to be and analogously These induce homomorphisms Using the pairing we see that also holds for , and the two properties of the star operator on still hold true here.
Statement of the Hodge theorem
Recall that gives a well-defined operator for each and , which together gives an operator As we have an inner product on the space , the adjoint operator of is worth considering. Let by the operator defined as Note that here actually acts on the space , and for some , we have and hence
We claim that is exactly the adjoint operator of with respect to the inner product . It suffices to verify this on each . Suppose and . We have Note that , we have , implying that where the last equality holds from the Stokes’ formula. As , we have implying that This proves our assertion.
Using and , we can define the Laplace-Beltrami operator This has some properties which are quite clear from its definition. First, is an operator preserving the type of a form, i.e., for each and we have Second, by the adjoint relation of and , we have showing that is self-adjoint with respect to . Moreover, using the above expression we can see that the kernel of is exactly the intersection of the kernel of and that of . The forms in the kernel of are called the harmonic forms, and the kernel of is often denoted by . The space of harmonic -valued -forms are denoted by .
Now we can state the main theorem of the Hodge theory – the Hodge theorem.
Theorem (Hodge theorem). Suppose is a compact hermitian manifold and is a holomorphic hermitian vector bundle on . Then
;
providing the orthogonal projection there exists a compact operator called the Green operator such that , and , where is the identity operator.
The proof of the Hodge theorem involves some functional analysis methods and will occupy the following several posts.