Linear Algebra and Hodge Theory

Consider a finite dimensional real vector space , endowed with an almost complex structure, i.e. an endomorphism such that . Note that such a has even dimension and that its complexification admits a natural decomposition. Indeed, let

and consider the -linear extension of to . It has eigenvalues and, according to its decomposition in eigenspaces, we have
where denotes the eigenspace of and the one of .

endowed with an almost complex structure, i.e. an endomorphism such that . First, note that has even dimension. Moreover, we can decompose its complexification in the obvious way. Indeed, let

and consider the -linear extension of to . It has eigenvalues and, according to its decomposition in eigenspaces, we have
where denotes the eigenspace of and the one of . Finally, complex conjugation on induces an isomorphism between and . This is an example of a pure (real) Hodge structure of weight 1. Besides, Hodge structures arise in many more complicated contexts. For instance, let be an -dimensional compact oriented Riemannian manifold and consider the sheaf of -forms on by , with the exterior derivative . Let us define the Laplacian operator
where and is the Hodge operator. Now, consider the set of harmonic forms
Then, there is a natural map
sending any harmonic form to its cohomology class. Thanks to the work of Hodge, Kodaira et al, this map turns out to be an isomorphism of vector spaces, hence
Take a complex manifold and consider the sheaf of complex -forms , which can be decomposed into the direct sum
where denotes the sheaf of -forms and , where and denote the Dolbeault operators. We may try to see if is even dimensional, but, unfortunately,
which means that does not act on closed forms. The picture becomes much nicer when we restrict to Kähler manifolds. Indeed, it turns out that
so acts on harmonic forms. Moreover, we have
This leads to a decomposition of into the direct sum of the spaces of -harmonic forms ,
where . When we restrict our attention to a compact Kähler manifold, the isomorphism
induces the (Hodge) decomposition
where acts by on each . Moreover, we get a decreasing (Hodge) filtration, which is an equivalent data to the decomposition above:
where
Note that the two data are equivalent since, given a filtration on , we recover the decomposition simply defining

Definition 11.1

An integral Hodge structure pure of weight is a finitely generated free abelian group , with a decomposition of ,

such that

 

Equivalently, as discussed above, we can replace the decomposition with a finite decreasing filtration of , such that

whenever . In the case of compact Kähler manifolds,
is a pure Hodge structure of weight .


Definition 11.2

Given and two pure -Hodge structures of weight and , then a morphism of Hodge structures of bidegree is a group homomorphism such that

 

We restrict our attention to morphisms of bidegree introducing the Tate twist.

Definition 11.3

Define the Tate-Hodge structure to be the pure Hodge structure of weight -2

Moreover, given a pure Hodge structure of weight and an integer , define the Tate twist to be the pure Hodge structure of weight , defined by

 

We refer to morphism of pure Hodge structures of weight and as a morphism of Hodge structures of bidegree between and .

The first example of morphism between pure Hodge structure comes again from Kähler manifolds. Given holomorphic map between compact Kähler manifolds, we get

preserving the weight! So is a morphism of Hodge structure. What about the Gysin morphism ? Note that does not preserve the weight, since
but induces a morphism of Hodge structures between Exercise. Prove that a morphism of Hodge structures (of bidegree ) is strict for the Hodge filtration, i.e. for all

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