Consider a finite dimensional real vector space $V$, endowed with an almost complex structure, i.e. an endomorphism $J:V\to V$ such that $J^{2}=-id$. Note that such a $V$ has even dimension and that its complexification admits a natural decomposition.
Indeed, let

$V_{\mathbb {C} }:=qV\otimes _{\mathbb {R} }\mathbb {C} ,$

and consider the

$\mathbb {C}$-linear extension of

$J$ to

$V_{\mathbb {C} }$. It has eigenvalues

$\pm i$ and, according to its decomposition in eigenspaces, we have

$V_{\mathbb {C} }=V^{1,0}\oplus V^{0,1},$

where

$V^{1,0}$ denotes the eigenspace of

$i$ and

$V^{0,1}$ the one of

$-i$.

endowed with an almost complex structure, i.e. an endomorphism $J:V\to V$ such that $J^{2}=-id$. First, note that $V$ has even dimension. Moreover, we can decompose its complexification in the obvious way. Indeed, let

$V_{\mathbb {C} }:=qV\otimes _{\mathbb {R} }\mathbb {C} ,$

and consider the

$\mathbb {C}$-linear extension of

$J$ to

$V_{\mathbb {C} }$. It has eigenvalues

$\pm i$ and, according to its decomposition in eigenspaces, we have

$V_{\mathbb {C} }=V^{1,0}\oplus V^{0,1},$

where

$V^{1,0}$ denotes the eigenspace of

$i$ and

$V^{0,1}$ the one of

$-i$.
Finally, complex conjugation on

$V_{\mathbb {C} }$ induces an isomorphism between

$V^{1,0}$ and

$V^{0,1}$.
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

$X$ be an

$m$-dimensional compact oriented Riemannian manifold and consider the sheaf of

$n$-forms on

$X$ by

${\mathcal {A}}_{X}^{n}$, with the exterior derivative

$d:{\mathcal {A}}_{X}^{n}\to {\mathcal {A}}_{X}^{n+1}$. Let us define the Laplacian operator

$\Delta _{d}:=qdd^{*}+d^{*}d,$

where

$d^{*}=(-1)^{m(n-1)-1}*d*:{\mathcal {A}}_{X}^{n}\to {\mathcal {A}}_{X}^{n-1}$ and

$*$ is the Hodge operator. Now, consider the set of harmonic forms

${\mathcal {H}}^{n}(X)=\left\{\alpha \in {\mathcal {A}}_{X}^{n}{\mbox{ such that }}\Delta _{d}\alpha =0\right\}.$

Then, there is a natural map

${\mathcal {H}}^{n}(X)\longrightarrow H^{n}(X,\mathbb {R} ),$

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

${\mathcal {H}}^{n}(X)\simeq H^{n}(X,\mathbb {R} ).$

Take a complex manifold

$X$ and consider the sheaf of complex

$n$-forms

$({\mathcal {A}}_{X}^{n},d)$, which can be decomposed into the direct sum

${\mathcal {A}}_{X}^{n}=\bigoplus _{i+j=n}{\mathcal {A}}_{X}^{i,j},$

where

${\mathcal {A}}_{X}^{i,j}$ denotes the sheaf of

$(i,j)$-forms and

$d=\partial +{\bar {\partial }}$, where

$\partial :{\mathcal {A}}_{X}^{i,j}\to {\mathcal {A}}_{X}^{i+1,j}$ and

${\bar {\partial }}:{\mathcal {A}}_{X}^{i,j}\to {\mathcal {A}}_{X}^{i,j+1}$ denote the Dolbeault operators. We may try to see if

$H^{1}(X)$ is even dimensional, but, unfortunately,

$[J,d]\neq 0,$

which means that

$J$ does not act on closed forms. The picture becomes much nicer when we restrict to Kähler manifolds. Indeed, it turns out that

$[\Delta _{d},J]=0,$

so

$J$ acts on harmonic forms. Moreover, we have

$\Delta _{d}=2\Delta _{\partial }=2\Delta _{\bar {\partial }}.$

This leads to a decomposition of

${\mathcal {H}}^{n}(X)$ into the direct sum of the spaces of

$(p,q)$-harmonic forms

$H^{p,q}(X)$,

${\mathcal {H}}^{n}(X)=\bigoplus _{p+q=n}H^{p,q}(X),$

where

${\overline {H^{p,q}(X)}}=H^{q,p}(X)$.
When we restrict our attention to a compact Kähler manifold, the isomorphism

$H^{n}(X,\mathbb {C} )\simeq {\mathcal {H}}^{n}(X)$

induces the (Hodge) decomposition

$H^{n}(X,\mathbb {C} )=\bigoplus _{p+q=n}H^{p,q}(X),$

where

$J$ acts by

$(i)^{p}(-i)^{q}$ on each

$H^{p,q}(X)$.
Moreover, we get a decreasing (Hodge) filtration, which is an equivalent data to the decomposition above:

$F^{0}H^{n}(X,\mathbb {C} )=H^{n}(X,\mathbb {C} )\supseteq F^{1}H^{n}(X,\mathbb {C} )\supseteq \cdots \supseteq F^{p}H^{n}(X,\mathbb {C} )\supseteq \cdots ,$

where

$F^{p}H^{n}(X,\mathbb {C} )=\bigoplus _{\begin{matrix}i+j=n\\i\geq p\end{matrix}}H^{i,j}(X).$

Note that the two data are equivalent since, given a filtration

$F^{\bullet }$ on

$H^{n}(X,\mathbb {C} )$, we recover the decomposition simply defining

$H^{p,q}=F^{p}\cap {\overline {F^{q}}}.$

**Definition 11.1**

An integral Hodge structure pure of weight $k\in \mathbb {Z}$ is a finitely generated free abelian group $V_{\mathbb {Z} }$, with a decomposition of $V_{\mathbb {C} }=V_{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {C}$,

$V_{\mathbb {C} }=\bigoplus _{p+q=k}V^{p,q},$

such that

${\overline {V^{p,q}}}=V^{q,p}.$

Equivalently, as discussed above, we can replace the decomposition with a finite decreasing filtration of $V_{\mathbb {C} }$, $F^{\bullet }V_{\mathbb {C} }$ such that

$F^{p}V_{\mathbb {C} }\cap {\overline {F^{q}V_{\mathbb {C} }}}=0{\mbox{ and }}F^{p}V_{\mathbb {C} }\oplus {\overline {F^{q}V_{\mathbb {C} }}}=V_{\mathbb {C} },$

whenever

$p+q=k+1$.
In the case of compact Kähler manifolds,

$V_{\mathbb {Z} }=H^{k}(X,\mathbb {Z} )/{\mbox{torsion}}$

is a pure Hodge structure of weight

$k$.

**Definition 11.2**

Given $(V_{\mathbb {Z} },V^{p,q})$ and $(W_{\mathbb {Z} },W^{p,q})$ two pure $\mathbb {Z}$-Hodge structures of weight $k$ and $k+2r$, then a morphism of Hodge structures of bidegree $(r,r)$ is a group homomorphism $\phi :V_{\mathbb {Z} }\to W_{\mathbb {Z} }$ such that

$\phi (V^{p,q})\subseteq W^{p+r,q+r},{\mbox{ or }}\phi (F^{p}V_{\mathbb {C} })\subseteq F^{p+r}W_{\mathbb {C} }.$

We restrict our attention to morphisms of bidegree $(0,0)$ introducing the Tate twist.

**Definition 11.3**

Define the Tate-Hodge structure $\mathbb {Z}$ to be the pure Hodge structure of weight -2

$(2\pi i\mathbb {Z} ,H^{-1,-1}),{\mbox{ where }}H^{-1,-1}=\mathbb {Z} \otimes \mathbb {C} .$

Moreover, given a pure Hodge structure

$(V,V^{p,q})$ of weight

$k$ and an integer

$c$, define the Tate twist to be the pure Hodge structure

$(V(-c),V(-c)^{p,q})$ of weight

$k+2c$, defined by

$V(-c)=V{\mbox{ and }}(V(-c))^{p,q}=V^{p-c,q-c}.$

We refer to morphism of pure Hodge structures $V,W$ of weight $k$ and $k+2r$ as a morphism of Hodge structures of bidegree $(0,0)$ between $V(-r)$ and $W$.

The first example of morphism between pure Hodge structure comes again from Kähler manifolds. Given $f:X\to Y$ holomorphic map between compact Kähler manifolds, we get

$f^{*}:H^{n}(Y,\mathbb {Z} )\to H^{n}(X,\mathbb {Z} ),$

preserving the weight! So

$f^{*}$ is a morphism of Hodge structure. What about the Gysin morphism

$f_{*}$? Note that

$f_{*}$ does not preserve the weight, since

$f_{*}:H^{n}(X,\mathbb {Z} )\to H^{n+2r}(Y,\mathbb {Z} ),{\mbox{ where }}r=\operatorname {dim} _{\mathbb {C} }(Y)-\operatorname {dim} _{\mathbb {C} }(X),$

but induces a morphism of Hodge structures between

$H^{n}(X,\mathbb {Z} )(-r)\to H^{n+2r}(Y,\mathbb {Z} ).$
**Exercise.** Prove that a morphism of Hodge structures

$\phi :V_{\mathbb {Z} }\to W_{\mathbb {Z} }$ (of bidegree

$(0,0)$) is strict for the Hodge filtration, i.e. for all

$p$$\operatorname {im} \phi \cap F^{p}W_{\mathbb {C} }=\phi (F^{p}V_{\mathbb {C} }).$