# Linear Algebra and Hodge Theory

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

${\displaystyle V_{\mathbb {C} }:=qV\otimes _{\mathbb {R} }\mathbb {C} ,}$
and consider the ${\displaystyle \mathbb {C} }$-linear extension of ${\displaystyle J}$ to ${\displaystyle V_{\mathbb {C} }}$. It has eigenvalues ${\displaystyle \pm i}$ and, according to its decomposition in eigenspaces, we have
${\displaystyle V_{\mathbb {C} }=V^{1,0}\oplus V^{0,1},}$
where ${\displaystyle V^{1,0}}$ denotes the eigenspace of ${\displaystyle i}$ and ${\displaystyle V^{0,1}}$ the one of ${\displaystyle -i}$.

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

${\displaystyle V_{\mathbb {C} }:=qV\otimes _{\mathbb {R} }\mathbb {C} ,}$
and consider the ${\displaystyle \mathbb {C} }$-linear extension of ${\displaystyle J}$ to ${\displaystyle V_{\mathbb {C} }}$. It has eigenvalues ${\displaystyle \pm i}$ and, according to its decomposition in eigenspaces, we have
${\displaystyle V_{\mathbb {C} }=V^{1,0}\oplus V^{0,1},}$
where ${\displaystyle V^{1,0}}$ denotes the eigenspace of ${\displaystyle i}$ and ${\displaystyle V^{0,1}}$ the one of ${\displaystyle -i}$. Finally, complex conjugation on ${\displaystyle V_{\mathbb {C} }}$ induces an isomorphism between ${\displaystyle V^{1,0}}$ and ${\displaystyle V^{0,1}}$. 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 ${\displaystyle X}$ be an ${\displaystyle m}$-dimensional compact oriented Riemannian manifold and consider the sheaf of ${\displaystyle n}$-forms on ${\displaystyle X}$ by ${\displaystyle {\mathcal {A}}_{X}^{n}}$, with the exterior derivative ${\displaystyle d:{\mathcal {A}}_{X}^{n}\to {\mathcal {A}}_{X}^{n+1}}$. Let us define the Laplacian operator
${\displaystyle \Delta _{d}:=qdd^{*}+d^{*}d,}$
where ${\displaystyle d^{*}=(-1)^{m(n-1)-1}*d*:{\mathcal {A}}_{X}^{n}\to {\mathcal {A}}_{X}^{n-1}}$ and ${\displaystyle *}$ is the Hodge operator. Now, consider the set of harmonic forms
${\displaystyle {\mathcal {H}}^{n}(X)=\left\{\alpha \in {\mathcal {A}}_{X}^{n}{\mbox{ such that }}\Delta _{d}\alpha =0\right\}.}$
Then, there is a natural map
${\displaystyle {\mathcal {H}}^{n}(X)\longrightarrow H^{n}(X,\mathbb {R} ),}$
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
${\displaystyle {\mathcal {H}}^{n}(X)\simeq H^{n}(X,\mathbb {R} ).}$
Take a complex manifold ${\displaystyle X}$ and consider the sheaf of complex ${\displaystyle n}$-forms ${\displaystyle ({\mathcal {A}}_{X}^{n},d)}$, which can be decomposed into the direct sum
${\displaystyle {\mathcal {A}}_{X}^{n}=\bigoplus _{i+j=n}{\mathcal {A}}_{X}^{i,j},}$
where ${\displaystyle {\mathcal {A}}_{X}^{i,j}}$ denotes the sheaf of ${\displaystyle (i,j)}$-forms and ${\displaystyle d=\partial +{\bar {\partial }}}$, where ${\displaystyle \partial :{\mathcal {A}}_{X}^{i,j}\to {\mathcal {A}}_{X}^{i+1,j}}$ and ${\displaystyle {\bar {\partial }}:{\mathcal {A}}_{X}^{i,j}\to {\mathcal {A}}_{X}^{i,j+1}}$ denote the Dolbeault operators. We may try to see if ${\displaystyle H^{1}(X)}$ is even dimensional, but, unfortunately,
${\displaystyle [J,d]\neq 0,}$
which means that ${\displaystyle J}$ does not act on closed forms. The picture becomes much nicer when we restrict to Kähler manifolds. Indeed, it turns out that
${\displaystyle [\Delta _{d},J]=0,}$
so ${\displaystyle J}$ acts on harmonic forms. Moreover, we have
${\displaystyle \Delta _{d}=2\Delta _{\partial }=2\Delta _{\bar {\partial }}.}$
This leads to a decomposition of ${\displaystyle {\mathcal {H}}^{n}(X)}$ into the direct sum of the spaces of ${\displaystyle (p,q)}$-harmonic forms ${\displaystyle H^{p,q}(X)}$,
${\displaystyle {\mathcal {H}}^{n}(X)=\bigoplus _{p+q=n}H^{p,q}(X),}$
where ${\displaystyle {\overline {H^{p,q}(X)}}=H^{q,p}(X)}$. When we restrict our attention to a compact Kähler manifold, the isomorphism
${\displaystyle H^{n}(X,\mathbb {C} )\simeq {\mathcal {H}}^{n}(X)}$
induces the (Hodge) decomposition
${\displaystyle H^{n}(X,\mathbb {C} )=\bigoplus _{p+q=n}H^{p,q}(X),}$
where ${\displaystyle J}$ acts by ${\displaystyle (i)^{p}(-i)^{q}}$ on each ${\displaystyle H^{p,q}(X)}$. Moreover, we get a decreasing (Hodge) filtration, which is an equivalent data to the decomposition above:
${\displaystyle F^{0}H^{n}(X,\mathbb {C} )=H^{n}(X,\mathbb {C} )\supseteq F^{1}H^{n}(X,\mathbb {C} )\supseteq \cdots \supseteq F^{p}H^{n}(X,\mathbb {C} )\supseteq \cdots ,}$
where
${\displaystyle F^{p}H^{n}(X,\mathbb {C} )=\bigoplus _{\begin{matrix}i+j=n\\i\geq p\end{matrix}}H^{i,j}(X).}$
Note that the two data are equivalent since, given a filtration ${\displaystyle F^{\bullet }}$ on ${\displaystyle H^{n}(X,\mathbb {C} )}$, we recover the decomposition simply defining
${\displaystyle H^{p,q}=F^{p}\cap {\overline {F^{q}}}.}$

Definition 11.1

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

${\displaystyle V_{\mathbb {C} }=\bigoplus _{p+q=k}V^{p,q},}$
such that
${\displaystyle {\overline {V^{p,q}}}=V^{q,p}.}$

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

${\displaystyle F^{p}V_{\mathbb {C} }\cap {\overline {F^{q}V_{\mathbb {C} }}}=0{\mbox{ and }}F^{p}V_{\mathbb {C} }\oplus {\overline {F^{q}V_{\mathbb {C} }}}=V_{\mathbb {C} },}$
whenever ${\displaystyle p+q=k+1}$. In the case of compact Kähler manifolds,
${\displaystyle V_{\mathbb {Z} }=H^{k}(X,\mathbb {Z} )/{\mbox{torsion}}}$
is a pure Hodge structure of weight ${\displaystyle k}$.

Definition 11.2

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

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

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

Definition 11.3

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

${\displaystyle (2\pi i\mathbb {Z} ,H^{-1,-1}),{\mbox{ where }}H^{-1,-1}=\mathbb {Z} \otimes \mathbb {C} .}$
Moreover, given a pure Hodge structure ${\displaystyle (V,V^{p,q})}$ of weight ${\displaystyle k}$ and an integer ${\displaystyle c}$, define the Tate twist to be the pure Hodge structure ${\displaystyle (V(-c),V(-c)^{p,q})}$ of weight ${\displaystyle k+2c}$, defined by
${\displaystyle V(-c)=V{\mbox{ and }}(V(-c))^{p,q}=V^{p-c,q-c}.}$

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

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

${\displaystyle f^{*}:H^{n}(Y,\mathbb {Z} )\to H^{n}(X,\mathbb {Z} ),}$
preserving the weight! So ${\displaystyle f^{*}}$ is a morphism of Hodge structure. What about the Gysin morphism ${\displaystyle f_{*}}$? Note that ${\displaystyle f_{*}}$ does not preserve the weight, since
${\displaystyle f_{*}:H^{n}(X,\mathbb {Z} )\to H^{n+2r}(Y,\mathbb {Z} ),{\mbox{ where }}r=\operatorname {dim} _{\mathbb {C} }(Y)-\operatorname {dim} _{\mathbb {C} }(X),}$
but induces a morphism of Hodge structures between ${\displaystyle H^{n}(X,\mathbb {Z} )(-r)\to H^{n+2r}(Y,\mathbb {Z} ).}$ Exercise. Prove that a morphism of Hodge structures ${\displaystyle \phi :V_{\mathbb {Z} }\to W_{\mathbb {Z} }}$ (of bidegree ${\displaystyle (0,0)}$) is strict for the Hodge filtration, i.e. for all ${\displaystyle p}$
${\displaystyle \operatorname {im} \phi \cap F^{p}W_{\mathbb {C} }=\phi (F^{p}V_{\mathbb {C} }).}$