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Hodge filtration

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aisejohan committed Sep 25, 2019
1 parent 2be4055 commit e0c8b04caa27b177e033c2203f43723b6b99c582
Showing with 194 additions and 7 deletions.
  1. +194 −7 derham.tex
@@ -224,7 +224,7 @@ \section{Cup product}
$f^* : R\Gamma(X, \Omega^\bullet_{X/S}) \to R\Gamma(X', \Omega^\bullet_{X'/S'})$
and
$f^* : H^i_{dR}(X/S) \longrightarrow H^i_{dR}(X'/S')$
are compatible with the cup product just defined.
are compatible with the cup product defined above.

\begin{lemma}
\label{lemma-cup-product-graded-commutative}
@@ -312,9 +312,9 @@ \section{Hodge cohomology}
S' \ar[r] & S
}
$$
of schemes, the are pullback maps
of schemes, there are pullback maps
$f^* : H^i_{Hodge}(X/S) \longrightarrow H^i_{Hodge}(X'/S')$
compatible with gradings and with the cup product just defined.
compatible with gradings and with the cup product defined above.



@@ -369,9 +369,8 @@ \section{Two spectral sequences}
H^0_{dR}(X/S) =
\Ker(\text{d} : H^0(X, \mathcal{O}_X) \to H^0(X, \Omega^1_{X/S}))
$$
Of course, this is also immediately clear from the fact that
the de Rham complex starts in degree $0$ with
$\mathcal{O}_X \to \Omega^1_{X/S}$.
Of course, this is also immediately clear from the fact that the
de Rham complex starts in degree $0$ with $\mathcal{O}_X \to \Omega^1_{X/S}$.

\medskip\noindent
The second spectral sequence is usually called
@@ -416,8 +415,66 @@ \section{Two spectral sequences}



\section{The Hodge filtration}
\label{section-hodge-filtration}

\noindent
Let $X \to S$ be a morphism of schemes. The Hodge filtration on $H^n_{dR}(X/S)$
is the filtration induced by the Hodge-to-de Rham spectral sequence
(Homology, Definition
\ref{homology-definition-filtration-cohomology-filtered-complex}).
To avoid misunderstanding, we explicitly define it as follows.

\begin{definition}
\label{definition-hodge-filtration}
Let $X \to S$ be a morphism of schemes. The {\it Hodge filtration}
on $H^n_{dR}(X/S)$ is the filtration with terms
$$
F^pH^n_{dR}(X/S) = \Im\left(H^n(X, \sigma_{\geq p}\Omega^\bullet_{X/S})
\longrightarrow H^n_{dR}(X/S)\right)
$$
where $\sigma_{\geq p}\Omega^\bullet_{X/S}$ is as in
Homology, Section \ref{homology-section-truncations}.
\end{definition}

\noindent
Of course $\sigma_{\geq p}\Omega^\bullet_{X/S}$ is a subcomplex of
the relative de Rham complex and we obtain a filtration
$$
\Omega^\bullet_{X/S} = \sigma_{\geq 0}\Omega^\bullet_{X/S} \supset
\sigma_{\geq 1}\Omega^\bullet_{X/S} \supset
\sigma_{\geq 2}\Omega^\bullet_{X/S} \supset
\sigma_{\geq 3}\Omega^\bullet_{X/S} \supset \ldots
$$
of the relative de Rham complex with
$\text{gr}^p(\Omega^\bullet_{X/S}) = \Omega^p_{X/S}[-p]$.
The spectral sequence constructed in
Cohomology, Lemma \ref{cohomology-lemma-spectral-sequence-filtered-object}
for $\Omega^\bullet_{X/S}$ viewed as a filtered complex of sheaves
is the same as the Hodge-to-de Rham spectral sequence constructed in
Section \ref{section-hdoge-to-de-rham}. Further the
wedge product (\ref{equation-wedge}) sends
$\text{Tot}(\sigma_{\geq i}\Omega^\bullet_{X/S} \otimes_{p^{-1}\mathcal{O}_S}
\sigma_{\geq j}\Omega^\bullet_{X/S})$ into
$\sigma_{\geq i + j}\Omega^\bullet_{X/S}$. Hence we get
commutative diagrams
$$
\xymatrix{
H^n(X, \sigma_{\geq j}\Omega^\bullet_{X/S}))
\times
H^m(X, \sigma_{\geq j}\Omega^\bullet_{X/S}))
\ar[r] \ar[d] &
H^{n + m}(X, \sigma_{\geq i + j}\Omega^\bullet_{X/S})) \ar[d] \\
H^n_{dR}(X/S) \times
H^m_{dR}(X/S)
\ar[r]^\cup &
H^{n + m}_{dR}(X/S)
}
$$
In particular we find that
$$
F^iH^n_{dR}(X/S) \cup F^jH^m_{dR}(X/S) \subset F^{i + j}H^{n + m}_{dR}(X/S)
$$



@@ -691,8 +748,95 @@ \section{First chern class in de Rham cohomology}
represents the cohomology class
$c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_a)$

\begin{remark}
\label{remark-truncations}
Here is a reformulation of the calculations above in more abstract terms.
Let $p : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an
invertible $\mathcal{O}_X$-module. If we view $\text{d}\log$ as a map
$$
\mathcal{O}_X^*[-1] \to \sigma_{\geq 1}\Omega^\bullet_{X/S}
$$
then using $\Pic(X) = H^1(X, \mathcal{O}_X^*)$ as above we find a
cohomology class
$$
\gamma_1(\mathcal{L}) \in H^2(X, \sigma_{\geq 1}\Omega^\bullet_{X/S})
$$
The image of $\gamma_1(\mathcal{L})$ under the map
$\sigma_{\geq 1}\Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}$
recovers $c_1^{dR}(\mathcal{L})$. In particular we see that
$c_1^{dR}(\mathcal{L}) \in F^1H^2_{dR}(X/S)$, see
Section \ref{section-hodge-filtration}. The image of $\gamma_1(\mathcal{L})$
under the map $\sigma_{\geq 1}\Omega^\bullet_{X/S} \to \Omega^1_{X/S}[-1]$
recovers $c_1^{Hodge}(\mathcal{L})$. Taking the cup product
(see Section \ref{section-hodge-filtration}) we obtain
$$
\xi = \gamma_1(\mathcal{L}_1) \cup \ldots \cup \gamma_1(\mathcal{L}_a) \in
H^{2a}(X, \sigma_{\geq a}\Omega^\bullet_{X/S})
$$
The commutative diagrams in Section \ref{section-hodge-filtration}
show that $\xi$ is mapped to
$c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_a)$
in $H^{2a}_{dR}(X/S)$ by the map
$\sigma_{\geq a}\Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}$.
Also, it follows
$c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_a)$
is contained in $F^a H^{2a}_{dR}(X/S)$. Similarly, the map
$\sigma_{\geq a}\Omega^\bullet_{X/S} \to \Omega^a_{X/S}[-a]$
sends $\xi$ to
$c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_a)$
in $H^a(X, \Omega^a_{X/S})$.
\end{remark}


\begin{remark}
\label{remark-log-forms}
Let $p : X \to S$ be a morphism of schemes. For $i > 0$
denote $\Omega^i_{X/S, log} \subset \Omega^i_{X/S}$ the abelian subsheaf
generated by local sections of the form
$$
\text{d}\log(u_1) \wedge \ldots \wedge \text{d}\log(u_i)
$$
where $u_1, \ldots, u_n$ are invertible local sections of $\mathcal{O}_X$.
For $i = 0$ the subsheaf $\Omega^0_{X/S, log} \subset \mathcal{O}_X$
is the image of $\mathbf{Z} \to \mathcal{O}_X$. For every $i \geq 0$ we
have a map of complexes
$$
\Omega^i_{X/S, log}[-i] \longrightarrow \Omega^\bullet_{X/S}
$$
because the derivative of a logarithmic form is zero. Moreover, wedging
logarithmic forms gives another, hence we find bilinear maps
$$
\wedge : \Omega^i_{X/S, log} \times
\Omega^j_{X/S, log} \longrightarrow \Omega^{i + j}_{X/S, log}
$$
compatible with (\ref{equation-wedge}) and the maps above.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
Using the map of abelian sheaves
$\text{d}\log : \mathcal{O}_X^* \to \Omega^1_{X/S, log}$
and the identification $\Pic(X) = H^1(X, \mathcal{O}_X^*)$
we find a canonical cohomology class
$$
\tilde \gamma_1(\mathcal{L}) \in H^1(X, \Omega^1_{X/S, log})
$$
These classes have the following properties
\begin{enumerate}
\item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical
map $\Omega^1_{X/S, log}[-1] \to \sigma_{\geq 1}\Omega^\bullet_{X/S}$
sends $\tilde \gamma_1(\mathcal{L})$ to the class
$\gamma_1(\mathcal{L}) \in
H^2(X, \sigma_{\geq 1}\Omega^\bullet_{X/S})$
of Remark \ref{remark-truncations},
\item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical
map $\Omega^1_{X/S, log}[-1] \to \Omega^\bullet_{X/S}$
sends $\tilde \gamma_1(\mathcal{L})$ to $c_1^{dR}(\mathcal{L})$ in
$H^2_{dR}(X/S)$,
\item the image of $\tilde \gamma_1(\mathcal{L})$ under the canonical
map $\Omega^1_{X/S, log} \to \Omega^1_{X/S}$
sends $\tilde \gamma_1(\mathcal{L})$ to $c_1^{Hodge}(\mathcal{L})$ in
$H^1(X, \Omega^1_{X/S})$,
\item the construction of these classes is compatible with pullbacks,
\item add more here.
\end{enumerate}
\end{remark}



@@ -863,6 +1007,49 @@ \section{de Rham cohomology of projective space}
$c_1^{Hodge}(\mathcal{O}(1))^p$ and the proof is complete.
\end{proof}

\begin{lemma}
\label{lemma-de-rham-cohomology-projective-space}
For $0 \leq i \leq n$ the de Rham cohomology
$H^{2i}_{dR}(\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$
with basis element $c_1^{dR}(\mathcal{O}(1))^i$.
In all other degrees the de Rham cohomology of $\mathbf{P}^n_A$
over $A$ is zero.
\end{lemma}

\begin{proof}
Consider the Hodge-to-de Rham spectral sequence of
Section \ref{section-hdoge-to-de-rham}.
By the computation of the Hodge cohomology of $\mathbf{P}^n_A$ over $A$
done in Lemma \ref{lemma-hodge-cohomology-projective-space}
we see that the spectral sequence degenerates on the $E_1$ page.
In this way we see that $H^i_{dR}(\mathbf{P}^n_A/A)$ is a free
$A$-module of rank $1$ for $0 \leq i \leq n$ and zero else.
Observe that $c_1^{dR}(\mathcal{O}(1))^i \in H^{2i}_{dR}(\mathbf{P}^n_A/A)$
for $i = 0, \ldots, n$ and that for $i = n$ this element is the
image of $c_1^{Hodge}(\mathcal{L})^n$ by the map of complexes
$$
\Omega^n_{\mathbf{P}^n_A/A}[-n]
\longrightarrow
\Omega^\bullet_{\mathbf{P}^n_A/A}
$$
This follows for example from the discussion in Remark \ref{remark-truncations}
or from the explicit description of cocycles representing these classes in
Section \ref{section-first-chern-class}.
The spectral sequence shows that the induced map
$$
H^n(\mathbf{P}^n_A, \Omega^n_{\mathbf{P}^n_A/A}) \longrightarrow
H^{2n}_{dR}(\mathbf{P}^n_A/A)
$$
is an isomorphism and since $c_1^{Hodge}(\mathcal{L})^n$ is a generator of
of the source (Lemma \ref{lemma-hodge-cohomology-projective-space}),
we conclude that $c_1^{dR}(\mathcal{L})^n$ is a generator
of the target. By the $A$-bilinearity of the cup products,
it follows that also $c_1^{dR}(\mathcal{L})^i$
is a generator of $H^{2i}_{dR}(\mathbf{P}^n_A/A)$ for
$0 \leq i \leq n$.
\end{proof}





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