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\begin{document}
\section{Certain sheaves associated to a variation of Hodge structures}
Let $X$ be a smooth quasiprojective variety over $\C$ and let $H$ be a $O_X$-coherent $D_X$-module with a descending filtration $F^\bullet H$ such that the filtration is compatible with the natural filtration on $D_X$. This situation arises when we have a projective smooth family $f:Y\To X$ and 
$H=\Rd^k f_* \Omega_{Y/X}^\bullet$
for some $k\ge 0$ equipped with the Hodge filtration and the Gauss-Manin connection. 

Denote the connection on $H$ as
\[
\nabla:H\To H\otimes \Omega^1_X.
\]
The connection induces the Kodaira-Spencer maps:
\[
KS_i(H):\Gr^iH\To \Gr^{i-1}H\otimes \Omega^1_X.
\]

We are going to consider certain sheaves in Zariski topology on $X$. Namely fix an integer $k\ge 0$. Then we define the following complexes of sheaves:
\[
F^k\Omega(H)^\bullet = F^kH\xrightarrow{\nabla} F^{k-1}H\otimes \Omega^1_X \xrightarrow{\nabla} F^{k-2}H\otimes \Omega^2_X \xrightarrow{\nabla} \dots,
\]
\[
B_k(H)^\bullet = H^\nabla\longrightarrow H/F^kH \xrightarrow{\nabla} H/F^{k-1}H\otimes \Omega^1_X\xrightarrow{\nabla} \dots.
\]
On the other hand we define 
\[
L_k(H)_\bullet = \dots\xrightarrow{\partial} D_X\otimes\tau_X\otimes F_{k-1}H^* \xrightarrow{\partial} D_X\otimes F_kH^* \xrightarrow{\mu} H^*,
\]
where $\tau_X$ is the tangent sheaf, $H^*=\hom_{O_X}(H, O_X)$ is a $D_X$-module with increasing filtration $F_iH^*=(F^iH)^\perp$ and for $d\in D_X(U)$, $v\in \tau_X(U)$, $h\in H^*(U)$, $U\subset X$ $\partial$ acts like
\[
\partial(d\otimes v\otimes h) = d\otimes v(h) - dv\otimes h
\]
and $\mu$ acts like
\[
\mu(d\otimes h) = d(h).
\]
Also we define
\begin{multline*}
S_k(H)_\bullet = \dots\xrightarrow{\partial} D_X\otimes\wedge^2\tau_X\otimes H^*/F_{k-2}H^*\xrightarrow{\partial} D_X\otimes\tau_X\otimes H^*/F_{k-1}H^*
\xrightarrow{\partial} \\D_X\otimes H^*/F_kH^*,
\end{multline*}
which is a quotient of the Spencer complex $D_X\otimes \wedge^\bullet\tau_X\otimes H^*$, which gives a projective resolution of $H^*$ over $D_X$.
\begin{defn}
If $H$ is a filtered $O_X$-coherent $D_X$-module with a compatible decreasing filtration we call $H$ $k$-admissible if $F_0H=H$, $F_lH=0$ for some $l$, $Gr^l H$ are locally free for all $l$ and $KS_l(H)$ are injective for $l\ge k$.
\end{defn}
We are going to prove the following theorem:
\begin{thm}
Suppose $H$ is $k$-admissible. Then the sheaves $H^0(F^k \Omega(H)^\bullet)$, $H^0(B_k(H)^\bullet)$, $H_0(L_k(H)_\bullet)$, $H_0(S_k(H)_\bullet)$ are zero. There is a natural commutative square
\[\tag{*}
\begin{CD}
H^1(B_k(H)^\bullet) @@>{\text{injection}}>>    H^1(F^k \Omega(H)^\bullet)\\
@@V{\text{injection}}VV                        @@V{\sim}VV\\
\hom_{D_X}(H_1(L_k(H)_\bullet), O_X) @@>{\sim}>>   \hom_{D_X}(H_1(S_k(H)_\bullet), O_X)
\end{CD}
\]
in which upper horizontal and left vertical arrows are injections, lower horizontal and right vertical arrows are isomorphisms.
\end{thm}
\begin{proof}
Recall that for a complex $C$ $C[1]$ denotes the shift to the left, i.e. $S_k(H)[1]_i=S_k(H)_{i-1}$.
We have the following exact sequence of complexes:
\[
0\To L_k(H)_\bullet \To L_\infty(H)_\bullet \To S_k(H)_\bullet[1] \To 0.
\]
Since the Spencer complex $D_X\otimes \wedge^\bullet\tau_X\otimes H^*$ gives a resolution of $H^*$ the complex $L_\infty(H)_\bullet$ is exact. It follows that $L_k(H)_\bullet$ is quasi-isomorphic to $S_k(H)_\bullet$. 

There is also the following exact sequence of complexes:
\[
0\To F^k\Omega(H)^\bullet\To B_\infty(H)^\bullet[1]\To B_k(H)^\bullet[1]\To 0.
\]
This gives a distinguished triangle in the derived category 
\[
B_\infty(H)^\bullet\To B_k(H)^\bullet \To F^k\Omega(H)^\bullet.
\]

We prove that $\mu$ is surjective. Since the filtration on $H^*$ is finite it is enough to prove that the associated graded map is surjective. This follows from the fact that for $l\le k$ the restriction of $\Gr_l\mu$ to $O_X\otimes \Gr_lH^*$ is simply the isomorphism and for $l>k$ the restriction of $\Gr_l\mu$ to $\tau_X^{l-k}\otimes \Gr_kH^*$ is surjective because it is the composite of Kodaira-Spencer maps.

Since $\mu$ is surjective $H_0(L_k(H)_\bullet)=0$. Hence $H_0(S_k(H)_\bullet)=0$. By duality and left exactness of the $\hom$ functor $H^0(F^k \Omega(H)^\bullet)$ and $H^0(B_k(H)^\bullet)$ are also zero.

Now we construct the commutative square. The vertical arrows are given by duality. The lower horizontal arrow is given by the quasi-iomorphism between $L_k(H)_\bullet$ and $S_k(H)_\bullet$. The upper horizontal arrow is given by the long exact cohomology sequence
\begin{multline*}
0\To H^1(B_\infty(H)^\bullet)\To H^1(B_k(H)^\bullet) \To H^1(F^k\Omega(H)^\bullet)\To \\H^2(B_\infty(H)^\bullet)\To H^2(B_k(H)^\bullet)\To\dots.
\end{multline*}
The upper horizontal arrow is injective because $H^1(B_\infty(H)^\bullet)=0$. 

It is left to prove that the right vertical arrow is an isomorphism. Since the map
\[
\partial:D_X\otimes\tau_X\otimes H^*/F_{k-1}H^*
\To D_X\otimes H^*/F_kH^*
\]
is surjective and $D_X\otimes H^*/F_kH^*$ is projective the map $\partial$ splits. By duality it follows that the map
\[
\nabla: F^kH\To F^{k-1}H\otimes \Omega^1_X
\]
also splits with $\coker\nabla=\hom_{D_X}(\ker\partial,O_X)$.
Consider the following exact sequence:
\[
D_X\otimes\wedge^2\tau_X\otimes H^*/F_{k-2}H^*\To\ker\partial\To H_1(S_k(H)_\bullet)\To 0.
\]
Applying $\hom_{D_X}(-,O_X)$ we obtain again an exact sequence
\[
0\To \hom_{D_X}(H_1(S_k(H)_\bullet), O_X)\To\coker\nabla\To F^{k-2}H\otimes \Omega^2_X,
\]
which implies that 
\[
\hom_{D_X}(H_1(S_k(H)_\bullet), O_X) = H^1(F^k \Omega(H)^\bullet).
\]
\end{proof}
\begin{rem}
What we have also proved is that in the derived category $S_k(H)\cong L_k(H)$. Since $S_k(H)$ is projective we also get
\[
F^k\Omega(H)\cong \Rd\hom_{D_X}(S_k(H),O_X) \cong \Rd\hom_{D_X}(L_k(H),O_X).
\]
On the other hand the complex $B_k(H)$ is isomorphic to $\hom_{D_X}(L_k(H),O_X)$ (not derived $\hom$).
\end{rem}
\begin{rem}
In the analytic topology de Rham complex $\Omega(H)^\bullet$ is a resolution of $H^\nabla$, so the complex $B_\infty(H)^\bullet$ is exact, so all arrows in the square (*) are isomorphisms. We are looking for a sheaf which would be the target of the Abel-Jacobi map from a higher Chow group for a family. In the analytic topology there are Abel-Jacobi maps to sheafs of the kind
\[
\ker\left(\frac{H}{F^kH+H^\nabla}\To \frac{H}{F^{k-1}H}\otimes\Omega^1_X\right)=H^1(B_k(H)^\bullet).
\]
In the algebraic setting the sheaf $H^1(B_k(H)^\bullet)$ does not work. So our expectation is that the sheaf $H^1(F^k\Omega(H)^\bullet)$ is a natural replacement and it is possible to construct the corresponding Abel-Jacobi maps algebraically.
\end{rem}
\begin{rem}
We have proved that the cokernel of the injection in the theorem is isomorphic to the kernel of the map
\[
H^2(B_\infty(H)^\bullet)\To H^2(B_k(H)^\bullet),
\]
that is, to the sheaf
\[
F^k H^1(\Omega(H)^\bullet).
\]
\end{rem}
\begin{example}
Let $X$ be $1$-dimensional. Let $H=O_X$, $F^0H=H$, $F^1H=0$. This is $1$-admissible sheaf. Let $k=1$. The sheaf $H^1(B_1(H)^\bullet)$ is $O_X/\C_X$, which is isomorphic to the image of $O_X$ in $\Omega^1_X$. On the other hand the sheaf $H^1(F^k\Omega(H)^\bullet)$ is $\Omega^1_X$. Clearly the first sheaf is strictly smaller then the second because in the stalk at the generic point the first sheaf does not contain differential forms with non-trivial residues. The Abel-Jacobi map in this case is the map 
\[
f \mapsto \frac{df}{f},
\]
which is the map from $O_X^\times$ to $\Omega^1_X$.
\end{example}

\section{Algebraic version of the relative Deligne cohomology}
Let $f:Y\To X$ be a projective smooth family. The ordinary relative Deligne cohomology is defined as the higher direct image of the relative Deligne complex:
\[
\Z_{Y/X,an}(k)_\D=\Z_{Y,an}(k)\To\Omega_{Y/X,an}^{<k},
\]
that is 
\[
\Z_{Y/X,an}(k)_\D=\cone(\Z_{Y,an}(k)\To\Omega_{Y/X,an}^{<k})[-1].
\]
There is a map from $\Z_{Y/X,an}(k)_\D$ to $\C_{Y/X,an}(k)_\D$, where
\[
\C_{Y/X,an}(k)_\D=\cone(\C_{Y,an}\To\Omega_{Y/X,an}^{<k})[-1].
\]
There is a natural quasi-isomorphism
\[
\C_{Y,an}(k)\To \Omega_{Y,an}
\]
and there is a natural map $\varphi$ from the complex $\Omega_{Y,an}$ to the complex $\Omega_{Y/X,an}^{<k}$, which induces the map
\[
\C_{Y,an}\To\Omega_{Y/X,an}^{<k}.
\]
So in the derived category
\[
\C_{Y/X,an}(k)_\D=\cone(\varphi:\Omega_{Y,an}\To \Omega_{Y/X,an}^{<k})[-1].
\]
Using properties of the cone this is isomorphic in the derived category to the kernel of $\varphi$. There are following exact sequences:
\[
0\To f^*\Omega_{X,an}^1\otimes \Omega_{Y/X,an}^{l-1} \To \Omega_{Y,an}^l \To \Omega_{Y/X,an}^l\To 0,
\]
which are compatible with the differentials. Hence we may further transform
\[
\C_{Y/X,an}(k)_\D=0\To f^*\Omega_{X,an}^1\otimes\Omega_{Y/X,an}^{<k-1}\To\Omega_{Y,an}^{\ge k}.
\]
This motivates the following definition:
\begin{defn}
The $k$-th algebraic relative Deligne complex is the following complex:
\[
\D_{Y/X}(k) = (f^*\Omega_X^1\otimes\Omega_{Y/X}^{<k-1}\To\Omega_Y^{\ge k})[-1].
\]
The algebraic relative Deligne cohomology is the higher direct image of this sheaf
\[
H_D^i(Y/X, k) = \Rd^i{f_*}\D_{Y/X}(k).
\]
\end{defn}

\end{document}@