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@%\input commons.tex
%\author{Anton Mellit}
%\title{Higher Chow groups and Abel-Jacobi maps}
%\begin{document}
%\bibliographystyle{alpha}
%\maketitle
\section{Notation}
The symbol "$\I$" denotes $\sqrt{-1}$ to distinguish it from "$i$", which will usually be used as an index.
\section{The Hodge theory}
Let $X$ be a smooth projective variety over $\C$. For each $k$ we denote the sheaf of smooth $k$-forms by $\A^k_X$. We have the usual decomposition
\[
\A^k_X=\bigoplus_{p+q=k}\A^{p, q}_X,
\]
where $\A^{p, q}_X$ is the sheaf of smooth $(p,q)$-forms on $X$. We have the Hodge filtration on $\A^k_X$ defined as
\[
F^j\A^k_X:=\bigoplus_{p+q=k,\, p\ge j}\A^{p, q}_X.
\]
One can compute the cohomology groups of $X$ by taking the cohomology of the complex of global forms:
\[
H^k(X, \C) = H^k(\A^\bullet(X)),
\]
where for a sheaf $\F$ and an open set $U$ $\F(U)$ denotes the sections of the sheaf over $U$ and we omit the subscript $X$ when we write $\A^\bullet(X)$. One obtains the Hodge filtration on $H^k(X, \C)$ as the one induced by the filtration on $\A^k_X$, i.e.
\[
F^jH^k(X,\C) = \frac{\kernel(d:\A^k(X)\To\A^{k+1}(X))\cap F^j\A^k(X)}{\image(d:\A^{k-1}(X)\To\A^k(X))\cap F^j\A^k(X)}.
\]
By the Hodge theory there is a canonical decomposition
\[
H^k(X,\C)=\bigoplus_{p+q=k} H^{p, q}(X,\C)
\]
with
\[
F^jH^k(X,\C)=\bigoplus_{p+q=k,\,p\ge j} H^{p, q}(X,\C).
\]
We will frequently use the following consequence of the Hodge theory:
\begin{prop}
If $\omega\in F^j\A^k(X)$ is exact, i.e. there exists $\eta\in\A^{k-1}(X)$ with $d\eta=\omega$, then $\omega=d\eta'$ for some $\eta'\in F^j\A^{k-1}(X)$.
\end{prop}
\begin{proof}
Let 
\[
\omega=\sum_{p+q=k,\,p\ge j} \omega_{p, q},\qquad \eta=\sum_{p+q=k}\eta_{p, q},\qquad\omega_{p, q},\eta_{p, q}\in\A^{p, q}(X).
\]
Put
\[
\eta^\#=\sum_{p+q=k,\,p\ge j} \eta_{p, q}.
\]
Then the following form is exact:
\[
\omega^\#=\omega-d\eta^\#=\omega_{j, k-j}-\bar\partial\eta_{j, k-j-1}.
\]
Since $d\omega=0$ and $\omega\in F^j\A^k(X)$ we have $\bar\partial\omega_{j, k-j}=0$. Therefore $\bar\partial\omega^\#=0$, so $\omega^\#$ gives some class in the Dalbeaut cohomology group $H^{j, k-j}_{\bar\partial}(X)$. This class is trivial because of existense of the Hodge decomposition and the fact that the class of $\omega^\#$ is trivial in $H^k(X,\C)$. Hence $\omega^\#=\bar\partial\eta'$ for some $\eta'\in\A^{p,q-1}(X)$. This implies that $\omega-d(\eta^\#+\eta')\in F^{j+1}\A^k(X)$, so we can use the induction on $j$ to complete the proof.
\end{proof}
Note that there is a canonical homomorphism
\[
\iota:H^k(X,\Z)\To H^k(X,\C).
\]
To simplify the notation we will sometimes write
\[
H^k(X,\Z)\cap F^j H^k(X,\C),\qquad H^k(X,\Z)+F^j H^k(X,\C),
\]
respectively, instead of
\[
H^k(X,\Z)\cap \iota^{-1}(F^j H^k(X,\C)),\qquad H^k(X,\Z)+\iota(F^j H^k(X,\C)).
\]

\section{Higher Chow groups}\label{chow_groups}
Let $X$ be a smooth complex projective variety of dimension $n$. Recall that the ordinary Chow group $CH^k(X)$ of codimension $k$ cycles is the quotient group
\[
CH^k(X):=Z^k(X)/B^k(X),
\]
where $Z^k(X)$ is the free abelian group generated by irreducible algebraic subvarieties of $X$ of codimension $k$ and $B^k(X)$ is the subgroup generated by principal divisors on subvarieties of $X$ of codimension $k-1$.

To define the first higher Chow group $CH^k(X,1)$ (see Gordon-Lewis) consider the group $C^k(X,1)$ which is the free abelian group generated by pairs $(W,f)$, where $W$ is an irreducible algebraic subvariety of $X$ of codimension $k-1$ and $f$ is a non-zero rational function on $W$, modulo the relations
\[
(W, f_1 f_2) = (W,f_1)+(W, f_2),
\]
where $f_1$ and $f_2$ are two rational functions on $W$. The group $Z^k(X,1)$ is defined to be the kernel of the map 
\[
C^k(X,1)\To B^k(X)
\]
sending $(W,f)$ to $\Div f$, the divisor of $f$. Define the group $B^k(X,1)$ as the subgroup of $Z^k(X,1)$ generated by elements of the form
\[
(\Div g, h|_{\Div g})-(\Div h, g|_{\Div h}),
\]
where $g$, $h$ are non-zero rational functions on some $V\subset X$ of codimension $k-2$ whose divisors have no component in common. Here we extend the notation $(W,f)$ to linear combinations of subvarieties, i.e. if $W=\sum_j n_j W_j$ is a linear combination of irreducible subvarieties with integer coefficients and $f$ is a non-zero rational function on some bigger subvariety which restricts to a non-zero rational function on each $W_j$, then 
\[
(W,f):=\sum_j (W_j,f^{n_j}|_{W_j}).
\]
We put
\[
CH^k(X,1):=Z^k(X,1)/B^k(X,1)
\]
so that any element of $CH^k(X,1)$ has the form
\[
\sum_i (W_i, f_i),
\]
where 
\[
\sum_i \Div f_i = 0.
\]
\begin{example}
If $k=1$ then the corresponding higher Chow group is simply the multiplicative group of complex numbers, $CH^1(X,1)=\C^\times$.
\end{example}
\begin{example}
The group $CH^{n+1}(X,1)$ is generated by pairs $(W,f)$ where $W$ is a point and $f$ is a non-zero complex number. In fact one can see that there is a surjective homomorphism of abelian groups
\[
CH^n(X)\otimes \C^\times \To CH^{n+1}(X,1).
\]
Using the Weil reciprocity law one can also check that there exists a homomorphism from $CH^{n+1}(X,1)$ to $\C^\times$ which sends $(W,f)$ to $f$.
\end{example}

\section{The Abel-Jacobi map}
\subsection{Abel-Jacobi for the ordinary Chow group}
Recall that for the ordinary Chow group we have the cycle class map
\[
\cl^k:CH^k(X) \To H^{2k}(X, \Z)\cap F^k H^{2k}(X,\C),
\]
which sends $V$, a subvariety of codimension $k$, to its class 
\[
[V]\in H_{2n-2k}(X,\Z)\cong H^{2k}(X,\Z).
\]
Denoting by $CH^k(X)_0$ the kernel of $\cl^k$ we have the Abel-Jacobi map
\[
AJ^k:CH^k(X)_0 \To \frac{H^{2k-1}(X, \C)}{F^k H^{2k-1}(X,\C)+H^{2k-1}(X,\Z)}.
\]
This is defined as follows. Let $\gamma$ be an algebraic cycle of codimension $k$ whose homology class is $0$. It follows that $\gamma=\partial \xi$ for some $2n-2k+1$-chain $\xi$. Choosing such $\xi$ we obtain a linear functional on the space of $2n-2k+1$-forms given by integrating a form against $\xi$. 

Let us show that this defines a linear functional on $F^{n-k+1}H^{2n-2k+1}$. Indeed, if $\omega$ is an exact form from $F^{n-k+1}\A^{2n-2k+1}$ then $\omega=d\eta$ for $\eta\in F^{n-k+1}\A^{2n-2k}$ by Hodge theory. Hence
\[
\int_\xi\omega = \int_\gamma\eta
\]
is zero. 

Choosing another $\xi'$ such that $\partial \xi'=\gamma$ we have $\xi-\xi'$ closed, so the corresponding functionals for $\xi$ and $\xi'$ differ by the functional induced by the corresponding element of $H_{2n-2k+1}(X, \Z)$. So we obtain a map
\begin{equation*}
AJ^k:CH^k(X)_0 \To \frac{(F^{n-k+1}H^{2n-2k+1}(X,\C))^*}{H_{2n-2k+1}(X, \Z)}
\cong \frac{H^{2k-1}(X, \C)}{F^k H^{2k-1}(X,\C)+H^{2k-1}(X,\Z)}.
\end{equation*}
\subsection{Abel-Jacobi for the first higher Chow group}
Let $x$ represent an element of $CH^k(X,1)$, $k\in Z^i(X,1)$ i.e.
\[
x=\sum_i (W_i, f_i)
\]
with
\[
\sum_i \Div f_i = 0.
\]
We denote the corresponding element in $CH^k(X,1)$ by $[x]$.
We choose a path $[0,\infty]\subset \C \PP^1$. Let us denote 
\[
\gamma_i = f_i^* [0,\infty],
\]
which is a $2n-2k+1$-chain on $X$ whose boundary is $-\Div f_i$. This implies that the chain
\[
\gamma = \sum_i \gamma_i
\]
is a cycle. By Poincare duality $\gamma$ has a class $[\gamma]\in H^{2k-1}(X,\Z)$. 
\begin{prop}
The map $x\rightarrow [\gamma]$ defines a cycle class map
\[
\cl^{k,1}:CH^k(X,1)\To H^{2k-1}(X, \Z) \cap F^k H^{2k-1}(X, \C).
\]
\end{prop}
\begin{proof}
We have to check two things:
\begin{enumerate}
\item For any $x\in Z^k(X,1)$ the image of $[\gamma]$ belongs to $F^k H^{2k-1}(X, \C)$.
\item If $x=(\Div g, h|_{\Div g})-(\Div h, g|_{\Div h})$
for $V$, $g$ and $h$ as in the definition of $B^k(X,1)$ in Section \ref{chow_groups}, then $\gamma$ is homologically trivial.
\end{enumerate}

For (i) it is enough to prove that the pairing of $\gamma$ with any element of $F^{n-k+1} H^{2n-2k+1}(X,\C)$ is zero. Take a closed form $\omega\in F^{n-k+1} \A^{2n-2k+1}(X)$. Recall that
\[
\int_\gamma \omega = \sum_i \int_{\gamma_i} \omega.
\]
Let $n(\gamma_i)$ be a small tubular neighbourhood of $\gamma_i$ in $W_i$. Then, up to terms which tend to $0$ as the radius of the neigbourhood tends to $0$, we have
\[
\int_{\gamma_i} \omega = \frac{1}{2\pi \I}\int_{\partial n(\gamma_i)} \omega \log f_i = -\frac{1}{2\pi \I}\int_{W_i-n(\gamma_i)} d(\omega \log f_i) = -\frac{1}{2\pi \I} \int_{W_i-n(\gamma_i)} \frac{d f_i}{f_i} \wedge \omega.
\]
The form in the last integral belongs to $F^{n-k+2}\A^{2n-2k+2}$ and $W_i$ has complex dimension $n-k+1$, so the integral is zero.

For proving (ii), if $x=(\Div g, h|_{\Div g})-(\Div h, g|_{\Div h})$, then the chain $-(g\times h)^* ([0,\infty]\times[0,\infty])$ has boundary $\gamma$.
\end{proof}

On the other hand since the complex conjugation acts trivially on the group $H^{2k-1}(X, \Z)$ and
\[
F^k H^{2k-1}(X,\C) \cap \overline{F^k H^{2k-1}(X,\C)} = \{0\},
\]
we have the following
\begin{prop}
For any $[x]\in CH^k(X,1)$ the class $\cl^{k,1} [x]$ is torsion.
\end{prop}
Thus the cycle class map is a map
\[
\cl^{k,1}:CH^k(X,1)\To H^{2k-1}(X, \Z)_{tors}.
\]
Let us denote the kernel of this map by $CH^k(X,1)_0$.
To construct the Abel-Jacobi map 
\[
AJ^{k,1}:CH^k(X,1)_0\To \frac{H^{2k-2}(X,\C)}{F^k H^{2k-2}(X,\C)+H^{2k-2}(X,\Z)}
\]
we first identify
\[
\frac{H^{2k-2}(X,\C)}{F^k H^{2k-2}(X,\C)+H^{2k-2}(X,\Z)}\cong
\frac{(F^{n-k+1}H^{2n-2k+2}(X,\C))^*}{H^{2k-2}(X,\Z)}.
\]
If $[x]\in CH^k(X,1)_0$ with $x\in Z^k(X,1)$ then $\gamma=\partial \xi$ for some $2n-2k+2$-chain $\xi$. Then for any $\omega\in F^{n-k+1}\A^{2n-2k+2}(X)$ with $d\omega=0$ we take the following number:
\[\tag{*}
\langle AJ^{k,1}[x], [\omega] \rangle= \frac{1}{2\pi\I}\sum_i\int_{W_i\setminus\gamma_i} \omega\log f_i+\int_\xi \omega,
\]
where the logarithm on $\C \PP^1$ is defined using the cut along the chosen path $[0,\infty]$.

To prove that this correctly defines a map 
\[
AJ^{k,1}:CH^k(X,1)_0\To \frac{F^{n-k+1}H^{2n-2k+2}(X,\C)^*}{H^{2k-2}(X,\Z)}
\]
we need to show that the construction does not depend on the following choices:
\begin{itemize}
\item the choice of the path $[0,\infty]$;
\item the choice of the branch of the logarithm on $\C \PP^1-[0,\infty]$;
\item the choice of the representative of $x$, which is defined up to an element of $B^k(X,1)$;
\item the choice of $\xi$, which is defined up to a $2n-2k+2$-cycle;
\item the choice of $\omega$, which is defined up to a coboundary.
\end{itemize}
We prove this in the series of propositions
\begin{prop}
The value of (*) does not depend on the choice of the path $[0,\infty]$.
\end{prop}
\begin{proof}
Let $p$ and $p'$ be two different paths on $\C \PP^1$ from $0$ to $\infty$. Let 
\[
\gamma_i = f_i^* p,\; \gamma_i'=f_i^*p',
\]
\[
\gamma=\sum_i \gamma_i,\; \gamma'=\sum_i \gamma_i',
\]
Choose a 2-chain $q$ on $\C \PP^1$ whose boundary is $p'-p$, let
\[
\eta_i = f_i^* q,\; \eta=\sum_i \eta_i.
\]
We choose $\xi$ such, that $\partial \xi = \gamma$ and put $\xi'=\xi+\eta$ so that $\partial \xi' = \gamma'$. Let $l$ be a branch of the logarithm on $\C \PP^1-p$. Then the function
\[
l'(t)=l(t)-2\pi\I{\mathbf 1}_q(t)
\]
is a branch of the logarithm on $\C \PP^1-p'$, where ${\mathbf 1}_q$ is the characteristic function of $q$. Then we compare
\[
\sum_i\int_{W_i-\gamma_i'} \omega \,l'(f_i) -
\sum_i\int_{W_i-\gamma_i} \omega \,l(f_i) = -2 \pi\I \int_\eta \omega,
\]
\[
\int_{\xi'}\omega - \int_{\xi}\omega = \int_\eta\omega,
\]
so the value of (*) does not change.
\end{proof}
\begin{rem}
In fact this proof also shows that we can even vary each $\gamma_k$ as long as its class in the homology $H_{2n-2k+1}(W_i,|\Div f_i|)$ stays the same.
\end{rem}
\begin{prop}
Changing the branch of the logarithm changes the value of (*) by an element from $H^{2k-2}(X,\Z)$.
\end{prop}
\begin{proof}
Changing the branch of the logarithm amounts to adding $2\pi\I m$ for $m\in \Z$, which changes the value of (*) by
\[
2\pi\I m \sum_i \int_{W_i}\omega,
\]
which is a functional induced by the image of $m W_i$ in $H^{2k-2}(X,\Z)$.
\end{proof}
\begin{prop}
If
\[
x=\Div g \otimes h|_{\Div g}-\Div h \otimes g|_{\Div h},
\]
for $V$, $g$ and $h$ as in the definition of $B^k(X,1)$ in Section \ref{chow_groups}, then the value of (*) is zero.
\end{prop}
\begin{proof}
We may take
\[
\xi = -(g\times h)^* ([0,\infty]\times[0,\infty])
\]
since
\[
\partial ([0,\infty]\times[0,\infty]) = [0,\infty]\times([0]-[\infty])-([0]-[\infty])\times [0,\infty].
\]
Inside $V$ we consider $\Div g$ which has real codimension $2$ and $\gamma_h=h^*[0,\infty]$ which has real codimension $1$.
Consider a small neighbourhood of the divisor of $g$ which we denote by $n(\Div g)$ and a small neighbourhood of $\gamma_g$ which we denote by $n(\gamma_g)$. We will use corresponding notation for $h$, i.e. $n(\Div h)$ and $n(\gamma_h)$. Then we may rewrite
\[
\int_{\Div g-\gamma_h} \omega\log h
=\frac{1}{2\pi\I}\int_{\partial n(\Div g)\cap n(\gamma_h)^c} \frac{dg}{g}\wedge\omega \log h.
\]
Take a chain $S=V\cap n(\Div g)^c \cap n(\gamma_h)^c$. Then 
\[
\partial S = -\partial n(\Div g)\cap n(\gamma_h)^c - \partial n(\gamma_h)\cap n(\Div g)^c,
\]
so using the Stokes formula we obtain
\[
-\int_{\partial n(\Div g)\cap n(\gamma_h)^c} \frac{dg}{g}\wedge\omega \log h - \int_{\partial n(\gamma_h)\cap n(\Div g)^c} \frac{dg}{g}\wedge\omega \log h = 
\int_{S} \frac{dh}{h}\wedge\frac{dg}{g}\wedge\omega. 
\]
Since $\omega\in F^{n-k+1}\A^{2n-2k+2}(X)$, $\frac{dh}{h}\wedge\frac{dg}{g}\wedge\omega\in F^{n-k+3}\A^{2n-2k+4}(S)$ and its integral is zero because $S\subset V$, which is a $n-k+2$-dimensional complex variety.
Hence
\begin{equation*}
\int_{\Div g-\gamma_h} \omega\log h = -\frac{1}{2\pi\I} \int_{\partial n(\gamma_h)\cap n(\Div g)^c} \frac{dg}{g}\wedge\omega \log h
=-\int_{\gamma_h\cap n(\Div g)^c} \frac{dg}{g}\wedge\omega.
\end{equation*}
We apply the Stokes formula again for $T=\gamma_h\cap n(\gamma_g)^c$ and $\omega\log g$. We have
\[
\partial T = -\Div h\cap n(\gamma_g)^c - (\partial n(\gamma_g)\cap\gamma_h),
\]
so
\[
-\int_{\Div h\cap n(\gamma_g)^c} \omega\log g-\int_{\partial n(\gamma_g)\cap\gamma_h} \omega\log g = \int_{\gamma_h-n(\Div g)} \frac{dg}{g}\wedge\omega.
\]
This implies that
\[
\int_{\Div g-\gamma_h} \omega\log h-\int_{\Div h-\gamma_g}\omega\log g = \int_{\partial n(\gamma_g)\cap\gamma_h} \omega\log g = 2\pi\I\int_{\gamma_g\cap\gamma_h} \omega.
\]
Since $\xi=-\gamma_g\cap\gamma_h$ the statement follows.
\end{proof}
\begin{prop}
Changing $\xi$ by a $2n-2k+2$-cycle changes the value of the functional (*) by an element of $H^{2k-2}(X,\Z)$.
\end{prop}
\begin{proof}
This is clear.
\end{proof}
\begin{prop}
Changing $\omega$ by a coboundary does not change the value of the functional (*).
\end{prop}
\begin{proof}
Indeed, if $\omega$ is a coboundary, then $\omega=d \eta$ with $\eta\in F^{n-k+1}\A(X)$ by the Hodge theory. This implies
\[
(*)=\sum_i\int_{W_i-\gamma_i} d(\eta \log f_i)-\sum_i\int_{W_i-\gamma_i}\frac{df_i}{f_i}\wedge\eta+2\pi\I\int_\gamma\eta.
\]
Since $\frac{df_i}{f_i}\wedge\eta\in F^{n-i+2}\A(X)$ and $W_i$ is a $n-k+1$-dimensional complex variety, the second summand is zero. Applying Stokes formula to the first summand we obtain
\[
\sum_i\int_{W_i-\gamma_i} d(\eta \log f_i)=-\sum_i\int_{\partial n(\gamma_i)} \eta \log f_i=-2\pi\I\sum_i\int_{\gamma_i}\eta,
\]
where $n(\gamma_i)$ denotes a small neighbourhood of $\gamma_i$ inside $W_i$. Hence the statement.
\end{proof}

\section{Special values of the Abel-Jacobi map}
Let $x=\sum_i (W_i, f_i)$ represent an element $[x]\in CH^k(X,1)_0$.
Then 
\[
AJ^{k,1} [x]\in \frac{F^{n-k+1}H^{2n-2k+2}(X,\C)^*}{H^{2k-2}(X,\Z)}.
\]
Given a subvariety $Z\subset X$ of codimension $n-k+1$ we may consider $\cl^{n-k+1} Z\in H^{2n-2k+2}(X,\Z)\cap F^{n-k+1} H^{2n-2k+2}(X,\C)$. Then
\[
(AJ^{k,1}[x], \cl^{n-k+1} Z)\in \C/\Z,
\]
so this number can be written as $\frac{1}{2\pi\I} \log\alpha$ for a unique $\alpha\in\C$. We now show how to construct this number in a different way.
Consider the cycle $S\subset X$ which is the union of all $|\Div f_i|$ and singular parts of $W_i$.
We say that $Z$ intersects $x$ properly if $Z$ properly intersects $S$ and all $W_i$. This means that $Z$ does not intersect $S$ and intersects each $W_i$ in several points. 
Note that by the moving lemma for any given $Z$ there exists $Z'$ which is rationally (hence homologically) equivalent to $Z$ and intersects $x$ properly.
\begin{thm}
Let $x=\sum_i (W_i,f_i)$ be a representative of $[x]\in CH^k(X,1)$.
Let $Z\subset X$ be a smooth subvariety of dimension $k-1$ intersecting $x$ properly.
Then
\[
2\pi\I (AJ^{k,1}[x], \cl^{n-k+1} Z) \equiv \log\prod_i \prod_{p\in W_i\cap Z} f_i(p)^{\ord_p(W_i\cdot Z)} \mod 2\pi\I.
\]
\end{thm}
\begin{proof}
By the definition (*) of $AJ^{k,1}$ we have
\[
2\pi\I (AJ^{k,1}[x], \cl^{n-k+1} Z) = \sum_i\int_{W_i\setminus\gamma_i}\omega\log{f_i}+2\pi\I\int_\xi\omega,
\]
where $\omega\in F^{n-k+1}\A^{2n-2k+2}(X)$ is a form whose class $[\omega]$ equals to the Poincare dual of the class $[Z]\in H_{2k-2}(X,\Z)$.

The current
\[
\omega - \delta_Z.
\]
is homologically trivial since both $\omega$ and $\delta_Z$ represent the same class in the cohomology. Hence there exists a current $\eta\in F^{n-k+1}\D^{2n-2k+1}(X)$, smooth outside $|Z|$, such that
\[
d\eta = \omega - \delta_Z.
\]
If we denote by $\eta_0$ the corresponding form on $X \setminus |Z|$, 
\[
\eta_0\in F^{n-k+1}\A^{2n-2k+1}(X\setminus |Z|),
\]
we obtain an identity
\[
d\eta_0 = \omega,
\]
which is true outside $|Z|$. 

In the definition of the Abel-Jacobi map we choose $\gamma_k$ and $\xi$ to be transversal to $|Z|$. This means that $\gamma_k$ does not intersect $|Z|$ for each $k$ and $\xi$ intersects $|Z|$ only in several points.

Choosing small neighbourhoods of $|\Div f_i|$, $|Z|\cap W_i$, $\gamma_i$ inside $W_i$ and $|Z|\cap \xi$ inside $\xi$ and denoting them by $n(\Div f_i)$, $n(|Z|\cap W_i)$, $n(\gamma_i)$, $n(|Z|\cap \xi)$ respectively we may write
\begin{multline*}
\sum_i\int_{W_i\setminus(n(\gamma_i)\cup n(|Z|\cap W_i))}\omega\log{f_i}+2\pi\I\int_{\xi\setminus n(|Z|\cap\xi)}\omega =\\ 
\sum_i\int_{W_i\setminus(n(\gamma_i)\cup n(|Z|\cap W_i))}d \eta_0\log{f_i}+2\pi\I\int_{\xi\setminus n(|Z|\cap\xi)}d \eta_0.
\end{multline*}
We transform the second term into
\[
2\pi\I\int_{\gamma} \eta_0 - 2\pi\I\int_{\partial n(|Z|\cap\xi)} \eta_0
\]
and the $i$-th summand in the first term into
\[
-\int_{\partial n(\gamma_i)}\eta_0\log{f_i}-\int_{\partial n(|Z|\cap W_i)}\eta_0\log{f_i}-\int_{W_i\setminus(n(\gamma_i)\cap n(|Z|\cap W_i))}\frac{d f_i}{f_i}\wedge\eta_0,
\]
where the last term equals to $0$ 
because $\frac{d f_i}{f_i}\wedge\eta_0\in F^{n-k+2}\A^{2n-2k+2}$.

Consider the integral
\[
\int_{\partial n(|Z|\cap W_i)}\eta_0\log{f_i}.
\]
Let $p$ be an intersection point of $|Z|$ and $W_i$. Then there is a neighbourhood $U$ of $p$ which is analytically isomorphic to the product of open balls $B_W$ and $B_Z$ and $W_i\cap U$ maps to $B_W\times \{0\}$,  $|Z|\cap U$ maps to $\{0\}\times B_Z$. Let $n(|Z|\cap W_i)$ have only one connected component in $U$ and this is $D_W\times \{0\}$ where $D_W\subset B_W$ is a closed ball. We extend $f_i$ to $U$ by means of the projection $U\To B_W$. Let $\chi_\varepsilon(t)$ be a family of smooth functions on $\R$ which approximate $\delta_0$ as in [Griffits-Harris]. For any current $T$ on $U$ we put
\[
T_\varepsilon=T*(\chi_\varepsilon^{2n})
\]
where $*$ denotes the convolution and 
\begin{equation*}
\chi_\varepsilon^{2n}(w_1, \dots, w_{2n-2k+2},b_1,\dots,b_{2k-2}) = \chi_\varepsilon(w_1)\dots\chi_\varepsilon(w_{2n-2k+2})\chi_\varepsilon(b_1)\dots\chi_\varepsilon(b_{2k-2}).
\end{equation*}
Then
\[
\int_{\partial D_W \times \{0\}} \eta_0\log{f_i} = \lim_{\varepsilon\rightarrow 0} \int_U (\delta_{\partial D_W \times \{0\}})_\varepsilon\wedge\eta_0\log{f_i}
\]
since $\eta_0$ is smooth outside $\{0\}\times B_Z$. Applying the identity
\[
(\delta_{\partial D_W \times \{0\}})_\varepsilon = -d (\delta_{D_W \times \{0\}})_\varepsilon
\]
the last expression equals to
\[
-\lim_{\varepsilon\rightarrow 0} \int_U d (\delta_{D_W \times \{0\}})_\varepsilon \wedge \eta_0 \log{f_i} = \lim_{\varepsilon\rightarrow 0}(\eta_0 \log{f_i}, d (\delta_{D_W \times \{0\}})_\varepsilon),
\]
where we treat $\eta_0 \log{f_i}$ as a current and $d (\delta_{D_W \times \{0\}})_\varepsilon$ as a form. This is, by the definition of the differential for currents,
\[
\lim_{\varepsilon\rightarrow 0} (d(\eta_0 \log{f_i}), (\delta_{D_W \times \{0\}})_\varepsilon).
\]
Now we expand
\[
d(\eta_0 \log{f_i}) = (\omega-\delta_Z)\log{f_i} + \frac{d f_i}{f_i}\wedge\eta_0,
\]
thus obtaining
\begin{multline*}
\lim_{\varepsilon\rightarrow 0} ((\omega-\delta_Z)\log{f_i} + \frac{d f_i}{f_i}\wedge\eta_0, (\delta_{D_W \times \{0\}})_\varepsilon)\\=\int_{D_W\times\{0\}} \omega \log{f_i} - \log{f_i(p)}\cdot\ord_p(Z\cdot W_i) + \lim_{\varepsilon\rightarrow 0}(\frac{d f_i}{f_i}\wedge\eta_0, (\delta_{D_W \times \{0\}})_\varepsilon),
\end{multline*}
the last summand being zero because 
\[
\eta_0\in F^{n-k+1}\D^{2n-2k+1},\;(\delta_{D_W \times \{0\}})_\varepsilon\in F^{k-1}\A^{2k-2}.
\]
Note that when the radius of the ball $D_W$ tends to zero the first summand tends to zero, so can be neglected. Therefore the limit value of 
\[
\int_{\partial n(|Z|\cap W_i)}\eta_0\log{f_i}
\]
is
\[
-\sum_{p\in W_i\cap |Z|} \log{f_i(p)}\cdot\ord_pZ.
\]

The sum
\[
\sum_i\int_{\partial n(\gamma_i)}\eta_0\log{f_i}
\]
annihilates (in the limit) the integral
\[
2\pi\I\int_{\partial\xi} \eta_0.
\]

The remaining summand
\[
-2\pi\I\int_{\partial n(|Z|\cap\xi)} \eta_0
\]
tends to $2\pi\I$ times the intersection number of $Z$ and $\xi$ according to a reasoning similar to the one used above.
\end{proof}

\subsection{Construction of the fundamental class}
We would like to produce another proof of this theorem without usage of currents. For this recall some cohomology constructions. What follows can be done for any smooth variety $X$ over $\C$.

Let $\Omega^\bullet_X$ be the holomorphic de Rham complex of $X$. For any integer $j$ denote by $F^j\Omega^\bullet_X$ the subcomplex
\[
F^j\Omega^i_X=\begin{cases}0& \text{if $i<j$,}\\ \Omega^i_X& \text{if $i\ge j$.}\end{cases}
\]
There are natural maps $F^j\Omega^\bullet_X\rightarrow\Omega^\bullet_X$ and $F^j\Omega^\bullet_X\rightarrow\Omega^j_X$.

Let $Y\subset X$ be a smooth subvariety of codimension $j$. Recall the construction of the fundamental class of $Y$ in $\HC^{2j}_Y(X,F^j\Omega^\bullet_X)$ (see \cite{groth:fga}, Expos\'e 149 and \cite{bloch:semireg}). 

We first construct the Hodge class $c^H(Y)\in H^j_Y(X,\Omega^j_X)$. There is a spectral sequence
\[
E^{p,q}_2=H^p(X,\SC_Y^q(\Omega^j_X))\Rightarrow H_Y^{p+q}(X,\Omega^j_X).
\]
Since $\Omega^j_X$ is locally free $\SC_Y^q(\Omega^j_X)=0$ for $q<j$. This implies that 
\[
H_Y^j(X,\Omega^j_X) = \Gamma(X,\SC_Y^j(\Omega^j_X)).
\]
Let $V$ be an open subset of $X$ on which $Y$ is a complete intersection, so there exist regular functions $f_1$, $f_2$,\dots, $f_j$ on $V$ which generate the ideal of $Y\cap V$. Put $V_i=V\setminus \{f_i=0\}$. Then $V_i$ form a covering of $V\setminus(V\cap Y)$. So we can consider the \v Cech cohomology and the section
\[
(2\pi\I)^{-j}\frac{d f_1}{f_1}\wedge\dots\frac{d f_j}{f_j} \in \Gamma(\bigcap V_i, \Omega_X^j)
\]
produces an element of $H^{j-1}(V\setminus(V\cap Y), \Omega_X^j)$. We obtain an element of $H^j_{V\cap Y}(V,\Omega_X^j)$ by applying the boundary map of the long exact sequence
\[
\dots\To H^{j-1}(V\setminus(V\cap Y), \Omega_X^j)\To H^j_{V\cap Y}(V,\Omega_X^j)\To H^j(V,\Omega_X^j) \To\dots.
\]
This element is a section of the sheaf $\SC_Y^j(\Omega^j_X)$ over $V$. These local sections glue together to produce a global section
\[
c^H(Y)\in H^j_Y(X,\Omega^j_X).
\]

The differential $d:\Omega^j_X\rightarrow\Omega^{j+1}_X$ induces the differential on cohomology $d:H^j_Y(X,\Omega^j_X)\rightarrow H^j_Y(X,\Omega^{j+1}_X)$. We have $d c^H(Y)=0$ since for $V$, $f_i$ as above
\[
d\left( (2\pi\I)^{-j}\frac{d f_1}{f_1}\wedge\dots\frac{d f_j}{f_j}\right) = 0.
\]
There is a spectral sequence for the hypercohomology
\[
E^{p,q}_2=H^p(H^q_Y(X,F^j\Omega_X^\bullet))\Rightarrow \HC^{p+q}_Y(X,F^j\Omega_X^\bullet),
\]
which shows that 
\[
\HC^{2j}_Y(X,F^j\Omega_X^\bullet)=H^j(H^j_Y(X,F^j\Omega_X^\bullet))
=\kernel(d: H^j_Y(X,\Omega_X^j)\rightarrow H^j_Y(X,\Omega_X^{j+1})).
\]
Therefore the natural map $\HC^{2j}_Y(X,F^j\Omega_X^\bullet)\rightarrow H^j_Y(X, \Omega_X^j)$ is an injection and $c^H(Y)$ lifts to a unique $c^F(Y)\in\HC^{2j}_Y(X,F^j\Omega_X^\bullet)$. The natural map 
\[
\HC^{2j}_Y(X,F^j\Omega_X^\bullet)\To \HC^{2j}_Y(X,\Omega_X^\bullet) \cong H^{2j}_Y(X,\C)
\]
sends $c^F(Y)$ to an element $c^{DR}(Y)\in H^{2j}_Y(X,\C)$. 

We will prove now that $c^{DR}(Y)$ is the Thom class of $Y$, i.e. that its value on a class in $H_{2j}(X,X\setminus Y)$ equals to the intersection number of this class with $Y$. Since the real codimension of $Y$ is $2j$, $\SC^p_Y(\C)=0$ for $p<2j$. Therefore $H^{2j}_Y(X,\C)=\Gamma(X,\SC^{2j}_Y(\C))$ and $c^{DR}(Y)$ can be described locally. Let $V$ be a neighbourhood of $X$ which is isomorphic to a product of unit disks, $V\cong \DD^n$, and such that $V\cap Y$ is given by equations $z_i=0$ for $i=1,\dots j$, where $z_i$ is the coordinate on $i$-th disk. Then 
\[
H^{2j}_{Y\cap V}(V,\C)=H^{2j}(\DD^n,(\DD^j\setminus\{0\}^j)\times \DD^{n-j};\C)=\C.
\]
So to evaluate the restriction $c^{DR}(Y)|_V$ it is enough to evaluate $c^{DR}(Y)|_V$ on the generator of the homology $H_{2j}(\DD^n,(\DD^j\setminus\{0\}^j)\times \DD^{n-j};\C)$, the transverse class $\DD^j\times \{0\}^{n-j}$. Since the class $c^{H}(Y)|_V$ is coming from the boundary map $H^{j-1}((\DD^j\setminus\{0\}^j)\times \DD^{n-j},\Omega^j)\rightarrow H^j(\DD^n,(\DD^j\setminus\{0\}^j)\times \DD^{n-j};\Omega^j)$
\[
\langle c^{DR}(Y)|_V, \DD^j\times \{0\}^{n-j}\rangle = 
\langle c^{H}(Y)|_V, \DD^j\times \{0\}^{n-j}\rangle = 
\langle c_0, \partial(\DD^j\times \{0\}^{n-j})\rangle,
\]
where $c_0$ is the corresponding class in $H^{j-1}((\DD^j\setminus\{0\}^j)\times \DD^{n-j},\Omega^j)$. The class $c_0$, in its turn, comes from the map
\[
H^0(\DD_0^j\times\DD^{n-j}, \Omega^j) \To H^{j-1}((\DD^j\setminus\{0\}^j)\times \DD^{n-j},\Omega^j).
\]
This map comes from \v Cech cohomology and can also be constructed using successive appication of Mayer-Vietoris exact sequences. Hence we have the dual map in homology
\[
H_{2j-1}((\DD^j\setminus\{0\}^j)\times \DD^{n-j},\C)\To
H_j(\DD_0^j\times\DD^{n-j},\C)
\]
and one can see that the image of $\partial(\DD^j\times \{0\}^{n-j})$ under this map is $\SB^j\times\{0\}^{n-j}$, where $\SB$ is the unit circle. Indeed, let 
\[
U_k=\DD_0^k\times(\DD^{j-k}\setminus\{0\}^{j-k})\times\{0\}^{n-j}.
\]
Then $U_0=(\DD^j\setminus\{0\}^j$ and $U_{j-1}=\DD_0^j\times\DD^{n-j}$. Put 
\[
X_k^1=\DD_0^k\times\DD^{j-k}\times\{0\}^{n-j},\qquad
X_k^2=\DD_0^{k-1}\times\DD\times(\DD^{j-k}\setminus\{0\}^{j-k})\times\{0\}^{n-j}.
\]
Then
\[
X_k^1\cap X_k^2 = U_k,\qquad X_k^1\cup X_k^2 = U_{k-1},
\]
so there is a boundary map in the Mayer-Vietoris sequence associated to $X_k^1$, $X_k^2$ which goes from $H_{2j-k}(U_{k-1})$ to $H_{2j-k-1}(U_{k})$. We are going to show, by induction, that the $k$-th iterated image of $\partial(\DD^j\times \{0\}^{n-j})$ under these maps is 
\[
\SB^k\times\partial(\DD^{j-k})\times\{0\}^{n-j}\in H_{2j-k-1}(U_{k},\C).
\]
The boundary map in the Mayer-Vietoris sequence can be decomposed (see \cite{dold}, p. 49) as
\[
H_{2j-k}(U_{k-1})\To H_{2j-k}(U_{k-1},X_k^2) \cong H_{2j-k}(X_k^1, U_k) \To H_{2j-k-1}(U_k).
\]
Writing
\[
\SB^{k-1}\times\partial(\DD^{j-k+1})\times\{0\}^{n-j} = \SB^k\times\DD^{j-k}\times\{0\} + \SB^{k-1}\times\DD\times\partial(\DD^{j-k})\times\{0\}^{n-j}
\]
we see that the second summand is contained in $X_k^2$, so is trivial in $H_{2j-k}(U_{k-1},X_k^2)$. The first summand belongs to $H_{2j-k}(X_k^1, U_k)$, so it remains to take its boundary, which is exactly $\SB^k\times\partial(\DD^{j-k})\times\{0\}^{n-j}$.

Therefore
\[
\langle c_0, \partial(\DD^j\times \{0\}^{n-j})\rangle = \int_{\SB^j} (2\pi\I)^{-j} \frac{d z_1}{z_1}\wedge\dots\frac{d z_j}{z_j} = 1,
\]
which means that the constructed class $c^{DR}(Y)$ is indeed the Thom class of $Y$.

It is clear now that the following theorem is true:
\begin{thm}\label{thm52}
Let $X$ be a smooth variety over $\C$ of dimension $n$. Let $Y\subset X$ be a smooth closed subvariety of $X$ of codimension $j$. Then there is a class $c^F(Y)\in \HC_Y^{2j}(X, F^j\Omega_X^\bullet)$ which satisfies the following conditions:
\begin{enumerate}
\item The image of $c^F(Y)$ in $H_Y^{2j}(X, \C)$, $c^{DR}(Y)$ is the Thom class of $Y$.
\item The class $c^H(Y)$, which is the image of $c^F(Y)$ in $H_Y^j(X,\Omega_X^j)$ is logarithmic, i.e. for any open set $V\subset X$ and a holomorphic function $f$ on $V$ which is zero on $Y\cap V$ the product $f\cdot c^H(Y)|_V$ is $0$.
\end{enumerate}
\end{thm}

We extend the definition of $c^F$, $c^DR$, $c^H$ to formal linear combinations of subvarieties in the obvious way.

\subsection{Dolbeault local cohomology}
We show how to interpret the results of the previous subsection using smooth forms. For this we show how local cohomology can be computed using Dolbeault resolutions. Let $X$ be a smooth variety over $\C$ and $Y$ be a subvariety of codimension $j$, $U=X\setminus Y$. Let $\j$ be the inclusion $U\rightarrow X$.

\begin{prop}
Let $S$ be a soft sheaf on $X$ which locally has no nonzero sections supported on $Y$. Then $H^p_Y(X,S)$ and $\SC^p_Y(S)$ are zero unless $p=1$,
\[
H^1_Y(X,S)=\cokernel(\Gamma(X,S)\To\Gamma(U,S)),\qquad \SC^1_Y(S)=\cokernel(S\To \j(S|_U)).
\]
\end{prop}
\begin{proof}
For any open set $V\subset X$ we have the long exact sequence for local cohomology:
\[
\dots\To H^p_{V\cap Y}(V,S) \To H^p(V,S)\To H^p(V\cap U,S)\To\dots.
\]
The groups $H^p(V,S)$ and $H^p(V\cap U,S)$ vanish for $p>0$. Hence $H^p_{V\cap Y}(V,S)$ vanish for $p>1$. This group also vanishes for $p=0$ by the condition on $S$. For $V=X$ this implies the statement about the groups $H^p_Y(X,S)$. Since $\SC^p_Y(S)$ is the sheaf associated to the presheaf $(V\rightarrow H^p_{V\cap Y}(V,S))$, this implies the statement for the groups $\SC^p_Y(S)$.
\end{proof}

Therefore one can compute local cohomology in the following way:
\begin{prop}
Let $F^\bullet$ be a bounded complex of sheaves on $X$ and $F^\bullet\rightarrow S^\bullet$ a bounded soft resolution, such that each $S^i$ locally has no nonzero sections supported on $Y$. Then
\[
\HC^i_Y(X, F^\bullet)\cong H^i(S^\bullet(X,U)),\qquad \SC^i_Y(F^\bullet)\cong H^i(S^\bullet_{X,U}),
\]
where $S^\bullet_{X,U}$ is the complex of sheaves defined as follows:
\[
S^i_{X,U}=S^i\oplus \j_*(S^{i-1}|_U), \qquad d(a,b) = (da, -db-a|_U) \qquad \text{for $a$, $b$~--- sections of $S^i$, $\j_*(S^{i-1}|_U)$;}
\]
and $S^\bullet(X,U)=\Gamma(X, S^\bullet_{X,U})$.
\end{prop}
\begin{proof}
Consider the following spectral sequence:
\[
E_2^{pq}=H^p(H^q_Y(X,S^\bullet))\Rightarrow \HC^{p+q}_Y(X,S^\bullet).
\]
By the proposition above the spectral sequence degenerates and 
\begin{multline*}
\HC^i(X,S^\bullet) \cong H^{p-1}(\cokernel(\Gamma(X,S^\bullet)\To \Gamma(U,S^\bullet))) =\\ H^p(\cone(\Gamma(X,S^\bullet)\To \Gamma(U,S^\bullet))[-1]) = H^p(S^\bullet(X,U)).
\end{multline*}
The statement about $\SC^i_Y(F^\bullet)$ can be proved similarly.
\end{proof}

In particular the proposition works for the following resolutions:
\[
\Omega_X^j\To (\A_X^{j\bullet}, \bar\partial),\qquad F^j\Omega^\bullet\To (F^j\A_X^\bullet, d).
\]

\subsection{A proof using relative cohomology}
\begin{proof}[A proof using relative cohomology]
Let $j=n-k+1$, the codimension of $Z$ and the dimension of $W_i$.
By the definition (*) of $AJ^{k,1}$ we have
\[
2\pi\I (AJ^{k,1}[x], \cl^{n-k+1} Z) = \sum_i\int_{W_i\setminus\gamma_i}\omega\log{f_i}+2\pi\I\int_\xi\omega,
\]
where $\omega\in F^{n-k+1}\A^{2n-2k+2}(X)$ is a form whose class $[\omega]$ equals to the Poincare dual of the class $[Z]\in H_{2k-2}(X,\Z)$.

In the definition of the Abel-Jacobi map we choose $\gamma_k$ and $\xi$ to be transversal to $|Z|$. This means that $\gamma_k$ does not intersect $|Z|$ for each $k$ and $\xi$ intersects $|Z|$ only in several points. Let $U=X\setminus |Z|$.

Let $(\omega,\eta)\in F^j\A^{2j}(X)$ be a representative of $c^F(Z)$. Then $(\omega,\eta)$ is also a representative of $c^{DR}(Z)$, the Thom class of $Z$. Since the Thom class maps to the Poincar\'e dual class $[Z]$, we may choose $\omega$ as a representative of $[Z]$.

Then
\[\tag{*}
2\pi\I (AJ^{k,1}[x], \cl^{n-k+1} Z) = \sum_i\int_{W_i\setminus\gamma_i}\omega\log{f_i}+2\pi\I\int_\xi\omega.
\]

Since $(\omega,\eta)$ represent the Thom class of $Z$, we have the following identity:
\[
\int_\xi\omega+\int_{\partial\xi}\eta = [\xi] \cdot Z,
\]
where $[\xi]\in H_{2j}(X, U)$ is the class of $\xi$.
Therefore
\[
2\pi\I\int_\xi\omega \equiv -2\pi\I\int_\gamma\eta \mod{2\pi\I\Z}
\]
and we have the following decomposition:
\[
2\pi\I (AJ^{k,1}[x], \cl^{n-k+1} Z) = \sum_i(\int_{W_i\setminus\gamma_i}\omega\log{f_i} - 2\pi\I\int_{\gamma_i}\eta).
\]
Consider each summand separately. Let $n(\gamma_i)$ be a small neighbourhood of $\gamma_i$ in $W_i$. Then, up to terms which tend to zero when the radius of the neighbourhood tends to zero, we may write
\[
2\pi\I\int_{\gamma_i}\eta = \int_{\partial n(\gamma_i)} \eta \log{f_i}
\]
and
\[
\int_{W_i\setminus\gamma_i}\omega\log{f_i} - 2\pi\I\int_{\gamma_i}\eta
=\int_{W_i\setminus n(\gamma_i)} \omega\log{f_i} + \int_{\partial(W_i\setminus n(\gamma_i))} \eta \log{f_i}.
\]
We see that this is nothing else than a pairing of classes
\[
[W_i\setminus n(\gamma_i)]\in H_{2j}(W_i\setminus \gamma_i, W_i\setminus (\gamma_i\cup Z_i)), \qquad
[(\omega|_{W_i}\log{f_i}, \eta|_{W_i\setminus Z_i}\log{f_i})]\in H^{2j}_{Z_i}(W_i\setminus \gamma_i),
\]
where $Z_i=|Z|\cap W_i$. Indeed, 
\[
d(\eta|_{W_i\setminus Z_i}\log{f_i}) = -\omega|_{W_i\setminus Z_i} \log{f_i} 
\]
since $\eta\in F^j(U)$, so the pair $(\omega|_{W_i}\log{f_i}, \eta|_{W_i\setminus Z_i}\log{f_i})$ defines a class in the local cohomology.

For each point $p\in W_i\cap |Z|$ we consider $n(p)$~--- a small neighbourhood of $p$ in $W_i$, in a way that $n(p)$ do not intersect each other and do not intersect $\gamma_i$. Then
\[
\int_{W_i\setminus n(\gamma_i)} \omega\log{f_i} + \int_{\partial(W_i\setminus n(\gamma_i))} \eta \log{f_i} = \sum_{p\in |Z|\cap W_i} (\int_{n(p)} \omega\log{f_i} + \int_{\partial n(p)} \eta\log{f_i}).
\]

Since the dimension of $n(p)$ is $j$, $F^j\Omega^\bullet_{n(p)}$ is just $\Omega^j_{n(p)}$ in degree $j$. Consider the multiplication by the function $\log{f_i}-\log{f_i(p)}$, which acts on $\Omega^j_{n(p)}$. Since the class $[(\omega, \eta)|_{n(p)}]\in H^j_p(n(p), \Omega^j)$ is the restriction of the class $c^H(Z)$, which is logarithmic, it is logarithmic itself, so the multiplication by $\log{f_i}-\log{f_i(p)}$ kills it. Hence
\[
\int_{n(p)} \omega\log{f_i} + \int_{\partial n(p)} \eta\log{f_i} = 
(\int_{n(p)} \omega + \int_{\partial n(p)} \eta)\log{f_i(p)} =
\ord_p(Z\cdot W_i)\log{f_i(p)}
\]
and the assertion follows.
\end{proof}

\begin{rem}
One could also consider the product $CH^k(X,1)\times CH^{n-k+1}(X) \rightarrow CH^{n+1}(X,1)$ which sends $\sum_i(W_i,f_i)$ , $Z$ to 
\[
\sum_i\sum_{p\in W_i\cap |Z|} \ord_p(Z\cdot W_i)(p, f_i(p)).
\]
The Abel-Jacobi map acts
\[
CH^{n+1}(X,1)\To \frac{H^0(X,\C)^*}{H^0(X,\Z)} \cong \C/\Z
\]
sending $(p, a)$, $p\in X$, $a\in \C^\times$ to $\frac{1}{2\pi\I}\log a$, so the theorem proved above simply means that the following diagram commutes:
\[
\begin{CD}
CH^k(X,1)\times CH^{n-k+1}(X) @@>>> CH^{n+1}(X,1)\\
@@V{AJ^{k,1}\times cl^{n-k+1}(X)}VV @@V{AJ^{n+1,1}}VV\\
\frac{(F^{n-k+1}H^{2n-2k+2}(X,\C))^*}{H^{2k-2}(X,\Z)}\times (F^{n-k+1}H^{2n-2k+2}(X,\C)\cap H^{2n-2k+2}(X,\Z)) @@>>> \C/\Z
\end{CD}
\]
There is a more general statement that the regulator map from higher Chow groups into the Deligne cohomology is compatible with products. This is mentioned in \cite{bloch:acbc}. The construction of higher Chow groups there is different from the one above, but it can be proved that they are canonically isomorphic and the regulator map corresponds to  the Abel-Jacobi map (see \cite{KLM}).
\end{rem}

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