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%\begin{document}
\section{Green's function at CM points}
\begin{enumerate}
\item Compute the residue of $g_{k,z_0}^{\HH/\Gamma}$.
\item Show that the class in $H^2(\Gamma, A)$ is zero.
\end{enumerate}

%\end{document}@
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������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/green-functions/conjecture.tex,v�����������������������������������������������������0000444�0001357�0001362�00000001532�10704165745�021671� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
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\begin{document}

\section{The conjecture}
Let $z_1, z_2 \in \HH$ be two different complex multiplication points. Let $Q_1$, $Q_2$ be the corresponding positive definite primitive quadratic forms. Let $D_1$, $D_2$ be their discriminants. Let $k=2,3,4,5$. The conjecture is
\begin{conjecture}
There exists an algebraic number $f$, such that the lifted value of the Green's function equals
\[
    \wt{G}_k^{\HH/PSL_2\Z}(z_1, z_2) = (D_1 D_2)^{\frac{1-k}2} \log f.
\]
\end{conjecture}
Our aim here is to make the conjecture more precise. In fact we would like to describe the conjectural prime decomposition of the fractional ideal generated by $f$.
\end{document}@
����������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/green-functions/derham.tex,v���������������������������������������������������������0000444�0001357�0001362�00000007352�10704165745�020776� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
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\bibliographystyle{alpha}
\section{De Rham complex}
Complex, filtrations.
\section{The cycle class}
\subsection{The construction of the cycle class map in \cite{bloch:semireg}.}
Here Bloch refers to \cite{groth:fga}, Expos\'e 149 and \cite{harts:rd}, p.176. Let $f:X\rightarrow S$ be smooth and projective. Let $Z\subset X$ be a local complete intersection of codimension $k$ (e.g. $X$ is nonsingular over a field and $Z$ is also nonsingular). Then there exists a canonical cycle class of $Z$ in 
\[
\{Z\}\in\HC_Z^{2k}(X, F^p\Omega_{X/S}^\bullet).
\]
The theorem in \cite{harts:rd} is the following:
\begin{thm}[Fundamental local isomorphism]
Let $i:Z\rightarrow X$ be a closed immersion of preschemes, where $Z$ is locally a complete intersection in $X$ of codimension $k$, and let $F$ be a sheaf of $O_X$-modules on $X$. Then there is a natural functorial isomorphism
\[
\iExt_{O_X}^k(O_Z,F)\xrightarrow{\sim} F\otimes_{O_X} \omega_{Z/X}.
\]
Furthermore, if $F$ is $i^*$-acyclic, then
\[
\iExt_{O_X}^j(O_Z,F)=0 \qquad\text{for $j\neq k$.}
\]
\end{thm}
Recall that by definition $\omega_{Z/X}=(\bigwedge^k(J/J^2))^\vee$, where $J$ is the sheaf of ideals defining $Z$. Substituting $F=\bigwedge^k(J/J^2)$ we obtain a canonical isomorphism
\[
\iExt_{O_X}^k(O_Z,\bigwedge^k(J/J^2))\xrightarrow{\sim} O_Z,
\]
which gives a canonical section
\[
\alpha_Z\in\Gamma(X, \iExt_{O_X}^k(O_Z,\bigwedge^k(J/J^2))).
\]
There is a spectral sequence
\[
E^{pq}_2=H^p(X, \iExt_{O_X}^q(O_Z,\bigwedge^k(J/J^2))) 
\]
\subsection{The construction of the cycle class map in \cite{elzein:compdual}.}
\subsection{Cycle class map via D-modules}
We try to follow the construction in \cite{saitom:IMHM}, 1.15.

For a separated and reduced algebraic variety $X$ over $\C$ we denote by $D^b_h(\D_X)$ the bounded derived category which consists of complexes of $\D_X$-modules with holonomic homologies. We have the usual functors $f_*$, $f_!$, $f^*$, $f^!$, $\DD$, $\boxtimes$, $\otimes$, inner hom $\ihom$. There are the adjoint relations:
\[
\Hom(f^*M, N)=\Hom(M, f_*N),\qquad \Hom(f_!M, N)=\Hom(M, f^!N).
\]

Let $k$ be a field. Then we have the standard $\D_{\spec k}$-module $k$. For each $X$ over $k$ with $a_X:X\rightarrow \spec k$ we define 
\[
k_X = a_X^* k.
\]
For $X$ smooth we obtain 
\[
k_X = a_X^* k = a_X^![-2d_X] = O_X[-2d_X].
\]
Note that
\[
\DD k_X = \DD a_X^* k = a_X^! \DD k = O_X = k_X[2d_X].
\]

The intersection complex is defined as
\[
IC_X = \image(j_! k_U[d_X], j_* k_U[d_X])
\]
for any $U\subset X$ which is smooth and open. If $X$ is itself smooth we obtain $IC_X=k_X[d_X]=O_X[-d_X]$. Note that
\[
\DD IC_X = \DD(k_X[d_X]) = (\DD k_X)[-d_X] = k_X[d_X] = IC_X.
\]

Suppose $i:Z\hookrightarrow X$ is a closed immersion. Suppose $Z$ is smooth. We have $k_Z=i^* k_X$. There is the identity morphism in $\Hom(k_Z, k_Z)$ which induces a morphism 
\[
i^\# :k_X\rightarrow i_* k_Z.
\]
We apply duality to obtain 
\[
\DD i^\#:\DD i_* k_Z \rightarrow \DD k_X.
\]
Consider $\DD i_* k_Z$:
\[
\DD i_* k_Z = i_! k_Z[2d_Z] = i_* k_Z[2d_Z]
\]
since $i$ is a closed immersion hence proper. Therefore the dual morphism acts as 
\[
\DD i^\#:i_* k_Z[2d_Z] \rightarrow k_X[2d_X]
\]
and we can make a composite
\[
\DD i^\# [-2d_Z] \circ i^\#: k_X \rightarrow k_X[2d_X-2d_Z].
\]
This gives an element of 
\[
\Hom(k_X, k_X[2d_X-2d_Z]) = \Hom(k, a_{X*}k_X[2d_X-2d_Z]) = H^{2d_X-2d_Z}(a_{X*} k_X).
\]

One could also take the identity in $\Hom(i^! k_X, i^! k_X)$ and obtain an element
\[
i^\flat:\Hom(i_!i^! k_X, k_X) = \Hom(i_*i^! k_X, k_X).
\]
\[
\Hom(i^* k_X, i^! k_X) = \Hom(k_X, i_* i^! k_X)
\]
\bibliography{refs}
\end{document}@
��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/green-functions/derivatives.tex,v����������������������������������������������������0000444�0001357�0001362�00000016330�10704165745�022057� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
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\section{Derivatives of the Abel-Jacobi map}
Suppose we are given a family of elliptic curves $\E\rightarrow S$ over a $1$-dimensional complex analytic base $S$. For a point $t\in S$ we denote the corresponding elliptic curve by $E_t$. We denote the $n$-wise fiber product of $\E$ with itself over $S$ by $\E^n$. The fibre over $t$ of the family $\E^n\rightarrow S$ is $E_t^n$.

Let $n=2k-2$. It makes sense to consider families $x_t\in Z^k(E_t^n, 1)$, depending on $t$.
\begin{defn}
By a family of elements of $Z^k(E_t^n, 1)$ we mean an element $x\in Z^k(\E^t, 1)$,
\[
x = \sum_i(W_i, f_i),
\]
with each $W_i$ intersecting each fiber $E_t^n$ transversally and each function $f_i$ having neither zero nor pole along any fiber $E_t^n$. For each $t\in S$ the intersection of $x$ with $E_t^n$ is denoted by $x_t\in Z^k(E_t^n, 1)$.
\end{defn}

Let us fix a family $x$. Then $cl^{k,1}[x_t]=0$ for each $t$ since homology groups of $E_t^n$ does not have torsion. Consider the corresponding fiberwise Abel-Jacobi map $AJ^{k,1}[x_t]$. It takes values in 
\[
\frac{H^n(E_t^n, \C)}{F^k H^n(E_t^n,\C)+H^n(E_t^n,\Z)}.
\]
Consider the holomorphic vector bundle $H^{2k-2}(E_t^n, \C)$ on $S$. By the Kunneth theorem
\[
H^\bullet(E_t^n, \C)=(H^\bullet(E_t, \C))^{\otimes n}.
\]
Let $\pr$ denote the natural projection to the subbundle 
\[
\Sym^n H^1(E_t,\C) \subset H^n(E_t^n,\C).
\]
Then by slight abuse of notation we may define
\[
\alpha_t:=\pr AJ^{k,1}[x_t] \in \frac{\Sym^n H^1(E_t,\C)}{\pr F^k H^n(E_t^n,\C)+ \pr H^n(E_t^n,\Z)}.
\]

\subsection{Description of cohomology classes by polynomials}
Let us choose a neighbourhood $U\subset S$ where $H_1(E_t, \Z)$ can be trivialized. Let over $U$
\[
H_1(E_t, \Z) = \Z c_1 \oplus \Z c_2,\qquad c_1\bullet c_2=1,
\]
where $\bullet$ is the intersection pairing.
Let us identify $H^1(E_t, \C)$ with the space of polynomials in one variable $X$ in such a way that 
\[
\langle 1, c_1\rangle = 0,\qquad \langle 1, c_2\rangle=1,\qquad \langle X, c_1\rangle = -1, \qquad \langle X, c_2\rangle=0.
\]
This means that $1$ is the Poincar\'e dual for $c_1$ and $X$ is the Poincar\'e dual for $c_2$.

If $\omega$ is a holomorphic differential $1$-form on $E_t$ then we denote
\[
\Omega_1(\omega)=\int_{c_1}\omega,\qquad \Omega_2(\omega) = \int_{c_2}\omega.
\]
One verifies that the cohomology class of $\omega$ is given as
\[
[\omega]= \Omega_2(\omega)-\Omega_1(\omega)X = \Omega_1(\omega)(z-X),\qquad z=\frac{\Omega_2(\omega)}{\Omega_1(\omega)},
\]
where $z$ does not depend on $\omega$ and depends only on the elliptic curve and the choice of $c_1$ and $c_2$. Note that $\Im z>0$.

Let us consider another choice of $c_1$, $c_2$. Let 
\begin{equation}\label{eqcc}
c_2'=a c_2 + b c_1,\qquad c_1'=c c_2 + d c_1, \qquad ad-bc=1.
\end{equation}
Then the corresponding values of $z$ and $\Omega_1$ are given as follows:
\[
z'=\gamma z, \qquad \Omega_1'(\omega) = (cz+d) \Omega_1(\omega),
\]
where
\[
\gamma = \begin{pmatrix} a & b \\ c & d\end{pmatrix}\in SL_2(\Z).
\]

In this setting, for $m\in \Z$, the space $\Sym^m H^1(E_t,\C)$ is the space $V_m$ of polynomials in $X$ of degree not greater than $m$. It is clear that
\[
\pr F^l H^m(E_t^m, \C) = \{p\in V_m: \text{$(z-X)^l$ divides $p$}\},
\]
\[
\pr H^m(E_t^m, \Z) = \{p\in V_m: \text{$p=\sum_j p_j X^j$, $\binom{m}{j}p_j\in \Z$}\}.
\]
Let $\gamma\in SL_2(\Z)$ and $c_1'$, $c_2'$ are given as in (\ref{eqcc}). Let $p\in V_m$ gives a cohomology class with respect to $c_1$, $c_2$. Then the corresponding polynomial $p'\in V_m$ with respect to $c_1'$, $c_2'$ is given as
\[
p'=\gamma^{-1} p,\qquad p'(X) = (cX+d)^m p\left(\frac{aX+b}{cX+d}\right).
\]

\subsection{Representation of cohomology classes by forms}
In the definition of the Abel-Jacobi map we represented cohomology classes by smooth forms. We are going to represent cohomology classes by forms of second kind. Let $E$ be a smooth projective curve over $\C$. Recall the definition:
\begin{defn}
By a form of second kind on $E$ we mean a meromorphic $1$-form on $E$ whose residue is $0$ at every point of $E$.
\end{defn}
It is clear that integrating such a form gives a cohomology class in $H^1(E,\C)$. In fact the whole space $H^1(E,\C)$ is generated over $\C$ by classes of such forms.

Let $\eta_1$, $\eta_2$, \dots, $\eta_n$ be forms of second kind on $E$. Let $\pi_m$ be the canonical projection to the $m$-th coordinate in the product $E^n$. Suppose we are given a smooth chain $S$ on $E^n$. We are going to define the integral
\[
\int_S \pi_1^*\eta_1 \wedge \dots \wedge \pi_n^*\eta_n.
\]

We first generalize the situation. 
\begin{defn}
We say that a meromorphic $m$-form $\Omega$ on $E^n$ is good if
\begin{enumerate}
\item $\Omega=\pi_1^*\phi_1\wedge\dots\pi_n^*\phi_n$ with each $\phi_j$ either a meromorphic function on $E$ or a form of second kind.
\item The sum of the rank
\end{enumerate}

The definition is given by induction on the number $l$ of forms among $(\pi_j^*\eta_j)$ whose poles intersect $S$. 
\begin{enumerate}
\item If $l=0$ then the definition coincides with the usual definition of the integral.
\item Suppose $l>0$. Let $j$ be the minimal index such that the poles of $\eta_j$ intersect $S$. Let $p_1$, $p_2$,\dots, $p_l$ be the poles of $\eta_j$. For each $i$ we take a small neighbourhood $U_i$ of $p_i$ and a meromorphic function $f_i$ with the property that $\eta_{i}'=\eta_j-d f_{i}$ is holomorphic on $U_i$. Take $U_0=E\setminus\{p_1,\dots,p_l\}$. The sets $\pi_j^{-1}U_i$ form a covering of $E^n$. Decompose $S$ as a sum $S=S_0+S_1+\dots+S_l$ with $|S_i|\subset\pi_j^{-1}U_i$. Then put
\begin{multline}
\int_S \pi_1^*\eta_1 \wedge \dots \wedge \pi_n^*\eta_n := \int_{S_0}\pi_1^*\eta_1 \wedge \dots \wedge \pi_n^*\eta_n + \sum_{i=1}^l \int_{S_i}\pi_1^*\eta_1\wedge\dots\pi_j^*\eta_{i}'\dots\wedge\pi_n^*\eta_n \\+ (-1)^{j-1} \sum_{i=1}^l \int_{\partial S_i} \pi_1^*\eta_1\wedge\dots\pi_j^*f_i\dots\wedge\pi_n^*\eta_n.
\end{multline}

\end{enumerate}

Given a form of second kind $\eta$ there is a smooth form $\omega_\eta$ which has the same cohomology class. This means that there exists a function $\phi_\eta$, smooth outside the set of poles of $\eta$ with the property 
\[
d \phi_\eta = \eta - \omega_\eta.
\]

Let $x\in Z^k(E^n, 1)$ be a higher cycle, $k>1$, $n=2k-2$,
\[
x = \sum_i (W_i, f_i),\qquad \dim_\C W_i=k-1,
\]

Recall that $\gamma_i = f_i^*[0,\infty]$, $\gamma=\sum_i\gamma_i$, $d \xi = \gamma$ (such $\xi$ exists since the cohomology of $E^n$ has no torsion.) Let $\eta_1$, $\eta_2$, \dots, $\eta_n$ be forms of second kind on $E$. Let $\pi_m$ be the canonical projection to the $m$-th coordinate in the product $E^m$. Then we consider the class
\[
C=\pi_1^*[\eta_1]\cup\pi_2^*[\eta_2]\cup\dots\cup\pi_n^*[\eta_n]\in H^n(E^n, \C).
\]
This class can be represented by a smooth form
\[
\Omega=\pi_1^*\omega_{\eta_1}\wedge\dots\wedge\pi_n^*\omega_{\eta_n}.
\]
Suppose at least $n-k+1=k-1$ among $\eta_j$ are holomorphic. Then $C\in F^{k-1}H^n(E^n,\C)$, so we may consider
\[
\langle AJ^{k,1}[x],C\rangle = \frac{1}{2\pi\I}\sum_i\int_{W_i\setminus\gamma_i}\Omega\log f_i+\int_\xi\Omega.
\]
Recall that $\dim_\R \xi = 2k-2$
%\bibliography{refs}
%\end{document}@
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date	2007.10.13.16.01.41;	author mellit;	state Exp;
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@%\input commons.tex
%\author{Anton Mellit}
%\title{Elliptic curves and modular forms}
%\begin{document}

%\bibliographystyle{alpha}
%\maketitle

\section{Certain power series}

Let $R=k[a,b]$ be the ring of polynomials in two variables $a$, $b$. Denote by $K$ the field of fractions of $R$. Let $G_m$ be the multiplicative group. Let $G_m$ act on $R$ by the law
\[
a \ra \lambda^4 a,\; b\ra \lambda^6 b\qquad (\lambda\in G_m).
\]

We consider the family over $R$ given by the equation
\[
y^2=x^3+ax+b.
\]
This can be 'compactified' to the projective variety $E$ over $R$ given by the homogeneous equation in $\wt x$, $\wt y$, $\wt z$:
\[
\wt y^2 \wt z = \wt x^3+a \wt x \wt z^2 + b \wt z^3.
\]

The action of $G_m$ extends to the action on $E$ in the following way:
\[
\wt x\ra \lambda^2 \wt x,\; \wt y\ra \lambda^3 \wt y,\; \wt z\ra \wt z\qquad (\lambda\in G_m).
\]
Therefore the affine chart $\wt z=1$ is stable under the action. We denote this chart by $U_0$. In fact $E$ is an elliptic curve outside the zero locus of the discriminant
\[
\Delta=-16(4 a^3+27 b^2).
\]

If a rational function $\phi$ on $E$ transforms according to 
\[
\phi\ra \lambda^k \phi\qquad (\lambda\in G_m),
\]
then we say that $\phi$ is of weight $k$. Let us denote the space of rational functions of weight $k$ by $F_k$. The action of $G_m$ gives rise to the vector field whose derivation is the Euler operator, $\delta_e$. This operator acts on homogeneous rational functions as follows:
\[
\delta_e f = k f \qquad(f\in F_k).
\]

We have the zero section $s_0:\spec R \ra E$ given by sending 
\[
\wt x \ra 0, \wt y\ra 1, \wt z \ra 0.
\]
Let $t=-x/y = -\wt x/ \wt y \in F_{-1}$. This is a local parameter at $s_0$. We can express $x$ and $y$ as Laurent series in $t$:
\begin{align*}
x &= t^{-2}-a t^2-b t^4-a^2t^6 - 3 a b t^8+O(t^{10}),\\
y=-t^{-1}x &= -t^{-3}+a t + b t^3 + a^2 t^5 + 3 a b t^7+O(t^9).
\end{align*}

The invariant differential form $\omega=\frac{dx}{2y}$ has expansion
\[
\omega=\frac{dx}{2y}=(1+2a t^4+3b t^6+6 a^2 t^8+20 a b t^{10}+O(t^{12})) dt.
\]
Consider the formal integral of $\omega$:
\[
z = \int\omega=t+ \frac{2a}5 t^5+\frac{3b}7 t^7+\frac{2 a^2}3 t^9+\frac{20 a b}{11} t^{11}+O(t^{13}).
\]
In fact $z$ is the logarithm for the formal group law of the elliptic curve. We can now take $z$ as a new local parameter and express $x$ and $y$ in terms of $z$:
\begin{align*}
x &= z^{-2}-\frac{a}5 z^2-\frac{b}7 z^4+\frac{a^2}{75}z^6 + \frac{3 ab}{385}z^8+O(z^{10}),\\
y=\frac{\partial}{2\partial z}x &= -z^{-3}-\frac{a}5 z -\frac{2b}7 z^3 + \frac{a^2}{25} z^5 + \frac{12 a b}{385} z^7+O(z^9).
\end{align*}

Let us fix an isomorphism between $R$ and the ring of modular forms in the following way:
\[
\mu(a) = -\frac{E_4}{2^4 3},\; \mu(b) =  \frac{E_6}{2^5 3^3}.
\]
Then the integral of $-x dz$ can be expressed as follows:
\begin{multline*}
v_0=-\int x dz = z^{-1} +\frac{a}{15}z^3 +\frac{b}{35} z^5 - \frac{a^2}{525}z^7 - \frac{ab}{1155} z^9\\
= z^{-1}-\frac{E_4}{720} z^3+\frac{E_6}{30240}z^5-\frac{E_4^2}{1209600} z^7+\frac{E_4 E_6}{47900160} z^9+O(z^{11})\\=z^{-1}+\sum_{k\geq 2} \frac{B_{2k}E_{2k}}{(2k)!} z^{2k-1}.
\end{multline*}
In fact, this follows from the corresponding identity over the complex numbers which can be proved using the Taylor expansion of the Weierstrass $\wp$-function. We define
\[
v=v_0+\frac{E_2}{12} z = z^{-1}+\sum_{k\geq 1} \frac{B_{2k}E_{2k}}{(2k)!} z^{2k-1}\in R[E_2]((z)).
\]

Note that for $a=-\frac{1}{2^4 3}$, $b=\frac{1}{2^5 3^3}$ (this corresponds to $E_4=1$, $E_6=1$ and the curve is degenerate) we can find expansions of $v_0$, $x$ and $y$ explicitly:
\[
v_0 = \frac{1}{e^z-1} + \frac{1}{2} - \frac{z}{12},\qquad
x = \frac{e^z}{(e^z-1)^2}+\frac{1}{12},\qquad
y = -\frac{e^{2z}+e^z}{2(e^z-1)^3}.
\]
This corresponds to the fact that the Fourier expansion of $E_{2k}$ starts with $1$.

\section{Periods of differentials of second kind}
Let us view $E$ as an elliptic curve over $K$. We will consider odd differential forms on $U_0$. Each such form has an expansion of the type
\[
\sum_{k\in \Z} a_{2k} z^{2k} dz\qquad (a_{2k}\in K).
\]
In fact such a form is determined by its coefficients $a_0, a_{-2}, a_{-4},\ldots$. Moreover, given a polynomial $P$ there is a unique form which has Laurent expansion starting with $P(z^{-2})dz$. The space of odd differential forms on $U_0$ has basis 
\[
\omega_k=\frac{x^k dx}{y} \qquad (k\geq 0).
\]

An odd function on $U_0$ is a function of the form $Q(x)y$. It has expansion
\[
\sum_{k\in\Z} a_{2k-1} z^{2k-1}\qquad (a_{2k-1}\in K).
\]
Such a function is determined by its coefficients $a_{-3}, a_{-5},\ldots$. Conversely, for each polynomial $P$ there exists a form which has Laurent expansion starting with $z^{-3}P(z^{-2})$.

It follows that the space of odd forms modulo the space of exact odd forms is $2$-dimensional with basis 
\[
\omega=\frac{dx}{2y},\qquad \eta=\frac{x dx}{2y}.
\]
This space is canonically isomorphic to the first de Rham cohomology group of $E$. We denote it by $H^1_K$.

\begin{prop} % some statement
Consider the map which sends an odd differential form $\kappa$ to the following element of $K[E_2]$:
\[
\mu(\kappa):=\res(\kappa v).
\]
This map vanishes on exact forms. Therefore it defines a map from $H^1_K$ to $K[E_2]$.
\end{prop}
\begin{proof}
Indeed, if $f=Q(x)y$, then
\begin{multline*}
\res(v df) = -\res(f dv)=\res(Q(x) y (x-\frac{E_2}{12}) dz)\\=\frac 12 \res(Q(x)(x-\frac{E_2}{12}) dx) =0.
\end{multline*}
\end{proof}

One has
\[
\mu(\omega)=1,\;\mu(\eta)=\frac{E_2}{12}.
\]
Therefore using $\mu$ one can build an isomorphism of algebras over $K$:
\[
\Sym H^1_K/(\omega - 1) \xrightarrow[\sim]{} K[E_2].
\]
The symbol $\Sym H^1_K$ denotes the algebra of symmetric tensors of $H^1_K$.

For any $G_m$-module $M$ which is $G_m$-equivariant over some $G_m$-field we denote by $M(1)$ the same module but with the twisted action. If $m^M$ is the action on $M$ then the action on $M(1)$ is defined as follows:
\[
m^{M(1)}_\lambda a = \lambda m^M_\lambda a\qquad(a\in M,\lambda\in G_m.)
\]

Using this notation the isomorphism constructed above can be made into a $G_m$-equivariant isomorphism:
\[
\mu:\Sym H^1_K(1)/(\omega - 1) \xrightarrow[\sim]{} K[E_2].
\]

In the following two propositions the value of a quasi-modular form $f$ of weight $k$ on a pair of numbers $\omega_1, \omega_2\in\C$ with $\tau=\frac{\omega_1}{\omega_2}\in\HH$ is defined as follows:
\[
f(\omega_1, \omega_2) := f(\tau) \omega_2^{-k}.
\]

\begin{prop}
Let $k=\C$. Let $a_0, b_0\in\C$ and $f\in K$ a rational function which is defined at the point $(a_0, b_0)$. Suppose $f$ has weight $k$. Choose a basis $\cc_1, \cc_2$ of the first homology for the curve $y^2=x^3+a_0 x+ b_0$. Let $\omega_i=\int_{\cc_i} \omega$ with $\tau=\frac{\omega_1}{\omega_2}\in\HH$. Then
\[
f(a_0,b_0) = \mu(f)\left(\frac{\omega_1}{2\pi\I}, \frac{\omega_2}{2\pi\I}\right).
\]
\end{prop}

\begin{prop}
Let $k=\C$. Let $a_0, b_0\in\C$ and $[\kappa]\in H^1_K$ represented by an odd differential form $\kappa$ which is defined at the point $(a_0, b_0)$. Suppose $\kappa$ has weight $k$. Choose a basis $\cc_1, \cc_2$ of the first homology for the curve $y^2=x^3+a_0 x+ b_0$. Let $\omega_i=\int_{\cc_i} \omega$ with $\tau=\frac{\omega_1}{\omega_2}\in\HH$. Then
\[
\int_{\cc_2} \kappa = \omega_2 \mu(\kappa)\left(\frac{\omega_1}{2\pi\I}, \frac{\omega_2}{2\pi\I}\right).
\]
\end{prop}

\section{Gauss-Manin connection}
Consider the Gauss-Manin connection on the module $H^1_K$. It is a map
\[
\nabla: H^1_K \ra \Omega^1(K/k)\otimes H^1_K.
\]
This map is equivariant with respect to the $G_m$-action. Consider $\nabla \omega$. This is an element of $\Omega^1(K/k)\otimes H^1_K$. Let us view this element as a map 
\[
	\nabla[\omega]:\Der(K/k)\ra H^1_K.
\]
Since both spaces have dimension $2$ and one can check that the map is surjective, it is an isomorphism. For example, one has
\[
\nabla[\omega](\delta_e) = -[\omega].
\]

We would like to compute $\nabla[\omega]$ for a general vector field $\partial$. Let $\partial\in \Der(K/k)$ be a derivation of the field $K$. By $\partial^*$ we denote a lift of $\partial$ to the affine set $U_0$. We assume that $\partial^*$ is even, so it is given by an even function $\partial^* x$ and an odd function $\partial^* y$. There is no canonical choice of this lift. On the other hand we let $\partial$ act on formal Laurent series in $z$ simply by setting $\partial z=0$. We obtain
\[
2 y (\partial y -\partial^* y) = (3 x^2 + a) (\partial x -\partial^* x).
\]
Therefore there exists a Laurent series $\alpha$ with the property
\[
\partial y = \partial^* y + \alpha y',\qquad \partial x = \partial^* x+ \alpha x',
\]
where $'$ denotes the derivative of a Laurent series with respect to $z$. Since $\partial$ commutes with $'$, we have
\[
2\partial y = \partial x' = (\partial^* x)' + \alpha x'' + \alpha' x'= (\partial^* x)' + 2\partial y - 2 \partial^* y + \alpha' x'.
\]
Therefore
\[
\alpha'x'=2\partial^* y - (\partial^* x)'.
\]
It is easy to see that the right hand side is a regular odd function on $U_0$, hence it is a product of $y$ and a polynomial of $x$. Since $x'=2y$, we obtain that $\alpha'$ is a polynomial of $x$. Consider the form
\[
\alpha' dz.
\]
It is a linear combination over $K$ of an exact form, $\omega$, and $\eta$. Therefore $\alpha$ is a linear combination of a regular odd function on $U_0$, $z$, and $v_0$. We obtain
\[
\partial x = P(x) + (A_\partial z + B_\partial v_0) x',\qquad (P\in K[x],\; A_\partial\in K,\; B_\partial\in K).
\]
Note that 
\[
\partial x = -\frac{\partial a}5 z^2-\frac{\partial b}7 z^4+O(z^{6}).
\]
Therefore $\deg P\leq 2$. Moreover, looking at the expansions
\[
\begin{array}{cccccccc}
x^2 & = & z^{-4} && -\frac{2a}{5} & - \frac{2b}{7} z^2 & +
    \frac{a^2}{15} z^4 & + O(z^6),\\ 
x & = & & z^{-2} & & -\frac{a}5 z^2 & -\frac{b}7 z^4 & +
    O(z^{6}),\\
v_0 x' & = & -2 z^{-4} && -\frac{8a}{15} & - \frac{22b}{35} z^2 & + \frac{2 a^2}{35} z^4 & +O(z^6),\\
z x' & = & & -2 z^{-2} & & -\frac{2 a}{5} z^2 & -\frac{4 b}{7} z^4 & +O(z^6),
\end{array}
\]
we see that 
\[
\partial x = A_\partial( z x'+ 2x) + B_\partial(v_0 x' + 2 x^2 + \frac {4a}3).
\]
The elements $A_\partial$, $B_\partial$ can be found from the equations:
\[
-\frac{\partial a}{5} = -\frac{4 a}{5} A_\partial - \frac{6 b}{5} B_\partial ,\;
-\frac{\partial b}{7} = -\frac{6 b}{7} A_\partial + \frac{4 a^2}{21} B_\partial,
\]
or
\[
\partial a = 4 a A_\partial + 6 b B_\partial,\qquad
\partial b = 6 b A_\partial - \frac{4 a^2}{3} B_\partial.
\]

We see that it is convenient to choose the following vector fields as a basis:
\begin{align*}
\delta_e: & \;\delta_e a = 4a,\; \delta_e b = 6b,\\
\delta_s: & \;\delta_s a = 6 b,\; \delta_s b = -\frac{4 a^2}{3}.
\end{align*}
The first one, $\delta_e$ is the Euler derivative, which was already defined. The second one is the Serre derivative. One can check that our definition coincides with the standard one, i.e.
\[
\delta_s E_4 = -\frac{E_6}{3},\qquad 
\delta_s E_6 = -\frac{E_4^2}{2}.
\]
In particular, we see that $\delta_e$ and $\delta_s$ define vector fields on $\spec R$. Choosing $\alpha_e=z$, $\alpha_s=v_0$ we also obtain liftings of these vector fields to regular vector fields on $U_0$:
\[
\delta_e^* = \delta_e - \alpha_e \frac{d}{dz},\qquad
\delta_s^* = \delta_s - \alpha_s \frac{d}{dz}.
\]
We have
\begin{align*}
\delta_e^* z& = -z,& \delta_e^* v_0& = v_0,& 
\delta_e^* x& = 2x,& \delta_e^* y& = 3y, \\
\delta_s^* z& = -v_0,& \delta_s^* v_0& = -y - \frac{a}{3}z,&
\delta_s^* x& = 2 x^2 + \frac{4a}3,& \delta_s^* y& = 3xy.
\end{align*}

It is now easy to compute $\nabla_\partial [\omega]$:
\[
\nabla_\partial [\omega] = [d \partial^* z] = -[\alpha_\partial' dz],
\]
so, we obtain:
\[
\nabla_{\delta_e}[\omega] = -[\omega], \qquad
\nabla_{\delta_s}[\omega] = [\eta].
\]
The derivatives of $[\eta]$ are given by
\[
\nabla_{\delta_e}[\eta] = [\eta], \qquad
\nabla_{\delta_s}[\eta] = \frac{a}{3}[\omega].
\]

Consider the expression
\[
\delta_\tau:=\frac{E_2}{12} \delta_e + \delta_s.
\]
We can view this as a derivation $R \ra R[E_2]$. By the Gauss-Manin connection on $H^1_K$ the operator $\delta_\tau$ defines a derivation
\[
\mu\circ\delta_\tau:H^1_K \xrightarrow{\delta_\tau} H^1_K[E_2] \xrightarrow{\mu} K[E_2].
\]
Moreover,
\[
\mu \delta_\tau [\omega] = 0.
\]
Therefore $\delta_\tau$ extends to a derivation
\[
\delta_\tau: K[E_2] \ra K[E_2].
\]
We already know that
\[
\delta_\tau E_4 = \frac{E_2 E_4 - E_6}{3},\qquad \delta_\tau E_6 = \frac{E_2 E_6 - E_4^2}2.
\]
To find $\delta_\tau E_2$ we need to compute
\[
\mu \delta_\tau[\eta] = 12 \mu \left(\frac{a}3[\omega] + \frac{E_2}{12}[\eta]\right)
=\frac{a}{3} + \frac{E_2^2}{12^2}.
\]
Thus
\[
\delta_\tau E_2 = 4a + \frac{E_2^2}{12}=\frac{E_2^2-E_4}{12}.
\]

Now we may think that the formula we have obtained for $\delta_\tau E_2$ is simply its definition so that we have a derivation of $K[E_2]$.

We may put
\[
\delta_\tau^* = \delta_\tau - \alpha_\tau \frac{d}{dz}, \qquad
\alpha_\tau = \frac{E_2}{12} \alpha_e + \alpha_s = v.
\]
Then we obtain
\begin{gather*}
\delta_\tau^* z = -v,\qquad
\delta_\tau^* v_0 = -y-\frac{a}{3}z + \frac{E_2}{12} v_0,\\ 
\delta_\tau^* x = 2 x^2 + \frac{E_2 x}{6} + \frac{4a}{3},\qquad
\delta_\tau^* y = 3xy+ \frac{E_2}{4}y.
\end{gather*}
Let us consider the substitutions
\[
v=v_0+\frac{E_2}{12} z, \qquad
\xx = -v'= x - \frac{E_2}{12}, \qquad
\yy = 2 \xx' = y.
\]
One can compute
\begin{gather*}
\delta_\tau^* z = -v,\qquad
\delta_\tau^* v = -\yy,\\
\delta_\tau^* \xx = 2 \xx^2 + \frac{E_2}{3} \xx + \frac{E_2^2}{12^2} + a,\qquad
\delta_\tau^* \yy = 3\xx \yy + \frac{E_2}{2} \yy.
\end{gather*}

%\bibliography{refs}
%\end{document}@
��������������������������������������������cvs-repository/green-functions/elliptic_curves.tex,v������������������������������������������������0000444�0001357�0001362�00000054604�10704165745�022734� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
access;
symbols;
locks; strict;
comment	@% @;

1.1
date	2007.10.13.16.01.41;	author mellit;	state Exp;
branches;
next	;

desc
@@

1.1
log
@first addition
@
text
@%\input commons.tex
%\author{Anton Mellit}
%\title{Elliptic curves and modular forms}
%\begin{document}

%\bibliographystyle{alpha}
%\maketitle

\section{Certain power series}

Using this notation the isomorphism constructed above can be made into a $G_m$-equivariant isomorphism:
\[
\mu:\Sym H^1_K(1)/(\omega - 1) \xrightarrow[\sim]{} K[E_2].
\]

In the following two propositions the value of a quasi-modular form $f$ of weight $k$ on a pair of numbers $\omega_1, \omega_2\in\C$ with $\tau=\frac{\omega_2}{\omega_1}\in\HH$ is defined as follows:
\[
f(\omega_1, \omega_2) := (2\pi \I)^k f(\tau) \omega_2^{-k}.
\]

\section{Derivations on modular forms}
Consider the Gauss-Manin connection on the module $H^1_K$. It is a map
\[
\nabla: H^1_K \ra \Omega^1(K/k)\otimes H^1_K.
\]
This map is equivariant with respect to the $G_m$-action. Consider $\nabla \omega$. This is an element of $\Omega^1(K/k)\otimes H^1_K$. Let us view this element as a map 
\[
	\nabla[\omega]:\Der(K/k)\ra H^1_K.
\]
Since both spaces have dimension $2$ and one can check that the map is surjective, it is an isomorphism. For example, one has
\[
\nabla[\omega](\delta_e) = -[\omega].
\]

We see that it is convenient to choose the following vector fields as a basis:
\begin{align*}
\delta_e: & \;\delta_e a = 4a,\; \delta_e b = 6b,\\
\delta_s: & \;\delta_s a = 6 b,\; \delta_s b = -\frac{4 a^2}{3}.
\end{align*}
The first one, $\delta_e$ is the Euler derivative, which was already defined. The second one is the Serre derivative. One can check that our definition coincides with the standard one, i.e.
\[
\delta_s E_4 = -\frac{E_6}{3},\qquad 
\delta_s E_6 = -\frac{E_4^2}{2}.
\]
In particular, we see that $\delta_e$ and $\delta_s$ define vector fields on $\spec R$. Choosing $\alpha_e=z$, $\alpha_s=v_0$ we also obtain liftings of these vector fields to regular vector fields on $U_0$:
\[
\delta_e^* = \delta_e - \alpha_e \frac{d}{dz},\qquad
\delta_s^* = \delta_s - \alpha_s \frac{d}{dz}.
\]
For reference we summarise the values of the derivations:
\begin{align*}
\delta_e^* z& = -z,& \delta_e^* v_0& = v_0,& 
\delta_e^* x& = 2x,& \delta_e^* y& = 3y, \\
\delta_s^* z& = -v_0,& \delta_s^* v_0& = -y - \frac{a}{3}z,&
\delta_s^* x& = 2 x^2 + \frac{4a}3,& \delta_s^* y& = 3xy, \\
z'& = 1,& v_0'& = -x,& x'& = 2 y, & y'& = 3 x^2 + a.
\end{align*}

It is also convenient to introduce differential forms $d_e$, $d_s$ on the base as the dual basis to the basis $\delta_e$, $\delta_s$.  Then we have
\begin{align}\label{dxdy}
dx &= 2y (dz + z d_e + v_0 d_s) + 2 x d_e + (2 x^2 + \frac{4a}3) d_s,\\
dy &= (3 x^2 + a) (dz + z d_e + v_0 d_s) + 3 y d_e + 3 x y d_s.
\end{align}

It is now easy to compute $\nabla_\partial [\omega]$:
\[
\nabla_\partial [\omega] = [d \partial^* z] = -[\alpha_\partial' dz],
\]
so we obtain:
\[
\nabla_{\delta_e}[\omega] = -[\omega], \qquad
\nabla_{\delta_s}[\omega] = [\eta].
\]
The derivatives of $[\eta]$ are given by
\[
\nabla_{\delta_e}[\eta] = [\eta], \qquad
\nabla_{\delta_s}[\eta] = \frac{a}{3}[\omega].
\]

\section{Representing cohomology classes}
In this section we consider in details how to represent cohomology classes on elliptic curves using the language of hypercovers (section \ref{hypercovers}). We would like to represent cohomology classes for families of elliptic curves by differential forms on the total space.  For an elliptic curve $E$ we consider two affine open sets $U_0$ and $U_1$. The set $U_0$ was mentioned before, it is the complement of the neutral element $[\infty]$ of $E$. The equation of $U_0$ is $y^2=x^3+ax+b$. The set $U_1$ can be chosen to be any affine open set which contains $[\infty]$. We will not fix $U_1$ since sometimes we need $U_1$ to be ``small enough''. The intersection is denoted $U_{int} = U_0\cap U_1$. The triple $U_0, U_1, U_{int}$ defines a hypercover of $E$ which is a particular case of a \v{C}ech hypercover, the corresponding abstract chain complex is the segment, $\Delta_1$.

\subsection{Hyperforms}
Next, $0$-hyperforms are triples $(f_0, f_1, f_{int})$ where $f_0$ and $f_1$ are functions on $U_0$, $U_1$ correspondingly, and $f_{int}$ is forced to be $0$.  $1$-hyperforms are triples $(\theta_0, \theta_1, \theta_{int})$ where $\theta_0$ and $\theta_1$ are differential $1$-forms on $U_0$, $U_1$ correspondingly, and $\theta_{int}$ is a function on $U_{int}$. $2$-hyperforms are triples $(\iota_0, \iota_1, \iota_{int})$ where $\iota_0$ and $\iota_1$ are differential $2$-forms on $U_0$, $U_1$ correspondingly, and $\iota_{int}$ is a differential $1$-form on $U_{int}$. Note that in the case of a single elliptic curve or relative forms for a family of elliptic curves $\iota_0$ and $\iota_1$ must be zero. However if we consider absolute differential forms on a family of elliptic curves this is not the case.

According to the formulae of section \ref{hypercovers} the differentials are given as follows:
\begin{align*}
d(f_0, f_1, 0) &= (d f_0, d f_1, f_1-f_0),\\ 
d(\theta_0, \theta_1, \theta_{int}) &= (d\theta_0, d\theta_1, d\theta_{int}-\theta_1+\theta_0).
\end{align*}

\subsection{The class $[\omega]$.}
We are going to write down some representatives for $H^1(E, \C)$. Consider the differential form $\frac{dx}{2y}$. We want to write it down as a regular form on the Weierstrass family. For this we recall that we lifted two vector fields $\delta_e$ and $\delta_s$ on the base to regular vector fields $\delta_e^*$, $\delta_e^*$ on $U_0$. Consider an arbitrary vector field $\partial$ on the base. Expressing it as a linear combination of $\delta_e$ and $\delta_s$ and using the lifts $\delta_e^*$ and $\delta_s^*$ we construct a lift of $\partial$, denoted $\partial^*$. Then we have
\[
2 y \partial^* y = (3 x^2 + a) \partial^* x + x\partial a + \partial b.
\]
Therefore
\[
i_{\partial^*} (dx\wedge dy) = \partial^* x dy - \partial^* y dx = (-x\partial a - \partial b) \frac{dx}{2y} \; \mod da, db.
\]
We see that if we choose $\partial=-\frac\partial{\partial b}$ we obtain a differential form which is regular on $U_0$ and represents the same relative form as $\frac{dx}{2y}$.

More explicitly, one can check that
\[
\frac\partial{\partial b} = 12\Delta^{-1}(4 a \delta_s - 6 b \delta_e),\;
\frac\partial{\partial a} = -12\Delta^{-1}(6 b \delta_s + \frac{4 a^2}{3} \delta_e).
\]
Then we compute
\[
i_{-\left(\frac\partial{\partial b}\right)^*} (dx\wedge dy) = 8 \Delta^{-1}((-12 a x^2 +18 bx- 8 a^2) dy + (18 a x y - 27 b y) dx).
\]

Let us compute the Laurent expansion of the form above. More generally, for any $\partial$ we have
\begin{multline*}
\partial^* x dy - \partial^* y dx = (\partial^* x y' - \partial^* y x') (dz + z d_e + v_0 d_s) + (\partial^* x \delta_e^*y - \partial^* y \delta_e^* x) d_e \\+ 
(\partial^* x \delta_s^*y - \partial^* y \delta_s^* x) d_s.
\end{multline*}
We have already seen that
\[
\partial^* x y' - \partial^* y x' = -x \partial a - \partial b.
\]
For the remaining part it is enough to compute
\[
\delta_s^* x \delta_e^* y - \delta_e^*x \delta_s^* y = 4 a y.
\]
Hence
\[
i_{-\left(\frac\partial{\partial b}\right)^*} (dx\wedge dy) = 
dz + z d_e + v_0 d_s - 12\Delta^{-1}(16 a^2 y d_e + 24 a b y d_s).
\]
Noting that $d a = 4 a d_e + 6 b d_s$ we obtain
\[
i_{-\left(\frac\partial{\partial b}\right)^*} (dx\wedge dy) = 
dz + z d_e + v_0 d_s - 48\Delta^{-1}a y da.
\]

We would like to construct forms which have as small order of pole at infinity as possible. Therefore we take as $\omega_0$ the form
\[
\omega_0 = i_{-\left(\frac\partial{\partial b}\right)^*} (dx\wedge dy) + 48\Delta^{-1}a y da = dz + z d_e + v_0 d_s.
\]

As the form $\omega_1$, since it is always possible to approximate power series in $z$ by rational functions, one can choose any form such that
\[
\omega_1 = dz + z d_e + O(z^N) d_e + O(z^N) d_s \;\text{for some $N>>0$.}
\]
Indeed, one may start with $t=x/y$. This is a function on a neighbourhood of infinity. Then express $dt = \alpha \frac{dx}{2y} \;\mod da, db$, where $\alpha$ is a rational function which equals to $-1$ at infinity. Therefore $\alpha^{-1} dt = dz \mod da, db$. One can add correction terms of the form $f da$, $f db$ to construct $\omega_1$ as needed.

Next since we want $(\omega_0, \omega_1, \omega_{int})$ to be in the $F^1$ of the Hodge filtration we put $\omega_{int}=0$.

In this way we have constructed a hyperform which represents the cohomology class $[\omega]$:
\[
\omega = (\omega_0, \omega_1, 0) = (dz + z d_e + v_0 d_s, dz + z d_e + O(z^N) d_e + O(z^N) d_s, 0).
\]

\subsection{The class $[\eta]$}
To construct a representative of $[\eta]$ we consider $x \omega_0$. Its Laurent expansion is
\[
x\omega_0 = x dz + x z d_e + x v_0 d_s.
\]
Here it is possible to remove the pole of the coefficient at $d_s$ smaller by adding functions regular on $U_0$. Indeed, it is easy to see that the Laurent series $x v_0 + y$ has no pole. Therefore we put
\[
\eta_0 = x \omega_0 + y d_s = x dz + x z d_e + (x v_0 + y) d_s.
\]

Next we choose $\eta_{int}$. Since we need $d \eta_{int} + \eta_0 - \eta_1 = 0 \mod da, db$ it is natural to take as $\eta_{int}$ some kind of formal integral of $-\eta_0$. Therefore we choose $\eta_{int}$ as any function which satisfies
\[
\eta_{int} = v_0 + O(z^N).
\]

We finally choose $\eta_1$ as follows:
\[
\eta_1 = -\frac{a}3 z d_s + O(z^N) d_e + O(z^N) d_s.
\]

Then the hyperform $\eta$ is defined as
\begin{multline*}
\eta = (\eta_0, \eta_1, \eta_{int}) = (x dz + x z d_e + (x v_0 + y) d_s, \\
-\frac{a}3 z d_s + O(z^N) d_e + O(z^N) d_s, v_0 + O(z^N)).
\end{multline*}

\subsection{Gauss-Manin derivatives}
Let us now compute the Gauss-Manin derivatives for the hyperforms constructed above. First we compute $d\omega$:
\[
d\omega_0 = -d_e\wedge dz + x d_s \wedge dz = -d_e\wedge \omega_0 + d_s \wedge \eta_0 \;\mod d_e\wedge d_s,
\]
\begin{multline*}
d\omega_1 = -d_e\wedge dz + O(z^{N-1}) d_e\wedge dz + O(z^{N-1}) d_s\wedge dz \\
= -d_e\wedge \omega_1 + d_s \wedge \eta_1 + O(z^{N-1}) d_e\wedge dz + O(z^{N-1}) d_s\wedge dz \;\mod d_e\wedge d_s.
\end{multline*}
\[
d\omega_{int} - \omega_1 + \omega_0 = v_0 d_s + O(z^N) d_e + O(z^N) d_s.
\]
This shows that
\begin{multline*}
d\omega = -d_e \wedge \omega + d_s \wedge \eta 
\\+ (0, O(z^N) d_e\wedge dz + O(z^N) d_s\wedge dz, O(z^N) d_e + O(z^N) d_s) \;\mod d_e\wedge d_s.
\end{multline*}

Similarly
\[
d\eta_0 = x d_e\wedge dz + \frac{a}3 d_s\wedge dz = d_e \wedge \eta_0 + \frac{a}3 d_s \wedge \omega_0 \;\mod d_e \wedge d_s.
\]
\[
d\eta_1 = \frac{a}{3} d_s\wedge dz + O(z^{N-1})d_e\wedge dz + O(z^{N-1})d_s\wedge dz  \;\mod d_e\wedge d_s.
\]
\[
d\eta_{int} - \eta_1 + \eta_0 = v_0 d_e + O(z^{N-1}) d_e + O(z^{N-1}) d_s.
\]
Therefore
\begin{multline*}
d\eta = d_e \wedge \eta + \frac{a}{3} d_s \wedge \eta 
\\+ (0, O(z^{N-1}) d_e\wedge dz + O(z^{N-1}) d_s\wedge dz, O(z^{N-1}) d_e + O(z^{N-1}) d_s) \;\mod d_e\wedge d_s.
\end{multline*}

Thus we have proved:
\begin{prop}
For hyperforms $\omega$, $\eta$ defined above the following identities hold up to hyperforms of type $(0, O(z^{N-1}), O(z^{N-1}) d_e + O(z^{N-1}) d_s)$ and hyperforms from the piece $G^1$ of the filtration $G^\bullet$:
\begin{align*}
\nabla_{\delta_e} \omega &= -\omega, &\nabla_{\delta_s} \omega &= \eta,\\
\nabla_{\delta_e} \eta &= \eta, &\nabla_{\delta_s} \eta &= \frac{a}3 \omega.
\end{align*}
\end{prop}

\subsection{Computation of the Poincar\'e pairing}
We would like to compute the Poincar\'e pairing of $\omega$ and $\eta$ to illustrate Corollary \ref{cor_3_1_22}. By definition
\[
\langle \omega, \eta \rangle = \int_E \omega \wedge \eta = \int_{\Delta_E\subset E\times E} \omega \times \eta.
\]
Therefore we need to compute the integral on the right hand side. The hyperform $\omega\times\eta$ has components indexed by pairs $\alpha, \beta$ where $\alpha, \beta \in \{0, 1, int\}$. According to section \ref{products}) 
\[
\omega\times\eta = \begin{pmatrix} 
\omega_0 \times \eta_0 & \omega_0 \times \eta_1 & \omega_0 \times \eta_{int}\\
\omega_1 \times \eta_0 & \omega_1 \times \eta_1 & \omega_1 \times \eta_{int}\\
-\omega_{int}\times \eta_0 & -\omega_{int} \times \eta_1 & \omega_{int} \times \eta_{int}
\end{pmatrix}.
\]
Recall that $\omega_{int}=0$. Therefore the last row is zero.

Using the construction of section \ref{construction_refinements} we obtain a nice refinement $\U'$ (the set $\Delta_E$ is declared special) of the hypercover on $E\times E$. Then
\[
\int_{\Delta_E} \omega \times \eta = \sum_{a\in \sigma_1} \res^{\Int}_{\U',a} (\omega\times\eta)_a.
\]
The sum on the right is decomposed according to flags in $\Fl_{L}(\Delta_E)$.  Since the complement of $U_0$ is a point, we have only one flag in $\Fl_{L}(\Delta_E)$ for each $L\subset\{1,2\}$, $|L|=1$. This flag is $\Delta_E, [\infty\times\infty]$. Let us denote this flag $\fl$. We obtain
\[
\sum_{a\in \sigma_1} \res^{\Int}_{\U',a} (\omega\times\eta)_a = \res^{\Int}_{\{1\}, \fl} \omega_{int} \eta_0 + \res^{\Int}_{\{2\}, \fl} \eta_{Int} \omega_0.
\]
The first residue on the right is zero since $\omega_{int}$ is zero. Finally we obtain
\[
\int_{\Delta_E\subset E\times E} \omega \times \eta = \res^{\Int}_{\{2\}, \fl} \eta_{int} \omega_0.
\]
According to Corollary \ref{cor_3_1_22} the residue on the right equals to
\[
2\pi\I \res_{[\infty]\times[\infty]} \eta_{int}\omega_0.
\]
The residue in the last expression is the algebraic residue which can be computed using Laurent series expansion:
\[
\res_{[\infty]\times[\infty]} \eta_{int}\omega_0 = \res v_0 dz = 1.
\]
Therefore we have proved that
\[
\langle \omega, \eta \rangle = 2 \pi \I.
\]

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@%\input commons.tex
%\author{Anton Mellit}
%\title{Elliptic curves and modular forms}
%\begin{document}

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\section{Certain power series}

Using this notation the isomorphism constructed above can be made into a $G_m$-equivariant isomorphism:
\[
\mu:\Sym H^1_K(1)/(\omega - 1) \xrightarrow[\sim]{} K[E_2].
\]

Now we may think that the formula we have obtained for $\delta_\tau E_2$ is simply its definition so that we have a derivation of $K[E_2]$.

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@\section{Examples of higher cycles}
In this section we construct some higher cycles in $Z^2(E\times E', 1)$ where $E$ and $E'$ are elliptic curves.

\subsection{Notations}
We consider the Weierstrass family $y^2 = x^3 + ax +b$. Denote the total space by $\E$. A fiber, i.e. a single elliptic curve is denoted by $E$.  We cover $E$ by two charts. The first one is 
\[
U_0=E\setminus \{[\infty]\},
\]
which was mentioned before. The second one is 
\[
U_1=E\setminus \{(y=0)\}.
\]
The coordinates on $U_1$ are $x, z$ and the equation is
\[
z = x^3 + a x z^2 + b z^3.
\]
The gluing maps are given as follows:
\[
U_0\ra U_1: \; (x,y) \ra (x y^{-1}, y^{-1}), \;\; U_1\ra U_0: \; (x,z) \ra (x z^{-1}, z^{-1}).
\]

\subsection{The first cycle}
The first cycle will be on $E\times E$ for $b\neq 0$. To denote the coordinates on the first $E$ in $E\times E$ we will use index $1$, for the second one we use $2$. So $U_0\times U_0$ is given by two equations in $4$ variables:
\[
y_1^2 = x_1^3 + a x_1 + b,\;\; y_2^2 = x_2^3 + a x_2 + b.
\]

Consider the subvariety $W$ which is the closure in $E\times E$ of the subvariety given by the equation $x_2 y_1 + \I x_1 y_2=0$. To compute the closure first consider $U_0\times U_1$. We have equations:
\[
y_1^2 = x_1^3 + a x_1 + b,\;\; z_2 = x_2^3 + a x_2 z_2^2 + b z_2^3,\;\; x_2 z_2^{-1} y_1 + \I x_1 z_2^{-1}=0.
\]
This gives 
\[
x_1 = \I y_1 x_2,\;\; y_1^2 = -\I x_2^3 y_1^3 + \I a x_2 y_1 + b.
\]
when $(x_2, z_2) = 0$ we obtain $(x_1, y_1) = (0, \pm \sqrt{b})$. So we see that $W$ contains two more points on $U_0\times U_1$. In fact $x_2$ is a local parameter on $W$ at these points, $z_2$ has order $3$ and $x_1$ has order $1$.

Analogously $W$ contains two more points on $U_1\times U_0$, namely $(x_2, y_2) = (\pm \sqrt{b},0)$.

It remains to look at $U_1\times U_1$. The equations are
\[
z_1 = x_1^3 + a x_1 z_1^2 + b z_1^3,\;\;z_2 = x_2^3 + a x_2 z_2^2 + b z_2^3,\;\; x_2 z_1^{-1} z_2^{-1} + \I x_1 z_1^{-1} z_2^{-1}=0.
\]
Therefore we have
\[
x_1 = \I x_2.
\]
One can check that the point $[\infty]\times[\infty]$ also belongs to $W$ and the local parameter there can be chosen as $x_1$ or $x_2$.

Let $f$ be the rational function on $W$ given as $y_1-\I y_2$. We first compute the divisor of $f$.  Since $y$ has a triple pole at $[\infty]$ and the projections $W\to E$ are unramified at $[\infty]$ we see that $f$ has triple pole at the points $[\infty]\times(0, \pm\sqrt{b})$ and $(0, \pm\sqrt{b})\times[\infty]$. To study the behaviour of $f$ at $[\infty]\times[\infty]$ we write $f$ as
\[
f = z_1^{-1} - \I z_2^{-1} = (z_2 - \I z_1) z_1^{-1} z_2^{-1}.
\]
Using the fact that $z = x^3 + a x^7 + b x^9 + \cdots$ on $U_1$ at $[\infty]$ we see that
\[
f\sim 2 b \I x_2^{3} \;\; \text{at $[\infty]\times[\infty]$,}
\]
so $f$ has a triple zero.

Finally we look for zeroes of $f$ on the set $U_0\times U_0$. We need to find all common solutions of the equations
\[
y_1^2 = x_1^3 + a x_1 + b,\;\; y_2^2 = x_2^3 + a x_2 + b,\;\;x_2 y_1 + \I x_1 y_2=0,\;\;y_1-\I y_2=0.
\]
We get
\[
y_1=\I y_2,\;\; (x_1+x_2) y_1 y_2 = 0.
\]
In the case $y_1=y_2=0$ we see that for $\lambda_1, \lambda_2, \lambda_3$~--- the distinct roots of the polynomial $x^3+ax+b$ the $9$ points $(\lambda_i,0)\times(\lambda_j,0)$ are solutions. The case $x_1+x_2=0$ does not give any solution unless $b=0$. We will not check that the $9$ zeroes are indeed simple since we already know that $f$ has total multiplicity of poles $12$ and triple zero was already found. Therefore
\begin{multline*}
\Div f = \sum_{i,j} (\lambda_i,0)\times(\lambda_j,0) + 3 [\infty]\times[\infty] - 3 [\infty]\times(0, \sqrt{b}) \\- 3 [\infty]\times(0, -\sqrt{b}) - 3 (0, \sqrt{b})\times[\infty] - 3 (0, -\sqrt{b})\times [\infty].
\end{multline*}

It is not difficult to correct $(W,f)$ and obtain a cycle. The following combination is a cycle in $Z^2(E\times E, 1)$:
\[
(W,f) - \sum_i((\lambda_i,0)\times E, y_2) - 3(E\times[\infty], y_1) + 3([\infty]\times E, x_2) + 3(E\times[\infty], x_1).
\]

\subsection{The second cycle}
The second cycle is also on $E\times E$, but it is given by a single summand. We take $W$ as the closure of the subvariety defined by equation $x_1+x_2=0$. It is clear from the equation that when $x_1$ is infinite, $x_2$ must be infinite as well and vice versa. Therefore the closure contains only one additional point $[\infty]\times[\infty]$.

The problem with this $W$ is that it is singular. Namely on $U_1\times U_1$ the equations are
\[
z_1 = x_1^3 + a x_1 z_1^2 + b z_1^3,\;\;z_2 = x_2^3 + a x_2 z_2^2 + b z_2^3,\;\; x_1 z_1^{-1} + x_2 z_2^{-1}=0.
\]
So we obtain the following equations for $W$ inside $E\times E$:
\[
x_1 z_2 = -z_1 x_2,\;\; x_1^2+x_2^2 = 2 b x_2^2 z_1^2.
\]
One can see that the function $x_1 x_2^{-1}$ belongs to the integral closure of the structure ring, but does not belong to the structure ring itself. It is easy to check that if we pass to the normalization of $W$ we obtain $2$ points over $[\infty]\times[\infty]$, namely the one with $x_1 x_2^{-1} = \I$ and the one with $x_1 x_2^{-1} = -I$.

The function $f$ remains the same,
\[
f = y_1 - \I y_2.
\]

It is clear that $f$ does not have zeroes or poles on $W\cap U_0\times U_0$. The only remaining point is $[\infty]\times[\infty]$. We use expansion $z = x^3 + a x^7 + b x^9 + \cdots$.
The second equation for $W$ gives
\[
x_1^2 + x_2^2 = 2 b x_2^2 (x_1^6 + 2 a x_1^{10} + 2 b x_1^{12}+\cdots).
\]
This implies
\[
x_2^2 = -x_1^2 - 2 b x_1^8 - 4 ab x_1^{12} - 8 b^2 x_1^{14}+\cdots.
\]
There are $2$ solutions of this equation:
\[
x_2 = \pm \I (x_1 + b x_1^7 + 2 a b x_1^{11} + \frac{7}{2} b^2 x_1^{13}+\cdots).
\]
The corresponding value of $z_2$ can be computed:
\[
z_2 = \mp \I (x_1^3 + a x_1^7 + 2 b x_1^9+\cdots).
\]

We can now compute the expansion of $f$. Along the branch $x_2\sim \I x_1$ we have
\[
f = \frac{z_2 - \I z_1}{z_1 z_2} \sim 2 x_1^{-3}.
\]
Along the branch $x_2\sim -\I x_1$ we have
\[
f = \frac{z_2 - \I z_1}{z_1 z_2} \sim b x_1^3.
\]
Therefore $\Div f = 0$ and $(W,f) \in Z^2(E\times E, 1)$.

\subsection{Equivalence of the first and the second cycles}
Denote by
\begin{align*}
D_1 &\;\; \text{the closure of the set of zeroes of} & x_2 y_1 + \I x_1 y_2,\\
D_2 &\;\; \text{the closure of the set of zeroes of} & x_1+x_2,\\
D_3 &\;\; \text{the closure of the set of zeroes of} & y_1-\I y_2,
\end{align*}
\begin{align*}
Z_1 &= [\infty]\times E,\\
Z_2 &= E\times [\infty],
\end{align*}
\begin{align*}
f_1 &= x_2 y_1 + \I x_1 y_2& (\Div f_1 = D_1 - 3 Z_1 - 3 Z_2),\\
f_2 &= x_1 + x_2& (\Div f_2 = D_2 - 2 Z_1 - 2 Z_2),\\
f_3 &= y_1-\I y_2& (\Div f_3 = D_3 - 3 Z_1 - 3 Z_2).
\end{align*}

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The first cycle will be on $E\times E$ for $b\neq 0$. To denote coordinates on the first $E$ in $E\times E$ we will use index $1$, for the second one we use $2$. So $U_0\times U_0$ is given by two equations in $4$ variables:
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@%\input commons.tex
%\author{Anton Mellit}
%\title{Necessary geometric constructions}
%\date{\today}
%\begin{document}
%\bibliographystyle{alpha}
%\maketitle

\section{Hypercovers}\label{hypercovers}
\subsection{Abstract cell complex}
\begin{defn}
An \emph{abstract cell complex} is a graded set
\[
\sigma = \bigcup_{i\geq 0} \sigma_i
\]
together with homomorphisms $d_i:\Z[\sigma_i]\To\Z[\sigma_{i-1}]$ such that $d_i\circ d_{i+1}=0$ and $\veps\circ d_0=0$, where $\veps$ is the \emph{augmentation map} $\Z[\sigma_0]\To\Z$. The elements of the set $\sigma_i$ are called \emph{cells} of dimension $i$ and the homomorphisms $d_i$ are called \emph{boundary maps}.
\end{defn}
\[
\begin{CD}
\cdots @@>d>>\Z[\sigma_2] @@>d>> \Z[\sigma_1] @@>d>> \Z[\sigma_0] \\
& & & & & & @@V{\veps}VV\\
& & & & & & \Z
\end{CD}
\]

If an abstract cell complex is given we denote it usually by $\sigma$ and encode the boundary maps by the coefficients $D_{ab}$:
\[
d a = \sum_{b\in\sigma_{i-1}} D_{ab} b, \;a\in\sigma_i.
\]

The basic example is the standard simplex of dimension $n$, $\Delta_n$. In this case $\sigma_i$ is the set of all subsets of $\{0,1,\ldots,n\}$ of size $i+1$ and, writing subsets as increasing sequences, 
\[
d_i(j_0,\ldots,j_i) = \sum_{k=0}^i (-1)^k (j_0,\ldots,\hat{j}_k,\ldots,j_i).
\]

Given two abstract cell complexes $\sigma$, $\sigma'$ the \emph{product} 
\[
\sigma'' = \sigma\times \sigma'
\]
is again an abstract cell complex in the following way:
\[
\sigma''_i = \bigcup_{j=0}^i \sigma_j\times\sigma'_{i-j},
\]
\[
d_i(a\times b) = (d_j a) \times b + (-1)^j a\times d_{i-j} b \qquad \text{for $a\in\sigma_j$, $b\in\sigma'_{i-j}$.}
\]
For example, the cube is defined as
\[
\square_n = \Delta_1\times\cdots\times\Delta_1,\qquad\text{the product has $n$ terms.}
\]

\subsection{Hypercover}
Let $X$ be a topological space and $\sigma$ an abstract cell complex.

\begin{defn}
A \emph{hypercover} of $X$ (indexed by $\sigma$) is a system of open subsets $\U=(U_a)$ indexed by elements $a\in\sigma$ such that:

\begin{enumerate}
\item Whenever a cell $b$ belongs to the boundary of a cell $a$, $U_a\subset U_b$.
\item For any point $x\in X$ consider all cells $a$ such that $x\in U_a$. These cells form a subcomplex of $\sigma$ by the first property, call it $\sigma_x$. We require the complex of abelian groups coming from $\sigma_x$ to be a resolution of $\Z$ with the augmentation map induced by $\varepsilon$.
\end{enumerate}
\end{defn}

\begin{example}
Put $\sigma=\Delta_n$ and let $X$ be a space covered by $n+1$ open subsets $U_0,\dots U_n$. For any sequence $(j_0,\dots,j_i)\in\sigma_i$ we put 
\[
U_{(j_0,\dots,j_i)}= U_{j_0}\cap\dots\cap U_{j_i}.
\]
This clearly gives a hypercover which is called the \v{C}ech hypercover.
\end{example}

If $\sigma$ and $\sigma'$ are two abstract chain complexes, $(U_a)$ and $(U'_{a'})$ are hypercovers of spaces $X$, $X'$ indexed by $\sigma$ and $\sigma'$ correspondingly, by making all products $(U_a\times U'_{a'})$ one gets a hypercover of $X\times X'$ indexed by $\sigma\times\sigma'$, the \emph{product} hypercover.

\subsection{Hypersection}
Let $X$ be a topological space, $\sigma$ an abstract chain complex and $\U=(U_a)$ a hypercover.

Let $\Z_a$ denote the constant sheaf with fiber $\Z$ on $U_a$ extended by zero to $X$. Let 
\[
\Z_{\U i} = \bigoplus_{a\in \sigma_i} \Z_a.
\]
If $U_a\subset U_b$, there is a canonical morphism $r_{ab}:\Z_a \rightarrow \Z_b$. We define $d_i:\Z_{\U i} \rightarrow \Z_{\U i-1}$. If $a\in\sigma_i$ then the morphism from $\Z_a$ to $\Z_{\U i-1}$ is 
\[
\sum_{b\in\sigma_{i-1}}  D_{ab} r_{a b},\qquad \text{where}
\]
\[
d a = \sum_{b\in\sigma_{i-1}} D_{ab} b.
\]
Denote the corresponding sequence of sheaves by $\Z_\U$. The augmentation map $\veps:\Z_{\U 0} \rightarrow \Z_X$ is defined as the sum of the canonical morphisms $\veps_a:\Z_a\rightarrow\Z_X$ for $a\in\sigma_0$.
\[
\begin{CD}
\cdots @@>d>> \Z_{\U 2} @@>d>> \Z_{\U 1} @@>d>> \Z_{\U 0} \\
& & & & & & @@V{\veps}VV \\
& & & & & & \Z_X
\end{CD}
\]

\begin{prop}
The sequence $\Z_\U$ is a resolution of $\Z_X$.
\end{prop}
\begin{proof}
For any point $x\in X$ the stalk of $\Z_\U$ at $x$ is simply the complex of abelian groups corresponding to $\sigma_x$. So the statement follows from the second condition of hypercover.
\end{proof}

In other words, we have obtained a \emph{quasi-isomorphism}
\[
\veps:\Z_{\U} \rightarrow \Z_X.
\]
For any sheaf or complex of sheaves $\F$ on $X$ this gives a morphism of complexes
\[
\veps^*: \HHom(\Z_X, \F) \rightarrow \HHom(\Z_\U, \F).
\]
Note that $\HHom(\Z_X, \F) = \Gamma(X, \F)$ and for any $a\in\sigma$ $\HHom(\Z_a, \F) = \Gamma(U_a, \F)$. We denote the complex $\HHom(\Z_\U, \F)$ by $\Gamma(\U, \F)$ and call elements of this complex \emph{hypersections}.

The complex of hypersections can be defined more precisely as follows. Let $\F$ be a complex of sheaves on $X$ written as
\[
\F^0\rightarrow\F^1\rightarrow\dots.
\]
Then we put
\[
\Gamma(\U,\F)^i = \prod_{j\geq0,a\in \sigma_j} \Gamma(U_a, \F^{i-j}).
\]
The coboundary of a hypersection $s=(s_a)\in \Gamma(\U,\F)^i$ is a hypersection $ds=(s'_a)\in \Gamma(\U,\F)^{i+1}$, where
\begin{equation}\label{dsa}
s'_a = d s_a + (-1)^{i-j+1} \sum_{b\in \sigma_{j-1}} D_{ab} s_b|_{U_a},\qquad a\in\sigma_j.
\end{equation}
One can check directly that $d^2=0$. Let $d^2s=(s''_a)$, $a\in\sigma_j$.
\begin{multline*}
s''_a = d s'_a + (-1)^{i-j} \sum_{b\in \sigma_{j-1}} D_{ab} s'_b|_{U_a}
=
(-1)^{i-j+1} \sum_{b\in\sigma_{j-1}} D_{ab} d s_b|_{U_a} +\\ (-1)^{i-j} \sum_{b\in \sigma_{j-1}} D_{ab} d s_b|_{U_a} + \sum_{b\in \sigma_{j-1}} \sum_{c\in \sigma_{j-2}} D_{ab} D_{bc} s_c|_{U_a}=0.
\end{multline*}

The augmentation morphism $\veps^*:\Gamma(X,\F)\rightarrow \Gamma(\U,\F)$ is defined by sending a section $s\in\Gamma(X,\F^i)$ to the hyperchain $(s'_a)$ where $s'_a$ is not zero only for $a\in \sigma_0$ and equals to the restriction of $s$ to $U_a$.

\begin{rem}
In the case of the \v{C}ech hypercover (corresponding to an open cover) we obtain the \v{C}ech complex.
\end{rem}
\begin{rem}
If I took the definition of $\HHom$ from \cite{harts:rd}, p. 64 the sign in (\ref{dsa}) would be $(-1)^{i+1}$. I assume that switching from a chain complex $\Z_\U$ to the corresponding cochain complex adds the multiplier $(-1)^j$.  My choice of sign is made in order to get the sign  correct in the case of the \v{C}ech hypercover.
\end{rem}

\subsection{Hyperchain}
We assume some class of topological chains on $X$ is given, i.e. semialgebraic chains. For any open set $U\subset X$ we denote by $C_i(U)$ the group of chains with support on $U$ of dimension $i$, which is the free abelian group generated by maps of the chosen class from the standard simplex of dimension $i$ to $U$. Suppose an abstract cell complex $\sigma$ and a hypercover $(U_a)$ is given.

The complex of \emph{hyperchains} is defined similarly to the complex of hypersections, but with a change of sign. We put
\[
C_i(\U) = \bigoplus_{j\geq 0} C_{i-j}(\U, j), \qquad\text{where}
\]
\[
C_{i}(\U,j) = \bigoplus_{a\in \sigma_j} C_{i}(U_a).
\]
Given a hyperchain $\xi\in C_{i-j}(U_a)\subset C_i(\U)$ for $a\in\sigma_j$ its boundary is 
\[
\partial_h \xi:= \partial\xi + (-1)^{i-j} \sum_{b\in\sigma_{j-1}} D_{a b} j_b(\xi).
\]
Here $\partial\xi\in C_{i-1-j}(U_a)$ is the ordinary boundary of $\xi$ and $j_b(\xi)$ denotes the same chain as $\xi$, but considered as an element of $C_{i-j}(U_b)$ if $U_a\subset U_b$. The definition of the boundary map is then extended to $C_i(\U)$ by linearity.

The augmentation morphism $\veps:C(\U)\rightarrow C(X)$ sends all chains in $C_{i-j}(U_a)$ with $j>0$ to $0$ and the ones with $j=0$ to themselves.

We have the following lifting property for lifting chains to hyperchains. Here the word \emph{subdivision} of a chain or of a hyperchain means some iterated barycentric subdivision of all of its simplices. It is clear that the operation of taking subdivision commutes with the boundary operation.
\begin{lem}
Let $\xi$ be a chain on $X$, $\eta=\partial \xi$, $\bar\eta$ a hyperchain such that $\veps\bar\eta=\eta$ and $\partial_h \bar\eta=0$. Then, after possibly replacing $\eta$, $\bar\eta$, $\xi$ with a subdivision, there exists a hyperchain $\bar\xi$ such that $\partial_h\bar\xi=\bar\eta$ and $\veps \bar\xi=\xi$.
\end{lem}
\[
\begin{CD}
\bar\xi\in C_i(\U) @@>{\partial_h}>> \bar\eta\in C_{i-1}(\U) \\
@@V{\veps}VV @@V{\veps}VV \\
\xi\in C_i(X) @@>{\partial}>> \eta\in C_{i-1}(X)
\end{CD}
\]
\begin{proof}
The complex $C_\bullet(\U)$ is the total complex of the bicomplex $C_\bullet(\U,\bullet)$ with the horizontal differential induced by the boundary map of chains in space $X$ and vertical one induced by the boundary map of the complex $\sigma$:
\[
\begin{CD}
& & C_{i-1}(\U, 1) @@>>> C_{i-2}(\U,1) \\
& & @@VVV @@VVV\\
C_i(\U,0) @@>>> C_{i-1}(\U,0) @@>>> C_{i-2}(\U,0)\\
@@V{\veps}VV @@V{\veps}VV\\
C_i(X) @@>>> C_{i-1}(X)
\end{CD}
\]
Clearly, it is enough to prove that the vertical complexes are exact (up to a subdivision). That is, we need to prove that if 
\[
\gamma=\sum_{a\in\sigma_{j}} \gamma_a \in C_i(\U,j) \;\text{is such that}
\]
\[
\sum_{a\in\sigma_{j}} D_{ab} \gamma_a=0 \; \text{for any $b\in\sigma_{j-1}$,}
\]
then there exists 
\[
\bar\gamma=\sum_{c\in\sigma_{j+1}} \bar\gamma_c \in C_i(\U,j+1) \;\text{such that}
\]
\[
\sum_{c\in\sigma_{j+1}} D_{ca} \bar\gamma_c = \gamma'_a \; \text{for any $a\in\sigma_{j}$,}
\]
where $\gamma'_a$ is a subdivision of $\gamma_a$.

It is enough to prove the statement for multiples of a single simplex. Let $s$ be a simplex in $X$ which enters $\gamma_a$ with coefficient $s_a\in\Z$. Then 
\[
\sum_{a\in\sigma_{j}} D_{ab} s_a=0 \; \text{for any $b\in\sigma_{j-1}$.}
\]
Therefore the cycle $\sum_a s_a a$ of $\sigma$ is closed. Since the simplex $s$ must belong to $U_a$ for all $a$ for which $s_a\neq 0$, the mentioned cycle is a closed cycle of $\sigma_x$ for all $x$ in the closure of $s$. For each $x$ one can therefore represent it as a boundary of some cycle of $\sigma_x$, say $t_x$,
\[
t_x = \sum_{c\in\sigma_{j+1}} t_{x c} c.
\]
Here $t_{x c}$ is zero unless $x\in U_c$. Let $V_x$ be the open set which is the intersection of all $U_c$ for which $t_{xc}$ is not zero. Then $V_x$ form a cover of the support of $s$, so there exists a finite subcover. Let it be $V_1=V_{x_1}$, $V_2=V_{x_2}$, \dots, $V_k=V_{x_k}$. One can subdivide the simplex $s$ so that each simplex of the subdivision belongs to one of the chosen open sets. Let us do this and denote the subdivision by $s'$ so that
\[
s' = \sum_{l=1}^k s'_l, \; |s'_l| \subset V_k.
\]
Now it is clear that one may put
\[
\bar\gamma_l = \sum_{c\in\sigma_{j+1}} t_{x_l c} j_c(s'_l)
\]
because $s'_l$ belongs to $U_c$ for every $c$ for which $t_{x_l c}$ is not zero. Then 
\[
\sum_{c\in\sigma_{j+1}} D_{ca} t_{x_l c} s'_l = s_a s'_l \; \text{for any $a\in\sigma_{j}$,}
\]
therefore $\bar\gamma:=\sum_{l=1}^k \bar\gamma_l$ satisfies the required condition.
\end{proof}

This immediately implies
\begin{cor}
If $\xi$ is a closed chain on $X$ then there exists a closed hyperchain $\bar\xi$ whose augmentation is a subdivision of $\xi$. The hyperchain $\bar\xi$ is unique up to a boundary of a hyperchain whose augmentation is $0$.
\end{cor}

\begin{rem}
One extends the construction of hyperchains to the cases of chains on an open subset of $X$ and to the relative chains. It is clear that the boundary maps commute with the natural projections from chains to relative chains.
\end{rem}

\subsection{Integration}
We again fix a space $X$, an abstract cell complex $\sigma$ and a hypercover $(U_a)$. Suppose some class of chains and some class of differential forms are given such that one can integrate form along a chain. We have the de Rham complex of sheaves of differential forms $\Omega^\bullet$:
\[
\Omega^0\ra\Omega^1\ra\dots.
\]
Also we have the complex of chains on $X$:
\[
\dots\ra C_1(X)\ra C_0(X).
\]

The hypersections of $\Omega^\bullet$ will be called the \emph{hyperforms}. Given a hyperform $\omega$ of degree $d$ and a hyperchain $\xi$ of degree $d$ we define the \emph{integral}:
\[
\int_\xi \omega = \sum_{a\in\sigma} \int_{\xi_a}\omega_a.
\]
Here $\omega_a \in\Gamma(U_a, \Omega^{d-\dim a})$ and $\xi_a\in C_{d-\dim a}(U_a)$ are the components of $\omega$ and $\xi$.

The main property is the Stokes theorem:
\begin{prop}
For a hyperform $\omega$ of degree $d$ and a hyperchain $\xi$ of degree $d+1$ one has
\[
\int_{\partial_h \xi} \omega = \int_{\xi} d\omega.
\]
\end{prop}
\begin{proof}
By the definition
\[
\int_{\xi} d\omega = \sum_{a\in\sigma} \int_{\xi_a} (d \omega_a + (-1)^{d-\dim a+1}\sum_{b\in\sigma_{\dim a - 1}} D_{ab} \omega_b).
\]
Applying the Stokes formula this is further equal to 
\[
\sum_{a\in\sigma}(\int_{\partial\xi_a}\omega_a + (-1)^{d-\dim a+1}\sum_{b\in\sigma_{\dim a - 1}} D_{ab} \int_{\xi_a} \omega_b) = \sum_{a\in\sigma} \int_{\partial \xi_a} \omega = \int_{\partial_h\xi} \omega.
\]
\end{proof}

\subsection{Products}\label{products}
For two spaces $X$, $X'$, two abstract complexes $\sigma$, $\sigma'$, two hypercovers $(U_a)$, $U_{a'}'$ we have the product hypercover. There is exterior product on chains and on sheaves. For example, consider the exterior product on chains. For two chains $c\in C_i(X)$, $c'\in C_{i'}(X)$ we obtain the product chain $c\times c'\in C_{i+i'}(X\times X')$. This operation satisfies the property
\[
\partial(c\times c')=(\partial c) \times c' + (-1)^i c\times \partial c'.
\]

Introduce an exterior product on hyperchains. Let $c\in C_{i-j}(U_a)\subset C_i(\U)$, $c'\in C_{i'-j'}(U'_{a'})\subset C_{i'}(\U')$. Here $\dim a = j$, $\dim a'=j'$. Then put
\[
c\times_h c':=(-1)^{j(i'-j')} c\times c'\in C_{i+i'-j-j'}(U_a\times U'_{a'}) \subset C_{i+i'}(\U\times\U').
\]
The superscript $h$ stands for "hyper".

One can check that this satisfies
\begin{prop}
\[
\partial_h(c\times_h c') = (\partial_h c) \times_h c' + (-1)^i c\times_h \partial_h c'.
\]
\end{prop}

On hypersections the exterior product is defined in a similar way. If $s=(s_a)\in \Gamma(\U, \F^\bullet)^i$ and $s'=(s'_{a'})\in \Gamma(\U', \F'_\bullet)^{i'}$, then
\[
(s\times s')_{a\times a'} = (-1)^{\dim a (i'-\dim a')} s_a\times s'_{a'}.
\]
One has a similar formula
\begin{prop}
\[
d(s\times s') = (ds) \times s' + (-1)^i s\times ds'.
\]
\end{prop}

\subsection{Residues}
Let $X$ be a projective algebraic variety over $\C$, $\sigma$ an abstract cell complex and $\U=(U_a)$ a hypercover of $X$ in the Zariski topology.

\begin{defn}
By a \emph{refinement} of $\U$ we understand any hypercover $\U'=(U_a')$ of $X$ in the analytic topology such that for any $a\in \sigma$ $U_a'\subset U_a$.
\end{defn}

Suppose a finite family $M$ of irreducible subvarieties of $X$ is given. These subvarieties will be called \emph{special}.

\begin{defn}
A refinement $\U'=(U_a')$ is called \emph{nice} if for any cell $a\in \sigma$ and any special subvariety $Z$ such that $\dim Z < \dim a$ one has $U_a'\cap Z = \varnothing$. 
\end{defn}

If we have a nice refinement $\U'=(U_a')$ we can make the following construction. Fix a special subvariety $Z$. Take the fundamental class of $Z$, represent it by a chain (of dimension $2\dim Z$) and lift it to a closed hyperchain $(c_a)$ with respect to the hypercover $(U_a' \cap Z)$. Note that for $a\in\sigma_{\dim Z+1}$ the set $U_a'\cap Z$ is empty. Therefore for every $a\in \sigma_{\dim Z}$ the chain $c_a$ is a closed chain of dimension $\dim Z$. Moreover, if we choose a different representation of the fundamental class or a different lift, the hyperchain will differ by a boundary. This implies that the chain $c_a$ for $a\in\sigma_{\dim Z}$ will change by a boundary. Thus the class of $c_a$ in the homology of $Z\cap U_a'$ does not depend on the choices made. Denote this class by $h_a(Z, \U')\in H_{\dim Z}(Z\cap U_a')$.

\begin{defn}
For any meromorphic differential form $\omega$ on $Z\cap U_a$ of degree $\dim Z$ which is regular outside special subvarieties of $Z$ we define its residue as
\[
\res_{a,\U'}^{\Int} \omega = \int_{h_a(Z, \U')} \omega.
\]
\end{defn}

\subsection{Construction of nice refinements}\label{construction_refinements}
We will construct some nice refinements in certain special situation and relate the corresponding residues with the ordinary residues.

Let $X$ be a product of smooth projective curves, $X=X_1\times\cdots\times X_n$. Suppose on each curve a finite open cover (in the Zariski topology) is given. As we have seen before, this gives a hypercover of $X_k$ indexed by the standard simplex of dimension, say $m_k$, the \v{C}ech hypercover. By taking products we obtain an abstract cell complex
\[
\sigma = \Delta_{m_1}\times \Delta_{m_2}\times\cdots\times\Delta_{m_n}
\]
and the product hypercover $\U$ of $X$ indexed by $\sigma$. For any finite family $M$ of irreducible subvarieties of $X$ we construct a nice refinement of $\U$. Here we describe the construction.

Without loss of generality we may assume that the set of special subvarieties $M$ satisfies the following conditions:
\begin{enumerate}
\item For any open set $U_a$ belonging to the hypercover the irreducible components of its complement are in $M$.
\item For any two sets in $M$ the irreducible components of their intersection is also in $M$.
\item If $M$ contains a set $Z$, then it also contains the singular locus of $Z$.
\item If $M$ contains a set $Z$, then for any subset $L\subset\{1,\ldots,n\}$ $M$ contains the irreducible components of the set where the projection from $Z$ to the product $\times_{k\in L} X_k$ is not \'etale (this may be the whole set $Z$).
\end{enumerate}

Let $X_k$ be one of the curves above. Choose a Riemannian metric on $X_k$. Let $n(S,\veps)$ denote the $\veps$-neighbourhood of a set $S$. For any $k$ we give a refinement of the hypercover of the curve $X_k$. This depends on two real numbers $\veps>\veps'>0$.

Suppose $X_k$ is covered by the open sets $U_{k,0},\ldots,U_{k,m_k}$. Denote the complement $X_k\setminus U_{k,0}$ by $S_k$. The set $S_k$ is a finite set of points $p_1,p_2,\ldots,p_r$.

Consider those points among $p_1,p_2,\ldots,p_r$ which are covered by $U_{k,1}$. Suppose they are $p_1,p_2,\ldots,p_{r_1}$. Then consider those points among the remaining ones which are covered by $U_{k,2}$. Suppose they are $p_{r_1+1},\ldots,p_{r_2}$, etc. In this way we obtain a decomposition of the set $S_k$ into $m_k$ subsets, some of them empty:
\[
S_k = \bigcup_{i=1}^{m_k} S_{k,i}.
\]
Let $\wt{S_k} = S_k\cup \{\eta_k\}$, where $\eta_k$ is the generic point of $X_k$. Let $S_{k,0} = \{\eta_k\}$.
For $p\in\wt{S_k}$ put 
\[
U^p_k(\veps, \veps') = \begin{cases} n(p, \veps) & \text{if $p$ is a closed point, $p\in S_k$,}\\ X_k\setminus \overline{n(S_k, \veps')} & \text{if $p$ is the generic point.}\end{cases}
\]
For $p\in S_k$ put
\[
R^p_k(\veps, \veps') = U^p_k(\veps, \veps') \cap U^{\eta_k}_k(\veps, \veps'),\; R_k(\veps, \veps') = \bigcup_{p\in S_k} R^p_k(\veps,\veps'),\; U^S_k(\veps,\veps')=\bigcup_{p\in S_k} U^p_k(\veps,\veps').
\]

We define the refinement in the following way. Put 
\[
U_{k,i}'(\veps, \veps') = \bigcup_{p\in S_{k,i}} U^p_k(\veps, \veps'),\;i=1,\ldots,m_k.
\]
This is a cover of $X_k$. We obtain $\U_k'(\veps,\veps')$ as the \v{C}ech hypercover associated to this cover. If $\veps$ is small enough, this is a refinement of the original cover, i.e. $U_{k,i}'\subset U_{k,i}$. Moreover, if $\veps$ is small enough, the open sets $U^p_k$ for $p\in S_k$ are non-intersecting disks. Let $q>0$ be a number such that as soon as $\veps<q$ these two conditions are satisfied.

For $p\in \wt{S_k}$ we define by $a_k(p)$ the $0$-cell $a$ such that $p\in S_{k,a}$. For $p\in S_k$ we define by $a_k(\eta_k,p)$ the $1$-cell $a$ which joins $a_k(\eta_k)$ and $a_k(p)$. Then for any cell $a$ of the standard simplex $\Delta_{m_k}$ the elements of the hypercover are given as follows:
\[
U_{k,a}'(\veps, \veps') = \begin{cases} 
  \bigcup_{p\in \wt{S_k},a_k(p)=a} U^p_k(\veps, \veps') & \text{if $\dim a=0$,}\\
  \bigcup_{p\in S_k, a_k(p, \eta_k) =a} R^p_k(\veps, \veps') & \text{if $\dim a=1$,}\\
  \varnothing & \text{otherwise.}
\end{cases}
\]

We consider vectors of real numbers $\vec\veps=(\veps_1,\ldots,\veps_n, \veps_1',\ldots,\veps_n')$ such that for each $k$, $q>\veps_k>\veps_k'>0$. Denote
\[
\U(\vec\veps) = \times_{k=1}^n \U_k(\veps_k,\veps_k').
\]
This is a refinement of $\U$. We denote for any $L\subset\{1,\dots,n\}$ 
\[
R_L(\vec\veps) = \times_{k\in L} R_k(\veps_k,\veps_k'),\; U^S_L(\vec\veps) = \times_{k\in L} U^S_k(\veps_k,\veps_k'),\; S_L=\times_{k\in L} S_k.
\]
For any $p\in S_L$ we denote $p_k=\pi_k p$ and put
\[
R_L^p(\vec\veps) = \times_{k\in L} R_k^{p_k}(\veps_k, \veps_k').
\]
We will simply write $R_L$, $U_L^S$, $R_L^p$ when there is no confusion.
We put 
\[
X_L=\times_{k\in L} X_k,\; \pi_L:X\ra X_L\;\text{the projection.}
\]

\begin{defn}
We say that the choice of real numbers $\veps_m,\ldots,\veps_n, \veps_m',\ldots,\veps_n'$ is \emph{good} if $q>\veps_k>\veps_k'>0$ for $k=m,\ldots,n$ and there is an increasing sequence of positive real numbers 
\[
\delta_m, \delta_m',\delta_{m+1},\delta_{m+1}',\ldots,\delta_n,\delta_n'
\]
such that the following conditions are satisfied for any special set $Z$, an index $k$ ($m\leq k\leq n$) and $p\in S_k$:
\begin{enumerate}
\item If $x\in Z$ is such that $\dist(\pi_k(x),p)\leq\veps_k$, then $\dist(x, Z')<\delta_k$.
\item If $x\in X$ is such that $\dist(x, Z)<\delta_{k-1}'$ and $\dist(\pi_k(x), p)<\veps_k$, then $\dist(x, Z')<\delta_k$.
\item If $x\in X$ is such that $\dist(x, Z')<\delta_{k-1}'$, then $\dist(\pi_k(x), p)\leq\veps_k'$.
\end{enumerate}
And the following condition is satisfied for any two special sets $Z_1$, $Z_2$ and an index $k$, $m\leq k\leq n$: 
\begin{enumerate}
\item[(iv)] If $x\in X$ is such that $\dist(x, Z_1)<\delta_k$ and $\dist(x, Z_2)<\delta_k$, then $\dist(x, Z_1\cap Z_2)<\delta_k'$.
\end{enumerate}
We have denoted by $\pi_k$ the projection $X\rightarrow X_k$ and by $Z'$ the intersection $Z\cap\pi_k^{-1} p$. The second and the third conditions are required only for $k>m$.
\end{defn}

Next we prove that good choices indeed exist. For this it is enough to show
\begin{lem}\label{lem:good_choices}
For every good choice of real numbers $\veps_{m+1},\ldots,\veps_n, \veps_{m+1}',\ldots,\veps_n'$ there exists a number $t>0$ such that for any $\veps_m, \veps_m'$ satisfying $t>\veps_m>\veps_m'>0$ the choice $\veps_{m},\ldots,\veps_n$, $\veps_{m}',\ldots,\veps_n'$ is good. Therefore good sequences exist.
\end{lem}
\begin{proof}
Suppose there is a sequence $\delta_{m+1},\ldots,\delta_n'$ as in the definition above. Consider $Z\in M$ and $Z'=Z\cap \pi_{m+1}^{-1} S_{m+1}$. We first show that we can choose $\delta_m'$ such that the second and the third conditions are satisfied for $k=m+1$.

Consider the set 
\[
V=\{x\in X: \dist(x, Z')<\delta_k\; \text{or}\; \dist(\pi_{m+1}(x), S_{m+1})> \veps_{m+1}\}.
\]
This is an open neighbourhood of $Z$ by the first condition for $k=m+1$. Therefore if $\delta_m'>0$ is small enough, then the $\delta_m'$-neighbourhood of $Z$ is also contained in $V$. Therefore the second condition is satisfied for $k=m+1$.

Since $Z'$ is a compact set and $\pi_{m+1}(Z')\subset S_{m+1}$, if $\delta_m'>0$ is small enough, the third condition is satisfied.

Therefore we can choose $\delta_m'>0$ such that $\delta_m'<\delta_{m+1}$ and both the second and the third conditions are satisfied for all $Z$ and $k=m+1$.

Next we choose $\delta_m$ such that the fourth condition is satisfied. To see that this can be done for $Z_1$, $Z_2$ we consider the compact set $X\setminus n(Z_1\cap Z_2, \delta_m')$. The two continuous functions $\dist(\bullet, Z_1)$ and $\dist(\bullet, Z_2)$ do not attain simultaneously value zero on this set. Therefore if $\delta_m$ is small enough, these functions cannot attain simultaneously value less than $\delta_m$. This is equivalent to the fourth condition.

Next consider $Z\in M$ and $Z'=Z\cap \pi_m^{-1} S_m$. The set
\[
\{x\in Z: \dist(x,Z')\geq \delta_m\} 
\]
is compact, hence the continuous function $d(x)=\dist(\pi_m(x), S_m)$ attains its minimum. Suppose $\epsilon_m$ is less than the minimal value of this function. It follows that if $x\in Z$ and $d(x)\leq \epsilon_m$, then $x$ cannot belong to the set above. Therefore $\dist(x,Z')< \delta_m$ and the first condition is satisfied. This implies existence of $t>0$ with the required properties.
\end{proof}

\subsection{Flags of subvarieties}
We consider flags of subvarieties of $X$. A flag of length $m$ is a sequence of irreducible subvarieties $Z_\bullet=(Z_0\supset Z_1 \supset \cdots \supset Z_m)$. We say that a flag $Z_\bullet$ \emph{starts} with $Z_0$ and \emph{ends} with $Z_m$. We require $Z_m$ to be not empty.

\begin{defn}
Let $L\subset\{1,\ldots,n\}$, $L=\{k_1,\ldots,k_l\}$. Let $p\in S_L$. We say that a flag $Z_\bullet=Z_0\supset\cdots\supset Z_l$ is $L,p$\emph{-special} if
\begin{enumerate}
\item $Z_i$ is special for $0\leq i\leq l$,
\item $Z_i$ is an irreducible component of $Z_{i-1}\cap \pi_{k_i}^{-1} p_{k_i}$ for $1\leq i\leq l$.
\end{enumerate}
\end{defn}

\begin{defn}
We say that a flag $Z_\bullet=Z_0\supset\cdots\supset Z_l$ is \emph{strict at index $i$} if $Z_i\neq Z_{i-1}$. We say that a flag is \emph{strict} if it is strict at all indices.
\end{defn}

Suppose $\vec\veps$ is good for a sequence of numbers $\vec\delta=(\delta_1,\delta_1',\ldots,\delta_n,\delta_n')$. The \emph{$\vec\delta$-neighbourhood} of an $L$-special flag $Z_\bullet$ is defined to be the set
\[
n(Z_\bullet, \vec\delta) = \{x\in X: \dist(x, Z_i)<\delta_{k_i},\;i\geq 1\}.
\]

\begin{prop}\label{prop:sf0}
Suppose $Z\subset X$ is special, $L\subset\{1,\ldots,n\}$, $x$ is a point for which either $x\in Z$ or $\dist(x, Z)<\delta_{\min L -1}'$ (if $\min L>1$). Suppose $p\in S_L$ and $\pi_L(x)\in U_L^p(\vec\veps)$. Then there exists an $L,p$-special flag $Z_\bullet$ starting with $Z$ such that $x$ belongs to its $\vec\delta$-neighbourhood.
\end{prop}
\begin{proof}
We construct the flag step by step. If 
\[
\dist(x,Z_{i-1})<\delta_{k_{i-1}}'\leq\delta_{k_i-1}',\; \dist(\pi_{k_{i}} x, p_{k_i})<\veps_{k_i},
\]
by the second property of good sequences we obtain 
\[
\dist(x,Z_{i-1}\cap \pi_{k_i}^{-1} p_{k_i})<\delta_{k_i}.
\]
Therefore on each step we can choose $Z_i$ as an irreducible component of $Z_{i-1}\cap \pi_{k_i}^{-1} p_{k_i}$ so that $\dist(x,Z_i)<\delta_{k_i}$.
\end{proof}

\begin{prop}\label{prop:sf}
Let $x$, $Z$, $L$ and $p$ be as in the proposition above. Take an $L,p$-special flag $Z_\bullet$ such that $x$ belongs to the $\vec\delta$-neighbourhood of $Z_\bullet$. If $L'\subset L$ is such that $\pi_{L'}(x)\in R_{L'}(\vec\veps)$, then $Z_\bullet$ is strict at all indices $i$ for which $k_i\in L'$.
\end{prop}
\begin{proof}
If $k_i\in L'$, then $\dist(\pi_{k_i}, p_{k_i})>\veps_{k_i}'$. By the third property of good sequences we obtain $\dist(x, Z_{k_i})\geq \delta_{k_i}'>\delta_{k_{i-1}}'$. Since $\dist(x, Z_{k_{i-1}})<\delta_{k_{i-1}}$, $Z_{k_{i-1}}\neq Z_{k_i}$.
\end{proof}

\begin{cor}
If $\vec\veps$ is a good choice of numbers, then for any special subvariety $Z$ of dimension less than $d$ and $L\subset\{1,\ldots,n\}$ of size $d$ the intersection $Z\cap \pi_L^{-1} R_L(\vec\veps)$ is empty. Therefore the refinement $\U'(\vec\veps)$ is nice.
\end{cor}
\begin{proof}
If the intersection was not empty, by Proposition \ref{prop:sf} there would exist a strict $L,p$-special flag which starts with $Z$. Therefore the dimension of $Z$ would be at least $d$. 
\end{proof}

\subsection{Decomposition of the residue according to flags}
Let $X$, $\sigma$, $\U$, $M$ be as in the previous section. Let $Z$ be a subvariety of $X$, $Z\in M$. Let $d=\dim Z$ and $\omega$ be a meromorphic differential form on $Z$ of degree $d$ which is regular outside the special subvarieties of $Z$. Let $a\in\sigma_d$. Let $\U'=\U(\vec\veps)$ be the nice refinement of $\U$ corresponding to a good choice of numbers $\vec\veps$. We will show how to compute
the residue $\res_{a,\U'} \omega$.

Recall that the residue was defined as the integral of $\omega$ along the class $h_a(Z, \U')\in H_d(Z\cap U_a')$. Since 
\[
\sigma=\prod_{k=1}^n \Delta_{m_k},
\]
the cell $a$ is given by a sequence of cells $a_1\in\Delta_{m_1},\ldots,a_n\in\Delta_{m_n}$. The open set $U_a'$ is the product of sets $U_{k,a_k}(\veps_k, \veps_k')$ for $k=1,\ldots,n$. Note that for $k=1,\ldots,n$ the set $U_{k,a_k}(\veps_k, \veps_k')$ is not empty only if either $a_k$ is a point or $a_k$ is the edge joining the $0$ vertex and some other vertex. Therefore we get a non-zero residue only if there is a set $L\subset\{1,\ldots,n\}$ of size $d$ and 
\[
\dim a_k= \begin{cases} 1 & \text{for $k\in L$,}\\ 0 & \text{otherwise.}\end{cases}
\]
Moreover in this case we have $U_a'\subset \pi_L^{-1} R_L$.

Consider the set of strict special $L,p$-flags for all $p\in S_L$, starting with $Z$. Denote this set by $\Fl_L(Z)$. This is a finite set. We have seen that any point of $Z\cap \pi_L^{-1} R_L$ belongs to the $\vec\delta$-neighbourhood of a flag from $\Fl_L(Z)$. In fact we have
\begin{prop}\label{prop:unfl}
Any point $x\in Z\cap \pi_L^{-1} R_L$ belongs to the $\vec\delta$-neighbourhood of a unique flag from $\Fl_L(Z)$. 
\end{prop}
\begin{proof}
Suppose $x$ belongs to the $\vec\delta$-neighbourhood of two flags $Z_\bullet$ and $Z_\bullet'$. Let $i$ be the minimal index for which $Z_i\neq Z_i'$. Since $\dist(x, Z_i)<\delta_{k_i}$ and $\dist(x, Z_i')<\delta_{k_i}$, by the fourth property of good sequences we have $\dist(x, Z_i\cap Z_i')<\delta_{k_i}'$. Let $Y$ be an irreducible component of $Z_i\cap Z_i'$ such that $\dist(x, Y)<\delta_{k_i}'$. Let $K$ be the subset of elements of $L$ which are greater than $k_i$. By Proposition \ref{prop:sf} one can construct a strict $K$-special flag starting from $Y$ whose $\vec\delta$-neighbourhood contains $x$. But one can see that the length of the flag equals to the dimension of $Z_i$ and is at least one more than the dimension of $Y$. Hence such flag does not exist and we obtain a contradiction.
\end{proof}

This gives us a possibility to decompose $Z\cap \pi_L^{-1} R_L$ according to the set $\Fl_L(Z)$.
\begin{cor}\label{cor:unfl}
The set $Z\cap \pi_L^{-1} R_L$ is the union of non-intersecting open sets $R_{L,Z_\bullet} = Z\cap \pi_L^{-1} R_L \cap n(Z_\bullet, \vec\delta)$, one for each $Z_\bullet\in \Fl_L(Z)$. Correspondingly we obtain the decomposition
\[
\res_{a,\U'}^{\Int}\omega = \sum_{Z_\bullet \in \Fl_L(Z)} \res_{a,\U',L,Z_\bullet}^{\Int}\omega,\;\text{where}\;
\res_{a,\U',L,Z_\bullet}^{\Int}\omega = \int_{h_a(Z,\U')\cap R_{L,Z_\bullet}}\omega.
\]
\end{cor}

Next we show, for a given flag $Z_\bullet\in \Fl_L(Z)$, how to determine the cell $a$ for which $R_{L,Z_\bullet}$ is a union of several connected components of $U_a'$. We want this cell to be the product of cells $\times_{k=1}^n a_k$ with $\dim a_k=1$ for $k\in L$ and $\dim a_k=0$ for $k\notin L$. We will see that such a cell will be unique.

The last variety in the flag, $Z_d$, is a point from $S_L$. 
\begin{prop}
Suppose $x\in R_L$. If $x\in R_{L,Z_\bullet}$ and $\pi_L(x)\in R_L^p$, then $p=Z_d$.
\end{prop}
\begin{proof}
Since $x\in Z\cap\pi_L^{-1}R_L^p$, one has an $L,p$-flag whose $\vec\delta$-neighbourhood contains $x$. By Proposition \ref{prop:unfl} this flag must be $Z_\bullet$. Hence $Z_\bullet$ is $L,p$-special. Thus $Z_d=p$.
\end{proof}

For an index $k\notin L$ and a flag $Z_\bullet\in \Fl_L(Z)$ we define an element $p_k$ of $\wt{S_k}$ as follows. Let $i$ be the maximal index for which $k_i<k$ or $0$ if $k<k_1$. If $\pi_k Z_i$ is a point in $S_k$ we let $p_k$ be this point. Otherwise we put $p_k=\eta_k$, the generic point.
This gives an element $p(Z_\bullet)\in\wt{S_{L^c}}$, where $L^c=\{1,\ldots,n\}\setminus L$.

On the other hand if $x\in Z\cap R_L$ then for any $k\in L^c$ $\pi_k(x)$ belongs to $U_k^{p_k'}(\veps,\veps')$ for exactly one $p_k'\in \wt{S_k}$. If it was not true, then $x$ would belong to $R_{L\cup\{k\}}$, which is a contradiction. This defines another point $p(x)\in\wt{S_{L^c}}$.
\begin{prop}
For any $x\in Z\cap R_L$ if $x\in R_{L,Z_\bullet}$, then $p(Z_\bullet)=p(x)$.
\end{prop}
\begin{proof}
Let $K$ be the set of such $k$ that either $k\in L$ or $p_k'\in S_k$ (i.e. $p_k$ is not the generic point). For $k\in L$ let $p_k'$ be such that $\dist(\pi_k(x), p_k')<\veps_k$. This defines a point $p'\in S_K$. Then for every $k\in K$ $\dist(\pi_k(x), p_k')<\veps_k$. Thus $\pi_K(x)\in U_L^{p'}$.

By Proposition \ref{prop:sf0} there exists a $K, p'$-special flag $Z'$ starting from $Z$ whose $\vec\delta$-neighbourhood contains $x$. This flag must be strict for all indices which correspond to elements of $L$. Since the number of this indices equals the dimension of $Z$ the flag must be not strict at all other indices. Therefore this flag defines a $L,p'$-special flag, which must be $Z$ by the Proposition \ref{prop:unfl}. At the same time this shows that $p_k=p_k'$ for all $k\in K$.

Let $k\notin K$. Then $\dist(\pi_k(x), S_k)>\veps_k'$. Suppose $i$ is the maximal index for which $k_i<k$ or $0$ if $k<k_1$. If $i\neq 0$, by the third property of good sequences 
\[
\dist(x, Z_{k_i}\cap\pi_k^{-1} S_k)\geq \delta_{k-1}'\geq\delta_{k_i}\geq\dist(x, Z_{k_i}).
\]
Therefore $Z_{k_i}\neq Z_{k_i}\cap\pi_k^{-1} S_k$ which means that $p_k=\eta_k$. The case $i=0$ is obvious.
\end{proof}

We see that 
\begin{cor}
For every flag $Z_\bullet\in \Fl_L(Z)$ there is a canonically defined cell $a_L(Z_\bullet)$ of dimension $d$. For any cell $a$ of dimension $d$ the set $Z\cap U_a'$ is the disjoint union of open sets $R_{L,Z_\bullet}$ where $Z_\bullet$ runs over all $Z_\bullet\in \Fl_L(Z)$ with $a_L(Z_\bullet)=a$.
\end{cor}
\begin{rem}
Therefore we may omit $a$ in the notation $\res_{a,\U',L,Z_\bullet}^{\Int}$ and write $\res_{\U',L,Z_\bullet}^{\Int}$ instead.
\end{rem}

The set $a_L(Z_\bullet)$ can be reconstructed from the points $\pi_L(Z_d)\in S_L$ and $p(Z_\bullet)\in \wt{S_{L^c}}$. In fact
\[
a_L(Z_\bullet) = \times_{k\in L} a_k(\eta_k, \pi_k(Z_d)) \times \times_{k\in L^c} a_k(p_k(Z_\bullet)).
\]

\subsection{Computation of the residue using iterated residues}
Note that $Z$ and $X_L$ have the same dimension. 
\begin{prop}
If the restriction of $\pi_L$ to $Z$ is not surjective, then the set $\Fl_L(Z)$ is empty.
\end{prop}
\begin{proof}
If not, then $\dim \pi_L(Z)<d$. Let $L=\{k_1,\ldots,k_d\}$. Let $Z_\bullet\in \Fl_L(Z)$. Consider the corresponding flag of irreducible subvarieties of $X_L$:
\[
\pi_L(Z)=\pi_L(Z_0)\supset \pi_L(Z_{1}) \supset \cdots \supset \pi_L(Z_{d}).
\]
Because of the dimension reasoning there must be an index $i$ with $\pi_L(Z_i)=\pi_L(Z_{i-1})$. This implies
\[
\pi_{k_i}(Z_{i-1}) = \pi_{k_i}(Z_i) \subset S_{k_i}.
\]
Therefore $Z_i=Z_{i-1}$, so the flag is not strict, which is a contradiction.
\end{proof}

We may therefore suppose without loss of generality that $\pi_L:Z\ra X_L$ is surjective. Hence the set of points on $Z$ where this map is not \'etale is a proper closed subvariety. The irreducible components of this subvariety have dimension $d-1$ or less and are special. Therefore $\pi_L^{-1} R_L\cap Z$ belongs to its complement. This means the following is true.
\begin{prop}
The projection
\[
\pi_L:\pi_L^{-1} R_L\cap Z\ra R_L
\]
is an unramified covering.
\end{prop}

Let $p=\pi_L Z_d\in S_L$.
Since for a flag $Z_\bullet\in \Fl_L(Z)$ the set $R_{L,Z_\bullet}$ is open and closed in $\pi_L^{-1} R_L\cap Z$ we get
\begin{prop}
The projection
\[
\pi_L:R_{L,Z_\bullet} \ra R_L^p
\]
is an unramified covering.
\end{prop}

\begin{rem}
By the proposition we see that since $R_L^p$ is a product of annuli, $R_{L,Z_\bullet}$ is a disjoint union of products of annuli.
\end{rem}

We are going to determine the cycle $h_a(Z,\U')\cap R_{L,Z_\bullet}\in H_d(R_{L,Z_\bullet})$. The $d$-th homology group of a product of $d$ annuli is $\Z$. Hence there is a canonical generator $h_c$ of $H_d(R_L^p)$. In fact $h_c$ can be defined as the product $c_1\times\cdots\times c_d$, where $c_i$ is the circle in $R_{k_i}^{p_{k_i}}$ going around $p_{k_i}$ counterclockwise.

\begin{prop}
The cycle $h_a(Z,\U')\cap R_{L,Z_\bullet}$ is the pullback of $(-1)^{\frac{d(d-1)}2}h_c$ via the projection $\pi_L:R_{L,Z_\bullet} \ra R_L^p$.
\end{prop}
\begin{proof}
\begin{rem}
It is clear that we can decrease numbers $\veps_k$ and increase numbers $\veps_k'$. The sequence obtained in this way will be also good. Moreover the open subsets of the new hypercover are contained in the corresponding open subsets of the old one. Therefore the residues computed with respect to these hypercovers are equal. Therefore one can assume that the projection $\pi_{L}:R_{L,Z_\bullet} \ra R_{L}^{p}$
extends to an unramified covering for the closures of $R_{L,Z_\bullet}$ in $Z$, $R_{L}^{p}$ in $X_L$.
\end{rem}

Consider the commutative diagram:
\[
\begin{CD}
H_{2d}(X) @@>>> H_{2d}(\ol{R_{L,Z_\bullet}}, \partial R_{L,Z_\bullet}) @@<{\pi_L^*}<< H_{2d}(\ol{R_L^p},\partial R_L^p) \\
@@V{h_a}VV @@V{h_a}VV @@V{h_a}VV\\
H_d(U_a') @@>>> H_d(R_{L,Z_\bullet}) @@<{\pi_L^*}<< H_d(R_L^p)
\end{CD}
\]
We see that it is enough to prove that the image of the fundamental class of $H_{2d}(\ol{R_L^p},\partial R_L^p)$ by the map $h_a$ in $H_d(R_L^p)$ is $(-1)^{\frac{d(d-1)}2}h_c$. There is a direct product decomposition $\ol{R_L^p}=\times_{k\in L} \ol{R_k^{p_k}}$. The hypercover on $\ol{R_L^p}$ is the product hypercover. For $k\in L$ let $\phi_k$ be the fundamental class of $H_2(\ol{R_L^p}, \partial R_L^p)$. Let us lift it to a hyperchain $\wt{\phi_k}$.

The hypercover of $\ol{R_L^p}$ is the \v{C}ech hypercover associated to the cover with two open sets:
\begin{align*}
V_0 &=\ol{n(p_k, \veps_k)}\setminus \ol{n(p_k, \veps_k)}\;& \text{corresponding to the cell}\;& a(\eta_k),\\
V_1 &=n(p_k, \veps_k)\setminus n(p_k, \veps_k)\;& \text{corresponding to the cell}\;& a(p_k),\\
V_0\cap V_1 &=n(p_k, \veps_k)\setminus\ol{n(p_k, \veps_k)}\;& \text{corresponding to the cell}\;& a(\eta_k,p_k).
\end{align*}

Let $\veps'<r<\veps$. Consider topological chains
\begin{multline*}
c^k_0 = \ol{n(p_k, \veps_k)}\setminus n(p_k,r)\in C_2(V_0),\;
c^k_1 = \ol{n(p_k, r)}\setminus n(p_k,\veps_k')\in C_2(V_1),\\
c^k_{01} = \partial n(p_k,r)\in C_1(V_0\cap V_1).
\end{multline*}
They define a hyperchain $c^k$. We have $c_0+c_1=\phi_k$. The hyperchain is closed, therefore $h_{01}(\phi_k)=c^k_{01}$, which is the canonical generator of $H_1(R_k^{p_k})$.

Since the product of $\phi_k$ is the fundamental class of $H_{2d}(\ol{R_L^p},\partial R_L^p)$, the product of the hyperchains $c=\times_{k\in L} c^k$ lifts the fundamental class. The term of $c$ at $\times_{k\in L}(\eta_k, p_k)$ is, by the definition of the product for hyperchains, $(-1)^{\frac{d(d-1)}2} \times_{k\in L} c^k_{01}$. This is exactly $(-1)^{\frac{d(d-1)}2} h_c$.
\end{proof}

Let $Z_\bullet$ be a strict $L,p$-special flag, $L=\{k_1,\ldots,k_d\}$, $p=(p_{k_1},\ldots,p_{k_d})$. Let $Z'=Z_1$, $k=k_1$, $L'=\{k_2,\ldots,k_d\}$. Let $Z_\bullet'$ be the flag $Z_1\supset\cdots\supset Z_d$, $Z_\bullet'\in \Fl_{L'}(Z')$. Let $t_i$ be a local parameter on $X_{k_i}$ at the point $p_{k_i}=\pi_{k_i} Z_d$. Let $p'=\pi_{L'} p$. For $Z'$ we have the projection 
\[
\pi_{L'}:R_{L',Z_\bullet'} \ra R_{L'}^{p'},
\]
which is also an unramified covering. Let 
\[
\wt Z = Z'\times_{X_{L'}} Z = (X_{k_1}\times Z') \times_{X_L} Z. 
\]
We obtain the canonical diagrams
\[
\begin{CD}
\wt Z @@>{\rho}>> Z  &\qquad & \wt Z @@>{\rho}>> Z\\
@@V{\tau}VV @@VVV     @@V{\wt\tau}VV @@VVV     \\
Z' @@>>> X_{L'}      & & (X_{k_1}\times Z') @@>>> X_L
\end{CD}
\]
The diagonal embedding $Z'\ra Z'\times_{X_{L'}} Z'$ induces a morphism $\Delta:Z'\ra \wt Z$, which is a section to the natural projection $\tau:\wt Z \ra Z'$. Let 
\[
R_Z=Z\cap\pi_L^{-1}(U_k^{p_k}\times R_{L'}^{p'}) \cap n(Z_\bullet, \vec\delta).
\]
Consider the fiber products over $U_k^{p_k}\times R_{L'}^{p'}$:
\[
\begin{CD}
\wt R @@>{\rho}>> R_Z \\
@@V{\wt\tau}VV @@V{\pi_L}VV     \\
U_k^{p_k}\times R_{L',Z_\bullet'} @@>>> U_k^{p_k}\times R_{L'}^{p'}
\end{CD}
\]
Again, we have a section $\Delta:R_{L',Z_\bullet'}\ra \wt R$. Let $\wt R'$ be the union of connected components of $\wt R$ which intersect the image of $\Delta$. Let $\rho'$ be the restriction of $\rho$ to $\wt R'$.
\begin{rem}
The set $R_Z$ is open and closed in $Z\cap\pi_L^{-1}(U_k^{p_k}\times R_{L'}^{p'})$ because Proposition \ref{prop:unfl} and the first sentence of Corollary \ref{cor:unfl} work if we replace $R_L$ by $U_k^{p_k}\times R_{L'}^{p'}$.
\end{rem}
\begin{rem}\label{rem:RZ-nice}
All special subsets of $Z$ which intersect $R_Z$ are contained in $Z'$. Therefore the map $\pi_L$ on the diagram is an unramified covering outside $\{p_k\}\times R_{L'}^{p'}$ and $\pi_k^{-1} p_k \cap R_Z = R_{L',Z'_\bullet}$.
\end{rem}

\begin{prop}
The map $\rho'$ is an analytic isomorphism.
\end{prop}
\begin{proof}
Since $\rho'$ is a base change of the unramified covering $\pi_{L'}:R_{L',Z'_\bullet}\ra R_{L'}^{p'}$, it is an unramified covering itself. Therefore it is enough to construct a continuous section $s':R_Z\ra \wt R'$ to $\rho'$ which extends the diagonal map. This is equivalent to constructing a retraction $s:R_Z\ra R_{L',Z'_\bullet}$ which respects the projection $\pi_{L'}$.

Take a compact connected set $A\subset R_{L'}^{p'}$ such that $\pi_{L'}^{-1}(A)\cap R_{L',Z'_\bullet}=A_1\cup\cdots\cup A_m$ is a disjoint union of spaces isomorphic to $A$.

One can choose disjoint open subsets $V_1,\ldots, V_m$ in $\pi_{L'}^{-1}(A)\cap R_Z$ such that $A_i\subset V_i$. The set $C=\pi_{L'}^{-1}(A)\cap R_Z\setminus(V_1\cup\cdots\cup V_m)$ is closed. Therefore $\pi_L(C)$ is closed. Take $\alpha>0$ such that $n(p_k,\alpha)\times A$ does not intersect $\pi_{L}(C)$. This means that the open set $\pi_L^{-1}(n(p_k,\alpha)\times A) \cap R_Z$ is contained in the union of the sets $V_i$.

Let $V_i'= \pi_L^{-1}(n(p_k,\alpha)\times A) \cap V_i$. Let us prove that $V_i'$ is connected for each $i$. If not, then $V_i'=B_1\sqcup B_2$ with $B_j$ open, closed and nonempty. Since $A_i$ is connected, for some $j$ $B_j$ does not intersect $A_i$. Therefore $\pi_L B_j$ is open, closed and nonempty. Hence it must be the whole $U_k^{p_k}\times A$. This contradicts the assumption that $B_j$ does not intersect $A_i$.

One can constract a deformation retract retracting $\pi_{L'}^{-1}(A)\cap R_Z$ inside $\pi_L^{-1}(n(p_k,\alpha)\times A)\cap R_Z$. Therefore there are exactly $m$ connected components of $\pi_{L'}^{-1}(A)\cap R_Z$, each containing exactly one $A_i$. Therefore there is a unique map $s_A:\pi_{L'}^{-1}(A)\cap R_Z\ra \pi_{L'}^{-1}(A)\cap R_{L',Z'_\bullet}$ which is identity on $\pi_{L'}^{-1}(A)\cap R_{L',Z'_\bullet}$ and makes the diagram below commutative.
\[
\begin{CD}
\pi_{L'}^{-1}(A)\cap R_Z  @@>>> U_k^{p_k}\times A\\
@@V{s_A}VV @@VVV\\
\pi_{L'}^{-1}(A)\cap R_{L',Z'_\bullet} @@>>> A
\end{CD}
\]
Patching these $s_A$ together gives $s$ as required proving the first statement of the proposition.
\end{proof}

Take a meromorphic form $\omega$ on $Z$ which is holomorphic outside the special subvarieties of $Z$. The form $\omega$ can be written as
\[
\omega = f d t_1 \wedge d t_2 \wedge \cdots \wedge d t_n,
\]
where $f$ is a rational function on $Z$. Let $K$ be the field of fractions of $Z'$. Then $Z\times \spec K$ is a curve and $\Delta(\spec K)$ is a point. Therefore the algebraic residue $\res_{\Delta(\spec K)} \rho^* f d t_1$ is defined. We have
\begin{prop}
Put
\[
\omega' = (\res_{\Delta(\spec K)} \rho^* f d t_1) d t_2 \wedge \cdots \wedge d t_n.
\]
Then
\[
\res_{\U',L,Z_\bullet}^{\Int} \omega = (-1)^{d-1} 2\pi\I \res_{\U',L',Z_\bullet'}^{\Int} \omega'.
\]
\end{prop}
\begin{proof}
By the definition
\[
\res_{\U,L,Z_\bullet} \omega = \int_{h_a\cap R_{L,Z_\bullet}} \omega.
\]
Since $\rho'$ is an isomorphism,
\[
\int_{h_a\cap R_{L,Z_\bullet}} \omega = \int_{\rho'^*(h_a \cap R_{L,Z_\bullet})} \rho^* \omega.
\]
We have 
\[
\rho'^* (h_a \cap R_{L,Z_\bullet}) = (-1)^{\frac{d(d-1)}2}\wt R'\cap \rho^* \pi_L^* h_c = (-1)^{\frac{d(d-1)}2}\wt R'\cap \wt{\tau^*}(h_{c k_1}\times (R_{L',Z'_\bullet}\cap\pi_{L'}^* h_c')),
\]
where $h_{c k_1}$ is the circle in $R_{k_1}^{p_{k_1}}$ and $h_c'$ is the product of circles in $R_{L'}^{p'}$.
By Fubini's theorem
\[
\int_{\wt R'\cap \wt{\tau^*}(h_{c k_1}\times (R_{L',Z'_\bullet}\cap\pi_{L'}^* h_c'))} \rho^* \omega = \int_{R_{L',Z'_\bullet}\cap\pi_{L'}^* h_c'} g(z') d t_2\wedge\cdots\wedge d t_d,
\]
where for $z'\in R_{L',Z'_\bullet}$
\[
g(z') = \int_{\wt R'\cap \tau^{-1} z' \cap \pi_{k_1}^{-1} h_{c k_1}} \rho^* f d t_1.
\]
The last integral is nothing else than 
\[
2\pi \I \res_{\Delta(z')} \rho^* f d t_1.
\]
Therefore
\[
g(z') d t_2\wedge\cdots\wedge d t_d = 2\pi\I\omega'
\]
Taking into account that
\[
R_{L',Z'_\bullet}\cap\pi_{L'}^* h_c' = (-1)^{\frac{(d-1)(d-2)}2}R_{L',Z'_\bullet}\cap h_{a',Z'},
\]
where $a'$ is the cell obtained from $a$ by replacing the component $a(\eta_{k_1},p_{k_1})$ with the component $a(p_{k_1})$, we obtain the statement.
\end{proof}

Let us denote by $\res_{L,Z_\bullet} \omega$ the iterated algebraic residue of $\omega$ with respect to the flag $Z_\bullet$. This is defined by induction on the dimension of $Z$ by the formula
\[
\res_{L,Z_\bullet}f d t_1\wedge\cdots\wedge d t_d = \res_{L',Z_\bullet'} (\res_{\Delta(\spec K)} \rho^* f d t_1) d t_2 \wedge \cdots \wedge d t_d.
\]
Thus we obtain a formula for our residue.
\begin{cor}\label{cor_3_1_22}
\[
\res_{\U',L,Z_\bullet}^{\Int} \omega = (-1)^{\frac{d(d-1)}2}(2\pi\I)^d \res_{L,Z_\bullet} \omega.
\]
\end{cor}

\subsection{Gauss-Manin}
Suppose we have a morphism of algebraic varieties $X\ra S$. Let $\sigma$ be an abstract cell complex and $\U=(U_a)$ a hypercover on $X$ with respect to Zariski's topology. We assume that $S=\spec R$ is affine and all the open sets of the hypercover are affine.

Associated to the de Rham complex on $X$ we have the complex of hypersections
\[
\Omega_X^0(\U)\ra\Omega_X^1(\U)\ra\cdots.
\]
Consider the complex of hypersections corresponding to the relative de Rham complex.
\[
\Omega_{X/S}^0(\U)\ra \Omega_{X/S}^1(\U)\ra\cdots.
\]
These are both complexes of $R$-modules.

\begin{prop}
The following natural sequence is exact for all $k\geq 0$:
\[
\Omega_S^1(S)\otimes_R\Omega_X^{k-1}(\U)\ra \Omega_X^k(\U)\ra \Omega_{X/S}^k(\U)\ra 0.
\]
\end{prop}
\begin{proof}
Since the corresponding sequence of sheaves is exact, it induces an exact sequence over any affine set.
\end{proof}

We define Gauss-Manin connection as follows. Let $\omega\in\Omega_{X/S}^k(\U)$ be closed. Lift it to $\ol\omega\in\Omega_{X}^k$. Then $d \ol\omega\in \kernel(\Omega_X^{k+1}(\U)\ra\Omega_{X/S}^{k+1}(\U))$. Choose $\ol\eta\in \Omega_S^1(S)\otimes_R\Omega_X^{k}(\U)$ which maps to $d \ol\omega$. Let $\eta$ be the projection of $\ol\eta$ in $\Omega_{X/S}^{k}(\U)\otimes_R\Omega_S^1(S)$. We say that $\eta$ is a Gauss-Manin derivative of $\omega$. Of course, this construction depends on several choices. Although $\eta$ is not well-defined, by abuse of notation we will write
\[
\eta=\nabla\omega
\]
if $\eta$ is a Gauss-Manin derivative of $\omega$.

Suppose we have a family of hyperchains $c_s\in C_k(\U) $, $s\in S$. This means that $c_s$ is a linear combination of simplices $c_s^i$ with each $c_s^i$ being a map from $\Delta\times S$ to $X$ which composed with $X\ra S$ gives the projection $\Delta\times S\ra S$. Here $\Delta$ denotes a standard simplex. We require the maps $c_s^i$ to be of the same type as we require for simplices, i.e. semi-algebraic. For $\omega\in\Omega_{X/S}^k(\U)$ consider the integral
\[
f(s)=\int_{c_s} \omega, \; s\in S.
\]
For a path $s_0 s_1$ in $S$ let $c_{s_0 s_1}\in C_{k+1}(\U)$ denote the corresponding hyperchain over this path. It provides a homotopy:
\[
\partial_h c_{s_0 s_1} + (\partial_h c)_{s_0 s_1}= c_{s_1}-c_{s_0}.
\]
\begin{lem}
Let $\omega\in\Omega_{X/S}^k(\U)$ be a closed relative hyperform and $c=(c_s)_{s\in S}$ be a family of hyperchains, $c_s\in C_k(\U)$.
If $\eta=\nabla\omega$ with $\ol\eta$ and $\ol\omega$ as in the definition of $\nabla$, then we have
\[
d \int_{c_s} \omega = \int_{c_s} \nabla\omega + R(\partial_h c, \ol\omega).
\]
Here $R$ is the bilinear operator which is constructed as follows. The value of $R$ on a vector in $S$ represented by a path $s_t$ is
\[
\langle [s_t], R(\partial_h c, \ol\omega) \rangle = \frac{\partial}{\partial t}\bigg\vert_{t=0} \int_{(\partial_h c)_{s_0 s_t}} \ol\omega.
\]
\end{lem}
\begin{proof}
For a fixed $s_0\in S$ let $s_t$ be a path in $S$ starting from $s_0$. We have
\[
f(s_t)-f(s_0)=\int_{\partial_h c_{s_0 s_t}+(\partial_h c)_{s_0 s_t}} \ol\omega.
\]
The first summand can be transformed as
\[
\int_{\partial_h c_{s_0 s_t}} \ol\omega = \int_{c_{s_0 s_t}} \ol\eta.
\]
Over the path we are considering $\Omega_S^1$ is generated by $dt$. Therefore $\ol\eta=dt\wedge \ol\eta_0$. We obtain
\[
\int_{c_{s_0 s_t}} \ol\eta=\int_{c_{s_0 s_t}} dt \wedge \ol\eta_0=
\int_{c_{s_0 s_t}} (d(t\ol\eta_0) - t d\ol\eta_0)=\int_{\partial_h c_{s_0 s_t}} t\ol\eta_0 - \int_{c_{s_0 s_t}} t d\ol\eta_0.
\]
Examining the second term we see
\[
{\frac{\partial}{\partial t}}\bigg\vert_{t=0} \int_{c_{s_0 s_t}} t d\ol\eta_0 =0.
\]
The first one transforms to
\[
\int_{\partial_h c_{s_0 s_t}} t\ol\eta_0 = \int_{c_{s_t}-c_{s_0}} t\ol\eta_0 -\int_{(\partial_h c)_{s_0 s_t}} t\ol\eta_0.
\]
The second term gives
\[
{\frac{\partial}{\partial t}}\bigg\vert_{t=0} \int_{(\partial_h c)_{s_0 s_t}} t\ol\eta_0=0.
\]
The first one gives
\[
{\frac{\partial}{\partial t}}\bigg\vert_{t=0} \int_{c_{s_t}-c_{s_0}} t\ol\eta_0 = \lim_{t\ra 0} \int_{c_{s_t}} \ol\eta_0=\int_{c_{s_0}} \ol\eta_0.
\]
\end{proof}

For example, if $c_s$ is closed:
\begin{cor}
If $c_s$ is a closed family of hyperchains and $\omega$ is a closed relative hyperform, then
\[
d \int_{c_s} \omega = \int_{c_s} \nabla\omega.
\]
\end{cor}
%\bibliography{refs}
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\author{Anton Mellit}
\title{Higher green's functions for modular forms}
\begin{document}
\maketitle

\section{Notations}

We will consider the group $SL_2(\R)$. Elements of this group will be usually denoted by $\gamma$, and matrix elements by $a$, $b$, $c$, $d$:
\[
\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\R).
\]
This groups acts on the upper half plane $\HH$.

\section{Representations of $SL_2(\R)$}

The group $SL_2(\R)$ naturally acts on the following linear spaces:
\begin{itemize}
\item $V=\C^2$~--- the space of column vectors of length 2.
\item $V_m$~--- symmetric $m$-th power of $V$.
\item $V^m$~--- the space of homogenious polynomials of degree $m$ on $V$.
\end{itemize}
Note, that $V_m$ and $V^m$ are mutually dual, $V_0=V^0=\C$. Of course, $V_m$ is isomorphic to $V^m$ as a representation of $SL_2(\R)$, but we would like to represent elements of this spaces in different ways, also these two spaces have different natural $\Z$- structures~--- that is why we distinguish them.

We explain the way we view elements of $V_m$, $V^m$. Start with $V^m$. Let $e_1$, $e_2$ be the natural basis on $V$. Let $e^1$, $e^2$ be the dual basis. Every homogenious polynomial on $V$ is a polynomial in $e^1$, $e^2$. If we substitute $e^1$ by a new variable $X$ and $e^2$ by $1$ we obtain a non-homogenious polynomial in one variable of degree less or equal $m$. We represent elements of $V^m$ as polynomials in the variable $X$ of degree less or equal $m$, i.e.
\[
p\in V^m, \; p(X) = p_0 + p_1 X + \dots + p_m X^m.
\]
The group acts on $V^m$ on the right by
\[
(p \gamma)(X) = (p |_{-m} \gamma)(X) = p(\gamma X) (cX + d)^m, \qquad \gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\R).
\]
There is a corresponding action on the left
\[
(\gamma p)(X) = (p \gamma^{-1})(X) = p(\gamma^{-1} X) (-cX + a)^m.
\]
We may also consider the corresponding homogenious polynomials in two variables
\[
p(X,Y) = p \left( \frac {X}{Y} \right) Y^m = p_0 Y^m + p_1 Y^{m-1} X + \dots + p_m X^m,
\]
the action of $SL_2(\R)$ is then given by
\[
(\gamma p)(X,Y) = p(dX - bY, -cX + aY).
\]

Now turn to $V_m$. Any $v\in V_m$ defines a functional on $V^m$. Suppose its value on $p$ is 
\[
v_0 p_0 + v_1 p_1 + \dots v_m p_m,
\]
then we represent $v$ as a raw vector $(v_0, v_1, v_2, \dots, v_m)$. For any two numbers $x, y \in \C$ we form a vector
\[
v_{x, y} = (y^m, y^{m-1} x, y^{m-2} x^2, \dots, x^m) \in V_m.
\]
Then for any $p\in V^m$
\[
(v_{x, y}, p) = p\left(\frac{x}{y}\right) y^m = p(x, y),
\]
hence
\[
\gamma v_{x, y} = v_{\gamma (x, y)} = v_{a x + b y, c x + d y}.
\]
In this way the action of $SL_2(\R)$ is given on a subset of $V_m$ which spans $V_m$.

The isomorphism between $V_m$ and $V^m$ can be given as follows. Define an invariant pairing between elements of the form $v_{x,y} \in V_m$
\[
(v_{x,y}, v_{x', y'}) = (x y' - x' y)^m = \sum_{i=0}^m (-1)^i \binom{m}{i} x^{m-i} y^i x'^i y'^{m-i}.
\]
Since it is a homogenious polynomial of degree $m$ in each pair of variables this induces an equivariant linear map from $V_m$ to $V^m$
\[
(v_0, v_1, \dots, v_m) \mapsto \sum_{i=0}^m (-1)^i \binom{m}{i} v_{m-i} X^i.
\]
This is an isomorphism of representations. It induces invariant pairings on $V^m$
\[
(\sum_{i=0}^m p_i X^i, \sum_{i=0}^m p_i' X^i) = \sum_{i=0}^m \frac {(-1)^i p_i p_{m-i}'}{\binom{m}{i}}
\]
and on $V_m$
\[
((v_i), (v_i')) = \sum_{i=0}^m (-1)^i \binom{m}{i} v_i v_{m-i}'.
\]

\section{Operations on functions with weight}

\subsection{Differential operators}
Let $S$ be a discrete subset of the upper half plane $\HH$ and $f(\z)$ be a function on $\HH-S$ with values in $\C$ and $w\in \Z$. Define differential operators
\begin{gather*}
Df = \frac{1}{2\pi i} \frac{\partial f}{\partial \z},\\
\delta_w f = Df + \frac{1}{2\pi i} \frac{w}{\z - \zc} f,\\
\delta_w^- f = 2 \pi i (\z - \zc)^2 \frac{\partial f}{\partial \zc},\\
\Delta f = (\z - \zc)^2 \frac {\partial}{\partial \z} \frac {\partial}{\partial \zc} f + w (\z - \zc) \frac {\partial}{\partial \zc}.
\end{gather*}
We think about $w$ as the weight, attached to the function $f$. The weight will be always clear from the context, so we will omit the subscript $w$. For example, if we have a function which satisfies
\[
f(\gamma \z) = f(\z) (c \z + d)^w,
\]
for $\gamma$ from some subgroup of $SL_2(\R)$ then we will assign to $f$ weight $w$. Another example would be an element of a vector valued function which satisfies some transformation property.

We will follow the following agreement: the operators $D$ and $\delta$ increase weight by $2$, the operator $\delta^-$ decreases weight by $2$ and the operator $\Delta$ leaves weight untouched. Taking into account this agreement the following identities can be proved:
\begin{gather*}
\delta^- \delta - \delta \delta^- = w,\\
\delta \delta^- = \Delta,\\
\delta^- \delta = \Delta + w.\\
\end{gather*}

\subsection{Complex conjugation}
Define operator $\eps$ as follows:
\[
(\eps f)(\z) = (2 \pi i)^w (\z - \zc)^w \overline{f(\z)}.
\]
Assign to $\eps f$ weight $-w$. We can check that 
\begin{gather}
\eps \eps f = f \\
\delta^- \eps f = -\eps \delta f \\
\delta \eps f = -\eps \delta^- f \\
\Delta \eps f = \eps (\Delta + w) f
\end{gather}

\subsection{Action of $SL_2(\R)$ on functions}
Let the group $SL_2(\R)$ act on functions of weight $w$ by the usual formula:
\[
(f|_w \gamma)(\z) = f(\gamma \z) (c\z + d)^{-w}.
\]
This is a right action. We also define corresponding left action
\[
(\gamma f)(\z) = (f|_w \gamma^{-1})(\z) = f(\gamma^{-1} \z) (-c\z + a)^{-w}.
\]
Note that this action commutes with operators $\delta$, $\delta^-$, $\Delta$, $\eps$ it maps functions defined on $\HH-S$ to functions defined on $\HH-\gamma S$.

\section{Eigenfunctions of the laplacian}\label{eigenvalues}
If $f$ has weight $w$ and is an eigenfunction of $\Delta$ with eigenvalue $x$, i.e. $\Delta f = xf$, then 
\begin{gather*}
\Delta \delta f = (x + w) \delta f, \\
\Delta \delta^- f = (x - w + 2) \delta^- f, \\
\Delta \eps f = (x + w) \eps f.
\end{gather*}
So, operators $\delta$, $\delta^-$ and $\eps$ preserve the property to be an eigenfunction of the laplacian. We can compose a number from the weight and the eigenvalue which does not change under applications of these operators:
\[
x_0 = x - \frac {w (w-2) }{4}.
\]
Let us fix some $k\in \Z$, $k>0$ and consider functions with fixed 
\[
x_0 = k (1-k).
\]
Denote by $F_w$ the space of functions with weight $w$ and given value of $x_0$, i.e. satisfying
\begin{gather*}
\delta \delta^- f = \Delta f = (k (1-k) + \frac {w (w-2) }{4}) f = (k - \frac {w}{2}) (1-k-\frac{w}{2}) f,\\
\delta^- \delta f = (k + \frac {w}{2}) (1-k+\frac{w}{2}) f.
\end{gather*}
\begin{lem}
The following properties of $\delta$, $\delta^-$, $\eps$ are true:
\begin{enumerate}
\item The operator $\eps$ maps $F_w$ to $F_{-w}$.
\item The operator $\delta$ maps $F_w$ to $F_{w+2}$. It is invertible for all values of $w$, except, possibly, $2k - 2$ and $-2k$.
\item The operator $\delta^-$ maps $F_w$ to $F_{w-2}$. It is invertible for all values of $w$, except, possibly, $2k$ and $2-2k$.
\end{enumerate}
\end{lem}
In fact, choosing an element in $F_w$ for $2-2k \leq w \leq 2k-2$ determines a one dimensional subspace in $F_{w'}$ for each $w'$ from the same interval and operators $\delta$, $\delta^-$ act between these one dimensional subspaces by non-zero maps.

Fix some $f\in F_{2-2k}$. Define a sequence of functions
\[
f_l = \frac {\delta^{k+l-1} f}{(k+l-1)!} , \qquad 1-k \leq l \leq k-1.
\]
This sequence satisfies:
\begin{enumerate}
\item $f_l \in F_{2l}$
\item $f_{1-k} = f$
\item $\delta f_{l} = (l+k) f_{l+1}$ for $1-k \leq l \leq k-2$
\item $\delta^- f_{l} = (l-k) f_{l-1}$ for $2-k \leq l \leq k-1$
\end{enumerate}
Note, that the lemma above implies that any element of the sequence completely determines the full sequence. We make several remarks:
\begin{enumerate}
\item $\eps f_{k-1}$ has weight $1-k$, so we can construct the corresponding sequence for $\eps f_{k-1}$. It is trivial to check that the resulting sequence is $(\eps f_{-l})$.
\item $\delta^- f$ is zero if and only if $f$ is holomorphic.
\item $\delta^{2k-1} f$ is holomorphic.
\end{enumerate}

\section{Functions with values in representations}\label{funcs_in_reps}
We are going to consider functions with values in $V_m$ or $V^m$. Obviously the notion of weight, operators $D$, $\delta$, $\delta^-$, $\Delta$ and their properties mentioned above extend to such functions. Let the complex conjugation act on $V_m$, $V^m$ in the obvious way. Then the operator $\eps$ can also be defined and the same properties are valid. The group acts on such functions by the rule:
\[
(\gamma f)(\z) = \gamma (f(\gamma^{-1} \z)) (-c\z+a)^{-w},
\]
and again this commutes with operators $D$, $\delta$, $\delta^-$, $\eps$.

For each $l$ such that $1-k \leq l \leq k-1$ define a function with values in $V^{2k-2}$
\[
Q_l(\z)(X) = (2 \pi i)^{k-1-l} \frac {(X - \z)^{k-1-l} (X - \zc)^{k-1+l}}{(\z - \zc)^{k-1+l}}.
\]
Assign to $Q_l$ weight $2 l$. One can check:
\begin{enumerate}
\item $\delta Q_{k-1} = 0$
\item $\delta^- Q_{1-k} = 0$
\item $\delta Q_{l} = (l+1-k) Q_{l+1}$ for $1-k \leq l \leq k-2$
\item $\delta^- Q_{l} = (l+k-1) Q_{l-1}$ for $2-k \leq l \leq k-1$
\item $\eps Q_{l} = Q_{-l}$
\item $\gamma Q_{l} = Q_{l}$ for all $\gamma \in SL_2(\R)$
\end{enumerate}
It follows that $Q_l$ are eigenvalues for $\Delta$ with the same value $x_0 = k (1-k)$.
We have introduced an invariant bilinear form on $V^{2k-2}$. Let us compute its values on $Q_l$.
\begin{lem}
\[
(Q_i, Q_j) = \begin{cases} 
0, & \text{if $i\neq -j$} \\
(-1)^{i+k-1} \frac{1}{\binom{2k-2}{i+k-1}} (2\pi i)^{2k-2}, & \text{if $i= -j$} 
\end{cases}
\]
\end{lem}
\begin{proof}
We prove by induction on $i$ starting from $1-k$.
Since 
\[
Q_{1-k} = (2\pi i)^{2k-2} (X-\z)^{2k-2},
\]
for any polynomial $p\in V^{2k-2}$ we have
\[
(Q_{1-k}, p) = (2\pi i)^{2k-2} p(\z).
\]
Hence $(Q_{1-k}, Q_j)$ is not zero only for $j=k-1$ and in this case
\[
(Q_{1-k}, Q_{k-1}) = (2\pi i)^{2k-2}.
\]
If the statement is true for $i$ then for any $j$ since the weight of $(Q_i, Q_j)$ is $i+j$
\[
0 = \delta(Q_i, Q_j) = (\delta Q_i, Q_j) + (Q_i, \delta Q_j).
\]
Hence
\[
(Q_{i+1}, Q_j) = \frac{1}{i+1-k}(\delta Q_i, Q_j) = -\frac{1}{i+1-k}(Q_i, \delta Q_j)
\]
\[
= -\frac{j+1-k}{i+1-k}(Q_i, Q_{j+1}).
\]
We see, that if $i+j\neq -1$ this is zero. If $i+j=-1$ this equals exactly
\[
-(-1)^{i+k-1}\frac{-i-k}{(i+1-k)\binom{2k-2}{i+k-1}}(2\pi i)^{2k-2} = (-1)^{i+k} \frac{1}{\binom{2k-2}{i+k}}(2\pi i)^{2k-2}.
\]
\end{proof}

Consider the following function:
\[
\wt f = \sum_{l=1-k}^{k-1} f_l Q_{-l}, \qquad \text{i.e.}
\]
\[
\wt f (\z)(X) = \sum_{l=1-k}^{k-1} (2 \pi i)^{k-1+l} f_l(\z) \frac {(X - \z)^{k-1+l} (X - \zc)^{k-1-l}}{(\z - \zc)^{k-1-l}},
\]
which is of weight $0$. Then we can apply our differential operators to $\wt f$ and obtain:
\begin{gather} \label{holimage}
\frac{1}{2\pi i} \frac{\partial \wt f}{\partial \z} = D \wt f = \delta \wt f = Q_{1-k} \delta f_{k-1} = (2 \pi i)^{2k-2} \frac {(X - \z)^{2k-2}}{(2 k - 2)!} \delta^{2 k - 1} f 
\\ \label{antihol}
2 \pi i (\z - \zc)^2 \frac{\partial \wt f}{\partial \zc} = \delta^- \wt f = Q_{k-1} \delta^- f_{1-k} = 2 \pi i (\z - \zc)^2 \left( \frac {X - \zc}{\z - \zc} \right)^{2k-2} \frac{\partial f}{\partial \zc}
\end{gather}
To prove this we have used Leibniz rule which holds for operators $\delta$, $\delta^-$.
For the operator $\eps$ we also have a simple formula:
\[
(\eps \wt f)(\z) = \wt{\eps f_{k-1}}(\z)
\]
\begin{lem} \label{ftilde}
The polynomial $f(\z)$ admits the following expression as a polynomial of $X-\z$:
\[
\wt f(\z)(X) = \sum_{n=0}^{2k-2} \frac{D^n f}{n!} (2 \pi i)^n (X-\z)^n,
\]
moreover
\[
\delta^{2k-1} f = D^{2k-1} f.
\]
\end{lem}
\begin{proof}
There exists a decomposition
\[
\wt f(\z)(X) = \sum_{n=0}^{2k-2} \frac{a_n}{n!} (2 \pi i)^n (X-\z)^n
\]
for some unknown functions $a_n$. Substituting $X=\z$ in the definition of $\wt f$ gives $a_0 = f$. Applying $D$ gives:
\[
\begin{split}
(D \wt f)(\z)(X) = \sum_{n=0}^{2k-3} \frac{D a_n - a_{n+1}}{n!} (2 \pi i)^n (X-\z)^n \\+ \frac{D a_{2k-2}}{(2k-2)!} (2 \pi i)^{2k-2} (X - \z)^{2k-2}.
\end{split}
\]
The identity proved above implies $a_{n+1} = D a_n$ and $\delta^{2k-1} f = D a_{2k-2}$
\end{proof}

\section{Holomorphic and antiholomorphic images}
Let $f$ be as in previous section. Since $\delta \wt f$ is holomorphic and the weight of $\wt f$ is zero
\[
\Delta \wt f = \delta^- \delta \wt f = \delta \delta^- \wt f = 0.
\]
This implies
\begin{lem}
\begin{enumerate}
\item $D \wt f = \delta \wt f$ is holomorphic. If $D \wt f = 0$, then $\wt f$ is antiholomorphic.
\item $\overline{D} \wt f = \overline{\delta \eps \wt f}$ is antiholomorphic. If $\overline{D} \wt f = 0$, then $\eps \wt f$ is antiholomorphic, i.e. $\wt f$ is holomorphic.
\item If both $D \wt f$ and $\overline{D} \wt f$ are $0$, then $\wt f$ does not depend on $\z$.
\end{enumerate}
\end{lem}
In the latter case $f(\z) = \wt f(\z)(\z)$ is a polynomial in $\z$ of degree not greater than $2k-2$. It is also true in the other way:
\begin{lem}\label{poly}
If $f(\z) = p(\z)$ is a polynomial of degree not greater than $2k-2$, then $f\in F_{2-2k}$ and $\wt f = p(X)$, thus $D \wt f = 0$ and $\overline{D} \wt f = 0$.
\end{lem}
\begin{proof}
The fact that $f \in F_{2-2k}$ holds since $\delta^- f$ = 0. To prove the rest use lemma \ref{ftilde} and the fact, that $f$ coincides with its Taylor expansion terminated at term of order $2k-2$.
\end{proof}

If a function has weight $2k$ and is holomorphic then it belongs to $F_{2k}$. Denote corresponding subspace of $F_{2k}$ by $F_{2k}^{hol}$. Denote the analogous subspace of $F_{2-2k}$ by $F_{2-2k}^{hol}$.
Denote by $\phi$ the map from $F_{2-2k}$ to $F_{2k}^{hol}$ which takes $f$ to 
\[
\phi(f)=D^{2k-1}f = \delta^{2k-1}f = \frac {(2k-2)! D \wt f}{(2\pi i)^{2k-2} (X-\z)^{2k-2}},
\]
see (\ref{holimage}). We call $\phi(f)$ the \emph{holomorphic image} of $f$.
\begin{lem}\label{locsurj}
The restriction of the map $\phi$ to $F_{2-2k}^{hol}$ is locally surjective.
\end{lem}
\begin{proof}
If $g \in F_{2k}^{hol}$ then we may take
\[
(2 \pi i)^{2k-2} \frac {(X - \z)^{2k-2}}{(2 k - 2)!} g,
\]
multiply by $2 \pi i d\z$, integrate it in some neighbourhood of a given point and obtain $h$, which is a function with values in $V^{2k-2}$. Put $f(\z) = h(\z)(\z)$ and assign to $f$ weight $2-2k$. Since $f$ is holomorphic $\delta^- f$ is zero, hence $f$ belongs to $F_{2-2k}$. Then both $h$ and $\wt f$ can be represented as polynomials in $X-\z$ as in lemma \ref{ftilde}. These polynomials have the same 'constant term' $f$ and both $D \wt f$, $D h$ are divisible by $(X-\z)^{2k-2}$. It follows, that $f = h$ (see the proof of lemma \ref{ftilde}), thus $\phi f = g$ in some neighbourhood of a given point.
\end{proof}

Since the kernel of the restriction of $\phi$ to $F_{2-2k}^{hol}$ is $V^{2k-2}$ as follows from the lemma \ref{poly} we obtain a locally exact sequence, i.e. an exact sequence of sheaves:
\[
0 \longrightarrow V^{2k-2} \longrightarrow F_{2-2k}^{hol} \xrightarrow{\;\,\phi\;\,} F_{2k}^{hol} \longrightarrow 0.
\]
Denote by $\overline{F_{2k}^{hol}}$ the space of functions whose complex conjugate belongs to $F_{2k}^{hol}$. Denote by $\phi'$ the map from $F_{2-2k}$ to $\overline{F_{2k}^{hol}}$ which sends $f$ to 
\[
\phi'(f)=\overline{D^{2k-1} \eps f_{k-1}} = \frac{(2k-2)! \overline{D} \wt f}{(2\pi i)^{2k-2} (X-\zc)^{2k-2}},
\]
see (\ref{holimage}). We call $\phi'(f)$ the \emph{antiholomorphic image} of $f$. Clearly
\[
\phi'(f) = \overline{\phi(\eps f_{k-1})}
\]
Using lemma \ref{locsurj} we conclude that $\phi'$ is also locally surjective. Its kernel is $F_{2-2k}^{hol}$. Therefore the following sequence is locally exact:
\[
0 \longrightarrow F_{2-2k}^{hol} \longrightarrow F_{2-2k} \xrightarrow{\;\,\phi'\;\,} \overline{F_{2k}^{hol}} \longrightarrow 0.
\]
Taking into account exactness of the above two sequences we conclude that the following sequence is locally exact:
\[
0 \longrightarrow V^{2k-2} \longrightarrow F_{2-2k} \xrightarrow{\phi+\phi'} F_{2k}^{hol} \oplus \overline{F_{2k}^{hol}}\longrightarrow 0.
\]
If we consider modular curve $X = \HH/\Gamma$ for some congruence subgroup $\Gamma \subset SL_2(\Z)$, or $X \subset \HH/\Gamma$ is an open subset then $V^{2k-2}$ gives a local system on $X$ and we get a long exact cohomology sequence:
\begin{multline}
0 \longrightarrow H^0(X, V^{2k-2}) \longrightarrow H^0(X, F_{2-2k}) \xrightarrow{\phi+\phi'} \\ H^0(X, F_{2k}^{hol}) \oplus \overline{H^0(X, F_{2k}^{hol})} \xrightarrow{\psi} H^1(X, V^{2k-2}) \longrightarrow \dots,
\end{multline}
$\psi$ is the boundary map.
Take $f\in H^0(X, F_{2-2k})$, i.e. a function from $F_{2-2k}$ which is invariant under $\Gamma$. Let $\phi(f) = g$ and $\phi'(f) = g'$. We know that $\psi(g) = -\psi(g')$. We see that the obstruction to represent $f$ as a sum of preimages of $g$ and $g'$ \emph{separately} is $\psi(g) \in H^1(X, V^{2k-2})$. %Note H^0(\Gamma

\section{Integrating}\label{integrating}
Consider a meromorphic modular form $f$ of weight $2k$ on $X = \HH / \Gamma$. Suppose the differential $f(\z) p(\z) d \z$ has zero residue for any polynomial $p\in V^{2k-2}$. Choose a basepoint $a \in \HH$. Integrating $f$ gives a cocycle with values in $V^{2k-2}$ in the following way:
\[
\sigma_a(\gamma) = (2 \pi i)^{2k-1} \int_{a}^{\gamma a} f(\z) (X - \z)^{2k-2} d \z.
\]
It satisfies 
\[
\sigma_a(\gamma \gamma') = \sigma_a(\gamma) + \gamma \sigma_a(\gamma').
\]
The integral of $f$ is a function with values in $V^{2k-2}$:
\[
I_a(x) = (2 \pi i)^{2k-1} \int_{a}^x f(\z) (X - \z)^{2k-2} d \z.
\]
This integral depends on the basepoint. The integrand $f(\z) (X - \z)^{2k-2} d\z$ is invariant w.r.t $\Gamma$. The function $I(x)$ is not, it has 'jumps' given by $\sigma$.
If we change basepoint from $a$ to $a'$ the integral and the cocycle change in the following way:
\[
I_a'(x) = I_a(x) + (2 \pi i)^{2k-1} \int_{a'}^a f(\z) (X - \z)^{2k-2} d \z,
\]
\begin{multline*}
\sigma_{a'}(\gamma) = \sigma_a(\gamma) + (2 \pi i)^{2k-1} \int_{a'}^a f(\z) (X - \z)^{2k-2} d \z -\\ (2 \pi i)^{2k-1} \int_{\gamma a'}^{\gamma a} f(\z) (X - \z)^{2k-2} d \z,
\end{multline*}
so if we introduce an element $v_{a a'}$ of $V^{2k-2}$ by 
\[
v_{a a'} = (2 \pi i)^{2k-1} \int_{a'}^a f(\z) (X - \z)^{2k-2} d \z,
\]
then
\[
I_a'(x) = I_a(x) + v_{a a'},
\]
\[
\sigma_{a'} = \sigma_a - \delta v_{a a'},
\]
where $\delta$ denotes the differential 
\[
(\delta v)(\gamma) = \gamma v - v.
\]
Suppose $\sigma$ is homologically trivial, i.e.
\[
\sigma_a = \delta v_a,
\]
for some $v_a \in V^{2k-2}$. Then it is natural to modify our definition of the integral as follows:
\[
I(x) = I_a(x) + v_a.
\]
We see that now the integral is defined up to addition of elements $v\in V^{2k-2}$ such that $\delta v = 0$. If $(V^{2k-2})^{\Gamma} = 0$, which holds if $k > 1$, then this gives a correct definition which does not depend on the choice of the base point. Summarizing

\begin{lem}
If the class of $f$ in $H^1(\Gamma, V^{2k-2})$ is zero and $k>1$, then $I(x)$ is a correctly defined $\Gamma$-equivariant meromorphic function with values in $V^{2k-2}$ satisfying
\[
D I = (2 \pi i)^{2k-2} f(\z) (X - \z)^{2k-2}.
\]
\end{lem}

To calculate $I(x)$ it is convenient to choose base point $a=x$. Then $I_a(x)=0$ and $I(x) = v_x$, where $v_x\in V^{2k-2}$ is a solution of the system of linear equations
\[
\gamma v_x - v_x = (2 \pi i)^{2k-1} \int_x^{\gamma x} f(\z) (X - \z)^{2k-2} d \z,
\]
$\gamma$ runs over a set of generators of $\Gamma$.

Let us allow $f(\z) p(\z) d \z$ to have residues which satisfy certain integrality property, we say that
\begin{defn}\label{integ_res}
The meromorphic modular form of weight $2k$ has integral residues if the differential form
\[
 (2 \pi i)^{2k-1} \binom{2k-2}{n} f(\z) \z^n d \z
\]
has integral residues for all $n$, $0\leq n \leq 2k-2$.
\end{defn}
\begin{rem}
The normalization can be modified to suit our needs.
\end{rem}
Let $V_{\Z}^{2k-2}\subset V^{2k-2}$ denote the abelian group of polynomials with integral coefficients. This group is stable under the action of $SL_2(\Z)$. We make the quotient 
\[
V_{\C/2\pi i}^{2k-2} = V^{2k-2} / V_{\Z}^{2k-2}.
\]
We can develop similar theory as above with coefficients in $V_{\C/2\pi i}^{2k-2}$. The only difference is that here for $k>1$ the group of invariants
\[
(V_{\C/2\pi i}^{2k-2})^\Gamma
\]
may be not trivial, but rather a finite torsion group. Denote its exponent by $e$. Then the integral may be not well defined, but the image of $e I(x)$ in $V_{\C/2\pi i}^{2k-2}$ is well defined. So, in this case the lemma holds:
\begin{lem}
If $f$ has integral residues, the class of $f$ in $H^1(\Gamma, V_{\C/2\pi i}^{2k-2})$ is zero and $k>1$, exponent of $(V_{\C/2\pi i}^{2k-2})^\Gamma$ is $e$ then $e I(x)$ is a correctly defined $\Gamma$-equivariant function with values in $V_{\C/2\pi i}^{2k-2}$ satisfying
\[
D e I = e (2 \pi i)^{2k-2} f(\z) (X - \z)^{2k-2}.
\]
Moreover, $I$ is holomorphic everywhere except poles of $f$.
\end{lem}

\section{Green's functions}
Consider Green's function $G(\z_1, \z_2)=G_{2k}^{\Gamma}(\z_1, \z_2)$ for $\Gamma \subset SL_2(\Z)$~--- congruence subgroup, $k$~--- positive integer greater than $1$. It is a real-valued function, defined on the set
\[
\{(\z_1, \z_2)| \z_i\in \HH, \z_1\neq \gamma\z_2,\; \text{for all}\; \gamma\in \Gamma\}
\]
satisfying the following properties:
\begin{enumerate}
\item $G(\gamma_1 \z_1, \gamma_2 \z_2) = G(\z_1, \z_2)$ for all $\gamma_1, \gamma_2 \in \Gamma$.
\item $\Delta_i G = k(1-k) G$, where $\Delta_i$ denotes Laplace operator with respect to $\z_i$.
\item When $\z_1$ tends to $\z_2$ $G$ has asymptotics $\log|\z_1-\z_2|^2$.
\item When $\z_1$ tends to infinity $G$ tends to $0$.
\end{enumerate}
There exists the following formula for $G$:
\[
G(\z_1, \z_2) = -2\sum_{\gamma\in\Gamma} Q_{k-1}(1-2\frac{|\z_1-\gamma \z_2|^2}{(\z_1-\zc_1)(\gamma\z_2-\gamma\zc_2)}),
\]
where $Q_{k-1}$ is the $k-1$-th Legendre function. 
Denote 
\[
t(\z_1, \z_2) = 1-2\frac{|\z_1-\z_2|^2}{(\z_1-\zc_1)(\z_2-\zc_2)}.
\]
It satisfies
\[
t(\z_1, \z_2)-1 = 2\frac{(\z_1-\z_2)(\zc_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)},
\]
\[
t(\z_1, \z_2)+1 = 2\frac{(\z_1-\zc_2)(\z_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)}.
\]
It follows, that
\[
G(\z_1, \z_2) = -2 Q_{k-1}(t(\z_1, \z_2)) + g(\z_1, \z_2),
\]
where $g$ is a function, smooth in the neighbourhood of the set $\{\z_1=\z_2\}$.
Denote 
\[
f(\z_1, \z_2) = -2 Q_{k-1}(t(\z_1, \z_2)).
\]
Note, that $f$ also satisfies properties of $G$ except the first property. Instead it satisfies 
\begin{enumerate}
\item[(i')] $G(\gamma \z_1, \gamma \z_2) = G(\z_1, \z_2)$ for all $\gamma\in SL_2(\R)$.
\end{enumerate}

Since the function $G$ is in the space $F_0$ with respect to each of the arguments in the sence of the section \ref{eigenvalues} we can calculate holomorphic and antiholomorphic images with respect to some of the variables, or with respect to both. The same applies to the function $f$. Denote by $G_h$, $G_a$, $f_h$, $f_a$ corresponding holomorphic and antiholomorphic images of $G$ and $f$ w.r.t. $\z_1$. Denote by $G_{hh}$, $G_{ha}$, $G_{ah}$, $G_{aa}$ holomorphic image w.r.t. $\z_2$ of $G_h$, antiholomorphic image w.r.t. $\z_2$ of $G_h$, e.t.c. The same for $f$.
\begin{lem}
For $f_h$ and $f_a$ we have the following formulas:
\[
f_h(\z_1, \z_2)=(k-1)!^2 (-1)^{k-1} (2 \pi i)^{-k} \frac{(\z_2-\zc_2)^k}{(\z_1 - \z_2)^k (\z_1 - \zc_2)^k},
\]
\[
f_a(\z_1, \z_2)=(k-1)!^2 (-1)^{k-1} (2 \pi i)^{-k} \frac{(\z_2-\zc_2)^k}{(\zc_1 - \z_2)^k (\zc_1 - \zc_2)^k}.
\]
\end{lem}
\begin{proof}
The function $f_h$ satisfies the following properties:
\begin{itemize}
\item $f_h$ is holomorphic w.r.t $\z_1$
\item $f_h(\gamma \z_1, \gamma \z_2) = f_h(\z_1, \z_2) (c\z_1 + d)^{2k}$ for all $\gamma\in SL_2(\R)$.
\end{itemize}
Hence the function
\[
\frac{f_h(\z_1, \z_2) (\z_1 - \z_2)^k (\z_1 - \zc_2)^k} {(\z_2-\zc_2)^k}
\]
is invariant under the diagonal action of $SL_2(\R)$ and is holomorphic in $\z_1$ on its domain of definition. Hence if we fix $\z_2$ the function as a function of $\z_1$ must be constant along circles centered in $\z_2$. This is impossible for non-constant holomorphic functions. Therefore
\[
f_h(\z_1, \z_2) = c_1 \frac{(\z_2-\zc_2)^k}{(\z_1 - \z_2)^k (\z_1 - \zc_2)^k} 
\]
for some constant $c_1\in\C$.
To find the constant recall that 
\[
f_h = (k-1)!\delta_1^k f,
\]
where $\delta_1$ denotes the operator $\delta$ applied w.r.t. the argument $\z_1$. Since when $\z_1$ approaches $\z_2$
\[
f(\z_1, \z_2) \sim \log|\z_1-\z_2|^2,
\]
modulo terms of smaller growth.
and the function $f$ is a sum of products of logarithms and rational functions in $\z_1$, $\z_2$, $\zc_1$, $\zc_2$ we can apply $\delta_1$ to both sides and obtain
\[
f_h(\z_1, \z_2) \sim (k-1)!^2 (-1)^{k-1} (2 \pi i)^{-k}(\z_1-\z_2)^{-k},
\]
which implies that 
\[
c_1 = (k-1)!^2 (-1)^{k-1} (2 \pi i)^{-k}.
\]
Since the function $f$ is real its antiholomorphic image is simply the complex conjugate of the holomorphic image, which implies the second formula.
\end{proof}
Put $k'=1-k$ then
\[
f_h(\z_1, \z_2) = (k-1)!^2 (-1)^{k-1} (2\pi i)^{k'-1} \frac{(\z_1-\z_2)^{k'-1}(\z_1-\zc_2)^{k'-1}}{(\z_2-\zc_2)^{k'-1}}
\]
\[
=(k-1)!^2 (-1)^{k-1} Q_{k'\,0}(\z_2)(\z_1),
\]
where $Q_{k'\,l}$ is defined in the same way as polynomials $Q_l$ in section \ref{funcs_in_reps}, with $k'$ instead of $k$ - it is not a polynomial, but a rational function in $X$. However, most of the properties remain true, in particular
\[
\delta^k Q_{k'\,0} = (1-k')(2-k')\dots(k-k') Q_{k'\,k} = \frac{(2k-1)!}{(k-1)!} (2\pi i)^{-2k} (X-\z)^{-2k},
\]
which implies
\[
f_{hh}(\z_1, \z_2) = (k-1)!^2 (2k-1)! (-1)^{k-1} (2\pi i)^{-2k} (\z_1-\z_2)^{-2k},
\]
\[
f_{ha}(\z_1, \z_2) = (k-1)!^2 (2k-1)! (-1)^{k-1} (2\pi i)^{-2k} (\z_1-\zc_2)^{-2k},
\]
\[
f_{ah}(\z_1, \z_2) = (k-1)!^2 (2k-1)! (-1)^{k-1} (2\pi i)^{-2k} (\zc_1-\z_2)^{-2k},
\]
\[
f_{aa}(\z_1, \z_2) = (k-1)!^2 (2k-1)! (-1)^{k-1} (2\pi i)^{-2k} (\zc_1-\zc_2)^{-2k}.
\]
For $G_{**}$ we can conclude, that 
\begin{lem}
\begin{itemize}
\item The function $G_{hh}$ is a meromorphic cusp form of weight $2k$ in both variables with pole along diagonal, such that
\[
G_{hh}(\z_1-\z_2) - (k-1)!^2 (2k-1)! (-1)^{k-1} (2\pi i)^{-2k}(\z_1-\z_2)^{-2k}
\]
is holomorphic in the neighbourhood of $\{\z_1=\z_2\}$.
\item The function $G_{ha}$ is a holomorphic cusp form of weight $2k$ in first variable and $\overline{G_{ha}}$ is a holomorphic cusp form of weight $2k$ in second variable.
\item $G_{aa}$, $G_{ah}$ are complex conjugates of $G_{hh}$, $G_{ha}$ correspondingly.
\end{itemize}
\end{lem}

Since the function $G$ is in the space $F_0$ with respect to each of the arguments in the sence of the section \ref{eigenvalues} we can apply construction of section \ref{eigenvalues} to $G$ and get a sequence of functions $G_{ij}(\z_1, \z_2)$ for $1-k \leq i,j \leq k-1$, which satisfies
\begin{enumerate}
\item $G_{ij}$ is in $F_{2i}$ w.r.t. $\z_1$ and $F_{2j}$ w.r.t. $\z_2$
\item $G_{0 0} = G$
\item $\delta_1 G_{ij} = (i+k) G_{i+1\,j}$ for $1-k \leq i \leq k-2$
\item $\delta_2 G_{ij} = (j+k) G_{i\,j+1}$ for $1-k \leq j \leq k-2$
\item $\delta_1^- G_{ij} = (i-k) G_{i-1\,j}$ for $2-k \leq i \leq k-1$
\item $\delta_2^- G_{ij} = (j-k) G_{i\,j-1}$ for $2-k \leq j \leq k-1$
\end{enumerate}
Here we denote by $\delta_i$, $\delta_i^-$ corresponding operators w.r.t. $\z_i$. We can construct a function with values in polynomials in two valiables $X_1$, $X_2$ as in section \ref{funcs_in_reps}:
\[
\wt G(\z_1, \z_2) = \sum_{i,j=1-k}^{k-1} G_{ij}(\z_1, \z_2) Q_{-i}(\z_1)(X_1) Q_{-j}(\z_2)(X_2).
\]
The function $G_h$ has weight $0$ w.r.t. $\z_2$, so if we make a sequence $G_{hl}$ from $G_h$ with respect to variable $\z_2$ we obtain
\[
D_1 \wt G(\z_1, \z_2) = \frac{(2\pi i)^{2k-2}(X_1-\z_1)^{2k-2}}{(2k-2)!} \sum_{l=1-k}^{k-1} G_{hl}(\z_1, \z_2) Q_{-l}(\z_2)(X_2)
\]
\[
=\frac{(2\pi i)^{2k-2}(X_1-\z_1)^{2k-2}}{(2k-2)!} \wt G_h(\z_1, \z_2)(X_2).
\]
Differentiating w.r.t. $\z_2$ gives
\[
D_2 D_1 \wt G(\z_1, \z_2) = \frac{(2\pi i)^{4k-4}(X_1-\z_1)^{2k-2}(X_2-\z_2)^{2k-2}}{(2k-2)!^2}G_{hh}(\z_1, \z_2).
\]
Note, that functions $G_{**}$ are holomorphic (antiholomorphic) modular forms, so we can actually hope to express them in terms of, say, Eisenstein series and $j$-invariant. In order to recover $G$ we need to integrate $G_{**}$, and this can be impossible in the class of modular forms if certain cohomology groups are not trivial. Let us make an additional assumption:
\begin{assumption}
Suppose, there are no cusp forms of weight $k$ for $\Gamma$.
\end{assumption}
One simplification comes immediately:
\begin{prop}
Functions $G_{ah}$ and $G_{ha}$ are identically zero.
\end{prop}
Recall, that for the function $\wt{G_h}$ this implies:
\begin{prop}
Function $\wt{G_h}$ is holomorphic in its domain of definition.
\end{prop}
Final step for recovering of $\wt G$ would be to integrate $\wt{G_h}$. Let us study residue of the following differential at the point $\z_1=\z_2$:
\[
w_G = (X_1-\z_1)^{2k-2} \wt{G_h}(\z_1, \z_2)(X_2) 2\pi i d \z_1.
\]
Clearly it is the same as the residue of 
\[
w_f = (X_1-\z_1)^{2k-2} \wt{f_h}(\z_1, \z_2)(X_2) 2\pi i d \z_1.
\]
The latter can be calculated explicitly. Since
\[
f_{hl}(\z_1, \z_2) = (k-1)!^2 (-1)^{k-1} Q_{k'\,l}(\z_2)(\z_1),
\]
in particular,
\[
f_{h\,1-k}(\z_1, \z_2) = (k-1)!^2 (-1)^{k-1} (2\pi i)^{-1} (\z_1-\z_2)^{-1}\left(\frac{\z_2-\zc_2}{\z_1-\zc_2}\right)^{2k-1}
\]
we see that the Laurent expansion of $f_{h\,1-k}$ in the variable $\z_1-\z_2$ (treating $\z_2$ as a parameter) is
\[
f_{h\,1-k}(\z_1, \z_2) = (k-1)!^2 (-1)^{k-1} (2\pi i)^{-1} (\z_1-\z_2)^{-1} + O(1).
\]
Hence
\[
D_2^n f_{h\,1-k}(\z_1, \z_2) = (k-1)!^2 (-1)^{k-1} n! (2\pi i)^{-1-n} (\z_1-\z_2)^{-1-n} + O(1).
\]
We use the lemma \ref{ftilde} to compute the Laurent expansion of $\wt f_h$:
\[
\wt f_h(\z_1, \z_2)(X_2) = (k-1)!^2 (-1)^{k-1} (2\pi i)^{-1} \sum_{n=0}^{2k-2} (X_2 - \z_2)^n (\z_1-\z_2)^{1-n} + O(1).
\]
Since 
\[
\res (X_1-\z_1)^{2k-2}(\z_1-\z_2)^{1-n} d\z_1 = (-1)^n\binom{2k-2}{n}(X_1-\z_2)^{2k-2-n}
\]
we can compute the residue term by term:
\[
\res w_f = (k-1)!^2 (-1)^{k-1} \sum_{n=0}^{2k-2} (-1)^n\binom{2k-2}{n} (X_2 - \z_2)^n (X_1-\z_2)^{2k-2-n}
\]
\[
= (k-1)!^2 (-1)^{k-1} (X_1-X_2)^{2k-2}.
\]
Hence
\begin{prop}
The residue at $\z_1=\z_2$ of the differential 
\[
w_G = (X_1-\z_1)^{2k-2} \wt{G_h}(\z_1, \z_2)(X_2) 2\pi i d \z_1
\]
equals
\[
(k-1)!^2 (-1)^{k-1} (X_1-X_2)^{2k-2}.
\]
\end{prop}
Let $p\in V^{2k-2}$ be a polynomial with integer coefficients.
Since $w_G$ is a polynomial in $X_2$ we can consider value of the bilinear form on $w_G$ and $p$, which will be denoted $(w_G, p)_{X_2}$. It is again a differential. The residue at $\z_1=\z_2$ is
\[
\res (w_G, p)_{X_2} = (k-1)!^2 (-1)^{k-1} p(X_1).
\]
Therefore
\begin{prop}
The following form has integral residues w.r.t. variable $\z_1$ (see definition \ref{integ_res}):
\[
\frac{1}{(2\pi i)^{2k-2}(k-1)!^2} (\wt{G_h}(\z_1, \z_2), p)_{X_2}.
\]
\end{prop}
Therefore using results from section \ref{integrating} we obtain a function $I(\z_1,\z_2, p)$ with values in $V^{2k-2}_{\C/2\pi i}$ which satisfies
\[
D_1 I(\z_1,\z_2,p) = e \frac{1}{(k-1)!^2} (X_1-\z_1)^{2k-2} (\wt{G_h}(\z_1, \z_2), p)_{X_2},
\]
$e$ is a positive integer, defined in section \ref{integrating}. This $I$ is holomorphic in $\z_1$, $\z_2$ for $\z_1\neq\gamma \z_2$ for all $\gamma\in\Gamma$. Let $q$ be another polynomial with integer coefficients. Let $K$ be the least common multiple of numbers $\binom{2k-2}{i}$. Then the pairing with $Kq$ is defined for elements of $V^{2k-2}_{\C/2\pi i}$ and takes values in $\C/2\pi i$. We introduce a function
\[
\hat G(\z_1, \z_2, q, p) = \frac{(I(\z_1, \z_2, p), K q)_{X_1}}{Ke}
\]
Note, that $\hat G$ takes values in $\C/\frac{1}{K e}2\pi i$ and
\[
D_1 \hat G(\z_1, \z_2, q, p) = \frac{1}{(k-1)!^2} q(\z_1) (\wt{G_h}(\z_1, \z_2)(X_2), p)_{X_2},
\]
which implies that
\[
((\wt G(\z_1, \z_2)(X_1, X_2),q)_{X_1},p)_{X_2} = \frac{(k-1)!^2 (2\pi i)^{2k-2}}{(2k-2)!} 2 \Re \hat G(\z_1, \z_2, q, p),
\]
here $\Re$ denotes the real part of a complex number. Let $x_1$, $x_2$ be two CM points on $\HH$. Let
\[
a_i x_i^2 + b_i x_i + c_i = 0
\]
be corresponding minimal equations for $x_i$ with integral coefficients. Let 
\[
p_i(x) = (a_i x^2 + b_i x + c_i)^{k-1}
\]
be two polynomials  with integer coefficients. We can express the value of $G$ on $x_1$, $x_2$ as follows:
\[
G(x_1, x_2) = (2\pi i)^{4-4k}\binom{2k-2}{k-1}^2((\wt G(x_1, x_2), Q_0(x_1))_{X_1}, Q_0(x_2))_{X_2},
\]
since $Q_l$ are orthogonal. 
\[
Q_0(x_i) = (2\pi i)^{k-1} \frac{(X-x_i)^{k-1}(X-\bar x_i)^{k-1}}{(x_i-\bar x_i)^{k-1}} = (2\pi i)^{k-1} \frac{p_i(X)}{D_i^{\frac{k-1}2}},
\]
where $D_i = b_i^2-4 a_i c_i$~--- the discriminant.
Therefore
\[
G(x_1, x_2) = (2\pi i)^{2-2k}\binom{2k-2}{k-1}^2 (D_1 D_2)^{\frac{1-k}2} ((\wt G(x_1, x_2), p_1)_{X_1}, p_2)_{X_2}
\]
\[
=\binom{2k-2}{k-1} (D_1 D_2)^{\frac{1-k}2} 2 \Re\hat G(x_1, x_2, p_1, p_2).
\]
\end{document}
@
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\author{Anton Mellit}
\title{Higher green's functions for modular forms, local study}
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\author{Anton Mellit}
\title{Higher Green's functions for modular forms}
\begin{document}

\bibliographystyle{alpha}
\maketitle

\tableofcontents

\chapter{Modular forms}
\input{prelim}
\input{integral}
\input{greenfunclocal}
\chapter{Higher Chow groups and Abel-Jacobi maps}
\input{higherchow}
\chapter{Derivative of the Abel-Jacobi map}
\input{geometry}
\input{products}
\chapter{Cohomology of elliptic curves}
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\section{Local study of Green's functions}

Let $k$ be an integer, $k>1$.
We consider the Green's function for the upper half plane of weight $2 k$, which we denote by $G_{k}^{\HH}$. It is the unique function which satisfies the following properties:

\begin{enumerate}
\item $G_k^\HH$ is a smooth function on $\HH\times\HH-\{\z_1=\z_2\}$ with values in $\R$.
\item $G_k^\HH(\gamma \z_1, \gamma \z_2) = G_k^\HH(\z_1, \z_2)$ for all $\gamma\in SL_2(\R)$.
\item $\Delta_i G_k^\HH = k(1-k) G_k^\HH$, where $\Delta_i$ denotes the Laplace operator with respect to $\z_i$.
\item $G_k^\HH = \log|\z_1-\z_2|^2 + O(1)$ when $\z_1$ tends to $\z_2$.
\item $G_k^\HH$ tends to $0$ when $\z_1$ tends to infinity.
\end{enumerate}

In this section we obtain two formulae. The first formula is for $\delta_1^n\delta_2^m G_k^\HH$, and it involves the hypergeometric series. The second formula is a particular case of the first for $n=k$, and the resulting expression for this case is a rational function of $\z_1, \z_2, \zc_2$. Note that because of the symmetry between $\z_1$ and $\z_2$ this case is similar to the case $m=k$.

Because of the second property the function $G_k^\HH$ is the function of the hyperbolic distance. Denote by $t(\z_1, \z_2)$ the hyperbolic cosine of the hyperbolic distance, i.e.
\[
\begin{split}
t(\z_1, \z_2) = 1 + 2\frac{(\z_1-\z_2)(\zc_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)} = -1 + 2\frac{(\z_1-\zc_2)(\z_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)}\\
=\frac{-2\z_1\zc_1 + (\z_1+\zc_1)(\z_2+\zc_2) - 2\z_2\zc_2}{(\z_1-\zc_1)(\z_2-\zc_2)}.
\end{split}
\]
Then 
\[
G_k^\HH(\z_1, \z_2) = -2 \calQ_{k-1}(t),
\]
where $\calQ_{k-1}$ is the Legendre's function of the second kind.

The function $\calQ_{k-1}$ has the following two expansions at infinity (see [???]):
\[
\begin{split}
\calQ_{k-1}(t) = \frac{2^{k-1} (k-1)!^2}{(2k-1)!} t^{-k} F(\frac{k}2, \frac{k+1}2; k+\frac12; t^{-2}) \\
= \frac{2^{k-1} (k-1)!^2}{(2k-1)!} (t+1)^{-k} F(k, k; 2k; \frac2{1+t}),
\end{split}
\]
here $F$ denotes the hypergeometric series.
We are going to compute various derivatives of $G_k^\HH$ using the second expansion. For this purpose we first compute:
\[
\delta_1 t = \frac{\partial t}{\partial\z_1} = 2\frac{(\zc_1-\z_2)(\zc_1-\zc_2)}{(\z_1-\zc_1)^2(\z_2-\zc_2)},
\]
\[
\delta_2 t = \frac{\partial t}{\partial\z_2} = 2\frac{(\z_1-\zc_2)(\zc_1-\zc_2)}{(\z_1-\zc_1)(\z_2-\zc_2)^2},
\]
noting that $\delta_1 t$ has weight $2$ in $\z_1$ and weight $0$ in $\z_2$ we compute:
\[
\delta_1^2 t = \delta_2^2 t = 0,
\]
\[
\delta_1 \delta_2 t = -2\left(\frac{\zc_1-\zc_2}{(\z_1-\zc_1)(\z_2-\zc_2)}\right)^2 = \frac{\delta_1 t \delta_2 t}{t+1}.
\]
We will use the following formula for the derivative of the hypergeometric series:
\[
\frac{\partial F(a,b;c;x)}{\partial x} = a \frac{F(a+1,b;c;x)-F(a,b;c;x)}{x}.
\]
We find that
\begin{equation*}
\frac{\partial ((t+1)^{-m} F(m, n; c; \frac2{t+1}))}{\partial t} = 
-m (t+1)^{-m-1} F(m + 1, n; c; \frac2{t+1}), 
\end{equation*}
so
\begin{multline*}
\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{n+1} 2^k \frac{(k-1)! (k+n-1)!}{(2k-1)!} \times \\ 
(t+1)^{-k-n} F(k+n,k;2k;\frac2{t+1}) (\delta_1 t)^n.
\end{multline*}
To apply $\delta_2^m$ we rewrite the last expression as
\begin{multline*}
\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{n+1} 2^k \frac{(k-1)! (k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k} F(k,k+n;2k;\frac2{t+1}) (\delta_1\delta_2 t)^n (\delta_2 t)^{-n},
\end{multline*}
so we can again apply the same formula, since $\delta_2$ of $\delta_1\delta_2 t$ and $\delta_2 t$ is zero:
\begin{multline*}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} 2^k \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k-m} F(k+m,k+n;2k;\frac2{t+1}) (\delta_1\delta_2 t)^n (\delta_2 t)^{m-n}.
\end{multline*}
To make the formula symmetric in $m$ and $n$ we rewrite it as
\begin{multline*}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} 2^k \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k-m-n} F(k+m,k+n;2k;\frac2{t+1}) (\delta_1 t)^n (\delta_2 t)^m.
\end{multline*}
Let us introduce the following function of weight $-2$ in $\z_1$ and $0$ in $\z_2$:
\[
Q_{\z_2}(\z_1) = \frac{(\z_1 - \z_2)(\z_1 - \zc_2)}{\z_2-\zc_2},
\]
there is a corresponding function $Q_{\z_1}(\z_2)$. One can check:
\[
\delta_1 t = \frac{t^2 - 1}2 Q_{\z_2}(\z_1)^{-1},
\]
\[
\delta_2 t = \frac{t^2 - 1}2 Q_{\z_1}(\z_2)^{-1},
\]
so our first formula is
\begin{multline}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
\left(\frac{t+1}2\right)^{-k} \left(\frac{t-1}2\right)^{m+n} F(k+m,k+n;2k;\frac2{t+1}) Q_{\z_2}(\z_1)^{-n} Q_{\z_1}(\z_2)^{-m}.
\end{multline}

In particular, when $n=k$ we obtain
\begin{multline*}
\delta_2^m\delta_1^k G_k^\HH(\z_1, \z_2) = (-1)^{m+k+1} 2^{-m} (k+m-1)! (t+1)^{-k} (t-1)^{k+m} \times\\
F(k+m, 2k; 2k; \frac2{t+1}) Q_{\z_2}(\z_1)^{-k} Q_{\z_1}(\z_2)^{-m},
\end{multline*}
and using the identity
\[
F(a, b; b; x) = (1-x)^{-a}
\]
we get the second formula
\begin{multline}
\delta_1^k \delta_2^m G_k^\HH(\z_1, \z_2) = (-1)^{m+k+1} (k+m-1)! \left(\frac{t+1}2\right)^m Q_{\z_1}(\z_2)^{-m} Q_{\z_2}(\z_1)^{-k}\\
= (-1)^{k-1} (k+m-1)! \frac{(\z_2-\zc_2)^{k-m}}{(\z_1-\z_2)^{k+m}(\z_1-\zc_2)^{k-m}}.
\end{multline}

\section{Global study of Green's functions}
Let $\Gamma$ be a congruence subgroup of $SL_2(\Z)$ and $k>1$. The Green's function on $\HH/\Gamma$ of weight $2k$ is the unique function $G_k^{\HH/\Gamma}$ with the following properties:

\begin{enumerate}
\item $G_k^{\HH/\Gamma}$ is a smooth function on $\HH\times\HH-\{\z_1=\gamma\z_2\ \,|\, \gamma\in\Gamma\}$ with values in $\R$.
\item $G_k^{\HH/\Gamma}(\gamma_1 \z_1, \gamma_2 \z_2) = G_k^{\HH/\Gamma}(\z_1, \z_2)$ for all $\gamma_1, \gamma_2\in \Gamma$.
\item $\Delta_i G_k^{\HH/\Gamma} = k(1-k) G_k^{\HH/\Gamma}$.
\item $G_k^{\HH/\Gamma} = \log|\z_1-\z_2|^2 + O(1)$ when $\z_1$ tends to $\z_2$.
\item $G_k^{\HH/\Gamma}$ tends to $0$ when $\z_1$ tends to a cusp.
\end{enumerate}

The series
\[
\sum_{\gamma\in\Gamma} G_k^\HH(z_1, \gamma z_2)
\]
is convergent and satisfies the properties above, so
\[
G_k^{\HH/\Gamma}(z_1, z_2) = \sum_{\gamma\in\Gamma} G_k^\HH(z_1, \gamma z_2).
\]

Consider the function $G_k^{\HH/\Gamma}(z, z_0)$ for a fixed $z_0\in\HH$. We put 
\begin{multline*}
\G_{k, z_0}^{\HH/\Gamma}(z, X) = \wt{G_k^{\HH/\Gamma}(z, z_0)} \\
= (-1)^{k-1} \binom{2k-2}{k-1}\sum_{l=1-k}^{k-1} (-1)^l \delta^{-l}(Q_z(X)^{k-1}) \delta^l G_k^{\HH/\Gamma}(z, z_0),
\end{multline*}
recall that negative powers of $\delta$ can be defined for eigenfunctions of the Laplacian. Note, that since $G_k^{\HH/\Gamma}(z, z_0)$ has real values and $Q_z(X)$ has imaginary values (we let complex conjugation act on $X$ identically), we have the following
\begin{prop}
The function $i^{k-1} \G_{k, z_0}^{\HH/\Gamma}(z, X)$ has real values.
\end{prop}
Since 
\[
\delta^k(Q_z(X)^{k-1}) = 0,
\]
it is easy to compute, that
\begin{multline*}
\frac{\partial \G_{k, z_0}^{\HH/\Gamma}(z, X)}{\partial z} = \delta \G_{k, z_0}^{\HH/\Gamma}(z, X) = \binom{2k-2}{k-1} \delta^{1-k}(Q_z(X)^{k-1}) \delta^k G_k^{\HH/\Gamma}(z, z_0) \\
= (X-z)^{2k-2} \frac{(-1)^{k-1}\delta^k G_k^{\HH/\Gamma}(z, z_0)}{(k-1)!}.
\end{multline*}
On the other hand, because of the proposition above
\[
\frac{\partial \G_{k, z_0}^{\HH/\Gamma}(z, X)}{\partial \zc} = (X-\zc)^{2k-2} \frac{\overline{\delta^k G_k^{\HH/\Gamma}(z, z_0)}}{(k-1)!}.
\]

Consider the function 
\[
g_{k, z_0}^{\HH/\Gamma}(z) = \frac{(-1)^{k-1} \delta^k G_k^{\HH/\Gamma}(z, z_0)}{(k-1)!}.
\]
This is a meromorphic modular form in $z$. There is a corresponding differential $1$-form with coefficients in $V_{2k-2}$
\[
(X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz.
\]
Let $V_{2k-2}^\R$ denote the space of polynomials in $V_{2k-2}$ which have real coefficients.
\begin{prop}\label{cohProp}
The class of the differential form
\[
(X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz
\]
in the cohomology group
\[
H^1(\Gamma, i^{k-1} V_{2k-2}^\R) = H^1(\Gamma, V_{2k-2}/i^k V_{2k-2}^\R)
\]
is trivial and the function 
\[
\frac12 \G_{k, z_0}^{\HH/\Gamma}(z)
\]
is an integral of $\omega$.
\end{prop}
\begin{proof}
Let us denote
\[
\omega = (X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz.
\]
We have proved before, that
\[
d \G_{k, z_0}^{\HH/\Gamma}(z, X) = \omega + (-1)^{k-1} \bar\omega,
\]
so, that
\[
i^{k-1} d \G_{k, z_0}^{\HH/\Gamma}(z, X) = i^{k-1}\omega + \overline{i^{k-1}\omega}.
\]
It implies, that the integral of $i^{k-1}\omega$ around a pole of $\omega$ is in $i V_{2k-2}^\R$, so the integral of $\omega$ around a pole is in $i^k V_{2k-2}^\R$, that is why $\omega$ satisifes the residue condition of section \ref{integrating} for $A=i^k V_{2k-2}^\R$. Hence the class of $\omega$ in $H^1(\Gamma, V_{2k-2}/i^k V_{2k-2}^\R)$ is correctly defined. Moreover, 
\[
\frac12 d \G_{k, z_0}^{\HH/\Gamma}(z, X) = \omega - \frac{(-1)^{k} \bar\omega + \omega}2,
\]
so
\[
\frac{\G_{k, z_0}^{\HH/\Gamma}(a, X) - \G_{k, z_0}^{\HH/\Gamma}(b, X)}2 \equiv
\int_{b}^a \omega \mod{i^k V_{2k-2}^\R},
\]
which implies that $\omega$ is integrable and
\[
\sigma_\omega^{i^k V_{2k-2}^\R, \Gamma} \equiv 0
\].
\end{proof}

\begin{thm}
For any $z_0\in\HH$ which is not an elliptic point the function $g_{k, z_0}^{\HH/\Gamma}(z)$ is the unique function, which satisfies the following properties:
\begin{enumerate}
\item It is a meromorphic modular form of weight $2k$ in $z$, whose set of poles is $\Gamma z_0$, which is zero at the cusps.
\item In a neighbourhood of $z_0$
\[
g_{k, z_0}^{\HH/\Gamma}(z) = Q_{z_0}(z)^{-k} + O(1)
\]
\item The class of the corresponding differential form in the cohomology group 
\[
H^1(\Gamma, V_{2k-2}/i^{k} V_{2k-2}^\R)
\]
is trivial as in Proposition \ref{cohProp}.
\end{enumerate}
\end{thm}
\begin{proof}
First we prove that the function $g_{k, z_0}^{\HH/\Gamma}(z)$ actually satisifes these conditions. The first two conditions follow from the local study. In fact, we have
\[
\delta^k G_k^{\HH/\Gamma}(z, z_0) = \sum_{\gamma\in\Gamma} \delta^k G_k^{\HH}(z, \gamma z_0),
\]
because for all $0\le l \le k$ 
\[
\delta^l G_k^{\HH/\Gamma}(z, z_0) = O(t(z, z_0)^{l-k} |Q_{z_0}(z)|^{-l}),
\]
and using the inequality
\[
|Q_{z_0}(z)| \geq \frac{t(z, z_0)-1}2 |z-\zc|
\]
we obtain that the series
\[
\sum_{\gamma\in\Gamma} \delta^l G_k^{\HH}(z, \gamma z_0)
\]
is locally uniformly majorated by the series
\[
\sum_{\gamma\in\Gamma} t(z, \gamma z_0)^{-k},
\]
which converges locally uniformly in $z$.

This already implies that the function $g_{k, z_0}^{\HH/\Gamma}(z)$ is meromorphic and has poles of the specified type. The transformation property follows from the invariance of $G_k^{\HH/\Gamma}(z, z_0)$. Since we can move any cusp to $\infty$ by an element of $SL_2(\Z)$ it is enough to check the cuspidality at $\infty$. This is clear from the following:
\begin{multline*}
\delta^k G_k^{\HH/\Gamma}(z, z_0) =\\
(-1)^{k-1} (k-1)! (z_0-\zc_0)^k \sum_{\gamma\in\Gamma} \frac1{(\gamma z- z_0)^k (\gamma z - \zc_0)^k (cz+d)^{2k}}.
\end{multline*}

The last condition is precisely the Proposition \ref{cohProp}.

Now suppose, there are two different functions, which satisfy the conditiones. Then their difference is a cusp form of weight $2k$, which has either purely real or purely imaginary cohomology depending on whether $k$ is odd or even. This contradicts the Eichler-Shimura theorem, which says, that
\[
H_{parabolic}^1(\Gamma, V_{2k-2}) \cong S_{2k} \oplus \overline{S_{2k}},
\]
so any nontrivial parabolic cohomology class which is either purely real or purely imaginary cannot be represented by a cusp form.
\end{proof}

The value of the Green's function can be recovered in the following way:
\begin{thm}
Let $k>1$, $z_0\in\HH$ and $g_{k, z_0}^{\HH/\Gamma}(z)$ be the function that satisfies the conditions of the theorem above. Put
\[
w = (X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z)
\]
and apply one of the two approaches, formulated in the theorem \ref{int_pairing} for $A = i^k V_{2k-2}^\R$, $B = i^{k-1} V_{2k-2}^\R$, $v=Q_z(X)^{k-1}$ to get an element 
\[
I^{i^k V_{2k-2}^\R, i^{k-1} V_{2k-2}^\R, \Gamma}(\omega, z, Q_z(X)^{k-1}) \in \C/i\R,
\]
for some $z\in \HH$.
Then the real part of this element equals to the number
\[
\frac12 G_k^{\HH/\Gamma}(z, z_0).
\]
\end{thm}
\begin{proof}
We note that the theorem \ref{int_pairing} can be applied since
\begin{enumerate}
\item The class of $\omega$ in $H^1(\Gamma, V_{2k-2}/A)$ is trivial by the third property of the function $g_{k, z_0}^{\HH/\Gamma}(z)$.
\item The whole homology group $H_0(\Gamma, B)$ is trivial because $k>1$.
\end{enumerate}

We can put 
\[
I^{A, \Gamma}(z) = \frac12 \G_{k,z_0}^{\HH/\Gamma}(z)
\]
because of the proposition \ref{cohProp}. This implies that
\[
I^{i^k V_{2k-2}^\R, i^{k-1} V_{2k-2}^\R, \Gamma}(\omega, z, Q_z(X)^{k-1}) \equiv (\frac12 \G_{k,z_0}^{\HH/\Gamma}(z), Q_z(X)^{k-1}) \mod \C/i\R,
\]
so the statement follows from the identity
\[
(\G_{k,z_0}^{\HH/\Gamma}(z), Q_z(X)^{k-1}) = G_k^{\HH/\Gamma}(z, z_0),
\]
which was proved in the section \ref{eigenvalues}.
\end{proof}

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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 $[Z]\in H_{2k-2}(X,\Z)$.

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}

%\bibliography{refs}
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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}

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\begin{document}
Let $z_1$ and $z_2$ be two points on the upper half plane $\HH$. There is an element of $SL_2(\R)$ which sends $z_1$ to $i$ and $z_2$ to $iu$, where $u$ is real and $u\geq1$. Let 
\[
t(z_1, z_2) = 1 - 2\frac{(z_1-z_2)(\zc_1-\zc_2)}{(z_1-\zc_1)(z_2-\zc_2)}.
\]
Since the cross-ratio of points $z_1, z_2, \zc_1, \zc_2$, which is defined as
\[
\frac{(z_1-z_2)(\zc_1-\zc_2)}{(z_1-\zc_1)(z_2-\zc_2)}
\]
is preserved under transformations from $SL_2(\R)$, $t$ is also preserved. It follows, that
\[
t(z_1, z_2) = t(i, iu) = 1 + \frac{(u-1)^2}{2 u} = \frac{u - u^{-1}}2.
\]
On the other hand, the hyperbolic distance from $z_1$ to $z_2$ can be computed as
\[
d(z_1, z_2) = d(i, iu) = \int_1^\infty \frac{dy}y = \log u,
\]
hence
\[
t(z_1, z_2) = \cosh d(z_1, z_2).
\]
\end{document}@
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\section{Integrating modular forms}\label{integrating}
Let $\Gamma$ be a congruence subgroup of $SL_2(\R)$, $S$~--- a finite union of orbits of $\Gamma$ in $\HH$, $U=\HH-S$. Consider a smooth closed differential $1$-form $\omega$ on $U$ with coefficients in $V_{2k-2}$, where $V_{2k-2}$ is the space of polynomials in one variable $X$ of degree not greater than $2k-2$. $V_{2k-2}$ is equipped with the natural action of $SL_2(\R)$ (even $SL_2(\C)$). Suppose moreover, that $\omega$ is equivariant for the action of $\Gamma$ and let $A$ be a $\Gamma$-submodule of $V_{2k-2}$. We are looking for a function 
\[
I_\omega^{A,\Gamma}: U\longrightarrow V_{2k-2}/A,
\]
which satisifies the following properties:
\begin{enumerate}
\item $I_\omega^{A,\Gamma}$ is smooth, which means that for any point $z\in U$ there is a neighbourhood $W$ of $z$ and a smooth function $g:W\longrightarrow V_{2k-2}$, such that $g$ coincides with $I_\omega^{A,\Gamma}$ modulo $A$.
\item $d I_\omega^{A,\Gamma} = \omega$,
\item $I_\omega^{A,\Gamma}$ is equivariant for the action of $\Gamma$.
\end{enumerate}
Our basic example of $\omega$ is 
\[
\omega = f(z) (X-z)^{2k-2} dz,
\]
for some meromorphic modular form $f$ of weight $2k$. The module $A$ may be the submodule of $V_{2k-2}$ consisting either of polynomials with real coefficients, polynomials with imaginary coefficients or of polynomials with coefficients in some discrete subgroup of $\C$.

It is clear, that for the existense of $I_\omega^{A,\Gamma}$ the following condition is necessary:
\begin{condition}[Residue condition]
For every point $s\in S$ the integral of $\omega$ along a small loop around $s$ belongs to $A$.
\end{condition}

Choose a basepoint $a \in U$. The differential $\omega$ defines a cocycle with values in $V^{2k-2}/A$ in the following way:
\[
\sigma_a(\gamma) = \int_{a}^{\gamma a} \omega.
\]
It satisfies 
\[
\sigma_a(\gamma \gamma') = \sigma_a(\gamma) + \gamma \sigma_a(\gamma').
\]
The integral of $\omega$ is a function with values in $V_{2k-2}/A$:
\[
I_a(x) = \int_{a}^x \omega.
\]
If we change the basepoint from $a$ to $a'$ the integral and the cocycle change in the following way:
\[
I_{a'}(x) = I_a(x) + \int_{a'}^a \omega,
\]
\[
\sigma_{a'}(\gamma) = \sigma_a(\gamma) + \int_{a'}^a \omega - \int_{\gamma a'}^{\gamma a} \omega,
\]
so if we introduce an element $v_{a a'}$ of $V_{2k-2}/A$ by 
\[
v_{a a'} = \int_{a'}^a \omega,
\]
then
\[
I_{a'}(x) = I_a(x) + v_{a a'},
\]
\[
\sigma_{a'} = \sigma_a - \delta v_{a a'},
\]
where $\delta$ denotes the differential 
\[
(\delta v)(\gamma) = \gamma v - v.
\]
Therefore $\sigma_a$ defines a class in $\sigma_\omega^{A,\Gamma}\in H^1(\Gamma, V_{2k-2}/A)$, which does not depend on the base point.

If such an $I_\omega^{A,\Gamma}$ as explained in the beginning of the section exists, it must differ from $I_a$ by a constant, say $v_a\in V_{2k-2}/A$. Let us describe all $v_a\in V_{2k-2}$ such, that $I_a+v_a$ is equivariant for the action of $\Gamma$. This means that for any $\gamma\in\Gamma$
\[
I_a(\gamma x) + v_a = \gamma (I_a(x)+v_a).
\]
Splitting the path of integration and using the equivariance of $\omega$ we conclude:
\[
I_a(\gamma x) = I_{\gamma a}(\gamma x) + \sigma_a(\gamma) = \gamma I_a(x) + \sigma_a(\gamma),
\]
so the last equation is equivalent to the following:
\[
\sigma_a(\gamma) = \gamma v_a - v_a = \delta v_a.
\]
Therefore
\begin{prop}
Suppose $\omega$ satisifies the residue condition. Then, the function $I_\omega^{A,\Gamma}$ satisifying the properties (i)-(iii) exists if and only if the class $\sigma_\omega^{A,\Gamma}\in H^1(\Gamma, V_{2k-2}/A)$ is trivial. If such a function exists, then it is unique up to the addition of an element of $H^0(\Gamma,V_{2k-2}/A)$ and satisfies
\[
\gamma I_\omega^{A,\Gamma}(z) - I_\omega^{A,\Gamma}(z) = \int_z^{\gamma z} \omega \mod A \qquad \text{for all $\gamma\in\Gamma$.}
\]
\end{prop}

For any $\Gamma$-module $M$ we denote by $C^1(\Gamma, M)$ the abelian group of cocycles, i.e. maps $\sigma:\Gamma \longrightarrow M$, such that
\[
\sigma(\gamma_1\gamma_2) = \sigma(\gamma_1) + \gamma_1\sigma(\gamma_2), \qquad \text{for all $\gamma_1, \gamma_2 \in \Gamma$.}
\]
Then we have the following exact sequence:
\[
0\longrightarrow H^0(\Gamma, M) \To M \xrightarrow{\;\,d\;\,} C^1(\Gamma, M) \To H^1(\Gamma, M)\to 0,
\]
where $d$ is given by the usual formula
\[
(d m)(\gamma) = \gamma m - m, \qquad m\in M,\gamma\in\Gamma
\]
Denote by $C_1(\Gamma, M)$ the abelian group of cycles, by which we mean the quotient group of $\Z\Gamma\otimes M$ by the subgroup, generated by elements of the form
\[
\gamma_1\gamma_2\otimes m - \gamma_2\otimes m - \gamma_1\otimes\gamma_2 m, \qquad \text{for $m\in M$, $\gamma_1, \gamma_2 \in \Gamma$.}
\]
We have the corresponding exact sequence for homology:
\[
0\longrightarrow H_1(\Gamma, M) \To C_1(\Gamma, M) \xrightarrow{\;\,d\;\,} M  \To H_0(\Gamma, M)\to 0,
\]
where $d$ is given by 
\[
d (\gamma\otimes m) = \gamma m - m, \qquad m\in M,\gamma\in\Gamma.
\]
Suppose we have an invariant biadditive pairing
\[
(\cdot,\cdot): M\otimes M' \To N,
\]
where $M$ and $M'$ are $\Gamma$-modules and $N$ is an abelian group.
Denote by $C^1(\Gamma, M)_0$ the group of cocycles, which map to zero in $H^1(\Gamma, M)$, and by $M'_0$ the group of elements in $M'$, which map to zero in $H_0(\Gamma, M')$.

Note that there is a canonical pairing 
\[
C^1(\Gamma, M)\otimes C_1(\Gamma, M') \To N,
\]
given by
\[
(\sigma, \gamma\otimes m) = (\sigma(\gamma^{-1}), m),
\]
since
\begin{multline*}
(\sigma, \gamma_1\gamma_2\otimes m - \gamma_2\otimes m - \gamma_1\otimes \gamma_2 m) =\\
(\sigma(\gamma_2^{-1}\gamma_1^{-1}), m) - (\sigma(\gamma_2^{-1}), m) - (\sigma(\gamma_1^{-1}), \gamma_2 m)= 0.
\end{multline*}

This pairing has the following property, which can be easily checked:
\begin{prop}
For any $m\in M$, $c\in C_1(\Gamma, M')$
\[
(dm, c) = (m, dc).
\]
\end{prop}

This implies, that $(\cdot, \cdot)$ induces pairings
\[
H^0(\Gamma, M)\otimes H_0(\Gamma, M') \To N, \qquad 
H^1(\Gamma, M)\otimes H_1(\Gamma, M') \To N.
\]

If we have $\sigma\in C^1(\Gamma, M)_0$ and $m'\in M'_0$ we can either represent $\sigma$ as a coboundary, i.e. $\sigma = d m_0$ and then consider $(m_0, m')$, or represent $m'$ as a boundary, $m' = d c$ and then consider $(\sigma, c)$. 
\begin{prop}
Either of these two approaches defines a pairing 
\[
C^1(\Gamma, M)_0 \otimes M'_0 \To N,
\]
moreover, the resulting two pairings coincide.
\end{prop}

We put $M=V_{2k-2}/A$, $M' = B$, where $B$ is a $\Gamma$- invariant subgroup of $V_{2k-2}$, and $N=\C / (A, B)$.

\begin{thm}\label{int_pairing}
Let $S\subset \HH$ be a finite union of orbits of $\Gamma$, $A$ and $B$ be $\Gamma$-invariant subgroups of $V_{2k-2}$, $\omega$ be a smooth closed invariant differential $1$-form on $U=\HH-S$ with coefficients in $V_{2k-2}$, whose integrals along small loops around points of $S$ belong to $A$, $z$ be a point in $U$, $v$ be an element in $B$, such that:
\begin{enumerate}
\item The class of $\omega$ in $H^1(\Gamma, V_{2k-2}/A)$ is $0$,
\item the class of $v$ in $H_0(\Gamma, B)$ is $0$.
\end{enumerate}
Then the following two approaches lead to the same element of $\C/(A, B)$, which we denote by $I^{A, B, \Gamma}(\omega, z, v)$:
\begin{enumerate}
\item First represent $\omega$ as a differential of an invariant $V_{2k-2}/A$-valued function $I^{A, \Gamma}_\omega$, and then put
\[
I^{A, B, \Gamma}(\omega, z, v) = (I_\omega^{A, \Gamma}(z), v).
\]
\item First represent $v$ as
\[
v = \sum_{i=1}^n (\gamma_i u_i - u_i),\qquad \text{for $\gamma_i\in\Gamma$, $u_i\in B$,}
\]
and then put
\[
I^{A, B, \Gamma}(\omega, z, v) = \sum_{i=1}^n (\int_z^{\gamma_i^{-1} z} \omega, u_i).
\]
\end{enumerate}
\end{thm}

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There are following sources on the Griffiths infinitesimal invariant.

Original ideas of Griffiths are contained here (1983):

[1] MR0720288 (86e:32026a)  Carlson, James;  Green, Mark;  Griffiths, Phillip; Harris, Joe. Infinitesimal variations of Hodge structure. I. Compositio Math.  50  (1983),  no. 2-3, 109--205.

[2] MR0720289 (86e:32026b)  Griffiths, Phillip;  Harris, Joe. Infinitesimal variations of Hodge structure. II. An infinitesimal invariant of Hodge classes. Compositio Math.  50  (1983),  no. 2-3, 207--265.

[3] MR0720290 (86e:32026c)  Griffiths, Phillip A.  Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions. Compositio Math.  50  (1983),  no. 2-3, 267--324.

Then goes a paper by Green (1989):

[4] MR0992330 (90c:14006)  Green, Mark L.  Griffiths' infinitesimal invariant and the Abel-Jacobi map. J. Differential Geom.  29  (1989),  no. 3, 545--555.

Expository papers of Green and Voisin (1994):

[5] MR1335239 (96m:14012)  Green, Mark L.  Infinitesimal methods in Hodge theory. Algebraic cycles and Hodge theory (Torino, 1993),  1--92, Lecture Notes in Math., 1594, Springer, Berlin,  1994.

[6] MR1335241 (96i:14007)  Voisin, Claire. Transcendental methods in the study of algebraic cycles. Algebraic cycles and Hodge theory (Torino, 1993),  153--222, Lecture Notes in Math., 1594, Springer, Berlin,  1994.

A book by Voisin on Hodge theory (2002-2003):

[7] MR1967689 (2004d:32020)  Voisin, Claire. Hodge theory and complex algebraic geometry. I. Translated from the French original by Leila Schneps. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge,  2002. x+322 pp. ISBN: 0-521-80260-1

[8] MR1997577 (2005c:32024b)  Voisin, Claire. Hodge theory and complex algebraic geometry. II. Translated from the French by Leila Schneps. Cambridge Studies in Advanced Mathematics, 77. Cambridge University Press, Cambridge,  2003. x+351 pp. ISBN: 0-521-80283-0

A survey paper by Griffiths (2004):

[9] MR2083750 (2005g:14023)  Griffiths, Phillip. Hodge theory and geometry. Bull. London Math. Soc.  36  (2004),  no. 6, 721--757.

The papers that seemed interesting:

[10] MR1625958 (99i:14010)  Westhoff, Randall F.  Computing the infinitesimal invariants associated to deformations of subvarieties. Pacific J. Math.  183  (1998),  no. 2, 375--397.

[11] MR1487220 (98m:14010)  Collino, A.  Griffiths' infinitesimal invariant and higher $K$-theory on hyperelliptic Jacobians. J. Algebraic Geom.  6  (1997),  no. 3, 393--415.

[12] MR1162437 (93f:14004)  Muller-Stach, Stefan. On the nontriviality of the Griffiths group. J. Reine Angew. Math.  427  (1992), 209--218.

[13] MR2061847 (2005e:14014)  Muller-Stach, Stefan ;  Saito, Shuji ;  Collino, A.  On $K\sb 1$ and $K\sb 2$ of algebraic surfaces. Special issue in honor of Hyman Bass on his seventieth birthday. Part I. $K$-Theory  30  (2003),  no. 1, 37--69.

Something related to Manin's example:

[14] MR1732409 (2001b:14086)  Manin, Yu. I.  Sixth Painleve equation, universal elliptic curve, and mirror of $\bold P\sp 2$. Geometry of differential equations,  131--151, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI,  1998.

[15] MR1896474 (2003g:14007)  del Angel, Pedro Luis;  Muller-Stach, Stefan J.  The transcendental part of the regulator map for $K\sb 1$ on a mirror family of $K3$-surfaces. Duke Math. J.  112  (2002),  no. 3, 581--598.

[16] MR2019146 (2005b:14018)  del Angel, Pedro Luis;  Muller-Stach, Stefan. Picard-Fuchs equations, integrable systems and higher algebraic $K$-theory. Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), 43--55, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI,  2003.

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@\subsection{Hyperchain}
We assume some class of topological chains on $X$ is given, i.e. semialgebraic chains. For any open set $U\subset X$ we denote by $C_i(U)$ the group of chains with support on $U$ of dimension $i$. Suppose an abstract cell complex $\sigma$ and a hypercover $(U_a)$ is given.

The complex of \emph{hyperchains} is defined similarly to the complex of hypersections but with change of sign. We put
\[
C_\sigma(X)_i = \bigoplus_{j\geq 0} C_{\sigma_j}(X)_{i-j}, \qquad\text{where}
\]
\[
C_{\sigma_j}(X)_{i-j} = \bigoplus_{a\in \sigma_j} C_{i-j}(U_a).
\]
Given a hyperchain $\xi\in C_{i-j}(U_a)\subset C_\sigma(X)_i$ its boundary is 
\[
\partial_\sigma \xi:= \partial\xi - (-1)^i \sum_{b\in\sigma_{j-1}} D_{a b} \xi_b.
\]
Here $\partial\xi\in C_{i-1-j}(U_a)$ is the ordinary boundary of $\xi$ and $\xi_b$ denotes the same chain as $\xi$, but considered as an element of $C_{i-j}(U_b)$ if $U_a\subset U_b$. The definition of the boundary map is then extended to $C_\sigma(X)_i$ by linearity.

The augmentation morphism $\epsilon:C_\sigma(X)\rightarrow C(X)$ sends all chains in $C_{i-j}(U_a)$ with $j>0$ to $0$ and the ones with $j=0$ to themselves.

We have the following lifting property:
\begin{prop}
Let $\xi$ be a chain on $X$, $\eta=\partial \xi$, $\bar\eta$ a hyperchain such that $\epsilon\bar\eta=\eta$ and $\partial_\sigma \bar\eta=0$. Then there exists a hyperchain $\bar\xi$ such that $\partial_\sigma\bar\xi=\bar\eta$ and $\epsilon \bar\xi=\xi$.
\end{prop}
\begin{proof}
The complex $C_\sigma(X)_\bullet$ is the total complex of the bicomplex $C_{\sigma_\bullet}(X)_\bullet$ with the horizontal differentials induced by the boundary maps of chains in the space $X$ and the vertical ones induced by the boundary maps of the complex $\sigma$:
\[
\begin{CD}
& & C_{\sigma_1}(X)_{i-1} @@>>> C_{\sigma_1}(X)_{i-2} \\
& & @@VVV @@VVV\\
C_{\sigma_0}(X)_i @@>>> C_{\sigma_0}(X)_{i-1} @@>>> C_{\sigma_0}(X)_{i-2}\\
@@V{\epsilon}VV @@V{\epsilon}VV\\
C(X)_i @@>>> C(X)_{i-1}
\end{CD}
\]
Clearly it is enough to prove that the vertical complexes are exact.

Each group $C_{\sigma_j}(X)_i$ is a free abelian group generated by certain types of simplices in open subsets of $X$. For each simplex $s$ in $X$ there is a corresponding direct summand of $C_{\sigma_j}(X)$ which is generated by simplices in $U_a$ which coincide with $s$ in $X$. This makes a decomposition of $C_{\sigma_j}(X)_i$ into a direct sum. The decomposition clearly commutes with are so is decomposable into a direct sum of abeliancopies of $Z$ indexed by simplices in $X$. and the vertical differentials 
\end{proof}

Let the dimension of $\xi$ be $i$. Let $\bar\eta=\sum_{a\in\sigma} \eta_a$ with $\eta_a\in C_{i-1-\dim a}(U_a)$. We construct $\bar\xi=\sum_{a\in\sigma} \xi_a$ with $\xi_a\in C_{i-\dim a}(U_a)$. The conditions they need to satisfy are:
\[
\eta_b = \partial \xi_b - (-1)^i\sum_{a\in\sigma_j} D_{ab}\xi_a, \qquad b\in\sigma_{j-1};
\]
\[
\xi = \sum_{a\in \sigma_0} \xi_a.
\]
On the step $0$ we simply split $\xi$ into pieces $\xi_a$, $a\in\sigma_0$ so that $\xi_a\in\C_i(U_a)$. This is possible since $U_a$ for $a\in\sigma_0$ cover $X$. Thus the second condition is satisfied. Then we choose $\xi_a$ for $a\in\sigma_1$ such that the first condition is satisfied for $j=1$. On step $k$ we construct $\xi_a$ for $a\in\sigma_k$ such that the first condition is satisfied for $j=k$. We prove that it is always possible.

Let us define for every $Z\in M$ and $L\subset\{1,\ldots,n\}$ a subvariety $Z_L\subset Z$. The definition is inductive. We put
\begin{align*}
Z_{\varnothing} &= Z,\\
Z_{L\cup \{k\}} &= Z_L \cdot \pi_k^{-1} S_k\;\text{for $\max L<k\leq n$},
\end{align*}
where $\cdot$ means the following operation: for every irreducible component $Y$ of $Z_L$ consider $Y'=Y\cap \pi_k^{-1} S_k$; take the union of those $Y'$ which are strictly smaller than $Y$.

It is clear that the subvarieties $Z_L$ satisfy the following properties:
\begin{prop}
For any $Z\in M$ and $L\subset\{1,\ldots,n\}$ the subvariety $Z_L$ is either empty or is the union of a finite number of varieties from $M$ of dimension $\dim Z - |L|$.
\end{prop}

The main property of good choices of numbers is the following:
\begin{prop}\label{prop:nhp}
If $\vec\veps=(\veps_1,\dots,\veps_n, \veps_1',\dots,\veps_n')$ is a good choice of numbers with the corresponding sequence $\delta_1<\cdots<\delta_n$, then for every $Z\in M$ and $L\subset\{1,\ldots,n\}$ the intersection $Z\cap \pi_L^{-1}R_L(\vec\veps)$ is contained in the $\delta_{\veps_{\max L}}$-neighbourhood of $Z_L$.
\end{prop}
\begin{proof}
The proof goes by induction on $|L|$ with the base of induction being obvious. Let $L\subset\{1,\ldots,n\}$ and $\max L<k\leq n$. Let $x$ be a point belonging to the intersection $Z\cap \pi_{L\cup\{k\}}^{-1}R_{L\cup\{k\}} (\vec\veps)$. By the assumption of induction 
$\dist(x, Z_L)<\delta_{\max L} \leq \delta_{k-1}$. Therefore there exists an irreducible component $Y$ of $Z_L$ such that $\dist(x, Y)<\delta_{k-1}$. On the other hand $\pi_k(x)\in R_k(\veps_k, \veps_k')$. This means $\veps_k'<\dist(\pi_k(x),S_k)<\veps_k$. Let $Y'=Y\cap \pi_k^{-1}S_k$. Since $Y\in M$ one has $\dist(x, Y')<\delta_k$ by the second property of good choices. If $Y'\neq Y$ then $Y$ is a subset of $Z_{L\cup\{k\}}$ and we are done. Otherwise we have $\dist(x, Y')=\dist(x,Y)<\delta_{k-1}$ and by the third property of good choices $\dist(\pi_k(x),S_k)\leq \veps_k'$, which is a contradiction.
\end{proof}

This immediately implies
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I have tried to find the formula for $\delta_1^n\delta_2^m G_k^\HH$ using the second expansion at infinity. However I only succeded in calculating $\delta_1^n G_k^\HH$.

Recall, that (see [WW]) $\calQ_{k-1}$ has the following expansion at infinity:
\[
\begin{split}
\calQ_{k-1}(t) = \frac{\pi^{1/2} \Gamma(k)} {(2 t)^k \Gamma(k+\frac12)} F(\frac{k}2, \frac{k+1}2; k+\frac12; t^{-2}) \\
= \frac{2^{k-1} (k-1)!^2}{(2k-1)!} t^{-k} F(\frac{k}2, \frac{k+1}2; k+\frac12; t^{-2}),
\end{split}
\]
here $F$ denotes the hypergeometric series.
We are going to compute various derivatives of $G_k^\HH$. For this purpose we introduce the following function of weight $-2$ in $\tau_1$ and $0$ in $\tau_2$:
\[
Q_{\tau_2}(\tau_1) = \frac{(\tau_1 - \tau_2)(\tau_1 - \bar\tau_2)}{\tau_2-\bar\tau_2},
\]
we also assign to $t(\tau_1, \tau_2)$ weight $0$ in both variables.
This function is invariant (taking the weight into account) with respect to the simultaneous action of $SL_2(\R)$ on both arguments. We compute:
\[
\delta_1 Q_{\tau_2}(\tau_1) = \frac{\partial Q_{\tau_2}(\tau_1)}{\partial\tau_1} - \frac{2}{\tau_1-\bar\tau_1} Q_{\tau_2}(\tau_1) = t,
\]
\[
\delta_1 t = \frac{\partial t}{\partial\tau_1} = 2\frac{(\bar\tau_1-\tau_2)(\bar\tau_1-\bar\tau_2)}{(\tau_1-\bar\tau_1)^2(\tau_2-\bar\tau_2)} = \frac{t^2 - 1}2 Q_{\tau_2}(\tau_1)^{-1},
\]
\[
\delta_1^2 t = t \frac{t^2 - 1}2 Q_{\tau_2}(\tau_1)^{-2} - t \frac{t^2 - 1}2 Q_{\tau_2}(\tau_1)^{-2} = 0.
\]
Using the following formula for the derivative of the hypergeometric series:
\[
\frac{\partial F(a,b;c;x)}{\partial x} = a \frac{F(a+1,b;c;x)-F(a,b;c;x)}{x},
\]
we find, that
\begin{multline*}
\frac{\partial t^{-m} F(\frac{m}2, \frac{m+1}2; c; t^{-2})}{\partial t} = 
-m t^{-m-1} F(\frac{m}2 + 1, \frac{m+1}2; c; t^{-2}) \\= -m t^{-m-1} F(\frac{m+1}2, \frac{m+2}2;c;t^{-2}), 
\end{multline*}
so
\begin{multline*}
\delta_1^n G_k^\HH(\tau_1, \tau_2) = (-1)^{n+1} 2^k \frac{(k-1)! (k+n-1)!}{(2k-1)!} \times \\ 
t^{-k-n} F(\frac{k+n}2,\frac{k+n+1}2;k+\frac12;t^{-2}) \left(\frac{t^2-1}2\right)^n Q_{\tau_2}(\tau_1)^{-n}.
\end{multline*}
In particular, when $n=k$ we have
\begin{multline*}
\delta_1^k G_k^\HH(\tau_1, \tau_2) = (-1)^{k+1} 2^k (k-1)! \times \\ 
t^{-2k} F(k, k+\frac12;k+\frac12;t^{-2}) \left(\frac{t^2-1}2\right)^k Q_{\tau_2}(\tau_1)^{-k},
\end{multline*}
and using the identity
\[
F(a, b; b; x) = (1-x)^{-a}
\]
we obtain
\begin{equation*}
\delta_1^k G_k^\HH(\tau_1, \tau_2) = (-1)^{k-1} (k-1)! Q_{\tau_2}(\tau_1)^{-k}.
\end{equation*}

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\section{Notations}

The group $SL_2(\R)$ naturally acts on the linear space of homogenious polynomials of degree $m$ in two variables. This is the same as the space of polynomials in one variable $X$ of degree not greater than $m$. We denote this space $V^m$. For
\[
p\in V^m, \; p(X) = p_0 + p_1 X + \dots + p_m X^m,
\]
we write the right action of the group on $p$ as follows:
\[
(p \gamma)(X) = (p |_{-m} \gamma)(X) = p(\gamma X) (cX + d)^m, \qquad \gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\R).
\]
The corresponding action on the left is
\[
(\gamma p)(X) = (p \gamma^{-1})(X) = p(\gamma^{-1} X) (-cX + a)^m.
\]

The space $V^m$ possess an invariant scalar product, which is given by 
\[
(\sum_{i=0}^m p_i X^i, \sum_{i=0}^m p_i' X^i) = \sum_{i=0}^m \frac {(-1)^i p_i p_{m-i}'}{\binom{m}{i}}.
\]

Let $S$ be a discrete subset of the upper half plane $\HH$ and $f(\z)$ be a function on $\HH-S$ with values in $\C$ and $w\in \Z$. Define differential operators
\begin{gather*}
\delta_w f = \frac{\partial f}{\partial z} + \frac{w}{\z - \zc} f,\\
\delta_w^- f = (\z - \zc)^2 \frac{\partial f}{\partial \zc},\\
\Delta f = (\z - \zc)^2 \frac {\partial}{\partial \z} \frac {\partial}{\partial \zc} f + w (\z - \zc) \frac {\partial}{\partial \zc}.
\end{gather*}
We think about $w$ as the weight, attached to the function $f$. The weight will be always clear from the context, so we will omit the subscript $w$. 
We will follow the following agreement: the operator $\delta$ increases weight by $2$, the operator $\delta^-$ decreases weight by $2$ and the operator $\Delta$ leaves weight untouched. Taking into account this agreement the following identities can be proved:
\begin{gather*}
\delta^- \delta - \delta \delta^- = w,\\
\delta \delta^- = \Delta,\\
\delta^- \delta = \Delta + w.\\
\end{gather*}

Let the group $SL_2(\R)$ act on functions of weight $w$ by the usual formula:
\[
(f|_w \gamma)(\z) = f(\gamma \z) (c\z + d)^{-w}.
\]
This is a right action. We also define the corresponding left action
\[
(\gamma f)(\z) = (f|_w \gamma^{-1})(\z) = f(\gamma^{-1} \z) (-c\z + a)^{-w}.
\]
Note that this action commutes with the operators $\delta$, $\delta^-$, $\Delta$, it maps functions defined on $\HH-S$ to functions defined on $\HH-\gamma S$.

It is also convenient to modify the complex conjugation for functions with weight to make it commuting with the action of the group. For $f$ of weight $w$ we put
\[
f^*(\z) = (\z - \zc)^w \overline{f(\z)}.
\]
Assign to $f^*$ weight $-w$. We can check that 
\begin{gather*}
f^{**} = (-1)^w f, \\
\delta (f^*) = (\delta^- f)^*, \\
\delta^- (f^*) = (\delta f)^*, \\
\Delta (f^*) = ((\Delta + w) f)^*, \\
\gamma (f^*) = (\gamma f)^*.
\end{gather*}
We remark, that for the weight $0$ the operator $*$ is the usual complex conjugation, the operator $\delta$ is the usual $\frac{\partial}{\partial z}$, and the operator $\Delta$ is the usual Laplace operator for the hyperbolic metric $-y^2(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})$.

We list several formulae, which are convenient to use in computations. We assume that the constant function $1$ has weight $0$. Consider functions
\[
X-z,\qquad \frac{X-\zc}{z-\zc},
\]
which are thought as functions in $z$ of weights $-1$ and $1$ respectively. Then
\begin{gather*}
\delta 1 = \delta^- 1 = 0, \\
(X-z)^* = \frac{X-\zc}{z-\zc},\qquad \left(\frac{X-\zc}{z-\zc}\right)^* = -(X-z), \\
\delta (X-z) = -\frac{X-\zc}{z-\zc},\qquad \delta \frac{X-\zc}{z-\zc} = 0, \\
\delta^- (X-z) = 0,\qquad \delta^- \frac{X-\zc}{z-\zc} = X-z, \\
\delta (X-z)^a \left(\frac{X-\zc}{z-\zc}\right)^b = -a (X-z)^{a-1} \left(\frac{X-\zc}{z-\zc}\right)^{b+1}, \\
\delta^- (X-z)^a \left(\frac{X-\zc}{z-\zc}\right)^b = b (X-z)^{a+1} \left(\frac{X-\zc}{z-\zc}\right)^{b-1}, \\
\Delta (X-z)^a \left(\frac{X-\zc}{z-\zc}\right)^b = -b (a+1) (X-z)^a \left(\frac{X-\zc}{z-\zc}\right)^b.
\end{gather*}

\section{Eigenfunctions of the laplacian}\label{eigenvalues}
Let $S$ be a discrete subset of $\HH$.
For integers $k, w$ we denote by $F_{k, w}$ the space of functions on $\HH-S$ with weight $w$ satisfying
\[
\Delta f = (k (1-k) + \frac {w (w-2) }{4}) f.
\]
It is easy to check the following properties of the spaces $F_{k, w}$:
\begin{prop}
\begin{enumerate}
\item The space $F_{k, w}$ is invariant under the action of the group $SL_2(\R)$ (meaning, of course, that $\gamma\in SL_2(\R)$ changes $S$ to $\gamma S$).
\item The operator $*$ maps $F_{k, w}$ to $F_{k, -w}$.
\item The operator $\delta$ maps $F_{k, w}$ to $F_{k, w+2}$. It is invertible for all values of $w$, except, possibly, $2k - 2$ and $-2k$ with the inverse given by 
\[
\delta_w^{-1} = \frac4{(w+2k)(w-2k+2)} \delta_{w+2}^-.
\]
\item The operator $\delta^-$ maps $F_{k, w}$ to $F_{k, w-2}$. It is invertible for all values of $w$, except, possibly, $2k$ and $2-2k$ with the inverse given by
\[
(\delta_w^-)^{-1} = \frac4{(w-2k)(w+2k-2)} \delta_{w-2}.
\]
\end{enumerate}
\end{prop}
We will occasionally use negative powers of $\delta$ and $\delta^-$ when the argument belongs to the space $F_{k, w}$.

Next we state some basic facts
\begin{prop}\label{prop2_2}
For integer numbers $k$, $l$ the function 
\[
(X-z)^{k-l-1} \left(\frac{X-\zc}{z-\zc}\right)^{k+l-1}
\]
belongs to $F_{k, 2l}$ as a function of $z$.
\end{prop}
\begin{proof}
Use formulae listed in the end of the first chapter.
\end{proof}

\begin{prop}
For $f$, $g$ in $F_{k,0}$ and $1-k\leq l \leq k-1$ we have
\[
\delta^{-l} f \delta^l g = (\delta^-)^l f (\delta^-)^{-l}g.
\]
\end{prop}
\begin{proof}
Note, that
\[
(\delta^-)^l f = (-1)^l \frac{(k+l-1)!}{(k-l-1)!} \delta^{-l} f,
\]
and analogously
\[
\delta^l g = (-1)^l \frac{(k+l-1)!}{(k-l-1)!} (\delta^-)^{-l} g,
\]
which implies the required identity.
\end{proof}

Let us introduce the following operation. For two functions $f$, $g$ from $F_{k,0}$ we put
\[
f * g = \sum_{l=1-k}^{k-1} \delta^{-l} f \delta^l g,
\]
which has weight $0$.
Note, that the previous proposition implies
\[
f*g = \sum_{l=1-k}^{k-1} (-1)^l (\delta^-)^{-l} f (\delta^-)^l g,
\]
so
\[
\overline{f*g} = \overline{f}*\overline{g}.
\]
It is also easy to see, that 
\[
\frac{\partial}{\partial z} (f*g) = (-1)^{k-1}(\delta^{1-k}f \delta^k g + \delta^k f \delta^{1-k} g).
\]
Consider a function $Q_z(X)^{k-1}$. By Proposition \ref{prop2_2} 
\[
    Q_z(X)^{k-1} \in F_{k, 0}, \qquad \text{as a function of $z$.}
\]
One can compute, that for $1-k \leq l \leq k-1$
\[
\delta^l Q_z(X)^{k-1} = (-1)^l \frac{(k-1)!}{(k-l-1)!} (X-z)^{k-1-l} \left(\frac{X-\zc}{z-\zc}\right)^{k-1+l}.
\]
For $f\in F_{k, 0}$ we denote
\begin{multline*}
\wt f = (-1)^{k-1} \binom{2k-2}{k-1} f * Q_z(X)^{k-1}  
\\= (-1)^{k-1} \sum_{l=1-k}^{k-1} \frac{(2k-2)!}{(k+l-1)!(k-1)!} (X-z)^{k-1+l} \left(\frac{X-\zc}{z-\zc}\right)^{k-1-l} \delta^l f.
\end{multline*}
It is easy to check, that
\[
\wt{\gamma f} = \gamma \wt f,\qquad \text{for $\gamma\in SL_2(\R)$,}
\]
where $\gamma$ acts on $\wt f$ by the simultaneous action on $z$ in weight $0$ and $X$ in weight $2-2k$. Also
\[
\overline{\wt f} = (-1)^{k-1} \wt{\overline f}.
\]
One can compute the scalar product of $\delta^i Q_z(X)^{k-1}$ and $\delta^j Q_z(X)^{k-1}$ as follows:
\begin{lem}
\[
(\delta^i Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1}) = \begin{cases} 
0, & \text{if $i\neq -j$} \\
(-1)^{k-1-i} \binom{2k-2}{k-1}^{-1} & \text{if $i=-j$.}
\end{cases}
\]
\end{lem}
\begin{proof}
We prove by the induction on $i$ starting from $1-k$.
Since 
\[
\delta^{1-k} Q_z(X)^{k-1} = (-1)^{k-1} \frac{(k-1)!}{(2k-2)!} (X-\z)^{2k-2},
\]
for any polynomial $p\in V_{2k-2}$ we have
\[
(\delta^{1-k} Q_z(X)^{k-1}, p) = (-1)^{k-1} \frac{(k-1)!}{(2k-2)!} p(\z).
\]
Hence $(\delta^{1-k} Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1})$ is not zero only for $j=k-1$ and in this case
\[
(\delta^{1-k} Q_z(X)^{k-1}, \delta^{k-1} Q_z(X)^{k-1}) = \binom{2k-2}{k-1}^{-1}.
\]
If the statement is true for $i$ then for any $j$, taking into account, that the weight of $(\delta^i Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1})$ is $2i+2j$
\begin{multline*}
0 = \delta(\delta^i Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1}) 
\\= (\delta^{i+1} Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1}) + (\delta^i Q_z(X)^{k-1}, \delta^{j+1} Q_z(X)^{k-1}).
\end{multline*}
Hence
\[
(\delta^{i+1} Q_z(X)^{k-1}, \delta^j Q_z(X)^{k-1}) = -(\delta^i Q_z(X)^{k-1}, \delta^{j+1} Q_z(X)^{k-1}).
\]
We see, that if $i+j\neq -1$ this is zero. If $i+j=-1$ this equals exactly
\[
-(-1)^{k-1-i} \binom{2k-2}{k-1}^{-1}.
\]
\end{proof}

Therefore the original function $f$ can be recovered as
\[
f = (\wt f, Q_z(X)^{k-1}).
\]

Note also that $\Delta \wt f=0$. This is true because 
\[
\delta^- \delta^k f = (\Delta+2 k - 2) \delta^{k-1} f = 0.
\]

Let us summarize.
\begin{thm}
Let $f\in F_{k, 0}$ for $k\geq 1$. Then the function $\wt f$ satisfies the following properties:
\begin{eqnarray*}
\wt f \in F_{0,0}\otimes V_{2k-2},\\
(\wt f, Q_z(X)^{k-1}) = f,\\
\frac{\partial \wt f}{\partial z} = (X-z)^{2k-2}\frac{(-1)^{k-1}\delta^k f}{(k-1)!},\\
\frac{\partial \wt f}{\partial \zc} = (X-\zc)^{2k-2}\frac{\bar{\delta}^k f}{(k-1)!}.
\end{eqnarray*}
\end{thm}

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\section{Hypercovers}
\subsection{Abstract cell complex}
\begin{defn}
An \emph{abstract cell complex} is a graded set
\[
\sigma = \bigcup_{i\geq 0} \sigma_i
\]
together with homomorphisms $d_i:\Z[\sigma_i]\To\Z[\sigma_{i-1}]$ such that $d_i\circ d_{i+1}=0$ and $\epsilon\circ d_0=0$, where $\epsilon$ is the \emph{augmentation map} $\Z[\sigma_0]\To\Z$. The elements of the set $\sigma_i$ are called \emph{cells} of dimension $i$ and the homomorphisms $d_i$ are called \emph{boundary maps}.
\end{defn}

If an abstract cell complex is given we denote it usually by $\sigma$ and encode the boundary maps by the coefficients $D_{ab}$:
\[
d a = \sum_{b\in\sigma_{i-1}} D_{ab} b, \;a\in\sigma_i.
\]

The basic example is the standard simplex of dimension $n$, $\Delta_n$. In this case $\sigma_i$ is the set of all subsets of $\{0,1,\dots,n\}$ of size $i+1$ and, writing subsets as increasing sequences, 
\[
d_i(j_0,\dots,j_i) = \sum_{k=0}^i (-1)^k (j_0,\dots,\hat{j}_k,\dots,j_i).
\]

Given two abstract cell complexes $\sigma$, $\sigma'$ the \emph{product} 
\[
\sigma'' = \sigma\times \sigma'
\]
is again an abstract cell complex in the following way:
\[
\sigma''_i = \bigcup_{j=0}^i \sigma_j\times\sigma'_{i-j},
\]
\[
d_i(a\times b) = (d_j a) \times b + (-1)^j a\times d_{i-j} b \qquad \text{for $a\in\sigma_j$, $b\in\sigma'_{i-j}$.}
\]
For example, the cube is defined as
\[
\square_n = \Delta_1\times\dots\times\Delta_1,\qquad\text{the product has $n$ terms.}
\]

\subsection{Hypercover}
Let $X$ be a topological space and $\sigma$ an abstract cell complex.

\begin{defn}
A \emph{hypercover} of $X$ (indexed by $\sigma$) is a system of open subsets $U_a$ indexed by elements $a\in\sigma$ such that:

\begin{enumerate}
\item Whenever a cell $b$ belongs to the boundary of a cell $a$, $U_a\subset U_b$.
\item For any point $x\in X$ consider all cells $a$ such that $x\in U_a$. These cells form a subcomplex of $\sigma$ by the first property, call it $\sigma_x$. We require the complex of abelian groups coming from $\sigma_x$ to be a resolution of $\Z$, with the augmentation map sending every cell to $1$.
\end{enumerate}
\end{defn}

If $\sigma$ and $\sigma'$ are two abstract chain complexes, $(U_a)$ and $(U'_{a'})$ hypercovers of spaces $X$, $X'$ indexed by $\sigma$ and $\sigma'$ correspondingly, by making all products $(U_a\times U'_{a'})$ one gets a hypercover of $X\times X'$ indexed by $\sigma\times\sigma'$, the \emph{product} hypercover.

\subsection{Hypersection}
Let $X$ be a topological space, $\sigma$ an abstract chain complex and $(U_a)$ a hypercover.

Let $\Z_a$ denote the constant sheaf with fiber $\Z$ on $U_a$ extended by zero to $X$. Let 
\[
\Z_{\sigma i} = \bigoplus_{a\in \sigma_i} \Z_a.
\]
If $U_a\subset U_b$, there is a canonical morphism $r_{ab}:\Z_a \rightarrow \Z_b$. We define $d_i:\Z_{\sigma i} \rightarrow \Z_{\sigma i-1}$. If $a\in\sigma_i$ then the morphism from $\Z_a$ to $\Z_{\sigma i-1}$ is 
\[
\sum_{b\in\sigma_{i-1}}  D_{ab} r_{a b},\qquad \text{where}
\]
\[
d a = \sum_{b\in\sigma_{i-1}} D_{ab} b.
\]
Denote the corresponding sequence of sheaves by $\Z_\sigma$. An augmentation map $\epsilon:\Z_{\sigma 0} \rightarrow \Z_X$ is the sum of canonical morphisms $\epsilon_a:\Z_a\rightarrow\Z_X$ for $a\in\sigma_0$.
\begin{prop}
The sequence $\Z_\sigma$ is a resolution of $\Z_X$.
\end{prop}
\begin{proof}
For any point $x\in X$ the stalk of $\Z_\sigma$ at $x$ is simply the complex of abelian groups corresponding to $\sigma_x$. So the statement follows from the second condition of hypercover.
\end{proof}

In other words, we have obtained a \emph{quasi-isomorphism}
\[
\epsilon:\Z_{\sigma} \rightarrow \Z_X.
\]
For any sheaf or complex of sheaves $\F$ on $X$ this gives a morphism of complexes
\[
\epsilon^*: \HHom(\Z_X, \F) \rightarrow \HHom(\Z_\sigma, \F).
\]
Note that $\HHom(\Z_X, \F) = \Gamma(X, \F)$ and for any $a\in\sigma$ $\HHom(\Z_a, \F) = \Gamma(U_a, \F)$. We denote the complex $\HHom(\Z_\sigma, \F)$ by $\Gamma_\sigma(X, \F)$ and call elements of this complex \emph{hypersections}.

The complex of hypersections can be defined more precisely as follows. Let $\F$ be a complex of sheaves on $X$ written as
\[
\F^0\rightarrow\F^1\rightarrow\dots.
\]
Then we put
\[
\Gamma_\sigma(X,\F)^i = \prod_{j\geq0,a\in \sigma_j} \Gamma(U_a, \F^{i-j}).
\]
The coboundary of a hypersection $s=(s_a)\in \Gamma_\sigma(X,\F)^i$ is a hypersection $ds=(s'_a)\in \Gamma_\sigma(X,\F)^{i+1}$, where
\begin{equation}\label{dsa}
s'_a = d s_a + (-1)^{i-j+1} \sum_{b\in \sigma_{j-1}} D_{ab} s_b|_{U_a},\qquad a\in\sigma_j.
\end{equation}
One can check directly that $d^2=0$. Let $d^2s=(s''_a)$, $a\in\sigma_j$.
\begin{multline*}
s''_a = d s'_a + (-1)^{i-j} \sum_{b\in \sigma_{j-1}} D_{ab} s'_b|_{U_a}
=
(-1)^{i-j+1} \sum_{b\in\sigma_{j-1}} D_{ab} d s_b|_{U_a} +\\ (-1)^{i-j} \sum_{b\in \sigma_{j-1}} D_{ab} d s_b|_{U_a} + \sum_{b\in \sigma_{j-1}} \sum_{c\in \sigma_{j-2}} D_{ab} D_{bc} s_c|_{U_a}=0.
\end{multline*}

The augmentation morphism $\epsilon^*:\Gamma(X,\F)\rightarrow \Gamma_\sigma(X,\F)$ is defined by sending a section $s\in\Gamma(X,\F^i)$ to the hyperchain $(s'_a)$ where $s'_a$ are not zero only for $a\in \sigma_0$ and are equal to the restriction of $s$ to $U_a$.

\begin{rem}
In the case of the \v{C}ech hypercover (corresponding to an open cover) we obtain the \v{C}ech complex.
\end{rem}
\begin{rem}
If I took the definition of $\HHom$ from \cite{harts:rd}, p. 64 the sign in (\ref{dsa}) would be $(-1)^{i+1}$. I assume that switching from a chain complex $\Z_\sigma$ to the corresponding cochain complex adds the multiplier $(-1)^j$.  My choice of sign is made in order to get the sign  correct in the case of the \v{C}ech hypercover.
\end{rem}

The complex of \emph{hyperchains} is defined similarly to the complex of hypersections, but with a change of sign. We put
\[
C_\sigma(X)_i = \bigoplus_{j\geq 0} C_{\sigma j}(X)_{i-j}, \qquad\text{where}
\]
\[
C_{\sigma j}(X)_{i} = \bigoplus_{a\in \sigma_j} C_{i}(U_a).
\]
Given a hyperchain $\xi\in C_{i-j}(U_a)\subset C_\sigma(X)_i$ for $a\in\sigma_j$ its boundary is 
\[
\partial_\sigma \xi:= \partial\xi + (-1)^{i-j} \sum_{b\in\sigma_{j-1}} D_{a b} j_b(\xi).
\]
Here $\partial\xi\in C_{i-1-j}(U_a)$ is the ordinary boundary of $\xi$ and $j_b(\xi)$ denotes the same chain as $\xi$, but considered as an element of $C_{i-j}(U_b)$ if $U_a\subset U_b$. The definition of the boundary map is then extended to $C_\sigma(X)_i$ by linearity.

The augmentation morphism $\epsilon:C_\sigma(X)\rightarrow C(X)$ sends all chains in $C_{i-j}(U_a)$ with $j>0$ to $0$ and the ones with $j=0$ to themselves.

We have the following a lifting property for lifting chains to the hyperchains. Here the word \emph{subdivision} of a chain or a hyperchain means some iterated barycentric subdivision of all of its simplices.
\begin{prop}
Let $\xi$ be a chain on $X$, $\eta=\partial \xi$, $\bar\eta$ a hyperchain such that $\epsilon\bar\eta=\eta$ and $\partial_\sigma \bar\eta=0$. Then, after possibly replacing $\eta$, $\bar\eta$, $\xi$ with a subdivision, there exists a hyperchain $\bar\xi$ such that $\partial_\sigma\bar\xi=\bar\eta$ and $\epsilon \bar\xi=\xi$.
\end{prop}
\begin{proof}
The complex $C_\sigma(X)_\bullet$ is the total complex of the bicomplex $C_{\sigma \bullet}(X)_\bullet$ with horizontal differential induced by the boundary map of chains in space $X$ and vertical one induced by the boundary map of the complex $\sigma$:
\[
\begin{CD}
& & C_{\sigma 1}(X)_{i-1} @@>>> C_{\sigma 1}(X)_{i-2} \\
& & @@VVV @@VVV\\
C_{\sigma 0}(X)_i @@>>> C_{\sigma 0}(X)_{i-1} @@>>> C_{\sigma 0}(X)_{i-2}\\
@@V{\epsilon}VV @@V{\epsilon}VV\\
C(X)_i @@>>> C(X)_{i-1}
\end{CD}
\]
Clearly, it is enough to prove that the vertical complexes are exact (up to a subdivision). That is, we need to prove that if 
\[
\gamma=\sum_{a\in\sigma_{j}} \gamma_a \in C_{\sigma j}(X)_i \;\text{is such that}
\]
\[
\sum_{a\in\sigma_{j}} D_{ab} \gamma_a=0 \; \text{for any $b\in\sigma_{j-1}$,}
\]
then there exists 
\[
\bar\gamma=\sum_{c\in\sigma_{j+1}} \bar\gamma_c \in C_{\sigma {j+1}}(X)_i \;\text{such that}
\]
\[
\sum_{c\in\sigma_{j+1}} D_{ca} \bar\gamma_c = \gamma'_a \; \text{for any $a\in\sigma_{j}$,}
\]
where $\gamma'_a$ is a subdivision of $\gamma_a$.

It is enough to prove the statement for multiples of a single simplex. Let $s$ be a simplex in $X$ which enters $\gamma_a$ with coefficient $s_a\in\Z$. Then 
\[
\sum_{a\in\sigma_{j}} D_{ab} s_a=0 \; \text{for any $b\in\sigma_{j-1}$.}
\]
Therefore the cycle $\sum_a s_a a$ of $\sigma$ is closed. Since the simplex $s$ must belong to $U_a$ for all $a$ for which $s_a\neq 0$, the mentioned cycle is a closed cycle of $\sigma_x$ for all $x$ in the closure of $s$. For each $x$ one can therefore represent it as a boundary of some cycle of $\sigma_x$, say $t_x$,
\[
t_x = \sum_{c\in\sigma_{j+1}} t_{x c} c.
\]
Here $t_{x c}$ is zero unless $x\in U_c$. Let $V_x$ be an open set which is the intersection of all $U_c$ for which $t_{xc}$ is not zero. Then $V_x$ form a cover of the support of $s$, so there exists a finite subcover. Let it be $V_1=V_{x_1}$, $V_2=V_{x_2}$, \dots, $V_k=V_{x_k}$. One can subdivide the simplex $s$ so that each simplex of the subdivision belongs to one of the chosen open sets. Let us do this and denote the subdivision by $s'$ so that
\[
s' = \sum_{l=1}^k s'_l, \; |s'_l| \subset V_k.
\]
Now it is clear that one may put
\[
t_l = \sum_{c\in\sigma_{j+1}} t_{x_l c} j_c(s'_l)
\]
because $s'_l$ belongs to $U_c$ for every $c$ for which $t_{x_l c}$ is not zero. Then 
\[
\sum_{c\in\sigma_{j+1}} D_{ca} t_{x_l c} s'_l = s_a s'_l \; \text{for any $a\in\sigma_{j}$,}
\]
therefore $t:=\sum_{l=1}^k t_l$ satisfies the required condition.
\end{proof}

The main property is the Stokes theorem:
\begin{prop}
For a hyperform $\omega$ of degree $d$ and a hyperchain $\xi$ of degree $d+1$ one has
\[
\int_{\partial \xi} \omega = \int_{\xi} d\omega.
\]
\end{prop}
\begin{proof}
By the definition
\[
\int_{\xi} d\omega = \sum_{a\in\sigma} \int_{\xi_a} (d \omega_a + (-1)^{d-\dim a+1}\sum_{b\in\sigma_{\dim a - 1}} D_{ab} \omega_b).
\]
Applying the Stokes formula this is further equal to 
\[
\sum_{a\in\sigma}(\int_{\partial\xi_a}\omega_a + (-1)^{d-\dim a+1}\sum_{b\in\sigma_{\dim a - 1}} D_{ab} \int_{\xi_a} \omega_b) = \sum_{a\in\sigma} \int_{\partial \xi_a} \omega = \int_{\partial\xi} \omega.
\]
\end{proof}

\subsection{Gauss-Manin}
\subsection{Residues}

\section{Abel-Jacobi map for products of curves}
By a curve we mean a smooth projective curve over $\C$.
Let $X_1$, $X_2$,\dots,$X_n$ be curves. Put 
\[
X = X_1\times X_2\times \dots \times X_n.
\]
Let $x\in Z^k(X,1)$ be a higher cycle,
\[
x = \sum_i (W_i, f_i).
\]
Recall that $\dim_\C W_i = n-k+1$, $f_i\in\C(W_i)$, $\gamma_i = f_i^*[0,\infty]$, $\gamma = \sum_i \gamma_i$, $\partial\xi = \gamma$. The Abel-Jacobi map was defined as
\begin{equation}\label{defaj}
\langle AJ^{k,1} [x], [w] \rangle = \frac{1}{2\pi\I}\sum_i\int_{W_i\setminus\gamma_i}\omega\log f_i+\int_\xi\omega
\end{equation}
for $w\in F^{n-k+1}\A^{2n-2k+2}(X)$, $dw=0$. 
\begin{rem}
Since $X$ is a product of curves its cohomology has no torsion. This implies that the class of $\gamma$ is trivial.
\end{rem}

\subsection{Triangulations}
In fact we should justify the construction of $\gamma_i$ and the integration in (\ref{defaj}). The problems are that $W_i$ are not necessarily smooth, the rational functions $f_i$ do not necessarily define maps to $\PP^1$ and $\log f_i$ is not bounded. To define all the objects we may embed $X$ as a semi-algebraic set into some $\R^N$ and  consider the semi-algebraic subsets $W_i$, $|\Div f_i|$, $f_i^{-1} [0,\infty]$. We may apply \cite{hironaka:tri} to get a triangulation of $\R^N$ which is compatible with all the sets mentioned above. This triangulation is semi-algebraic and smooth on the interiors of simplices. Therefore we can integrate smooth forms over simplices. Moreover we have the necessary bounds on the growth of $\log f_i$ restricted to any simplex which belongs to $W_i$ and does not belong to $|\Div f_i|$. In fact any such simplex intesects $|\Div f_i|$ only along the boundary and $\log f_i$ grows not faster than some multiple of the logarithm of the distance to the boundary. We will only consider simplices obtained by a linear subdivision of simplices of the constructed triangulation. Formal linear combinations of simplices of equal dimension are called chains.

For each $i$ let us consider the space $\overline{W_i}$ constructed from $W_i$ by cutting out $\gamma_i$ and attaching two copies of $\gamma_i$ glued together along the boundary. Let us denote the two copies of $\gamma_i$ by $\gamma_i^+$ and $\gamma_i^-$ and suppose that they are attached in such a way that the function $\log f_i$ extends to $\gamma_i^+$ and $\gamma_i^-$, $\partial \overline{W_i} = \gamma_i^+ - \gamma_i^-$, and the value of $\log f_i$ on $\gamma_i^-$ is $2\pi\I$ plus the value on $\gamma_i^+$. Denote by $\iota_i^+$, $\iota_i^-$ the natural isomorphisms $\gamma_i\To\gamma_i^+$, $\gamma_i\To\gamma_i^-$. Denote by $p_i$ the natural projection $\overline{W_i}\To W_i$. The space $\overline{W_i} $is naturally endowed with the triangulation coming from the triangulation of $W_i$.

Let $l\geq 0$ and $\Delta$ be the standard simplex of dimension $l$,
\[
\Delta = \{(x_0, x_1, \dots, x_l)| \sum_{j=0}^l x_j = 0, x_j\geq 0\}.
\]
For $\epsilon>0$ denote
\[
\Delta_\epsilon = \{(x_0, x_1, \dots, x_l)\in\Delta| x_j\geq \epsilon\}.
\]
Let $\sigma:\Delta \To \overline{W_i}$ be a simplex in $\overline{W_i}$. Let $\omega$ be a smooth $l$-form on a neighbourhood of $\sigma(\Delta)$. 
\begin{defn}
Suppose $\sigma(\Delta)$ is not contained in $|\Div f_i|$. Put
\[
\int_\sigma \omega \log f_i = \lim_{\epsilon\To 0} \int_{\Delta_\epsilon} \sigma^* \omega \log f_i.
\]
\end{defn}
\begin{defn}
A simplex $\sigma$ is called good if it is not contained in $|\Div f_i|$ for all $i$ and any simplex of its boundary is not contained in $|\Div f_i|$ for all $i$. A chain is called good if it is a linear combination of good simplices.
\end{defn}
One can check that the Stokes formula holds:
\begin{prop}
If $\sigma$ is a good simplex and $\omega$ is a smooth $l-1$-form on a neighbourhood of $\sigma(\Delta)$, then
\[
\int_\sigma d\omega \log f_i = \int_{\partial \sigma} \omega \log f_i - \int_\sigma \frac{d f_i}{f_i}\wedge \omega.
\]
\end{prop}

\subsection{Chain systems}
For any $d$ we will consider $d$-dimensional chains on $\overline{W_i}$ and on $X$. In such considerations for any chain $c$ on $X$ with support on $\gamma_i$ we identify $c$ with $\iota_i^{-*} c - \iota_i^{+*} c$. We denote the group of such chains modulo the identification by $C_d$. For example in this group we have 
\[
\partial \overline{W_i} = -\gamma_i,
\]
so the sum 
\[
S:=\xi+\sum_i\overline{W_i}
\]
is closed.
If $\omega$ is a differential form of degree $d$ and $c$ is a $d$-dimensional chain on $X$ the integral $\int_c \omega$ is defined in the usual way. If $c$ is a chain on $\overline{W_i}$ we define the integral
\[
\int_c\omega := \frac{1}{2\pi\I}\int_c p_i^* \omega \log f_i.
\]
We see that if $c$ is a chain on $X$ and it has support on $\gamma_i$ then
\[
\int_{\iota_i^{-*} c - \iota_i^{+*} c} \omega = \int_c \omega.
\]
Now the Abel-Jacobi map can be written as
\[
\langle AJ^{k,1} [x], [w] \rangle = \int_S\omega.
\]
On can easily see that the Stokes formula holds when we put certain restriction on forms:
\[
\int_{\partial c} \omega = \int_c \omega, \qquad \omega\in F^{n-k+1} \A^d(U), \; c\in C_d, |c|\subset U
\]
for an open set $U$.

\begin{defn}
Let $\CC=((c_m)_{m\in\Lambda}, (D_{ml})_{m,l\in\Lambda})$ be a sequence of good chains with $c_m\in C_{d_m}$ and a sequence of natural numbers with the property that the boundary of any chain in $\CC$ is the integral combination of chains in $\CC$ with coefficients given by numbers $D_{ml}$, i.e.
\[
\partial c_m = \sum_l D_{m l} c_l.
\]
Suppose the square of the matrix $D_{m l}$ is zero and $D_{m l}\neq 0$ only when the chain $c_m$ has dimension $1$ bigger than that of $c_l$.
We call such an object a chain system.
\end{defn}

Suppose $X$ is covered by two open sets $U_1$, $U_2$ and $\CC$ is a chain system. The refined chain system $\CC'$ is constructed in the following way: 
\begin{enumerate} 
\item For each chain $c_m\in C_{d_m}$ of $\CC$, $\CC'$ has chains $c_m^1\in C_{d_m}$, $c_m^2\in C_{d_m}$ and $c_m^\partial\in C_{d_m-1}$ with the boundary operator acting as
\[
\partial c_m^1 = \sum_l D_{m l} c_l^1 - c_m^\partial,\qquad
\partial c_m^2 = \sum_l D_{m l} c_l^2 + c_m^\partial,
\]
\[
\partial c_m^\partial = -\sum_l D_{m l} c_l^\partial.
\]
\item For each $m\in\Lambda$ $c_m = c_m^1 + c_m^2$.
\item For each $m\in\Lambda$ $|c_m^1|\subset U_1$, $|c_m^2|\subset U_2$, $|c_m^\partial|\subset U_1\cap U_2$.
\item Suppose $c_m^1$, $c_m^2$, $c_m^\partial$ are constructed for all $m$ such that the dimension of $c_m$ is less then $d$. Take $c_m$ with dimension $d$. Decompose it as $c_m = c_m^1 + c_m^2$ with $|c_m^1|\subset U_1$, $|c_m^2|\subset U_2$. Put 
\[
c_m^\partial = \sum_l D_{m l} c_l^1 - \partial c_m^1.
\]
It is clear that $|c_m^\partial|\subset U_1$. On the other hand
\[
c_m^\partial = - \sum_l D_{m l} c_l^2 + \partial c_m^2.
\]
Therefore $|c_m^\partial|\subset U_2$. It is clear that
\[
\partial c_m^\partial = \partial (\sum_l D_{m l} c_l^1) = \sum_{l_1, l_2} D_{m l_1} D_{l_1 l_2} c_{l_2}^1 - \sum_l D_{ml} c_l^\partial
\]
with the first term being zero because the square of the matrix $(D_{ml})$ is zero.
\end{enumerate}

\begin{defn}
Let $\CC$ be a chain system, $\CC=((c_m)_{m\in\Lambda}, (D_{ml})_{m,l\in\Lambda})$. Let $d$ be an integer. By a system of forms of level $d$ attached to $\CC$ we mean a sequence of differential forms $(\omega_m)_{m\in\Lambda}$ such that $\omega_m$ is a smooth differential form on an open neighbourhood of $|c_m|$ of rank $\dim c_m - d$ and the following condition is satisfied on an open neighbourhood of $|c_m|$:
\begin{equation}\label{formsrel}
d \omega_m + \sum_{l} D_{l m} \omega_l = 0.
\end{equation}
\end{defn}

Suppose $(\omega_m)$ is a system of forms as above. If $\Omega$ is a closed form which belongs to $F^{n-k+1} \A^d(X)$, then we consider the following sum:
\[
I(\CC,(\omega_m), \Omega) = \sum_m \int_{c_m}\omega_m\wedge\Omega.
\]

\begin{defn}
Let $(\omega_m)$ be a system of forms of level $d$ attached to $\CC$. Choose an index $m\in\Lambda$. Let $\dim c_m = d_m$. Let $\phi$ be a smooth form of rank $d_m-d-1$ which is defined on a neighbourhood of $|c_m|$ and on neighbourhoods of those $|c_l|$ which satisfy $D_{ml}\neq 0$. Let us add $d \phi$ to the form $\omega_m$ and subtract $D_{m l} \phi$ from the form $\omega_l$ for each $l$. Any such move is called a natural transformation.
\end{defn}

It is clear that a natural transformation does not change the value of $I(\CC, (\omega_m), \Omega)$ introduced above.

The value of $I$ also does not change if we replace the chain system by the refined chain system with respect to some covering and define forms by restriction for $c_m^1$, $c_m^2$ and by zero for $c_m^\partial$.

Let $(\omega_m)$ be a system of forms of level $d$ attached to $\CC$ consisting only of forms which are meromorphic on $X$. Let $\eta$ be a closed $1$-form on $X$. Suppose there is an open covering $X=U_1 \cup U_2$ and meromorphic on $X$ forms $\eta_1$, $\eta_2$, a meromorphic function $\phi$ such that $\eta_1$ is holomorphic on $U_1$, $\eta_2$ on $U_2$ , $\phi$ on $U_1\cap U_2$, 
\[
\eta = \eta_1 + d \phi_1 = \eta_2 + d \phi_2, \qquad \phi_2-\phi_1 = \phi,
\]
where $\phi_1$ is a smooth function on $U_1$, $\phi_2$ is a smooth function on $U_2$.

Let us take the system $(\omega_m\wedge \eta)$, take the refinement $\CC'$ of $\CC$ with respect to $U_1$ and $U_2$, and take the corresponding system of forms on $\CC'$. Then 
\begin{prop}
The resulting system of forms can be transformed to a system of meromorphic forms by natural transformations.
\end{prop}
\begin{proof}
Recall that for the refinement we have chains $c_m^1$, $c_m^2$, $c_m^\partial$. Let us apply the natural transformation for $-(-1)^{d_m-d}\phi_1\omega_m$ on $c_m^1$ and $-(-1)^{d_m-d}\phi_2\omega_m$ on $c_m^2$ for each $m$. The resulting form on $c_m^1$ is
\begin{multline}
\omega_m\wedge\eta - (-1)^{d_m-d}d(\phi_1 \omega_m) - (-1)^{d_m-d} \sum_l D_{lm}\phi_1 \omega_l \\
= \omega_m\wedge\eta-(-1)^{d_m-d}d\phi_1\wedge\omega_m = \omega_m\wedge\eta_1.
\end{multline}
Analogously, the resulting form on $c_m^2$ is $\omega_m\wedge\eta_2$.

The resulting form on $c_m^\partial$ is 
\[
-(-1)^{d_m-d} \phi_1\omega_m + (-1)^{d_m-d} \phi_2\omega_m = (-1)^{d_m-d} \phi\omega_m
\]
\end{proof}

Let $n=2k-2$. Let $\eta^1$, $\eta^2$, \dots, $\eta^n$ be forms of second kind on $X_1$, $X_2$, \dots, $X_n$ correspondingly such that at least $k-1$ among them are holomorphic. We obtain a cohomology class 
\[
\alpha = \bigcup_{j=1}^n \pi_j^* [\eta^j] \in F^{k-1} H^{2k-2}(X,\C).
\]
For $j=1,\dots,n$ let $\eta_s^j=\eta^j$ if $\eta^j$ is holomorphic and let $\eta_s^j$ be a smooth form representing the same class of cohomology as $\eta^j$ otherwise. We start with the expression
\[
\langle AJ^{k,1} [x], \alpha \rangle = \langle AJ^{k,1} [x], [\bigwedge_{j=1}^n \pi_j^* \eta_s^j] \rangle = I(S, 1, \bigwedge_{j=1}^n \pi_j^* \eta_s^j).
\]
Let $j$ be such that $\eta^j$ is not holomorphic.

Denote by $U_1^j$ the set $X_j$ without small closed neighbourhoods of the poles of $\eta^j$ and let $U_2^j$ be a union of small open neighbourhoods of the poles of $\eta^j$ over which $\eta^j_2$ is holomorphic such that $U_1^j\cup U_2^j = X_j$.

Let $U^j_1$ be the set $X_j$ without small closed neighbourhoods of the poles of $\eta^j$. There is a meromorphic function $\phi$ on $X_j$ such that the set of poles of $\eta^j-d\phi$ is contained in $U^j_1$. Let $U_2^j$ be a union of small open neighbourhoods of the poles of $\eta^j$ over which $\eta^j-d\phi$ is holomorphic such that $U_1^j\cap U_2^j = X_j$. We put $\eta^j_1 = \eta^j$, $\eta^j_2 = \eta^j-d\phi$. One can apply the construction above for open sets $\pi_j^* U^j_1$, $\pi_j^* U^j_2$ and forms $\pi_j^*\eta_s^j$, $\pi_j^*\eta^j_1$, $\pi_j^*\eta^j_2$ for each $j$ and obtain
\[
\langle AJ^{k,1} [x], \alpha \rangle = I(\CC, (\omega_m), \Omega) = I(\CC, (\omega_m\wedge\Omega), 1).
\]
Here $\CC$ is a system of chains, $(\omega_m)$ is a system of forms attached to $\CC$ which contains only meromorphic forms and $\Omega$ is the wedge product of all the holomorphic forms among $\pi_j^*\eta_j$.
This proves the following theorem:
\begin{thm}
For any product of curves $X=\times_{j=1}^n X_j$, $n=2k-2$, an element $x\in Z^k(X,1)$ and a form of second kind $\eta_j$ on $X_j$ for each $j$ such that at least $k-1$ among $\eta_j$ are holomorphic the following holds:
\[
\langle AJ^{k,1} [x], \bigcup_{j=1}^n \pi_j^* [\eta^j]\rangle = I(\CC, (\omega_m), \Omega),
\]
where $\CC$ is a system of chains, $(\omega_m)$ is a system of forms attached to $\CC$ of level $d$ which contains only meromorphic forms and $\Omega$ is a holomorphic form of rank $d$. The number $d$ is the number of non-holomorphic forms among $\eta_j$.
\end{thm}

\subsection{Families}
Let $S$ be a smooth algebraic variety of dimension $1$ over $\C$.
Suppose we now have families of curves over $S$ $\X_1\To S$, \dots $\X_n\To S$. Denote 
\[
\X = \X_1\times_S \X_2 \times_S \dots \times_S \X_n.
\]
Choose a point $s\in S$. Suppose in the fiber over $s$ we have curves $X_1$,\dots, $X_n$ and their product $X$. Let $s'\in S$ be a point close to $s$ with curves $X_1'$,\dots, $X_n'$ and the product $X'$. Let $p:[0,1]\To S$ be a path joining $s$ and $s'$. Let $\wt X$ denote the union of the fibres over the points of $p$. Suppose we have $\wt{x}\in Z^k(\X,1)$ with fibres $x$ and $x'$ at $s$ and $s'$. Suppose the whole system is triangulated in such a way that the triangulation respects the projection to $S$. Let $t$ be a local coordinate on $S$ at $s$.

Let $n=2k-2$. Let $\eta^1$, $\eta^2$, \dots, $\eta^n$ be meromorphic forms on $\X_1$, $\X_2$, \dots, $\X_n$ which are fiberwise of second kind and at least $k-1$ of them are fiberwise holomorphic. For each $j$ we put $\eta^j_1=\eta^j$ and find rational functions $\phi^j$, $g^j$ such that the poles of the form
\[
\eta^j_2 = \eta^j - d\phi^j + g^j d t
\]
do not intersect the poles of $\wt{\eta^j}$ over some neighbourhood of $s$. If $\eta^j$ is fiberwise holomorphic we require $\phi^j=0$. We suppose that $s'$ and $p$ are contained in this neighbourhood.

Denote by $U_1^j$ the set $\wt{X_j}$ without small closed neighbourhoods of the poles of $\eta^j$ and let $U_2^j$ be a union of small open neighbourhoods of the poles of $\eta^j$ over which $\eta^j_2$ is holomorphic such that $U_1^j\cap U_2^j = \wt{X_j}$.

For any chain $\wt{C}$ on $\wt X$ of dimension $d+1$ which intersects $X$, $X'$ properly denote 
\[
\partial_0 \wt{C}:=\partial \wt{C} - (-1)^d(\wt C\cap X' - \wt C\cap X).
\]
Then $\partial_0 \wt{C}$ also intersects $X$, $X'$ properly and
\begin{equation}\label{comm1}
\partial_0^2=0,\qquad \partial(\wt{C} \cap X) = (\partial_0 \wt{C})\cap X,\qquad \partial(\wt{C} \cap X') = (\partial_0 \wt{C})\cap X'.
\end{equation}
For any chain $C$ of dimension $d+2$ on $\X$ which together with $\partial C$ intersects $X$, $X'$, $\wt X$ properly we have the formula
\[
\partial (C\cap\wt{X}) = (\partial C)\cap\wt{X} + (-1)^d (C\cap X' - C\cap X),
\]
hence
\begin{equation}\label{comm2}
\partial_0(C\cap\wt{X}) = (\partial C)\cap\wt{X}.
\end{equation}

Suppose
\[
\wt{x} = \sum_i (\W_i, f_i),\qquad \wt{W_i}=\W_i\cap\wt{X}.
\]
This gives us $\wt{\gamma_i}$, $\wt{\gamma}$, $\wt\xi$ laying over $p$ with properties
\[
\partial_0 \wt{\gamma} = 0, \qquad \partial_0\wt\xi = \wt\gamma.
\]
Denote
\[
\gamma = \wt\gamma\cap X,\qquad \gamma' = \wt\gamma\cap X',\qquad
\xi = \wt\xi\cap X, \qquad \xi' = \wt\xi\cap X.
\]
 The space obtained by cutting $\wt{W_i}$ along $\wt{\gamma_i}$ (which corresponds to $\overline{W_i}$ in the fiber) will be denoted by $\widehat{W_i}$.

We define families of chain systems on $\wt{X}$ in the same way as chain systems, but using the operator $\partial_0$ instead of the boundary operator. We also require all chains to intersect $X$, $X'$ properly. If we have a family of chain systems $\wt\CC$, then intersecting with $X$ and $X'$ we obtain families of chain systems $\CC$, $\CC'$ by the equation (\ref{comm1}).

Let us start as in previous section from the system $\wt\xi + \sum_i \widehat{W_i}$ and apply successive refinements with respect to pairs of sets $U^j_1$, $U^j_2$. Eventually we obtain systems $\wt\CC$, $\CC$, $\CC'$. Denote their chains by $\wt{c_m}$, $c_m$, $c_m'$. Coefficients $D_{m l}$ for the three systems are the same. Naturally we have also obtain a system of meromorphic forms $(\omega_m)$. The only difference for the family case is that instead of relations (\ref{formsrel}) we get
\begin{equation}\label{formsrel2}
d \omega_m + \sum_{l} D_{l m} \omega_l = (-1)^{d_m} h_m \wedge dt,
\end{equation}
where $h_m$ is a form of the same rank as $\omega_m$ and $d_m$ is the dimension of $c_m$.

Thus it is natural to define systems of forms attached to a family of chain systems as those which satisfy condition (\ref{formsrel2}).

\begin{prop}
For a system of forms $(\omega_m)$ which satisfy relations (\ref{formsrel2}) the forms $(h_m)$ satisfy relations 
\[
d h_m + \sum_{l} D_{l m} h_l = (-1)^{d_m} q_m \wedge dt
\]
for some other forms $q$.
\end{prop}
\begin{proof}
Write symbolically (\ref{formsrel2})
\[
d \omega + D \omega = (-1)^{d_m} h\wedge dt,
\]
and applying $d$ and $D$ to both sides we get
\[
d D \omega = (-1)^{d_m} dh\wedge dt,\qquad d D\omega = -(-1)^{d_m} Dh\wedge dt.
\]
Hence
\[
(dh+Dh)\wedge dt = 0.
\]
This implies that there are forms $q_m$ with
\[
dh+Dh = q \wedge dt.
\]
\end{proof}

Let us look at the difference of the integrals
\[
\sum_m \int_{c_m'} \omega_m - \sum_m \int_{c_m} \omega_m.
\]
We have the following proposition:
\begin{prop}
Suppose $\wt{c_m}$ is contained in $\wt{X}$ (not on $\widehat{W_i}$). Then
\[
\int_{c_m'} \omega_m - \int_{c_m} \omega_m = \int_{\wt{c_m}} h_m\wedge dt + \sum_l (A_{ml} - A_{lm}) + B_m,
\]
where
\[
A_{ml} = -(-1)^{d_m} \int_{\wt{c_l}}D_{ml}\omega_m,
\]
\[
B_m = \sum_i\int_{p_{i*}(\wt{c_m}\cap\widehat{W_i})} \omega_m\wedge\frac{d f_i}{f_i}.
\]
\end{prop}
\begin{cor}
\[
I(\CC', (\omega_m)_m) - I(\CC, (\omega_m)_m) = I(\wt{\CC}, (h_m\wedge dt)_m) + \sum_i I(\wt{\CC_i}, (\omega_m\wedge \frac{df_i}{f_i})_m),
\]
where $\wt{\CC_i} = p_{i*}(\wt\CC\cap\widehat{W_i})$.
\end{cor}
\begin{proof}
By the Stokes theorem
\[
\int_{\wt{c_m}} d\omega_m + \sum_i\int_{p_{i*}(\wt{c_m}\cap\widehat{W_i})} \frac{d f_i}{f_i} \wedge \omega_m = \int_{\partial \wt{c_m}} \omega_m.
\]
The first term in the lefthand side equals
\[
\int_{\wt{c_m}} d\omega_m = -\sum_l \int_{\wt{c_m}} D_{l m} \omega_l + (-1)^{d_m} \int_{\wt{c_m}} h_m\wedge dt.
\]
The righthand side equals
\[
\int_{\partial \wt{c_m}} \omega_m = \int_{\partial_0 \wt{c_m}} \omega_m + (-1)^{d_m} (\int_{c_m'} \omega_m - \int_{c_m} \omega_m)
\]
with
\[
\partial_0 \wt{c_m} = \sum_l D_{m l} c_l.
\]
Putting everything together we obtain the statement.
\end{proof}

\begin{prop}
Let $\wt{c_m}^t = \wt{c_m} \cap X_{[0,t]}$, where $X_{[0,t]}$ is the union of all fibers of $\X$ over points $p(t')$, $t'\in[0,t]$. Then 
\[
\frac{\partial}{\partial t}\vert_{t=0} \int_{\wt{c_m}^t} h_m\wedge dt = \int_{c_m} h_m.
\]
\end{prop}
\begin{proof}
Let $c_m^t = \wt{c_m} \cap X_t$, where $X_t$ is the fiber of $\X$ over the point $p(t)\in S$, $t\in[0,1]$. We have
\[
\int_{\wt{c_m}^t} d(h_m t) = \int_{\partial \wt{c_m}^t} h_m t.
\]
This means
\[
\int_{\wt{c_m}^t} h_m \wedge dt = \int_{c_m^t} h_m t - \int_{c_m^0} h_m t + (-1)^{d_m}(\int_{\partial_{(0,t)}\wt{c_m}^t} h_m t - \int_{\wt{c_m}^t} (dh_m)t),
\]
where 
\[
\partial_{(0,t)}\wt{c_m}^t = \partial \wt{c_m}^t - (-1)^{d_m}(c_m^t-c_m^0) = \partial_0\wt{c_m} \cap X_{[0,t]}.
\]
If we divide both sides by $t$ and tend $t$ to zero, the summands
\[
\int_{\partial_{(0,t)}\wt{c_m}^t} h_m t, \qquad \int_{\wt{c_m}^t} (dh_m)t
\]
will give zero and we will obtain desired identity.
\end{proof}
\subsection{Pure subvarieties}
We choose sets $U_1^j$, $U_2^j$ in some nice way so that some of $B_m$ will vanish and the remaining terms will be equal to certain residues. Let us consider a metric on each $X_j$, say, the one coming from some projective embedding. For each $j$ we choose positive numbers $\epsilon_1^j$, $\epsilon_2^j$ satisfying certain conditions and then put 
\[
U_1^j = \wt{X_j} \setminus \bigcup_{p\in \pole(\eta^j)}\ol{\B(p, \epsilon_1^j)},\qquad U_2^j = \bigcup_{p\in \pole(\eta^j)}\B(p, \epsilon_2^j)
\]
where $\pole(\eta^j)$ is the set where $\eta^j$ has a pole, $\B(p,r)$ is the open ball of radius $r$ with center at $p$.

For each subset $L\subset \{1,\dots,n\}$ and a closed subvariety $Y$ of $X$ we define a set of subvarieties $C_{L}(Y)$ inductively as follows:
\[
C_{\varnothing}(Y) = \{\text{all the irreducible components of $Y$}\},
\]
for $L\subset \{1,\dots,j-1\}$
\[
C_{L\cup\{j\}}(Y) = \bigcup_{Z\in C_L} \{\text{all the irreducible components of $Z\cap \pi_j^{-1} \pole(\eta^j)$}\}.
\]

Let $M$ be the set containing as elements varieties $W_i$, $|\Div f_i|\cap X$, the singular locus of $W_i$ and the ramification loci of $W_i\To\times_{j\in J}X_j$ for each $i$, $J\subset\{1,\dots,n\}$, $|J|=n-k+1$.

\begin{prop}
For any $\delta>0$ there exist sequences of positive numbers $\epsilon_1^j$, $\epsilon_2^j$ and an increasing sequence of positive numbers $\epsilon^j$ with $\epsilon^n<\delta$ which satisfy the following properties:
\begin{enumerate}
\item $\epsilon_1^j<\epsilon_2^j$ for all $j=1,\dots,n$.
\item $\eta_1^j$ has no poles on $U_1^j$ and $\eta_2^j$ has no poles on $U_2^j$.
\item The disks which constitute $U_2^j$ do not intersect each other for every $j=1,\dots,n$.
\item $Z \cap \ol{\pi_j^{-1}(U_2^j)} \subset n_{\epsilon^j}(Z\cap\pi_j^{-1} \pole\eta_j)$,
\item $n_{\epsilon^{j-1}}(Z) \cap \pi_j^{-1}(U_2^j) \subset n_{\epsilon^j}(Z\cap\pi_j^{-1} \pole\eta_j)$,
\item $n_{\epsilon^{j-1}}(Z\cap \pi_j^{-1}\pole(\eta_j))\subset \pi_j^{-1}(X_j\setminus U_1^j)$
for any $j=1,\dots,n$, set $L\subset\{1,\dots,j-1\}$, variety $Y\in M$ and variety $Z\in C_L(Y)$. $n_\epsilon$ denotes the $\epsilon$-neighbourhood of a set.
\end{enumerate}
\end{prop}
\begin{proof}
We start with $j=n$ and proceed decreasing $j$. We choose $\epsilon_n$ as any positive number smaller then $\delta$. At each step we choose $\epsilon_1^j$, $\epsilon_2^j$ so that the conditions (i)-(iv) are satisfied with given $\epsilon^j$. Then we consider all the elements $Z\in C_L(Y)$ for all $L\subset\{1,\dots,j-1\}$, $Y\in M$ and choose $\epsilon^{j-1}<\epsilon^j$ small enough so that the conditions (v), (vi) are satisfied.
\end{proof}

\begin{defn}
We say that $Z\in C_L(Y)$ is pure if its codimension in $Y$ equals to the size of $L$.
\end{defn}

\begin{prop}
Let for some $\delta$ the sequences $\epsilon_1^j$, $\epsilon_2^j$ and $\epsilon^j$ are chosen according to the proposition above. Then 
for any $L\subset\{1,\dots,n\}$ and $Y\in M$ the set 
\[
U^L(Y)=Y\cap \bigcap_{j\in L} \pi_j^{-1} U_1^j \cap \bigcap_{j\in L} \pi_j^{-1} U_2^j
\]
is contained in the $\delta$-neighbourhood of the union of all the pure varieties of $C_L(Y)$.
\end{prop}
\begin{proof}
We fix $Y\in M$ and prove by induction on $\max L$ the following statement:

For any $L\subset\{1,\dots,n\}$ the set $U^L(Y)$ is contained in the $\epsilon^{\max L}$-neighbourhood of the union of all pure elements of $C_L(Y)$.

When $L=\varnothing$ this is obvious. Take $j\in\{1,\dots,n\}$ and $L\subset\{1,\dots,j-1\}$. Take any point $p\in U^{L\cup\{j\}}(Y)$. Since $p\in U^{L}(Y)$ and the sequence $(\epsilon^{j'})$ is increasing, there is a pure element $Z$ of $C_L(Y)$ with $p\in n_{\epsilon^{j-1}}(Z)$. The property (v) implies $p\in n_{\epsilon^j}(Z\cap\pi_j^{-1}\pole\eta_j)$. Hence there is an irreducible component $Z'$ of $Z\cap\pi_j^{-1}\pole\eta_j$, $Z'\in C_{L\cup\{j\}}(Y)$ with $p\in n_{\epsilon^j} (Z')$. Suppose $Z'$ is not pure. This means that $\dim Z' = \dim Z$, hence $Z'=Z$. By the property (vi) it follows that
\[
p\in n_{\epsilon^{j-1}}(Z) = n_{\epsilon^{j-1}}(Z')\subset \pi_j^{-1}(X_j\setminus U_1^j).
\]
This is a contradiction. Therefore $Z'$ is pure and we are done.
\end{proof}

\begin{cor}\label{zero_int}
If $|L|>\dim Y$ for $Y\in M$, then $U^L(Y)=\varnothing$.
\end{cor}

The integral
\[
I(\wt{\CC_i}, (\omega_m\wedge\frac{df_i}{f_i}))
\]
is the sum of integrals of the following kind:
\[
\int_{\wt{c_m}} \omega_m\wedge\frac{df_i}{f_i}.
\]
Let us consider the corresponding chain $c_m$ in $\CC_i$ of dimension $1$ less then $\wt{c_m}$. We see that when $\dim c_m>n-k+1$, the integral is zero because the form we integrate is holomorphic, so its restriction to $\wt{W_i}$ is zero. If $\dim c_m<n-k+1$ the integral is zero because $c_m\subset U^L(W_i)$ for $|L|=2n-2k+2-\dim c_m>n-k+1$ and Corollary \ref{zero_int} implies that $c_m=0$. Therefore $\wt{c_m}$ intersects $X$ in codimension at least $2$ and the derivative with respect to $t$ of the corresponding integral is zero. If $\dim c_m=n-k+1$ then the same corollary implies $\partial c_m = 0$. Moreover, we can write
\[
\omega_m\wedge\frac{df_i}{f_i} = \omega_m'\wedge dt.
\]
This means that the derivative can be computed as follows:
\[
\frac{\partial}{\partial t}|_{t=0} \int_{\wt{c_m}} \omega_m\wedge\frac{df_i}{f_i} = \int_{c_m} \omega_m',
\]
so we obtain certain residue as a result. Let us denote
\[
\overrightarrow{I}(\CC_t^i, (\omega_m\wedge\frac{df_i}{f_i})_m) = \sum_{m:\dim c_m=n-k+1} (\int_{c_m} \omega_m') dt.
\]

Summarizing
\begin{thm}
\[
d I(\CC_t, (\omega_m)_m) = I(\CC_t, (h_m)_m) dt + \sum_i\overrightarrow{I}(\CC_t^i, (\omega_m\wedge\frac{df_i}{f_i})_m).
\]
\end{thm}

For a sequence of differential forms $(\omega_m)_m$ we denote 
\[
d (\omega_m)_m = (d \omega_m - \sum_l D_{lm}\omega_l)_m.
\]
It is easy to see that this is a system of forms which is equivalent to zero system via natural transformations. We have
\begin{thm}
\[
I(\CC, d(\omega_m)_m) = -\sum_i I(\CC_i, \frac{df_i}{f_i} \wedge\omega_m)
\]
\end{thm}
\begin{proof}
By the Stokes theorem 
\[
\int_{c_m} d\omega_m + \sum_i\int_{p_{i*}(c_m\cap W_i)}\frac{df_i}{f_i} \wedge\omega_m = \int_{\partial c_m} \omega_m.
\]
The righthand side is
\[
\int_{\partial c_m} \omega_m = \sum_{l} D_{ml} \int_{c_l} \omega_m.
\]
This implies
\[
I(\CC, d(\omega_m)_m) + \sum_i I(p_{i*}(\CC\cap W_i), \frac{df_i}{f_i} \wedge\omega_m) = 0.
\]
\end{proof}

\subsection{Differential operators}

\subsection{Residues}
Let us choose $\delta$ such that $\delta$-neighbourhoods of all pure points in $C_{L}(W_i)$ do not intersect for $|L|=n-k+1$. For a point $p\in W_i$, $L\subset\{1,\dots,n\}$, $|L|=n-k+1$, a holomorphic differential $n-k+1$-form $\omega$ on $U^L(W_i)$ we put
\[
\res_{L, i, p}\omega = \begin{cases} \int_{\B(p,\delta)\cap \CC} \omega& \text{if $\{p\}\in C_L(W_i)$,}\\ 0 & \text{otherwise.} \end{cases}
\]
Here all components of $\CC$ of dimension $n-k+1$ split into non-overlapping chains each belonging to $\B(p,\delta)$ for some $p$, so by $\B(p,\delta)\cap \CC_L$ we mean the chain that belongs to $\B(p,\delta)$. Each of these chains has zero boundary by Corollary \ref{zero_int}.

Let $X$ be a projective algebraic variety of dimension $n$ over $\C$. Let $\pi_i:X\To C_i$ be a map to a curve for each $i=1,\dots, n$. Let $p_i$ be a point on $C_i$ for each $i$. We define recursively the set $S_i$ of critical subvarieties of $X$ of codimension $i$. $S_0$ is simply the set of irreducible components of $X$. Given $S_{i-1}$, $S_{i}$ is the set of irreducible components of codimension $i$ of intersections $v\cap \pi_i^{-1} p_{i}$ for $v\in S_{i-1}$.

Let $\Sigma\subset X$ be a subvariety of $X$ which contains ramification loci of all $\pi_i$, singular locus of $X$ and $\pi_i^{-1} p_i$.

We would like to understand better the meaning of the sum 
\[
\sum_m B_m,
\]
where $B_m$ are coming from the proposition above. 
\bibliography{refs}
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For each $i$ let us consider the space $V_i$ constructed from $W_i$ by cutting out $\gamma_i$ and attaching two copies of $\gamma_i$ glued together along the boundary. Let us denote the two copies of $\gamma_i$ by $\gamma_i^+$ and $\gamma_i^-$ and suppose that they are attached in such a way that the function $\log f_i$ extends to $\gamma_i^+$ and $\gamma_i^-$, $\partial V_i = \gamma_i^+ - \gamma_i^-$, and the value of $\log f_i$ on $\gamma_i^-$ is $2\pi\I$ plus the value on $\gamma_i^+$. Denote by $\iota_i^+$, $\iota_i^-$ the natural isomorphisms $\gamma_i\To\gamma_i^+$, $\gamma_i\To\gamma_i^-$. Denote by $p_i$ the natural projection $V_i\To W_i$. The space $V_i$ is naturally endowed with the triangulation coming from the triangulation of $W_i$.

Let $l\geq 0$ and $\Delta$ be the standard simplex of dimension $l$,
\[
\Delta = \{(x_0, x_1, \dots, x_l)| \sum_{j=0}^l x_j = 0, x_j\geq 0\}.
\]
For $\epsilon>0$ denote
\[
\Delta_\epsilon = \{(x_0, x_1, \dots, x_l)\in\Delta| x_j\geq \epsilon\}.
\]
Let $\sigma:\Delta \To V_i$ be a simplex in $V_i$. Let $\omega$ be a smooth $l$-form on a neighbourhood of $\sigma(\Delta)$. 
\begin{defn}
Suppose $\sigma(\Delta)$ is not contained in $|\Div f_i|$. Put
\[
\int_\sigma \omega \log f_i = \lim_{\epsilon\To 0} \int_{\Delta_\epsilon} \sigma^* \omega \log f_i.
\]
\end{defn}
\begin{defn}
A simplex $\sigma$ is called good if it is not contained in $|\Div f_i|$ for all $i$ and any simplex of its boundary is not contained in $|\Div f_i|$ for all $i$. A chain is called good if it is a linear combination of good simplices.
\end{defn}
One can check that the Stokes formula holds:
\begin{prop}
If $\sigma$ is a good simplex and $\omega$ is a smooth $l-1$-form on a neighbourhood of $\sigma(\Delta)$, then
\[
\int_\sigma d\omega \log f_i = \int_{\partial \sigma} \omega \log f_i - \int_\sigma \frac{d f_i}{f_i}\wedge \omega.
\]
\end{prop}

\subsection{Using hypercovers}
Let us choose a Zariski affine cover on each of the curves $X_j$ this gives a hypercover $\U_j$ on $X_j$ indexed by $\sigma_j$, hence a product hypercover $\U$ on $X$ indexed by $\sigma$. This induces hypercovers on $V_i$. Let us lift chains $\Div f_i$ to hyperchains $\wt{\Div f_i}$, then lift the chains $\gamma_i$ to hyperchains $\wt\gamma_i$, then lift $V_i$ to $\wt V_i$ and $\xi$ to $\wt\xi$. Lifting $\gamma_i$ determines a lifting of $\gamma_i^+$, $\gamma_i^-$. These liftings are denoted $\wt\gamma_i^+$, $\wt\gamma_i^-$. We choose a nice analytic refinement $\U'$ of $\U$ and assume that all the hyperchains we are considering are in fact hyperchains on $\U'$. The following relations hold:
\[
\partial_h\wt\gamma_i = -\wt{\Div f_i},\; \partial_h \wt\xi=\sum_i\wt\gamma_i,\; \partial_h\wt V_i = \wt\gamma_i^+-\wt\gamma_i^-.
\]
For $\omega\in F_{n-k+1}\A^{2n-2k+2}(X)$, $d\omega=0$ we can denote by the same letter $\omega$ the corresponding hyperform. We get
\[
\langle AJ^{k,1}[x],[\omega]\rangle=\frac{1}{2\pi\I}\sum \int_{\wt V_i} \omega \log f_i + \int_{\wt\xi}\omega.
\]
Let $L\subset\{1,\ldots,n\}$, $L'\subset\{1,\ldots,n\}$ such that $L\cap L'=\varnothing$, $|L|=n-k+1$, $|L'|=n-k+1$. Let $\omega_k$ be a holomorphic $1$-form on $X_k$ for $k\in L$ (which is considered as a hyperform) and a closed algebraic $1$-hyperform on $X_k$ for $k\in L'$. Recall that $1$-hyperform on $X_k$ is a collection of $1$-forms on the open sets of the cover of $X_k$ and functions on the pairwise intersections. Saying algebraic we require the functions and the $1$-forms to be regular. For $k\in L'$ by the Hodge theory one can choose a smooth $0$-hyperform $g_k$ such that $\omega_k'=\omega_k-d g_k$ is a smooth $1$-form. Put $\omega_k'=\omega_k$ for $k\in L$. We obtain a smooth closed form on $X$
\[
\omega'=\times_{k\in L\cup L'} \omega_k'\in F_{n-k+1}\A^{2n-2k+2}(X),
\]
and an algebraic closed hyperform
\[
\omega = \times_{k\in L\cup L'} \omega_k \in \Omega^{2n-2k+2}(\U)
\]
which is zero on $U_a$ if $\dim a>n-k+1$. Moreover they differ by a coboundary,
\[
\omega'-\omega=d g, \; g\in \Omega_{\smooth}^{2n-2k+1}(\U),
\]
with $g$ zero on $U_a$ if $\dim a>n-k$.

\begin{prop}
One can compute $AJ^{k,1}$ using algebraic forms:
\[
\langle AJ^{k,1}[x],[\omega]\rangle=\frac{1}{2\pi\I}\sum_i \int_{\wt V_i} \omega \log f_i + \int_{\wt\xi}\omega.
\]
\end{prop}
\begin{proof}
By the definition
\begin{multline*}
\langle AJ^{k,1}[x],[\omega']\rangle=\frac{1}{2\pi\I}\sum_i \int_{\wt V_i} \omega' \log f_i + \int_{\wt\xi}\omega' \\= \frac{1}{2\pi\I}\sum_i \int_{\wt V_i} \omega \log f_i + \int_{\wt\xi}\omega + \frac{1}{2\pi\I}\sum_i \int_{\wt V_i} (dg) \log f_i + \int_{\wt\xi} dg.
\end{multline*}
The summand in the third term gives
\[
\frac{1}{2\pi\I} \int_{\wt V_i} (dg) \log f_i = \frac{1}{2\pi\I} \int_{\wt\gamma_i^+-\wt\gamma_i^-} g \log f_i - \frac{1}{2\pi\I} \int_{\wt V_i} \frac{d f_i}{f_i} \wedge g.
\]
The second integral is zero. Indeed, if $a\in \sigma$ is such that $\dim a>n-k$, then $(\frac{d f_i}{f_i} \wedge g)_a$ is zero as it was mentioned above. Otherwise $(\frac{d f_i}{f_i} \wedge g)_a\in\Omega^{2n-2k+2-\dim a}(U_a)$. Therefore the degree of this form is at least $n-k+2$, which is greater than the dimension of $V_i$. The first integral can be further transformed to
\[
\frac{1}{2\pi\I} \int_{\wt\gamma_i^+-\wt\gamma_i^-} g \log f_i=-\int_{\wt\gamma_i} g
\]
which cancels the corresponding summand in the term
\[
\int_{\wt\xi} dg=\sum_i\int_{\wt\gamma_i} g.
\]
\end{proof}

\subsection{Differentiating the Abel-Jacobi map}
Suppose we have families of curves $X_i\ra S$ with $S$ affine and 
\[
X = X_1\times_S X_2\times_S \dots \times_S X_n.
\]
Suppose we have a family of cycles $x_s\in Z^k(X_s,1)$. Let $\omega\in\Omega_{X/S}^{2n-2k+2}(\U)$. Suppose $\omega_a=0$ if $\dim a>n-k+1$. Let us compute 
\[
d \langle AJ^{k,1}[x_s],[\omega]\rangle.
\]
We first generalize it. Let for any $\omega\in \Omega_{X/S}^{2n-2k+2}(\U)$
\[
I(\omega)=\frac{1}{2\pi\I}\sum_i \int_{\wt V_i} \omega \log f_i + \int_{\wt\xi}\omega.
\]
We also define a new operation on hyperforms. Let $\ol\omega\in\Omega_{X}^{2n-2k+2}(\U)$. For any index $i$ consider the hyperform $\frac{d f_i}{f_i}\wedge\ol\omega$. It is zero on all $U_a\cap W_i$ with $\dim a<n-k+1$ and is a form of maximal degree on $U_a\cap W_i$ with $\dim a=n-k+1$. Therefore there exists an element $\ol \phi_{f_i} \omega\in \Omega^1(S)\otimes \Omega_{X}^{2n-2k+2}(\U)$ such that the difference $\frac{d f_i}{f_i}\wedge\ol\omega-\ol \phi_{f_i} \omega$ is zero on all $U_a\cap W_i$ with $\dim a\leq n-k+1$. Its image in $\Omega^1(S)\otimes \Omega_{W_i/S}^{2n-2k+2}(\U)$ will be denoted by $\phi_{f_i} \omega$.
\begin{prop}
We have 
\[
d I(\omega) = I(\nabla\omega) + \frac{1}{2\pi\I} \int_{\wt W_i} \phi_{f_i} \omega.
\]
\end{prop}
\begin{proof}
Let $\eta=\nabla \omega$. Then
\[
d \int_{\wt\xi}\omega = \int_{\wt\xi}\eta + \sum_i R(\wt\gamma_i,\ol\omega).
\]
Consider the hyperform $\omega\log f_i$ on $\U\cap V_i$. The restriction of $\omega\log f_i$ to $\U'\cap V_i$ is closed simply because $\Omega_{W_i/S}^{2n-2k+3}(\U'\cap W_i)=0$. We have 
\[
d(\ol\omega \log f_i)= \ol\eta\log f_i + \frac{d f_i}{f_i} \wedge \ol\omega.
\]
The sum $\eta\log f_i + \phi_{f_i}\omega$ gives a Gauss-Manin derivative of $\omega\log f_i$ on $V_i$ with respect to the hypercover $\U'$. Therefore
\[
d \int_{\wt V_i} \omega \log f_i=R(\wt\gamma_i^+-\wt\gamma_i^-,\ol\omega\log f_i) + \int_{\wt V_i} \eta\log f_i + \int_{\wt V_i} \phi_{f_i}\omega.
\]
The first summand equals to 
\[
R(\wt\gamma_i^+-\wt\gamma_i^-,\ol\omega\log f_i) = -2\pi\I R(\wt\gamma_i,\ol\omega).
\]
The third summand equals to
\[
\int_{\wt V_i} \phi_{f_i}\omega=\int_{\wt W_i} \phi_{f_i}\omega.
\]
\end{proof}

We also can compute $I(\omega)$ for exact hyperform.
\begin{prop}
Let $\omega=d\eta$ with $\eta\in\Omega_{X/S}^{2n-2k+1}(\U)$. Then
\[
I(\omega)=-\frac{1}{2\pi\I} \sum_i\int_{\ol W_i}\frac{d f_i}{f_i}\wedge \eta
\]
\end{prop}
\begin{proof}
By the definition
\[
I(\omega)=\frac{1}{2\pi\I}\sum_i \int_{\wt V_i} \omega \log f_i + \int_{\wt\xi}\omega.
\]
We have $\omega\log f_i=d(\eta \log f_i)-\frac{d f_i}{f_i}\wedge\eta$. Applying the Stokes formula we obtain the result.
\end{proof}

Let us denote 
\[
J_i(\omega)=(2\pi\I)^{-n+k-1} \int_{\wt V_i} \omega.
\]
for any $\omega\in \Omega^{2n-2k+2}_{W_i/S}(\U)$. We see that the following relations hold:
\[
d I(\omega) = I(\nabla\omega) + (2\pi\I)^{n-k} \sum_i J_i(\phi_{f_i}\omega),\; I(d\eta)=-(2\pi\I)^{n-k} \sum_i J_i(\frac{d f_i}{f_i}\wedge\eta).
\]
Note that the value of $J_i$ can be expressed as a certain sum of iterated residues. Therefore it is possible to compute $J_i$.

\subsection{Extensions of $\D$-modules}
We let $S=\spec R$. Suppose $S$ is smooth and $\Omega(R)$ is free. Let $\D$ be the ring of differential operators on $S$. Consider $R$-modules $\Omega_X^i(\U)$. We have two filtrations on $\Omega_X^i(\U)$. The first is the Hodge filtration. Elements of $F^j\Omega_X^i(\U)$ are those hyperforms which have as components only forms of rank at least $j$. We have another filtration defined as $G^j\Omega_X^i(\U):=\Omega^j(R)\wedge\Omega_X^{i-j}(\U)$. The exterior derivative respects these filtrations. We have 
\[
\Omega_{X/S}^i(\U)=\Omega_X^i(\U)/G^1\Omega_X^i(\U).
\]

To understand the results of the previous section we introduce two operations on hyperforms. The first one is $\Psi_0:\Omega_X^{2n-2k+1}\ra R$ defined as
\[
\Psi_0(\eta)=(2\pi\I)^{-n+k-1}\sum_i\int_{\wt W_i}\frac{d f_i}{f_i}\wedge \eta,\; \eta\in\Omega_X^{2n-2k+1}(\U).
\]
The second one is $\Psi_1:\Omega_X^{2n-2k+2}\ra \Omega(R)$ defined as
\[
\Psi_1(\omega)=(2\pi\I)^{-n+k-1}\sum_i\int_{\wt W_i}\phi_{f_i}\omega,\; \omega\in\Omega_X^{2n-2k+2}(\U).
\]
\begin{rem}
The integral with respect to $\wt W_i$ can be expressed as a sum of iterated residues as is proved in the previous chapter. Therefore the operations $\Psi_0$ and $\Psi_1$ can be defined purely algebraically
\end{rem}

\begin{prop}
The first operation satisfies the following properties:
\begin{enumerate}
\item $\Psi_0$ is $R$-linear.
\item $\Psi_0|_{G^1\Omega_X^{2n-2k+1}(\U)}=0$.
\item $\Psi_0|_{F^{n-k+1}\Omega_X^{2n-2k+1}(\U)}=0$.
\item $\Psi_0(\eta)=-(2\pi\I)^{-n+k} I(d\eta)$ for $\eta\in\Omega_X^{2n-2k+1}(\U)$.
\item If $d\eta\in G^1\Omega_X^{2n-2k+2}(\U)$, then $\Psi_0(\eta)=0$.
\end{enumerate}
\end{prop}

\begin{prop}
The second operation satisfies the following properties:
\begin{enumerate}
\item $\Psi_1$ is $R$-linear.
\item $\Psi_1|_{G^2\Omega_X^{2n-2k+2}(\U)}=0$.
\item $\Psi_0|_{F^{n-k+2}\Omega_X^{2n-2k+2}(\U)}=0$.
\item $\Psi_1(u\wedge\eta)=-u\Psi_0(\eta)$ for $u\in\Omega(R),\eta\in\Omega_X^{2n-2k+1}(\U)$.
\item $\Psi_1(\omega)=(2\pi\I)^{-n+k} (d I(\omega)-I(\nabla\omega))$ for $\omega\in\Omega_X^{2n-2k+2}(\U)$ such that $d\omega\in G^1\Omega_X^{2n-2k+3}(\U)$, and $\nabla\omega$ is defined as the class of $d\omega$ in $\Omega(R)\otimes\Omega_X^{2n-2k+2}(\U)$.
\item $\Psi_1(d \eta)=d \Psi_0(\eta)$ for $\eta\in\Omega_X^{2n-2k+1}(\U)$.
\end{enumerate}
\end{prop}

Let $B^{2n-2k+2}\subset Z^{2n-2k+2}\subset \Omega_{X/S}^{2n-2k+2}(\U)$ be defined as 
\[
B^{2n-2k+2}=\image d_{X/S},\; Z^{2n-2k+2}=\kernel d_{X/S}.
\]
Let 
\[
H^{2n-2k+2}=Z^{2n-2k+2}/B^{2n-2k+2}.
\]
Note that the Hodge theory implies 
\[
F^i \Omega_{X/S}^{2n-2k+2} \cap B^{2n-2k+2}= d_{X/S}(F^i\Omega_{X/S}^{2n-2k+1}).
\]
We have the following diagram:
\[
\begin{CD}
0 @@>>>B^{2n-2k+2} @@>>> Z^{2n-2k+2} @@>>> H^{2n-2k+2} @@>>> 0\\
& & @@V{\Psi_0}VV @@VVV @@| \\
0 @@>>>R @@>>> M @@>>> H^{2n-2k+2} @@>>> 0
\end{CD}
\]
Here $M$ is defined as 
\[
M=(R\oplus Z^{2n-2k+2})/\image(\Psi_0,d_{X/S}).
\]
Note that there is a canonical section $s:F^{n-k+1} H^{2n-2k+2}\ra M$ which is defined by sending the class $[\omega]\in F^{n-k+1} H^{2n-2k+2}$ ($\omega\in Z^{2n-2k+2}$) to $(0,\omega)$.

We have $R$ and $H^{2n-2k+2}$ modules over $\D$. On $M$ we have the following $\D$-module structure:
\[
v(r, \omega)=(v(r)-\langle v, \Psi_1(\ol\omega)\rangle , \langle v, \nabla \ol\omega\rangle),
\]
where $r\in R$, $v\in\Der(R)$, $\omega\in Z^{2n-2k+2}$, $\ol\omega\in\Omega_X^{2n-2k+2}(\U)$ represents $\omega$.
One can check that this indeed gives a $\D$-module structure. The section $s$ is compatible with the $\D$-module structure so that the diagram commutes:
\[
\begin{CD}
F^{n-k+2} H^{2n-2k+2} @@>{\nabla}>> \Omega(S)\otimes F^{n-k+1} H^{2n-2k+2} \\
@@V{s}VV @@V{\id_S\otimes s}VV\\
M@@>>> \Omega(S)\otimes M.
\end{CD}
\]
Let us choose $S$ smaller so that $H^{2n-2k+2}$ is free and $F^{n-k+1} H^{2n-2k+2}$ is a direct summand. Extend the section $s$ to a section $\wt s:H^{2n-2k+2}\ra M$. This provides an isomorphism of $R$-modules $M\cong R\oplus H^{2n-2k+2}$. We define the homomorphism $\Psi\in \Hom_R(H^{2n-2k+2},\Omega(R))$ in the following way. For any $[\omega]\in H^{2n-2k+2}$ put
\[
\Psi([\omega])=\nabla \wt s([\omega]) - (\id\otimes \wt s)(\nabla[\omega])\in\Omega(R).
\]
One can see that $\Psi|_{F^{n-k+2}H^{2n-2k+2}}=0$ and $\Psi$ is correctly defined up to the differential of an element of $\Hom_R(H^{2n-2k+2},\Omega(R))$. The structure of $\D$-module on $R\oplus H^{2n-2k+2}$ induced from $M$ can be recovered as follows:
\[
v(r,h)=(v(r)+\Psi(h),\nabla_v h),\qquad (h\in H^{2n-2k+2}, r\in R, v\in\Der(R)).
\]

Let $N$ be the kernel of the multiplication homomorphism
\[
N:=\kernel(\D\otimes_R H^{2n-2k+2} \ra H^{2n-2k+2}).
\]
It inherits the filtration $F^\bullet$ from the one on $H^{2n-2k+2}$. We have the canonical homomorphism of $\D$-modules $\Psi':F^{n-k+1} N\ra R$ defined as follows.  
\[
\Psi'(\sum_j \alpha_j \otimes h_j)=\sum_j \alpha_j s(h_j)\in R,\qquad(\sum_j \alpha_j \otimes h_j\in N,h_j\in F^{n-k+1} N,\alpha_j\in\D).
\]

On the other hand we could have defined $\Psi'':F^{n-k+1} N\ra R$
\[
\Psi''(\sum_j \alpha_j \otimes h_j)=\sum_j \alpha_j \langle AJ^{k,1}[x], h_j\rangle,
\]
but the properties of $\Psi_0$, $\Psi_1$ imply:
\begin{cor}
We have $\Psi''=(2\pi\I)^{n-k}\Psi'$.
\end{cor}

\subsection{Products of elliptic curves}
Suppose we have a family of elliptic curves $\pi:E\ra S$ and $X_i=E$. We have $S=\spec R$. We suppose that $R$ is a $1$-dimensional domain and its field of fractions is denoted as $R_0$. Let 
\[
V = R^1\pi_*\Omega_{E/S}^\bullet.
\]
This is a locally free module over $R$ of rank $2$. The Hodge filtration has two pieces, each a rank $1$ locally free $R$-module. Denote
\[
M^1=F^1 V.
\]
This is a line bundle of modular forms of weight $1$. We denote $M^j=(M^1)^{\otimes j}$ for $j\in\Z$. We have the canonical pairing
\[
F^1 V \otimes_R (V/F^1) \ra R.
\]
With the help of this pairing we identify
\[
V/F^1 \cong M^{-1}.
\]

Consider the Kodaira-Spencer map 
\[
KS: M^1\ra \Omega(R)\otimes M^{-1}.
\]
Suppose that the family $E$ is not constant and $S$ is small enough, so that $KS$ is an isomorphism. Therefore we have $\Omega(R)\cong M^2$. Let $H^1=R^1\pi_*\Omega_{E/S}^\bullet$, which is a $\D$-module of rank $2$ over $R$.

Suppose $n=2k-2$. Then, using the notations of the previous section,
\[
M^n=F^n H^{2n-2k+2}=F^n H^n.
\]
Let $H^n_s$ denote the direct summand of $H^n$ which corresponds to 
\[
H^n_s=\Sym^n V.
\]

We have the following fact:
\begin{prop}
There is a unique element $B\in M^{n+2}\otimes_R F^n N$ which contains only differential operators of degree at most $n+1$ and the corresponding symbol in $M^{n+2}\otimes_R \Der(R)^{\otimes (n+1)}\otimes_R M^n\cong R$ is $1$. 
\end{prop}
\begin{proof}
Let us prove uniqueness. Let $\D^n$ denote the differential operators of order at most $n$. The following map is a monomorphism: 
\[
m:\D^n\otimes_R F^n H^n\ra H^n.
\]
To prove this look at the filtration by the order of differential operator on $\D^n\otimes_R F^n H^n$ and the Hodge filtration on $H^n$. The graded pieces of the map are the Kodaira-Spencer maps which are injective.

Existence follows from the fact that the image of $m$ is $\Sym^n H^1$. So we can pick any element of $M^{n+2}\otimes_R \D^{n+1}\otimes_R M^n$ with symbol $1$ and then subtract an element of $M^{n+2}\otimes_R \D^n\otimes_R M^n$ which maps to the same element in $M^{n+2} \otimes_R \Sym^n H^1$.
\end{proof}

One can apply $\Psi'$ to $B$ and get an element 
\[
(2\pi\I)^{k-n}\Psi''(B)=\Psi'(B)\in M^{n+2}.
\]
Therefore we have the canonical modular form constructed from a family of cycles $x_s\in Z^k(X_s,1)$.

\subsection{Analytic computations}
Suppose we have a family of elliptic curves $\pi:E\ra S$. The elliptic curve over a point $s\in S$ is denoted $E_s$. Let $U$ be an analytic subset in $S$ homeomorphic to a disk. Choose families of $1$-cycles $c_1$, $c_2$ over $U$ such that $\cc_1(s)$ and $\cc_2(s)$ generate $H_1(E_s, \Z)$ and the intersection number is $\cc_1\cdot \cc_2=1$.

Any other choice $\cc_1'$, $\cc_2'$ can be obtained from the choice $\cc_1$, $\cc_2$ by a transformation 
\[
\gamma=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in SL_2\Z:\qquad \cc_2'=a \cc_2 + b \cc_1,\; \cc_1'=c \cc_2 + d \cc_1.
\]

Let $\omega$ be a closed relative differential $1$-form on $E$. We denote
\[
\Omega_1(\omega)=\int_{\cc_1} \omega,\;\;\Omega_2(\omega)=\int_{\cc_2} \omega.
\]
The cup product provides a pairing:
\[
(\omega_1,\omega_2) = \int_{E_s} \omega_1\wedge \omega_2.
\]
Let us integrate $\omega_1=df$ over the universal cover of $E_s$. Then 
\[
(\omega_1,\omega_2) = \int_{\partial \wt E_s} f \omega_2 = \Omega_1(\omega_1)\Omega_2(\omega_2)-\Omega_1(\omega_2) \Omega_2(\omega_1),
\]
where $\wt E_s$ denotes a fundamental domain of the universal cover of $E_s$.

If $\omega$ is holomorphic we put
\[
z=\frac{\Omega_2(\omega)}{\Omega_1(\omega)}.
\]
This locally defines an isomorphism between $S$ and the upper half plane. Indeed,
\[
\Im z = \frac{\Omega_2(\omega)\ol\Omega_1(\omega)-\Omega_1(\omega) \ol\Omega_2(\omega)}{2\I \Omega_1(\omega)\ol\Omega_1(\omega)} = -\frac{\Im \int_{E_s} \omega \wedge\ol\omega}{2 \Omega_1(\omega)\ol\Omega_1(\omega)}.
\]
If we represent $E_s$ as a quotient $\C/\Lambda$ with $\omega=dx+\I dy$, then
\[
\omega\wedge\ol\omega = -2 \I dx\wedge dy,
\]
therefore
\[
\Im z = \frac{\vol_\omega E_s}{|\Omega_1(\omega)|^2}>0,
\]
where $\vol_\omega E_s$ is the volume of $E_s$ defined with the help of the form $\omega$. For another choice $\cc_1'$, $\cc_2'$ we obtain
\[
z' = \frac{a z + b}{c z + d}.
\]
We define the canonical isomorphism of the analytic version of the sheaf $M^1$ as defined in the previous section, and the pullback via $z$ of the usual sheaf of modular forms of weight $1$ on the upper half plane:
\[
\omega \ra \Omega_1(\omega).
\]

Let $X$ be a formal variable. We identify $H^1(E_s, \C)$ with the space of polynomials of degree not greater than $1$ in $X$ in the following way:
\[
\langle 1, \cc_2\rangle=-1,\; \langle 1, \cc_1\rangle=0,\; \langle X, \cc_1\rangle=1,\; \langle X, \cc_2\rangle=0.
\]

Let $\omega$ be a closed differential $1$-form. Then the corresponding polynomial is 
\[
[\omega]_\cc=\Omega_1(\omega) X-\Omega_2(\omega).
\]
In particular, if $\omega$ is holomorphic,
\[
[\omega]_\cc=\Omega_1(\omega) (X-z).
\]
If $\cc'=\gamma \cc$, then one can check that
\[
[\omega]_{\cc'} = \gamma(\Omega_1(\omega) X - \Omega_2(\omega)),
\]
where the action on polynomials is defined as
\[
\gamma(p) = p|_{-1} \gamma^{-1}\;\; (p=p_1 X + p_0).
\]
Therefore the map
\[
\omega \ra f_\omega=\frac{\Omega_1(\omega) X-\Omega_2(\omega)}{X-z} = \Omega_1(\omega) + \frac{\Omega_2(\omega)-z \Omega_1(\omega)}{z-X}
\]
defines an isomorphism between the sheaf $V=H^1(E,\C)$ and the pullback of the sheaf of quasi-modular forms of weight $1$ and depth $1$. \emph{Quasi-modular forms} of weight $w$ and depth $d$ are functions of the form
\[
f(z, X) = \sum_{i=0}^d \frac{f_i(z)}{i!(z-X)^i}
\]
which transform like modular forms of weight $w$ in $z$ and weight $0$ in X:
\[
f(\gamma(z), \gamma(X)) (c z+d)^{-w} = f(z, X).
\]

One can see that the \emph{pairing} can be written as
\[
(aX+b, a' X + b') = - a b' +a' b.
\]
Therefore if $a(X-z)\in F^1 V$ and $a'X+b'\in V$, then
\[
(a(X-z), a'X+b') = -a (a' z + b').
\]
If $\omega$ is a holomorphic differential and $\eta$ an arbitrary closed differential, then
\[
(\omega, \eta) = f_\omega (f_{\eta})_1,
\]
where $(f_{\eta})_1$ denotes the coefficient at $(z-X)^1$ of $f_{\eta}$ and is a modular form of weight $-1$. Therefore the isomorphism $V/F^1 V\ra M^{-1}$ is given by sending $f$ to $f_1$.

\emph{The Gauss-Manin derivative} with respect to the parameter $z$ sends the cohomology class with periods $\Omega_1$, $\Omega_2$ to the cohomology class with periods $\frac{\partial \Omega_1}{\partial z}$, $\frac{\partial \Omega_2}{\partial z}$. Therefore on the level of quasi-modular forms it can be written as 
\[
\frac{\partial}{\partial z} + \frac{1}{z-X}.
\]
If we take a modular form $f$ of weight $1$, the Gauss-Manin derivative of the corresponding family of cohomology classes will be given by
\[
\left(\frac{\partial f}{\partial z} + \frac{f}{z-X}\right) dz.
\]
Therefore the \emph{Kodaira-Spencer} map sends 
\[
f\ra f dz.
\]
If $t$ is a local parameter on $S$ then the Kodaira-Spencer map acts as
\[
f \ra \frac{f}{\left(\frac{dt}{dz}\right)} dt,
\]
therefore the isomorphism $\Omega(S)\ra M^2$ acts by sending $dt$ for a function $t$ to the modular form $\frac{dt}{dz}$.

Next we construct the canonical differential operator from the previous section. Let $U$ be an open subset in $S$ such that there exist modular forms $f$ of weight $n$ and $g$ of weight $n+2$ with non-zero values on $U$. Let $D(f,g)$ be the operator
\[
\phi\ra \frac{1}{g} \left(\frac{\partial}{\partial z}\right)^{n+1} \frac{\phi}{f}.
\]
This is a differential operator which sends functions to functions, therefore $D(f,g)\in \D^{n+1}(U)$. Moreover $D(f,g) f = 0$, therefore 
\[
B_n(U)=g \otimes D(f,g)\otimes f
\]
defines a section of $M^{n+2}\otimes_R F^n N$. Its symbol is
\[
g \frac{1}{f g dz^{n+1}} f,
\]
which goes to $1$ under the isomorphism $M^{n+2}\otimes_R \Der(R)^{\otimes (n+1)}\otimes_R M^n\cong R$.

Let us consider the natural map
\[
\frac{H^{2k-2}}{F^k H^{2k-2} + H^{2k-2}_\Z} \ra \frac{H^{2k-2}}{F^1 H^{2k-2} + H^{2k-2}_\Z} \cong M^{2-2k}/M^{2-2k}_\Z,
\]
where $H^{2k-2}_\Z\subset H^{2k-2}$ is the subsheaf of integral cohomology classes and for any $j\geq 0$ $M^{-j}_\Z$ is the subsheaf of $M^{-j}$ generated by $1,z,\ldots,z^j$. The image of the section $AJ^{k,1}[x]$ under this map will be denoted by $AJ^{k,1}_m[x]$ ($m$ stands for modular). It is clear that the operator 
\[
\left(\frac{\partial}{\partial z}\right)^{2k-1}: M^{2-2k}\ra M^{2k}
\]
vanishes on $M^{2-2k}_\Z$. Therefore it is defined on $M^{2-2k}/M^{2-2k}_\Z$ and it is clear that 
\[
\Psi''(B_{2k-2}) = \left(\frac{\partial}{\partial z}\right)^{2k-1} AJ^{k,1}_m[x]\in M^{2k}.
\]

Also we have the canonical (non-holomorphic) section of $F^{k-1}H^{2k-2}$. This is defined by the polynomial
\[
Q^{k-1}=\left(\frac{(X-z)(X-\ol z)}{z-\ol z}\right)^{k-1},
\]
or the non-holomorphic quasi-modular form
\[
\left(\frac{X-\ol z}{(X-z)(z-\ol z)}\right)^{k-1}.
\]
Let us consider the canonical section $1\in M^{2-2k}\otimes H^{2k-2}$. We have the differential operators of non-holomorphic derivatives
\[
\delta_w:M^w\ra M^{w+2}, \;\; \delta_w = \frac\partial{\partial z} + \frac{w}{z-\ol z}.
\]
For $j\geq 0$ the following expression is a section of $M^{2-2k+2j}\otimes_R H^{2k-2}$:
\[
Q^j=\left(\frac{X-\ol z}{(X-z)(z-\ol z)}\right)^{j}.
\]
The operator $\delta_{2-2k+2j}:M^{2-2k+2j}\ra M^{4-2k+2j}$ and the $\D$-module structure on $H^{2k-2}$ induces an operator $M^{2-2k+2j}\otimes H^{2k-2} \ra M^{4-2k+2j}\otimes H^{2k-2}$ and we compute:
\begin{multline*}
\delta_{2-2k+2j}\left(\frac{X-\ol z}{(X-z)(z-\ol z)}\right)^{j} = \left(\frac{\partial}{\partial z} + \frac{2-2k+2j}{z-\ol z} + \frac{2k-2}{z-X}\right)\left(\frac{X-\ol z}{(X-z)(z-\ol z)}\right)^{j} \\= 
(2k-2-j) \left(\frac{X-\ol z}{(X-z)(z-\ol z)}\right)^{j+1}.
\end{multline*}
Therefore
\[
Q^{k-1} = \frac{(k-1)!}{(2k-2)!}\delta_{-2}\cdots\delta_{2-2k} 1.
\]
This implies
\[
\langle AJ^{k,1}[x], Q^{k-1}\rangle = \frac{(k-1)!}{(2k-2)!} \delta_{2-2k}^{k-1} AJ^{k,1}_m[x],
\]
where $\delta_{2-2k}^{k-1}=\delta_{-2}\cdots\delta_{2-2k}$. Since $\I^{k-1} Q^{k-1}$ is real, the following function is well-defined:
\[
G^{k,1}[x]=2\Re(\I^k\langle AJ^{k,1}[x], Q^{k-1}\rangle) = 2\Re(\I^k \frac{(k-1)!}{(2k-2)!} \delta_{2-2k}^{k-1} AJ^{k,1}_m[x]).
\]
Moreover, it satisfies the following relation:
\[
\delta^k_0 G^{k,1}[x] = \frac{(k-1)!}{(2k-2)!}\I^k \delta^{2k+1}_{2-2k} AJ^{k,1}_m[x] = \frac{(k-1)!}{(2k-2)!}\I^k \Psi''(B_{2k-2}).
\]
Indeed, we have
\[
\delta^k_0 \ol{\langle AJ^{k,1}[x], Q^{k-1}\rangle} = \ol{\ol{\delta^k_0} \langle AJ^{k,1}[x], Q^{k-1} \rangle}
\]
and since 
\[
\ol{\delta^j_0} Q^{k-1} = \ol{\I^{k-1}\delta^j_0 Q^{k-1}},
\]
we have 
\[
\ol{\delta^k_0} \langle AJ^{k,1}[x], Q^{k-1} \rangle = \langle AJ^{k,1}[x], \ol{\delta^k_0} Q^{k-1}\rangle
\]
and 
\[
\ol{\delta^k_0} Q^{k-1} = 0.
\]

1.1
date	2007.10.13.16.01.42;	author mellit;	state Exp;
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@\input commons.tex
\author{Anton Mellit}
\title{Research plan for stay at MPI during May, 2007~--- August, 2007.}
\begin{document}
\bibliographystyle{alpha}
\maketitle
\section{Higher Green's functions and conjectures on special values}
Let $\HH$ be the upper half plane.
Let $\Gamma$ be a congruence subgroup of $SL_2(\Z)$ and $k>1$. The Green's function on $\HH/\Gamma$ of weight $2k$ is the unique function $G_k^{\HH/\Gamma}$ with the following properties:

We have the following formula:
\[
G_k^{\HH/\Gamma}(z_1, z_2) = \sum_{\gamma\in\Gamma} -2 \calQ_{k-1}\left(1 + 2\frac{(\z_1-\gamma \z_2)(\gamma \zc_2-\zc_1)}{(\z_1-\zc_1)(\gamma\z_2-\gamma\zc_2)}\right).
\]

A particular case of the conjecture formulated in \cite{GKZ} is as follows:
Let $z_1, z_2 \in \HH$ be two different complex multiplication points. Let $Q_1$, $Q_2$ be the corresponding positive definite primitive quadratic forms. Let $D_1$, $D_2$ be their discriminants. Let $k=2,3,4,5,7$. The conjecture is
\begin{conjecture}
There exists an algebraic number $f$, such that the value of the Green's function equals
\[
    \wt{G}_k^{\HH/PSL_2\Z}(z_1, z_2) = (D_1 D_2)^{\frac{1-k}2} \log f.
\]
\end{conjecture}

\section{My result}
In my PhD thesis I prove the conjecture for the case $k=2$, $z_1=\sqrt{-1}$. The proof is composed from the following steps:

\begin{enumerate}
\item Let $g$ be a function on $\HH$ which is invariant under a congruence subgroup $\Gamma$, satisfies equation $\Delta g = k(1-k) g$, has only logarithmic singularities and small growth at infinity. Then $\partial^k g$ is a meromorphic modular form for $\Gamma$ of weight $2k$, where $\partial$ denotes the nonholomorphic derivative. The modular forms $\partial^k g$ and $\partial^k \bar{g}$ determine $g$.
\item Let $\E\To X$ be a family of elliptic curves over $X$, a smooth quasi-projective variety over $\C$. Let $E_t$ denote the fiber over $t\in X$. Let $x_t$ for each $t\in X$ be an element of the group $CH^2(E_t\times E_t,1)$, that is a sequence of curves $W_i\subset E_t\times E_t$, rational functions $f_i\in \C(W_i)$ with $\sum \Div f_i = 0$. Suppose $x_t$ form a "family", which means that $W_i$ is the intersection of a fixed subvariety of $\E\times_X\E$ with $E_t\times E_t$, and $f_i$ is the restriction of some fixed rational function to $W_i$. Then the Abel-Jacobi map gives for each $t\in X$ an element
\[
\epsilon_t\in \frac{\Hom_\C(F^1 H^2(E_t\times E_t,\C), \C)}{H^2(E_t\times E_t,\Z)}.
\]
\item Take the canonical element $\theta_t\in F^1 H^2(E_t \times E_t,\C)$ given by the form
\[
\frac{\omega \otimes \bar\omega + \bar\omega \otimes \omega}{\int_{E_t} \omega\wedge\bar{\omega}},
\]
where $\omega$ is a holomorphic differential form on $E_t$. Evaluating $\epsilon_t$ on $\theta_t$ gives a multi-valued function on $X$. I prove that the operator $\partial^2$ applied to this function gives a meromorphic modular form on $X$. Moreover I obtain a formula for this modular form which allows explicitly compute it for any given family $x_t$
\item For the function $g(z)=G^{\HH/SL_2(\Z)}_2(\sqrt{-1}, z)$ I find a family $x_t$ such that $\partial^2 \langle\epsilon_t, \theta_t\rangle$ and $\partial^2 g$ give the same modular form (up to a multiple of $2\pi \sqrt{-1}$. I conclude that for any $t\in X$ $g(z)=Re \langle\epsilon_t, \theta_t\rangle$, where $z$ correspond to the elliptic curve $E_t$.
\item Let $S$ be a smooth projective variety, $n=\dim S$, $i\in \Z$. Take the Abel-Jacobi map 
\[
AJ^{i,1}:CH^i(S,1)_0\To \frac{\Hom_\C(F^{n-i+1}H^{2n-2i+2}(S,\C), \C)}{H^{2i-2}(S,\Z)}.
\]
I prove that the value of $AJ^{i,1}$ on a cycle defined over a number field evaluated at a cohomology class representable by an algebraic subvariety defined over a number field gives a number of the form $\frac{1}{2\pi\sqrt{-1}} \log f$ with $f\in \bar\Q$. For any $t$ for which $E_t$ has complex multiplication the class $\theta_t$ is a simple multiple of a class representable by an algebraic subvariety defined over a number field. This implies the statement.

The parts (i), (ii), (v) are already written and are contained in the two preprints I attach. Parts (iii) and (iv) are present only in drafts, so their writing is still in progress.
\end{enumerate}
\section{Future development}
It would be interesting to generalize the proof, or at least some parts of it, in the following directions:
\begin{enumerate}
\item The case of arbitrary group $\Gamma$ or arbitrary weight $k$. Then the conjecture can be formulated for certain linear combinations of values of the Green function.
\item Different choices of the complex multiplication point (which was $\sqrt{-1}$).
\end{enumerate}
The generalization of parts (i), (ii) and (v) is straightforward.

For the part (iii) the problem is to obtain the formula for $\partial^k\langle\epsilon_t, \theta_t\rangle$ for higher weights (product of more that two elliptic curves). This involves studying cycle map to the de Rham cohomology and its generalization for higher Chow groups. The goal is to obtain some concrete formulae for the case of powers of families of elliptic curves. By concrete we mean some rational expressions involving coefficients of equations defining $x_t$.

For the part (iv) we need to construct a different family $x_t$ for each complex multiplication point. We have constructed one for $\sqrt{-1}$. We may try to construct one for $\frac{-1+\sqrt{-3}}2$, $\sqrt{-2}$ or $\frac{-1+\sqrt{-7}}2$ since in this cases the complex multiplication has small degree ($1$ or $2$).
\bibliography{refs}
\end{document}@
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\begin{document}
\section{Quasi-modular forms}
Let $f_0, f_1, \dots, f_n$ be functions of $z$ on $\HH$. We consider
\[
f(z, X) = \sum_i \frac{f_i(z)}{i!(z-X)^i}.
\]
An element of the group $SL_2(\R)$ acts in weight $w$ in $z$ and weight $0$ in $X$:
\begin{multline*}
f(z, X)|_w \gamma = \sum_i \frac{f_i(\gamma z) (cz+d)^{i-w} (cX+d)^i}{i! (z-X)^i}\\
=\sum_{i, j} \frac{f_i(\gamma z) (cz+d)^{i-w} \binom{i}{j} (cX-cz)^{i-j} (cz+d)^j}{i! (z-X)^i}\\
=\sum_{i, j} \frac{(-1)^{i-j}}{(i-j)!} f_i(\gamma z) (cz+d)^{i+j-w} c^{i-j} \frac{1}{j!(z-X)^j} \\
=\sum_{i, j} (-1)^{i-j} f_i(z)|_{w-2i} \gamma \frac{1}{(i-j)!}\left(\frac{c}{cz+d}\right)^{i-j} \frac{1}{j!(z-X)^j}.
\end{multline*}
Therefore, expanding the result as a polynomial in $(z-X)^{-1}$ we get
\[
f(z, X)|_w \gamma = \sum_i \frac{g_i(z)}{i!(z-X)^i},
\]
where
\[
g_i(z) = \sum_j f_{i+j}(z)|_{w-2i-2j} \gamma \frac{(-1)^{j}}{j!} \left(\frac{c}{cz+d}\right)^j.
\]
Therefore we define the action of the group on tuples of functions as follows:
\[
(f_0, f_1, \dots, f_n)|_w \gamma = (g_0, g_1, \dots, g_n),
\]
where $g_i$ are given by the formula above. It is clear, that $g_i$ depends only on $f_i$, $f_{i+1}$, \dots, $f_n$,
so this action is an iterated exension of the usual action on functions.
\begin{defn}
A quasi-modular form of weight $w$ and depth $n$ for group $\Gamma$ is a holomorphic function $f$ on $\HH$, which has moderate growth at the cusps and such, that there exist functions $f_1$, \dots, $f_n$ such, that the tuple $(f, f_1, \dots, f_n)$ is invariant under the action of $\Gamma$ in weight $w$.
\end{defn}
\begin{rem}
It implies, that the each function $f_i$ is again a quasi-modular form of weight $w-2i$ and depth $n-i$.
\end{rem}
Looking at the definition of the action we can express $f_i(z)|_{w-2i}\gamma$ in terms of $g_i(z)$:
\[
f_i(z)|_{w-2i} \gamma = \sum_j g_{i+j}(z) \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j.
\]
We prove this as follows:
\begin{multline*}
\sum_j g_{i+j}(z) \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j \\
= \sum_{j, l} \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j f_{i+j+l}|_{w-2i-2j-2l}\gamma \frac{(-1)^l}{l!} \left(\frac{c}{cz+d}\right)^l \\
= \sum_{m} f_{i+m}|_{w-2i-2m}\gamma \left(\frac{c}{cz+d}\right)^m \sum_{l=0}^m \frac{(-1)^l}{l!(m-l)!} = f_i|_{w-2i} \gamma.
\end{multline*}
So, one can prove, that
\begin{prop}
The function $f$ is quasi-modular of weight $w$, depth $n$ for a group $\Gamma$ if and only if is holomorphic, of moderate growth at the cusps and
\[
f(z)|_w\gamma = \sum_{i=0}^n f_i(z) \frac{1}{i!} \left(\frac{c}{cz+d}\right)^i
\]
for some functions $f_i$ and for all $\gamma\in\Gamma$. Moreover, if this is true, than the functions $f_i$ are the functions, which are needed in the definition and $f_0=f$.
\end{prop}
Let $f_0$, $f_1$, \dots, $f_n$ be again functions of $z$ and
\[
f(z, X) = \sum_i \frac{f_i(z)}{i!(z-X)^i}.
\]
We put 
\[
\delta_w f(z, X) = \frac{\partial f(z, X)}{\partial z} + \frac{w f(z, X)}{z-X},
\]
this is a derivation (taking the weight into account), and it commutes with the action of the group.
The coefficients of $\delta_w f(z, X)$ are given by
\[
(\delta_w f(z, X))_i = \frac{\partial f_i(z)}{\partial z} + i(w+1-i) f_{i-1}(z),
\]
we see, in particular, that the first coefficient is simply the derivative of the original first coefficient, and the depth is increased by $1$.
Another differential operator is 
\[
\delta_w^- f(z, X) = (z-X)^2 \frac{\partial f(z, X)}{\partial X},
\]
it also is a derivation and commutes with the action of the group. The coefficients are
\[
(\delta_w^- f(z, X))_i = f_{i+1},
\]
so this operator 'forgets' the first coefficient and decreases depth by $1$. Their commutator is
\[
(\delta_{w+2}^-\delta_w - \delta_{w-2}\delta_w^-) f(z, X) = w f(z, X).
\]
On the level of quasi-modular forms the first operator is simply the derivative with respect to $z$, and the second maps $f$ to $f_1$ (see the definition).
Since there is a group action on tuples $(f_0, f_1, \dots, f_n)$ we have an action of the Lie algebra. For a function of $z$ and $X$ one can compute:
\[
L_wf = \frac{\partial}{\partial t}|_{t=0} (f|_w\begin{pmatrix}1&t\\0&1\end{pmatrix}) = \frac{\partial f}{\partial z} + \frac{\partial f}{\partial X},
\]
\[
M_wf = \frac{\partial}{\partial t}|_{t=0} (f|_w\begin{pmatrix}1&0\\t&1\end{pmatrix})
= -w z f - z^2 \frac{\partial f}{\partial z} - X^2 \frac{\partial f}{\partial X}.
\]
\[
(L_wM_w - M_w L_w)f = -wf - 2z\frac{\partial f}{\partial z} - 2X\frac{\partial f}{\partial X}.
\]

\end{document}@
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@\documentclass[12pt]{amsart}
\usepackage{amsfonts}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ax}{Axiom}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]

\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
\newtheorem*{notation}{Notation}

%\numberwithin{equation}{section}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\HH}{\mathfrak H}
\newcommand{\wt}{\widetilde}
\newcommand{\ol}{\overline}
\newcommand{\eps}{\epsilon}
\renewcommand{\labelenumi}{(\roman{enumi})}

\author{Anton Mellit}
\title{Some calculations}
\begin{document}
\maketitle

For $f$ we obtain
\[
f_{hl}(\tau_1, \tau_2) = (k-1)!^2 (-1)^{k-1} Q_{k'\,l}(\tau_2)(\tau_1),
\]
\[
D_1 \wt f(\tau_1, \tau_2) = (k-1)!^2 (-1)^{k-1}\frac{(2\pi i)^{2k-2}(X_1-\tau_1)^{2k-2}}{(2k-2)!} \sum_{l=1-k}^{k-1} Q_{k'\,l}(\tau_2)(\tau_1) Q_{k\,-l}(\tau_2)(X_2)
\]
\[
=(k-1)!^2 (-1)^{k-1}\frac{(2\pi i)^{2k-2}(X_1-\tau_1)^{2k-2}}{(2k-2)!} \times
\]
\[
\sum_{l=1-k}^{k-1} (2\pi i)^{-1} (\tau_1-\tau_2)^{-k-l} (\tau_1-\bar\tau_2)^{-k+l} (X_2-\tau_2)^{k-1+l} (X_2-\bar\tau_2)^{k-1-l} (\tau_2-\bar\tau_2)
\]
\[
= (k-1)!^2 (-1)^{k-1}\frac{(2\pi i)^{2k-2}(X_1-\tau_1)^{2k-2}}{(2k-2)!} \frac{(\tau_2-\bar\tau_2)}{2\pi i (\tau_1-\tau_2) (\tau_1-\bar\tau_2)} \times
\]
\[
\sum_{i=0}^{2k-2}\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^i \left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-2-i}
\]
\[
= \frac{(-1)^{k-1}(k-1)!^2 (2\pi i)^{2k-3}}{(2k-2)!} \frac{(X_1-\tau_1)^{2k-2}}{X_2-\tau_1}\left(\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}-\left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-1}\right)
\]
\section{May be we go another way}
Then we have, for example the following property:
\[
D_1 D_2 \wt G = (2 \pi i)^{4k-4} \frac {(X_1 - \tau_1)^{2k-2}(X_2 - \tau_2)^{2k-2}}{(2 k - 2)!^2} G_{hh},
\]
where $G_{hh}=\phi_1\phi_2 G$ is the holomorphic image w.r.t. $\tau_1$ of the holomorphic image w.r.t. $\tau_2$ of $G$. $G_{hh}$ is a function in $\tau_1$, $\tau_2$, which is holomorphic in its domain of definition. Define analogously $G_{ha}$, $G_{ah}$, $G_{aa}$, where the letter $a$ denotes the antiholomorphic image, and the letter $h$ denotes the holomorphic image. Define also $f_{hh}$, $f_{ha}$, $f_{ah}$, $f_{aa}$ analogously starting from the function $f$ instead of $G$. We want to describe singularity of functions $G_{**}$, $*\in\{a,h\}$. We describe functions $f_{**}$ instead. Then we use the fact that the difference $G_{**}-f_{**}$ is smooth near $\{\tau_1=\tau_2\}$.
\begin{lem}
For functions $f_{**}$ we have the following explicit formulas:
\[
f_{hh}(\tau_1, \tau_2) = \frac{(k-1)!^2 (2k-1)! (-1)^{k-1}}{(2\pi i)^{2k} (\tau_1-\tau_2)^{2k}},
\]
\[
f_{aa}(\tau_1, \tau_2) = \frac{(k-1)!^2 (2k-1)! (-1)^{k-1}}{(2\pi i)^{2k} (\bar\tau_1-\bar\tau_2)^{2k}},
\]
\end{lem}
\begin{proof}
Consider $f_{hh}$. It is holomorphic and satisfies
\[
f_{hh}(\gamma \tau_1, \gamma \tau_2) = f_{hh}(\tau_1, \tau_2) (c\tau_1+d)^{2k}(c\tau_2+d)^{2k},
\]
because the action of the group commutes with differential operators which we used to construct $f_{hh}$ and the weight of $f_{hh}$ is $2k$ in each variable. It implies that the function
\[
f_{hh}(\tau_1, \tau_2) (\tau_1-\tau_2)^{2k}
\]
is also holomorphic and invariant under the simultaneous action of $SL_2(\R)$ on $\tau_1$, $\tau_2$. Hence it is a constant function. Hence
\[
f_{hh}(\tau_1, \tau_2) = \frac {c_1} {(\tau_1-\tau_2)^{2k}}
\]
for some $c_1\in\C$. To find the constant $c_1$ recall, that 
\[
f(\tau_1, \tau_2) \sim \log|\tau_1 - \tau_2|^2.
\]
By definition, 
\[
f_{hh} = (k-1)!^2 \delta_1^k \delta_2^k f,
\]
so
\[
f_{hh}(\tau_1, \tau_2) \sim \frac{(k-1)!^2 (2k-1)! (-1)^{k-1}}{(2\pi i)^{2k} (\tau_1-\tau_2)^{2k}},
\]
which proves the first formula.
Since $f$ is real-valued, it is invariant under $\eps$, hence 
\[
f_{aa} = \overline{f_{hh}},
\]
and the formula for $f_{aa}$ follows.
For $f_{ha}$ we use trick similar to the one for $f_{hh}$. We have
\[
f_{ha}(\tau_1, \tau_2) = \frac {c_2} {(\tau_1-\bar\tau_2)^{2k}}
\]
for some unknown constant $c_2\in\C$. In particular, we see that $f_{ha}$ is smooth. Look at $f_h = \phi_1 f$. It is a function holomorphic in $\tau_1$ and has weight $0$ in $\tau_2$. Consider sequence $f_{h i}$, $1-k \le i \le k-1$ such, that 
\begin{enumerate}
\item $f_{h i}$ is in $F_{2i}$ w.r.t. $\tau_2$
\item $f_{h 0} = f_h$
\item $\delta_2 G_{hi} = (i+k) G_{h\,i+1}$ for $1-k \leq i \leq k-2$
\item $\delta_2^- G_{hi} = (i-k) G_{h\,i-1}$ for $2-k \leq i \leq k-1$
\end{enumerate}
Then $D_2^{2k-1} f_{h\,1-k} = f_{hh}$. Denote by $f'_{h\,1-k}$ the function
\[
f'_{h\,1-k}(\tau_1, \tau_2) = \frac{(2\pi i)^{2k-1} c_1}{(2k-1)! (\tau_1-\tau_2)}.
\]
Note, that $f'_{h\,1-k}$ is holomorphic and $\phi_2 f'_{h\,1-k} = f_{hh}$. Correspondingly let
\[
f''_{h\,1-k}(\tau_1, \tau_2) = \frac{(2\pi i)^{2k-1} \bar c_2}{(2k-1)! (\bar\tau_1-\tau_2)}.
\]
$f''_{h\,1-k}$ is holomorphic in $\tau_2$ and $\phi_2 f''_{h\,1-k} = \ol{f_{ha}}$. Construct corresponding $\wt{f'}_h$, $\wt{f''}_h$. By lemma \ref{ftilde}
\[
\wt{f'}_h(\tau_1, \tau_2)(X_2) = c_1 \sum_{n=0}^{2k-2} \frac{(2\pi i)^{2k-1} (X_2-\tau_2)^n}{(2k-1)!(\tau_1-\tau_2)^{n+1}}
\]
\[
=c_1 \frac{(2\pi i)^{2k-1}}{(2k-1)! (\tau_1-\tau_2)}\frac{1-\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}}{1-\frac{X_2-\tau_2}{\tau_1-\tau_2}}
\]
\[
=c_1 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{1-\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}}{\tau_1 - X_2}.
\]
Analogously
\[
\wt{f''}_h(\tau_1, \tau_2)(X_2) = \bar c_2 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{1-\left(\frac{X_2-\tau_2}{\bar\tau_1-\tau_2}\right)^{2k-1}}{\bar\tau_1 - X_2}
\]
Consider the function
\[
\wt{f^*}_h = \wt{f'}_h + \ol{\wt{f''}_h} = 
c_1 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{1-\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}}{\tau_1 - X_2}
-
c_2 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{1-\left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-1}}{\tau_1 - X_2}
\]
It follows from the construction, that
\[
\phi_2(\wt{f^*}_h) = \phi_2(\wt f_h),\qquad \phi'_2(\wt{f^*}_h) = \phi'_2(\wt f_h),
\]
Hence the difference $\wt{f^*}_h-\wt f_h$ does not depend on $\tau_2$. Rewrite $\wt{f^*}_h$ as
\[
\wt{f^*}_h = 
c_1 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{\left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-1}-\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}}{\tau_1 - X_2}
\]
\[
+(c_1-c_2) \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{1-\left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-1}}{\tau_1 - X_2}
\]
\[
=c_1 \frac{(2\pi i)^{2k-1}}{(2k-1)!} \frac{\left(\frac{X_2-\bar\tau_2}{\tau_1-\bar\tau_2}\right)^{2k-1}-\left(\frac{X_2-\tau_2}{\tau_1-\tau_2}\right)^{2k-1}}{\tau_1 - X_2}
\]
\end{proof}
\end{document}@
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\begin{document}
\section{Signs of topological operations}
The dual of a cohomology class $x$ is $x\cap o$. If $y$ is a complementary cohomology class then
\[
(x\cap o)\bullet(y\cap o) = (x\cup y) \cap o, \qquad\text{page 337 in Dold.}
\]
Applying p.239 
\[
\langle 1, (x\cup y) \cap o\rangle = \langle x\cup y, o\rangle = \langle x, y\cap o\rangle.
\]
So for any cohomology class $x$ and cohomology class $\xi$
\[
\langle x, \xi\rangle = (x\cap o) \bullet \xi.
\]
\end{document}@
�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/pari-python/�������������������������������������������������������������������������0000775�0001357�0001362�00000000000�10705402633�015702� 5����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/pari-python/pari.cpp,v���������������������������������������������������������������0000444�0001357�0001362�00000025460�10705402623�017605� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.3;
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date	2007.10.17.12.52.35;	author mellit;	state Exp;
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@#include <pari.h>
#include <Python.h>

/* ----------------------------------------------------- */
extern "C" {

static PyTypeObject* type_ptr;
static PyObject *pari_error;
PyObject* pari_module;
static long pari_prec;
#define SET_ERROR PyErr_SetString(pari_error, errmessage[pari_errno])
#define PRINT_ERROR printf("Pari error: %s\n", errmessage[pari_errno])

struct PyGEN {
	PyObject_HEAD
	
	GEN val; //this value must be on the heap (always use gclone)
};

static PyGEN* create_gen(GEN initVal) { // the initVal is cloned and disposed
	if(initVal==NULL) return NULL;
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		GEN val = gclone(initVal);
		PyGEN* self;
		self = PyObject_New(PyGEN, type_ptr);
		self->val = val;
		cgiv(initVal);
		return self;
	} ENDCATCH
}

static void
gen_dealloc(PyObject* _self) {
	PyGEN* self = (PyGEN*) _self;
	CATCH(CATCH_ALL) {
		PRINT_ERROR;
	} TRY {
		gunclone(self->val);
	} ENDCATCH
	PyObject_DEL(self);
}

static PyObject*
gen_str(PyObject* _self) {
	PyGEN* self = (PyGEN*) _self;
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		char* res = GENtostr(self->val);
		//printf("string returned %s\n", res);
		PyObject* ret = Py_BuildValue("s", res);
		free(res);
		return ret;
	} ENDCATCH
}

static GEN genFromLong(PyObject* obj) {
	int sign = _PyLong_Sign(obj);
	int bits = _PyLong_NumBits(obj);
	if(PyErr_Occurred()) return NULL;
	int byte_length = bits/8+1; // include the sign bit
	int long_length = (byte_length - 1) / BYTES_IN_LONG + 1;
	byte_length = long_length*BYTES_IN_LONG;
	unsigned char* byte_arr = (unsigned char*) malloc(byte_length);
	_PyLong_AsByteArray((PyLongObject*) obj, byte_arr, byte_length, true, true);
	if(PyErr_Occurred()) {
		free(byte_arr);
		return NULL;
	}
	CATCH(CATCH_ALL) {
		free(byte_arr);
		SET_ERROR;
		return NULL;
	} TRY {
		GEN res = cgeti(long_length+2);
		res[1] = evalsigne(_PyLong_Sign(obj)) | evallgefint(long_length+2);
		GEN ptr = int_LSW(res);
		int shift = 0;
		for(int i = 0; i<byte_length; i+=BYTES_IN_LONG) {
			*ptr=0;
			for(int j = 0; j<BYTES_IN_LONG; j++) {
				long val = 0xFFL&byte_arr[i+j];
				if(sign<0) {
					val = 0xFFL&(-shift - val);
					shift = (val>=1-shift) ? 1 : 0;
				}
				*ptr += val<<(8*j);
			}
			ptr = int_nextW(ptr);
		}
		res = int_normalize(res, 0);
		free(byte_arr);
		return res;
	} ENDCATCH
}

static GEN
gen_cast(PyObject* obj) {
	GEN res = NULL;
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		if(obj->ob_type==type_ptr) {
			res = ((PyGEN*) obj)->val;
		} else if(obj->ob_type==&PyLong_Type) {
			res = genFromLong(obj);
		} else if(obj->ob_type==&PyInt_Type) {
			res = stoi(PyInt_AsLong(obj));
		} else if(obj->ob_type==&PyFloat_Type) {
			res = dbltor(PyFloat_AsDouble(obj));
		} else {
			res = stoi(PyInt_AsLong(obj));
		}
	} ENDCATCH
	if(PyErr_Occurred()) return NULL;
	return res;
}

typedef GEN(*gunary) (GEN);

static PyObject*
gen_unary(gunary op, PyObject* _obj1) {
	PyGEN* obj1 = (PyGEN*) _obj1;
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		return (PyObject*) create_gen(op(obj1->val));
	} ENDCATCH
}

#define GEN_UNARY(op) \
static PyObject* \
gen_##op(PyObject* _obj1) { \
	return gen_unary(g##op, _obj1); \
}

typedef GEN (*gbinary) (GEN, GEN);

static PyObject*
gen_binary(gbinary op, PyObject* _obj1, PyObject* _obj2) {
	pari_sp __av = avma;
	GEN gen1 = gen_cast(_obj1);
	GEN gen2 = gen_cast(_obj2);
	if(gen1==NULL || gen2==NULL) {
		avma = __av;
		return NULL;
	}
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		PyObject* res = (PyObject*) create_gen(op(gen1, gen2));
		avma = __av;
		return res;
	} ENDCATCH
}

#define GEN_BINARY(op) \
static PyObject* gen_##op(PyObject* _obj1, PyObject* _obj2) { \
	return gen_binary(g##op, _obj1, _obj2); \
}

GEN gpow2(GEN x, GEN y) {
	return gpow(x, y, pari_prec);
}

GEN gabs1(GEN x) {
	return gabs(x, pari_prec);
}

GEN_BINARY(add)
GEN_BINARY(sub)
GEN_BINARY(mul)
GEN_BINARY(div)
GEN_BINARY(mod)
GEN_BINARY(pow2)
GEN_UNARY(neg)
GEN_UNARY(abs1)

static PyObject*
gen_pow_tern(PyObject* _obj1, PyObject* _obj2, PyObject*) {
	return gen_pow2(_obj1, _obj2);
}

static PyNumberMethods gen_as_number = {
    gen_add, // binaryfunc nb_add;
    gen_sub, //binaryfunc nb_subtract;
    gen_mul, //binaryfunc nb_multiply;
    gen_div, //binaryfunc nb_divide;
    gen_mod, //binaryfunc nb_remainder;
    0, //binaryfunc nb_divmod;
    gen_pow_tern, //ternaryfunc nb_power;
    gen_neg, //unaryfunc nb_negative;
    0, //unaryfunc nb_positive;
    gen_abs1, //unaryfunc nb_absolute;
    0, //inquiry nb_nonzero;
    0, //unaryfunc nb_invert;
    0, //binaryfunc nb_lshift;
    0, //binaryfunc nb_rshift;
    0, //binaryfunc nb_and;
    0, //binaryfunc nb_xor;
    0, //binaryfunc nb_or;
    0, //coercion nb_coerce;
    0, //unaryfunc nb_int;
    0, //unaryfunc nb_long;
    0, //unaryfunc nb_float;
    0, //unaryfunc nb_oct;
    0 //unaryfunc nb_hex;
};

static PyTypeObject pari_GEN = {
	PyObject_HEAD_INIT(NULL)
	0,
	"pari.GEN", //char *tp_name; /* For printing */
	sizeof(PyGEN), 0, //int tp_basicsize, tp_itemsize; /* For allocation */
	gen_dealloc, //destructor tp_dealloc;
    0, //printfunc tp_print;
    0, //getattrfunc tp_getattr;
    0, //setattrfunc tp_setattr;
    0, //cmpfunc tp_compare;
    gen_str, //reprfunc tp_repr;

/* Method suites for standard classes */

&gen_as_number, //PyNumberMethods *tp_as_number;
	0, //PySequenceMethods *tp_as_sequence;
	0, //PyMappingMethods *tp_as_mapping;

/* More standard operations (here for binary compatibility) */

0, //hashfunc tp_hash;
    0, //ternaryfunc tp_call;
    gen_str, //reprfunc tp_str;
    0, //getattrofunc tp_getattro;
    0, //setattrofunc tp_setattro;

/* Functions to access object as input/output buffer */
    0, //PyBufferProcs *tp_as_buffer;

/* Flags to define presence of optional/expanded features */
    Py_TPFLAGS_DEFAULT | Py_TPFLAGS_CHECKTYPES, //long tp_flags;

0, //char *tp_doc; /* Documentation string */

};

static PyObject *
ex_foo(PyObject *self, PyObject *args)
{
	printf("Hello, world\n");
	
	Py_INCREF(Py_None);
	return Py_None;
}

static PyObject *
pari_eval(PyObject *self, PyObject *args) {
	char* s;
	if(!PyArg_ParseTuple(args, "s", &s)) return NULL;
	//printf("parsed %s\n", s);
	GEN g = gp_read_str(s);
	//printf("GP executed %X\n", g);
	char* res = GENtostr(g);
	printf(" *** result: %s\n", res);
	free(res);
	Py_INCREF(Py_None);
	return Py_None;
}

static PyObject *
pari_parse(PyObject *self, PyObject *args) {
	char* s;
	if(!PyArg_ParseTuple(args, "s", &s)) return NULL;
	//printf("parsed %s\n", s);
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		GEN g = gp_read_str(s);
		PyObject *ret = (PyObject*) create_gen(g);
		return ret;
	} ENDCATCH
}

static PyObject *
pari_cast(PyObject *self, PyObject *args) {
	PyObject* obj;
	if(!PyArg_ParseTuple(args, "O", &obj)) return NULL;
	//printf("parsed %s\n", s);
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		return (PyObject*) create_gen(gen_cast(obj));
	} ENDCATCH
}

static PyObject *
pari_var(PyObject *self, PyObject *args) {
	char* s;
	if(!PyArg_ParseTuple(args, "s", &s)) return NULL;
	//printf("parsed %s\n", s);
	CATCH(CATCH_ALL) {
		SET_ERROR;
		return NULL;
	} TRY {
		long v = fetch_user_var(s);
		GEN g = pol_x[v];
		PyObject *ret = (PyObject*) create_gen(g);
		if(ret!=NULL) {
			if(PyModule_AddObject(pari_module, s, ret)<0) {
				return NULL;
			}
		}
		return ret;
	} ENDCATCH
}

static struct PyMethodDef pari_methods[] = {
	{"eval", pari_eval, METH_VARARGS, "Evaluates expression in pari and prints the result"},
	{"parse", pari_parse, METH_VARARGS, "Evaluates string expression in pari and returns an object"},
	{"cast", pari_cast, METH_VARARGS, "Transforms a Python object, e.g. a number, to a pari object"},
	{"var", pari_var, METH_VARARGS, "Creates a variable in PARI and stores the object with the same name in the module"},
	{"foo", ex_foo, METH_VARARGS, "foo() doc string"},
	{NULL, NULL}
};

PyMODINIT_FUNC
initpari()
{
	pari_GEN.ob_type = &PyType_Type;
	type_ptr = &pari_GEN;
	// init PARI
	pari_init(64000000, 500000);
	// init module
	pari_module = Py_InitModule("pari", pari_methods);
	// create exception type
	pari_error = PyErr_NewException("pari.error", NULL, NULL);
	Py_INCREF(pari_error);
	PyModule_AddObject(pari_module, "error", pari_error);
	// default precision
	pari_prec=20;
}

}
@

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@d5 1
d7 8
a14 1
typedef struct {
d18 16
a33 1
} PyGEN;
d36 7
a42 2
gen_dealloc(PyGEN* self) {
	gunclone(self->val);
d46 173
a218 1
static struct PyTypeObject pari_GEN = {
d228 1
a228 1
    0, //reprfunc tp_repr;
d232 1
a232 1
	0, //PyNumberMethods *tp_as_number;
d240 1
a240 1
    0, //reprfunc tp_str;
d248 1
a248 1
    0, //long tp_flags;
a251 35
    /* Assigned meaning in release 2.0 */
    /* call function for all accessible objects */
    0, //traverseproc tp_traverse;

/* delete references to contained objects */
    0, //inquiry tp_clear;

/* Assigned meaning in release 2.1 */
    /* rich comparisons */
    0, //richcmpfunc tp_richcompare;

/* weak reference enabler */
    0, //long tp_weaklistoffset;

/* Added in release 2.2 */
    /* Iterators */
    0, //getiterfunc tp_iter;
    0, //iternextfunc tp_iternext;

/* Attribute descriptor and subclassing stuff */
    0, //struct PyMethodDef *tp_methods;
    0, //struct memberlist *tp_members;
    0, //struct getsetlist *tp_getset;
    0, //struct _typeobject *tp_base;
    0, //PyObject *tp_dict;
    0, //descrgetfunc tp_descr_get;
    0, //descrsetfunc tp_descr_set;
    0, //long tp_dictoffset;
    0, //initproc tp_init;
    0, //allocfunc tp_alloc;
    0, //newfunc tp_new;
    0, //destructor tp_free; /* Low-level free-memory routine */
    0, //PyObject *tp_bases;
    0, //PyObject *tp_mro; /* method resolution order */
    0  //PyObject *tp_defined;
d271 1
a271 2
	//printf("string returned %s\n", res);
	PyObject* ret = Py_BuildValue("s", res);
d273 2
a274 2
	Py_INCREF(ret);
	return ret;
d278 1
a278 1
pari_pari(PyObject *self, PyObject *args) {
d282 42
a323 2
	GEN g = gp_read_str(s);
	
d327 4
a330 2
	{"eval", pari_eval, METH_VARARGS, "Evaluates expression in pari and returns a string"},
	{"pari", pari_pari, METH_VARARGS, "Evaluates expression in pari and returns an object"},
d338 3
d342 10
a351 1
	Py_InitModule("pari", pari_methods);
@

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d113 9
d124 1
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//	freopen("CONIN$","rb",stdin);   // reopen stdin handle as console window input
//	freopen("CONOUT$","wb",stdout);  // reopen stout handle as console window output
//	freopen("CONOUT$","wb",stderr); // reopen stderr handle as console window output
//	function_that_is_not_used_anywhere();
@

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/pari-python/setup.py,v���������������������������������������������������������������0000444�0001357�0001362�00000001002�10704410571�017642� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
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@from distutils.core import setup, Extension

module1 = Extension('pari',
					include_dirs = ['/usr/include/pari'],
					libraries = ['pari'],                    
                    sources = ['pari.cpp'])

setup (name = 'Pari',
       version = '1.0',
       description = 'Pari interface',
       ext_modules = [module1])

@
������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������cvs-repository/pari-python/install,v����������������������������������������������������������������0000444�0001357�0001362�00000000324�10704422206�017525� 0����������������������������������������������������������������������������������������������������ustar  �mellit��������������������������mellit�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������head	1.1;
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@python setup.py install
@
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